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MATHEMATICSAND5D

Introduction.

TheFractalUniverse:itsbio-topo-logicproperties,its5Dimotions&s=Tsymmetries.

BookI.Thespatialview.Geometry.ItsS=Tsymmetry.Numbers

I.∞MindSpaces.Itsdifferentp.o.v.sArtisticGeometries.

II.PhilosophyofMathematics:5sub-disciplinesasmirrorsof¬∆@st.3Ages.S-point=∆-numberssymmetry.

III.∆-scales:Numbertheory.S=TDualities:SpacePoints=Scalenumbers.Polygons.Primes.Closure:N,Z,R,Q,C.

IV.Algebra:ItsoperandsasDimotions.TheFractalGenerator.ExistentialAlgebra.

V.Fractalpoints:¬EPoints.¡logicGeometry.Its5Non-EPostulatesasMirrorsofscalargrowth.

VI.VitalTopology.The3±¡varietiesofreality.

VII. Time Geometry:3 ages: 2D Geometry->Analytic & Differential Geometry ->5D hyperbolic geometry &Topology.

VIII. Space Geometry: Entanglement. Trinity and Pentalogic. Its 3 Classes: S@:Mental; S=T; Topologic & 5DSpaces.

IX.ScalarGeometryofthe5thDimension.Networkspaces.

X.1stAge.Greek,linealGeometryinthebidimensionalplane.

XI.2ndAge.Analyticgeometry:S=Tsymmetries.Conics&Curves.

XII.2ndAge.DifferentialGeometry:Motionsofapoint.

XIII.2ndAge.VectorialSpaces

XIV.3rdAge.∆±¡:Hyperbolicgeometry

XV.3rdAge:Metricspaces:Riemann’sr=evolution.FromphasespacestoHilbertSpaces.

XVI. ∆±¡ Age: The Future of Geometry. Pangeometry of Existential actions. Ethonomics. Its vital frame ofreference.

XVII.∆±¡Age:Evolutionofvitalgeometricspace-time:howmentalspacesandvitaltopologiesconstructtheworld.

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BookII.

INTRODUCTION.5DTheoreticalminimumtounderstand¬AlgebraasamirroroftheUniverse.

Non-Euclideangeometry:Thefractalpoint.ThelinealnetworkVitalTopology.OrganicSpacetimeplanes.

∆ST.ThefractalUniverse:Its5Dmetric,S=Trelativity.

Minds=Monads:ItsSupœrganisms,actionsandterritoriesoforder.

¬Ælgebraofexist¡ence:Itsworldcycle&fractalgenerator.¡tsentropiclimits.

BOOKII.THESCALARVIEWOFMATHEMATICS.¬ALGEBRAOFNUMBERS.

I.Philosophyof®Mathematics.

Itssubdisciplines:Non-EuclideanGeometryandNon-Aristotelian¬Algebra

Ameasureof¬∆@ST,stœps,Dimotions&Symmetries.

¬AlgebraasamirrorlanguagewithinMathematicswithinthefractalUniverse.

II.3Agesof¬Algebra

1stage:Arithmetic,numbers.

2ndAge:operandsandfunctions.Analysis.

3rdAge:Groups,setsandfunctionals

±¡Ages:Existential¬ÆlgebraandBoolean¬Algebra.

III.1styoung∆-1age:Numbertheory,Operands,thecomplexplane.Polynomials.

Numbersasregularforms.Prime,evenandoddnumbers.Constants.

Thefivefamiliesofnumbers.

TheSymmetricinvertedOperands=Dimotionsofspace-timetranslatedinto¬Algebra.

Dimensionalgrowth.Equations.PolynomialFunctions&its3operands:±;x÷;xª

Sinusoidalfunctions.

ThenewoperandsofCalculus:Derivativesandintegrals.Itsfunctions.

Finitesimals&PowerSeries:fromArchimedestoNewton.

Finitesimalsintime,space&scale.Itswholes.

Thefundamentaltheoremof¬Algebra.Complexnumbers.

IV.2ndclassicƼageofequations.Thefunctionofexistience.

Operandsofequivalenceandrelationsoforder.

Thefunctionofexistience.Itscurvature,probabilities&statistics.

Pentalogiconthefunctionofexistenceits5Dimotions.

ItsStœpsanda(nti)symmetries:S=T

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V.EquationsofsimplexDimotions.Polynomials.Statistics=Probabilities.

PolynomialsapproximatingcalculusofDimotionsand∆±¡Planes.

Probabilities:discretefinitesimalsintime.

Statistics:Discretewholesinspace.

VI.3rdinformativeage.∆+1Wholes:§œt.Groups,functionals.

Grouptheory.Present,entangled∆@st-symmetries.Groupsinspace,timeandscale.

Sets.FromCantor’sSetstoexistential§œ.Ts.Itsproperuseandhyperbolic,∞egocy.

Functionals:Probingnewplanesofexistence.HilbertSpaces&quantumphysics.

Philosophyof¬Algebrainits3rdinformative,oldage:Hilbert'saxiomaticmethod.Itsincompleteness.

VII.The2futures.-¡:DigitalThought:AI-lgebra.Transhumanism.

Humind’sentropicdeath.Themetalmind

Its1stage:Binary,Boolean¬Algebra.Reticulesofthought.

Its2ndage:AI:AlgorithmsofInformation.Its2genderbrain.

Its3rdage:Aconsciousmetalmind.¡tsfunctionofexistence.

2036:chipHomoctonos,2ndera.ApentalogicAI-system.

BOOKIII:EXISTENTIALCALCULUS.

I.Existentialcalculus.

∆STreproduction:theGalileanparadox.

Finitesimalsanditsintegralworldcycles.

Finitesimalsinspace:Curvature.Thedisomorphicmethod.

Finitesimalsintime:TheWorldcycleofexistence&itsactions=dimotions.

Finitesimalinscale.Networks.Itssymmetries.

II.Analysis.

Its3ages.

1stAge:Calculus:PartsofWholes.

FromGreecetoLeibniz.Finitesimalsofwholes.Series.

TrilogiconCalculus.Curvatureofspace=changeintime=finitesimalinscale.

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2ndAge.Analysis:Sentencesofmotions.

ODEsbetween3planesofexistence.

PDEsbetweenSpaceandTimeparameters.Multiplevariables.

Nt.WehavereducedtheIIBook,eliminatingmostofCalculus,andexistentialalgebraonthe5Dimotionsofreality, and its closermathematical mirror, as the whole calculus will make the paper exceed the 1000pages.

Sothispaperwilldealafteranintroductionto∆STlaws,withGeometry,Philosophyofmathematics,thefirst age of Algebra, arithmetics and the trinity of social operands and its ‘sentences’, polynomials,probabilityandstatistics;andthe3rdagegroup,settheoryandBooleanalgebras,withemphasisonGrouptheoryatanintroductorylevelastheaimofthosepapersistoshowthecorrespondencebetweenclassicscienceaandthe lawof∆STtheymirror, in thecaseofmathemagticswithscalarnumbers,spatialpointsanddimotionaloperands.

Asalwaysthisisaworkinprogresscopycatfirstfrom30yearsofresearchin5ÐSTsciences,correctedandexpandedastimegoesby,fromheretomyeternity.

Somyapologiesforallpartsnotyetcorrecteduptothestandardsofscholarship.

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BOOKI.¬E-GEOMETRYAbstract.Thisarticle introducesmathematicsasanexperimentalmirror languageofthe5DUniverse,and its3∆STessentialelements, studiedby its3main sub-disciplines. Spatial fractalpoints studiedbygeometry;Scalarnumbers studied by Algebra, and time=change dimotions studied by analysis. As the 3 elements, ∆ST areentangledinto‘planesofspace-time’,soarethe3sub-disciplinesofscience.Asscalesofspace-timeincludetheother two elements, so Algebra includes all other elements and numbers become the main substance ofmathematics.Asentanglementmeansahugenumberofdifferentpossible,¡logicmirrorsymmetriesbetweenallthoseelements,whena‘mind’createsalanguagetoobserveit,mathematicshasmultipleformstoperceive∆st.

Howtoputasequentialordertoallthis?Wedon’tbeyondusingthesimplest‘agescheme’ofgrowingcomplexitythatprovidesthebestsequentialprogressionthatmimicstheworldcycleofanyexistence,mentalorphysical.Butthe true ‘jump’ of understanding is to force yourmind into parallel, synchronous, simultaneous thinking andperceivethe‘symmetries’,dualities,trinities,pentalogicstructuresthatcrossbetweendisciplines.

Considertheoperandsthatgrowincomplexity,from±socialsums,toproductsandinversedivisions,to√xa,tothe crown of ∫∂ operands that include all others to study change between time-space planes of finitesimal,derivativepartsand integralwholes…Wecannotstudythusthemisolated,butwewill inthebooksonalgebraandanalysisbuildasa stairof growing complexitybecause theUniversealsobuilds thedifferentdimotions insequencesofgrowingcomplexitytoachievethe‘higherscalar’dimotionsofreproductionandsocialevolution.

Moreover ∆ST real elements of Nature, its co-existing planes of space-time, which form every 3 ∆±¡ units asupœrganism,areaccessednotfullybeyondourownT.œ(Timespacesupœrganism)butthrough‘limitedminds’,which select information,byentropically erasing ‘dark spaces’ intoa continuum; reducing innerproperties intopointswithoutparts,etc.

Soweaddtotherealityof∆St,the¬(entropy)and@-mindmirrorstoform,¬∆@st,the5elementsofreality.

Sowemustregainbacktheeraseinformation–thoseforgottenpropertiestoimprovetheÆ-mirror(Aristotelian,single time causality and Euclidean points) into ¬Æmaths,with the handicap of being this a single humind indeclininghealth.SoIwillalsoasan¬@reducewhatIsayandrepeatasoldmendotheessenceofit.Sinceatthebeginningandtheendofall,thereisindeedjustaninfiniterepetitionof¬∆@st.

Thusthisasallpapersarejustafinalseedforfutureresearcherstogrowonbetterfoundationshuman‘stiences’of timespace planes. To do so you have first to forget sequential, lineal single time causality, as reality is anentanglementof5causes,whichcanbeseeninmanymirrorviews.Thesimplestonethoughisasatugofwarofthemathematicalunit–afractalpoint,which‘holdsamindinitself’(aLeibnizianmonad,ornon-euclideanpointtraversedbyinfiniteparallels).ThisFractalpointinstillmind-moodisSS,andittriesto‘stop’theflowofentropictime, TT, to formmind images ofmomentum, Ts, dominant in Time-motion and information, St, dominant inSpatialmental form. Andwhen it achieves it, St=Ts, a balanced S=Twave of energy is born,which ‘connects’fractal points into networks, which become vital topological planes, according to its degree of congruence,similarity,enhancedbytheconstanttransferofSt-informationandTs-momentum.Thisisthegameofexist¡ence,whichaneasy-upgradingoftheAxiomsandpostulateofEuclideanGeometrystudiesindepth.

Assuch¬EGeometryisbyfarthesimplest,easiestmodetomirrortheworldofthose5‘dimensionalmotions’ofspaceandtimeanditsscales,andSS-minds/pointsthatgaugeitsSt-informationwithintheentropiclimitsoftheirinnerworlds;reasonwhyitisourfirstpaperon5DMathematics.

Butthewhole∆Stisrepresentedinmathematicsby‘algebraofscalarnumbers’,‘geometryofspatialpoints’,and‘analysisofthebest∫∂timeoperands’anditsentropicinversefunctions,putinmentalspacethrough@-framesofreference.Allthis,asIassumereaderswillknowmorethanIdoofmathematics,istheessenceofit.Irepeat–mytaskistoestablishasimplerCopernicanmoretruthfulfoundationsforfuturescholarstorecastallwhatthey

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knowbetterthanIdointhemorefocusedtransparentimage.Forgivethenmyerrors,drinkonthestrengthsofthemirror,becauseifyoudoasthosewhodevelopedtheseedofPlanck,thenewfieldof5Disginormousandlargelyunexplored.Ican’tevenorderthehugeGBsof30yearsofresearchnowindecliningphysicalandmentalhealth.I’lljustdowhatIcanfromheretomyentropiceternity.ThissaidthisisthebestthingthathashappenedtoMathematicssinceRiemannandLobachevski, jumpingoverMr.Cantor’s idealist sets, justa lessermirrorofthe∆-planesofspace-timethat include itall (reasonwhysetscan include itall,butwhytousea foggymirrorwhenweshallprovide therealone,ofscalaralgebraicnumbers, spatial fractalpointswithpartsanddimotionoperands?) Ifthemirrorweofferisnotgoodenoughismyfault,NOTthefaultofthesubstanceofthemirror–rememberthat;blametheoldman,notthevisionheoncehadandyoucanimprovewithyouthandenergyIlack.

WeconcentrateontheS@,spatialmentalgeometricelements,keepingforasecondvolumethestudyoftimealgebra.Ouraim is toprove theexperimentalnatureofmathematics,whichalong logic is themainscienceofspaceand time. Its content is a smallpartofmy research in the field inadiscipline that if ever completedbyfuture researcherswill recast the entire subject ofmathematics as set theory did in the XX c. in terms of itsexperimentalpowertodescribetheultimatelawsof‘mentalspaces’and‘timemotions’.

Scalarspacetimehas3units,timeoperands,studiedinalgebra,socialscalarnumbersandNon-Euclideanfractalspace points,whose symmetry is study by non-E geometry completeswith 5 postulates of points, lines andplanes with breath and relative congruence. Once we complete ¬E geometry; we define the main dualitybetweensubjectivementalphasespacesasstillmirrorsoftheworldthatselecttheinformationmindsneedtoactandsurviveinthegameofexist¡ence,thatcreatesobjective‘vital,topologicorganisms’andprojectingitsgeometricmindinitsterritoriallocalorderasgeometry‘informs’theunderlyingtimespaceweareallmadeof.

5Dmathematicsreturnstothematureexperimentalageofthedisciplinethatwentasallsystemsdothrough3agesfromasimplelinealageofGreekGeometrytoitscurvedrealistageofcalculusthataddedmotion&scalesof 5D finitesimal derivatives and 4D wholes to better mirror the fractal Universe, to a 3rd age of excessiveinformativefictions,whenitabandonsitsrealistfoundations,whenduringtheGermancreationistego-tripofHilbert and his Cantorian paradises, he says, 'I imagine points, lines and planes' thinking he shares the onlylanguage 'God' uses to create reality. Hencemathematics is no longer considered an a posteriori mirror ofscalartimespacebutits‘generator’andasthefirst‘Aristoteliancause’,needsnoexperimentalproofbutrathertheopposite: realityexistsonly if it canbecasted intomathematicalmodels (‘onlywhatwemeasure is real’Planck). Somathematicians abandon space points, scale numbers and timeoperands as the 3mathematicalimagesgeneratedbythe∆stUniverse,usinginstead‘Cantorian’setsasits'imagined'unitsnowheretobeseenin reality. And Hilbert affirms the self-contained Axiomatic method of proof, NOT connected to the world,despite Gödel’s incompleteness theorem- a baroque inward looking 3rd age of excessive form proper of allsystemswe abandon to return to its empirical foundations, now formulated in terms of those 5 structuralelements that create reality, mimicked by the main mathematical elements. Since mathematics reflect thepropertiesof scalar spaceand cyclical time in itsmainelements,pointsnumbersandoperandsentangled infeed-back≤=≥equations.Sowedefinemathematicsasanexperimental languagethatmirrors inasimplifiedmannerasalllanguagesdotofitthemind,theelementsandstructuresofthegenerationalspace-time,weareall made of. As such mathematics is only 2nd to ¡logic and its fractal, ternary Universal grammar, as thefundamentalformal,experimentallanguageoftheUniverseanditsgenerationaltimespacethatcreatesallitssuper-organisms (ab. T.œs). TheMath’spromustbehumble to value thiswork that is not somuchonnewtheorybutonentanglingmathswiththevitalRealityitmirrors&settheorycutofffrom.

Westudyhowmindlanguagesmirrorsupœganismsmadeofscalarspaceandcyclicaltime.Soweintroducethemetricequationthatdescribesthescalesof the5thDimensionalUniverse:SxT (max.s=t)=C.Thenwecouplethe 5D elements of reality (entropy, scales,minds, space and time) in its 3 ‘relative scales of size and timeduration’, as fast 5 Dimotions=actions (short view), superorganisms tracing worldcycles (medium view) and¬∆@stelements(longview)withthe5disciplinesofmathematicsasexperimentalmirrorsofthose5elements:

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Space=geometry,∆:Scales:Number theory,S<=>TDimotionsandSymmetries=algebra;whichanalysis isbestfor timemotions;@-minds=Analyticgeometry (Framesof reference)andphilosophyofmathematics;whileentropy,negationofinformationappearsinthelimitsofcalculus,inverseoperandsandexponentialfunctions.

Since,asscalarspace includesscales& ‘stillmindmappings’andTimetheentropic limitsofdeathandall itsmotions;wecanreducemathematicstospatialgeometryandtemporalalgebra,whichindeedweretheoriginaldisciplines, fromwhere all others branched, and to itsminimal units, points of space and numbers of time,whichappearedevenearlierinthehumanconsciousness.WeshallthusconsidertheS=TsymmetriesbetweenfractalNon-Euclideanspacepoints (ab.•)andnumbers(ab.Nº);and latteronbetweenTopologyandAlgebra;andcouple the internalelementsofS-geometry (dimensions)andT-algebra (operands)with the5Dimotions(dimensionalmotions)ofthefractalUniverse.

Somathematics is only 2nd to ¡logic in the quality of its experimentalmirror expressing the Disomorphisms(equaldimensionallaws)ofallsystemsofNature.Aseachofits2->5subdisciplinesreflectthose5DimotionsoftheUniverse,which all stiences expresses in different forms. Inmathematics geometry does so through 3±¡topologic bidimensional varieties and classic 3 dimensions (1D: height, 2D: Length, 3D: width; 4-5D: fractaldimensions and topologic networks). While algebra mirrors the 5 Dimotions with its 5 operands & inverseentropic functions (1D: sin/cos, 4D:±, 3D: x÷, 4D: xª √, 5D:∫∂). So finally we focus on themedium view of‘entangled supœrganisms, in simultaneous space’, as expressed by vital topology, tracing ‘worldcycles’ asexpressedbyExistentialalgebra,whichbecomethenew‘integrated’2polardisciplinesof5DmathematicsthatshouldmakeitabettermirrorofthefractalUniverse.

VitaltopologicGeometryvs.mentalsubjectivespaces.

The essential duality of geometry is betweenmental spaceswhich are continuous, as they erase the holesbetweenforms,stillgeometrythateliminatemotionsandscalarplanes,reducedtoamindandwithadistorted@-centeredframeofreferencevs.realvitalgeometriesofdiscontinuousT.œsandfractalscales:

Mentalgeometry:singleplane,nomotion,0-pointcentervs.Vitaltopology,∆±¡co-existingplanes,motion.

The paper thus deals with vital geometry, the experimental stience that describes the fundamental laws ofTime-SpaceSupœrganisms,seeninsimultaneityasasynchronous,stableformofspace.

Whendealingwithspace,wedohavetodifferentiatetwomaintypeofspacesandlaws:

- Objective, vital topological spaces, made of the 3 bidimensional varieties of topology as adjacent organs,whichwecandefinewiththe‘Generatorequation’ofTimespaceorganisms,whichingeometrictermswrites:

Lineal-limbs/fields<Ø-hyperbolicBodywaves>O-sphericalParticle/heads

As those 3 geometriesmaximize the efficiency ofmotion, the line being the shortest distance between twopoints, iteration, the hyperbole the more complex form summoning all others, and the sphere the highestvolume ofminimal perimeter. The example shows the essence of vital topologic spaces: The forms that aremoresymmetricwiththemotionandsurvivalaction=functiontheyrepresentintherealworldsurviveandarethereforerepeated incloneT.œs(Timespacesupœrganisms).Theobjectiverulesofgeometryarethusaboutvitalspace,aboutthesymmetrybetweenS-formandT-function,S=Tandmergeabstractgeometryandbiologiclawsofsurvivalandreproductionandphysicallawsofmotion.Theyareeasytounderstandandthetautologicaltruthderivedfromthemisindeedtheobvioustruththatwearethevitalspaceweoccupywhosefunctionsintimethegeometriclawsofvitalnon-Euclideantopologymaximize.

Subjective,mentalspaces,mirrorsoftheouterworld.ThissecondgreatfieldofgeometrywasillunderstoodtilltheLobachevski->RiemannR=evolution,whichdefinitelyunderstoodthat‘spaces’aredistortedmindmappingsinsimultaneityofthe‘flowsoftimespacecyclesandorganisms’perceivedoutsideourmembrain.Butagain,the

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vitallawsofsurvivalplayhereaclearrole,asmindmirrorswhichareunfocusedanddistortedintheirjudgmentoftheouterformsdonotsurviveasacorrectperceptionofrealityisneededtomakeitintheexistentialgame.

A key difference between both type of geometries however is the fact that the laws of non-euclidean vitalgeometrythatdescribetheouterworldareallthesameregardlessoftheobserver,astheyareobjectivelawsofconstructionofefficientsuperorganisms.Butthemindspacesareinfinite,oneforeach‘monad’,eachpointthat is a world in itself (Leibniz) even when they belong to the same species, because their self-centeredperspectivewilldiffer.ThedualityofobjectivesingleUniversev.subjectiveindividualmind-pointmakesthoughtlanguagesinflationaryinitskaleidoscopicmultiplicationofdifferentviewsoverthesameobjectsuchas:

Objectivelawsofvitaltopology>∑∝(relativeinfinitysymbol)mentalmirrorspaces.

It is important to stress from the beginning this inflationary nature of mind spaces, as many of them are‘entropy’, disordered unfocused mirrors that do not survive and shouldn’t be study. This is not the case.Mathematicianssponsoranegocyparadox(ego=idiocy)asallhumindsdo,believingallmathematicalformsareworthtoexplore.Butthatisnotreallythecase.AsinBorges’BabylonianLibrarywheremonkeystype∞bookswritingbychanceaBible;allbooksexistinSets,butwhatmatterstousistodistinguishmathematialformsthatdoexist,toapply themwithout errors tothenext layerofmathematicalunderstandingof theUniverse, thepraxisofmathematicalphysicsandanyapplicationofmathematicstostiencethatreinforceswitha3rdlayerofexperimentalevidencethe1st-∆STlawsofgenerationalspace-timeand2ndlayerofnon-AEmathematics.

Anexamplewillsuffice.Thereare infinitecurvesdescribedbypolynomialsbutallcurvesofthesecondordercanbereducedtothecanonicalconics,whichwefindinNaturebecausetheyareefficient.

Sotheveryessenceof5Dmathematics,inthe2introductorycoursesofthosepapers;istoconnectthelawsofvitalspaceandcyclicaltimewiththeformsofthemathematicalmirror,spatialgeometryandtemporalalgebra,whiletimepermittedthe3rdpartofthismethod–theexamplesinallsciences,notablymathematicalphysicsofthoselaws,willcomeinfurtherpapers1.

Westartthenwithgeometrybecauseitisfarmoreevidenttothevisualmindofthemodernagethattemporalalgebra, and so we shall introduce first the very basic of 5D space-time laws and the consider the basicupgrading 5D makes of geometry – the rewriting of the postulates and axioms of Euclidean geometry,completingtheworkofGauss,LobachevskiandRiemannwhoonlyupgradedthe5thpostulate.Andthenwiththose more solid basis, put in relationship to the Gst laws of fractal organisms, slowly review the simplestelements of Geometry at the level of an Introductory university course of mathematics, with those newpostulates.MygoalistointerestenoughafewprofessionalmathematicsforthisCopernicanrevolutionnottodiewithmeasIdon’tknowhowlongIwillremaininexist¡ence.

Youhavethentounderstandthenatureofscientificr=evolutions,whosepioneersalwaysstartwiththebasics,eveniftheyseemverysimpleforascholarwhohaslearnedapreviousmodel,whichdespiteitsrelativelylessaccurate firstprinciplesandpostulatehasbuiltamirror-imagethatsufficestohandlereality,becauseoneofthemarvels of the fractalworld of infinitemonads ordering a flowof continuous entropic, indistinguishabletimemotionisthatallmirrorsareimperfect,reducedimagesofthewhole;soevenaPtolemaicsystemwiththeaddedepicyclescandescribereality,butitisalwaysbesttokeepimprovingtheinitialpostulates,asCopernicusdid putting the sun in the center, to simplify and better understand those orbital ellipses. This is what thecompletionofNon-Euclideangeometrymeans. Ina fewgenerationswhenthemodel isdeveloped in full thebeautyof those renewed firstprincipleswillbe themarvelofhigh school students. As theywillunderstandexperimentalmathematics,whichnowtheyhatebecauseoftheaxiomatic,settheoryandpedanticdiscourse.

Nt.1 Idoapologizeforhavingfailed inmy initialplan,30yearsagocomingoutofColumbiaUniversity,whenmymindwasfreshmyenthusiasmfor5Dandmyhopes formankindhuge,ofgatheringagroupofoutstandingscholarsofall fieldstocompletethetaskofrenewinghumind’sknowledgeoftheUniversewithexhaustive5Dupgradinginallsciences,withmeasthe‘orchestradirector’…Itwasnotallmyfault,asthestaunchrejectionorrathercensorshipof5Dsocialscienceshinderedanychancetogetinstitutionalhelp,andI

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couldnotethicallyhidethelawsof5Dhistoryduringtheearly90swhenIpursuedhelpforacademicresearch.ItistrulyamiraclegiventhehugerejectionIhadallmylifetomyworkthatIcanmanageinmy3rdagestilltoorderthosepapersthatIseeasmytestamenttoalifeinpursuitoftruthandthesurvivalofmankind.

INTRODUCTION.5D:AFRACTALUNIVERSEMADEOFB¡O-TOPO-LOGICTIME§PACEBEINGS.

“Henceforthspacebyitself,andtimebyitself,aredoomedtofadeawayintomereshadows,andonlyakindofunionofthetwowillpreserveanindependentreality.HermannMinkowski

Whenwegooglethe5thdimensiononegetssurprisedbythequantityofspeculativeanswerstoaquestion,which is no longer pseudo-science, but has been for twodecades a field of research in systems sciencesratherthanphysics(:no,theanswersofgoogle,consideringthefifthdimensiontheupper-selfetc.seemtobe very popular, but are to the scienceof the 5th dimensionmore like amedium in earlier XX c. talkingaboutthe4thdimensionasastrologicalawareness,forlackofunderstandingofEinstein’smetricequationsofthe4thdimension).

Thisisthekeywordthatdifferentiatespseudo-sciencefromaproperscientificdescriptionofadimensionofspace-time, theexistenceofametricequation thatdescribesadimensionandallows to travel through it.Why the 5th dimensionmetrics are not well known inmodern science has to do with the fact it is notresearched in physics but systemics, themother discipline of all sciences of information, far less popularthanphysics;andtheproprietaryfeelingphysicistsacquiredonspace-timematterssinceGalileodefinedits3D metric equation v=s/t completed with Einstein’s 4D formalism, which makes difficult to spread theknowledge on space-time acquired on other disciplines. The arguments still raging about evolution, thefundamentaltheoryoftimeintermsofinformation,asthe‘arrowthatdefines’thefutureofspeciesbuthasnothingtodowithRelativityandlocomotionisaclearcaseofthatdifficulty.

Indeed,weknowsincetheXIXc.thatthecreationofthe‘futuretime’ofanexistentialentity isnotONLYmediatedby the arrowof locomotion andentropy studiedbyphysicistswithRelativityMetrics (Galileo’sV=s/t and Einstein’s more complex formalism), but there is a second arrow that defines the ‘future’ ofexistential species - the evolution of its information. So time – the changes=motions that defines theexistenceofanyspecies,hasatleast2dimensions,locomotionororderedtranslationinspaceandamoredisorderedversion,entropy(scatteredmotionthat‘dissolves’theinnerformofthesystem,akintodeath)…

And in-form-ation, generationof form, inverse toentropyas it requires the social gatheringofparts intowholes;happeningwithoutexternal locomotions,asaninternaltrans-formationofform.ThisevolutionoforganicformasopposedtoexternalchangeiswhatSystemicscallsthefifthdimensionoftimethatappliestoallsciences.

In the graph the Universe is a fractal that reproduces 'forms with motion’,informations and then organizes them in networks and systems that evolve intolargerorganicsystemscreatingthescalarstructureofreality. Thuswecallthesumofallthoseco-existingscalesofpartsandwholesthefifthdimension.

Thenitisnecessarytofindametricequationtodefinethisnewdimensionofspace-time.Sinceadimensiononlyexistswhenwecanwriteamathematicalsimplemetricthat leaves thedimension invariantwhenwechangeourparametersof spaceandtime - hence we travel through it. (Klein). This equation, as all space-timemetricequations, is simple; since metric equations are meant to represent measures of‘covariant’motioninagivenspace-timedimensionthatleavetheotherdimensions

unchanged.Sowewriteusingðforcyclictimeinsteadoft,foramotionthatchangestherelativesizeandspeedofclocksofasystem(measuredwithfrequency):

5DMetric:S(LinealSize/Volumeinspace)xð(cyclicspeedofitstimeclocks)=Constant.

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According to thosemetrics, smaller systems in space have faster time clocks. As information is stored in thefrequencyandformofthosecycles,smallersystemshavemoreinformation,codinglargerones:genescodecells,memessocietiesandparticles'quantumnumberscodeatomsandmolecules.

Thisequationand itsuseto improveourknowledgeofspaceandtime inall sciences,withanemphasis inourmodels of physical systems will be the theme of this paper. Even if physicists stubbornly refuse to treatinformationwiththesamevaluethanentropy.Sotheycallitnegentropy,andwhenyougiveaconferenceonthefifth dimension – the dimension of ‘creation of social forms of information, of organicwholes’ - there are nophysicistsonattendance;andlikelynophysicistswillbereadingthispost…Let’sthenusethemetricsofthe5thscalardimensiontoexplainthefractal,nestedUniverseanditsscales,showninthegraph:

Themetricequationofthefifthdimensionofspace-time(ab.∆)defines3knownscalesofphysicalsystems,withdifferentquantityofinformationaccordingto5Dmetrics,Se(sizeinspace)xTi(volumeofinformation)=k.Sinceaswebecomesmaller in spaceparadoxicallyour timeclocksaccelerate,andsince information is stored in thecyclicalpatternsand frequenciesof thoseclockssmaller systemscodemore information, soquantumparticlescode atoms, genes organisms, memes civilizations and chips machines establishing the essential symbiosisbetween∆-1scalesand∆ºsuperorganisms,inscribedinanslower∆+1world.

Thosemetricsmeans information ishigher in thesmaller ‘quantumplane’ than in the largergravitationalone,andinverselythesizeofitsphysicalpartsislargerinstheGravitationalcosmological‘plane’thaninthequantumone,withthehumanthermodynamicscalein-between.

As there is no reason to stop the scales of the fractal Universe in particles and galaxies, there is a ‘potential’fourth,∆±4organicplanedefined‘above’thegalaxy,(∆+4,darkenergyworld)andbelowthequantumworld(∆-4,Bohm’squantumpotential),whichrepresentsthelargercosmos.Furtheron,accordingtothefractal,nestedprincipleanylargerorganicsystem,enclosessmallernestedsystems.Thusthe∆±4cosmoscontains∆±3galaxies,whichcontain∆±2solar systemsandplanets,whichcontain∆±1 thermodynamicorganismsandmatter states,describedbythehuman∆ºmindlanguages,containedonourbrains,whichaccordingtothosemetricswillhaveamuchdensercontentofinformationbecominga‘linguisticMind-Mirror’ofthewhole.

RECAP.Ænthropichumindsreducethemultipleclocksoftimeandvitalspacesofrealitytothesinglehumanclockandspatialscale,andrejecttheorganicpropertiesofotherUniversalsystems.InrealitytheUniverseisafractalorganismoftime=motionandspace=form,whosepurposeistoreproducethoseformalmotions,andpatternsofcyclicalreproduction;withitsfundamentalmetriclawofbalancebetweenspaceandtime,asthe

guidanceofthosemotions.

Theoutcomeofthoseprocessesofreproductionofform,andsymbiosisbetweenthedifferentscalesandsynchronoustimecyclesofitsspeciesisthecreationoforganicsuperorganisms.Itfollowsfromthatnestedstructureand5Dmetricsthatspeedupinformationprocessinginsmallerspaces,asymbioticrelationship

between∆-¡,thesmallerpartsoffastertimeclocks,whichcarrymorein-form-ationintheformandfrequencyofitslogiccycles,andactaslanguagesthatcodethelargersystems:genescodecells,memescodesocieties,

quantumnumberscodeatomsandlanguagescodelargerwholes.

Theunderlyingquestionoftime§pacephysics:AbsoluteorRelational,GenerationalSpace-Time

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Thefundamentalquestionphysicistswonderedforcenturiesregardingthenatureofspaceandtimeunfortunatelywasresolvedasusualinfavorofthesimplerview:itisspaceandtimeanabsoluteabstract

backgroundoftheUniverse(Mr.Newton’sview)orarewemadeof‘vitalspace’thatlastsatimeduration,sowearegeneratedbythebio-topo-logicpropertiesofscalarspaceandcyclicaltime?Thisisthechoiceof5D

‘stiences’.Anditssimplerversionwascalledrelationalspacetime,sponsoredbyMr.Leibniz.Arealistinterpretationoftheworldwelivein,whichhasnevershowninanyscaleofrealitysuch'background'-

ultimatelyamathematicalgraphusedinabstractbyhumanscientists-considersthatweAREthevitalspaceweoccupywithourcells,andweLIVEacyclictimedurationbetweenbirthandextinction.Sowearespaceand

timeandmustextractour'existentialpropertiesfromboth.

InNewton’scosmos,spaceandtimeprovideafixed,immutableandeternalbackground,throughwhichparticlesmove.Spaceandtimearethestageofintersectinglinessketchedintheillustration.Factisthis‘mathematicalartifact’madewithpenandpaperbyearlierphysicists,calledtheCartesiangraph,usefultomeasure'translationinspace'isnowheretobeseeninreality.Unfortunatelyastimewentbythegraphbecamesomehow'real'asscientists'feltthe'mathematicallanguage’createdreality.Itmeantalsothe

inventionofanabsolute'continuousspace'andasingle'linealtime'thatextendstoinfinitycontradictingtheobviousfactthatall‘spaces’arebroken,dividedbymembranes,andallbeingshaveafinitetimeduration.

Furtheron,aswekeptexploringsmallerscalesofreality,weneverfoundthedrawingsofGod,notevenasolidstillsubstance,butalways'motions'tracingclosedtime-spacecycles;sinceevenparticlesturnedtobealso

'vorticesoftime-spacemotions'.

Thetrue,soundexperimentalandlogictheorywasLeibniz’s,whoconsideredabsolutespaceandtimeanabstraction,andsohecoinedtheconceptofrelationalspace-merelytheadjacentpeggingofsimilarformsinsimultaneousspaceandrelationaltime-thesequenceofeventswhichwerelatecausallywithreasonoriginofthe‘Generationalspace-time’modelof5Dinwhicharethespaceweoccupyandthetimewelast–asinthegraphwherethereisnolongerabstractbackgroundlines.WhatNewtoncalledabsolutespace-timeISNOT.

Sospaceisthesumofallthediscontinuousvitalspaces,occupiedbydifferentbeings:∑s=S.Andlinealtime,Tthesumofallthe

finitelife-deathcyclesofallbeingsT=∑t.

Sincespace&timedoexistiftheyarenotinthebackgroundwe‘are’vitalspaceandcyclictime.Thesimpleideabehindthe

structureofthefractalUniverseistoconsidertime=change=motionandTopologic,formalspace=extensionthe2elementsofwhichallbeingsaremade.

WE ARE space and time, merely of a different kind to that of Newton: Organic scalar spaces, andcyclical,discontinuous times who ‘live'… worldcycles (no longerworldlines as we have a 2nd arrow of information) ofexist¡ence (as all species follow the common laws of space and time). As cyclical time that explains theinformative repetitive patterns or Laws Nature and its multiple space-time clocks. Why a Universe made ofspace-timebeingsisessentialtoaphilosophyofmathematicsisobvious.Becausethemainexperimentalscienceconcernedwithspaceismathematicsandthemainscienceofcausaltimeislogic,ifwearemadeoffractalspace& cyclic time, both mathematics & logic become experimental sciences, reflecting the properties of those 2primary substances, as mirror-languages of maximal synoptic information and minimal size (Sxð=C), whoseunderlying laws emerge in all other larger scales of the fractal Universe of bigger size and less information,provingalsowhytheyapplytoallofthem

Cyclictime:ð-thecausalrepetitivelawsof‘stiences’

AUniverseof∞timeclocksofdifferentsizeandspeeddiffersfromitshumandescriptionwithasinglemechanical

clock-timetowhichalltimeclocksoftheuniverseare

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equalized,elongatedintoalineal'second-minute-hour-day-year'systemofequalizedtimeclocks(oflightwaves,mechanicalclocks,earth'sastronomicalclocks).AsGalileanphysics,bornofballistics,simplifiedthenatureofcyclesoftime-spaceintolinealdurations,tomeasurebestthelocomotionsofcannonballs:Timeis

cyclical,allclocksoftimeandlawsofsciencearebasedinthecyclicalpatternsofnature.Butphysicistsdevelopedballisticsanddeniedtheobvioustruththatwecanknowthefuturebecauseitwillrepeatthe

causalityofthepast,andwecanchangeitbychangingthatcausality,inHistorybyrepressingthelethalmemesofthetreeofmetalandenhancethewelfarememesthatmakeussurvive.Ofcourse,linealandcyclicaltime

renderthesameequationsasoneistheinverseoftheother,measuredbyfrequency,T=1/ƒ,butthephilosophicalimplicationsofcyclicaltime,areginormousandthein-form-ationprovidedbythosecycles,erasedbylinealtime,ahandicapforhumanstotrulyunderstandthecyclesofhistoryandeconomics,the‘deeptime’

scalesofthefifthdimension,andthewholeworkingsofsuperorganismsanditsphysiologicalstructures.

Akeyfactthatofatimecycleistobreakreality(1stknottheorem)inanouterandinnerregion,creatingarepetitivemotionthatbecomesanisolatingmembranethatenclosesavitalspace,the‘substanceofwhichwe

areallmade’.

Asecondkeyelementistobemadeof3‘relativepi-diameters’,whichthereforedetermine‘3agesoftime’.

LocalPast=Entropy,Present=IterationandFuture=Informationinzero-sumworldcycles.

‘Theseparationbetweenpast,presentand future isanillusion’.Einstein

Ofalltheconsequencesofcyclicaltime,themostimportantistheexistenceofinfinitelocaltimeclocksofwhichweareallmade,whichthereforeimplytheexistenceofinfinitelocalpast,presentandfuturestates.

Pastthenmeansasystemwithless‘form’,lessinformation,whichslowlyacquiresadimensionofheight-form,asitcompletes its cycle to return back in the moment of death to an age of no information. This ‘worldcycle’ ofexistence,which creates and erases information becomes then the equation of ‘trinity’, the 3 local ages of life,whicheachofusfollowsasatime-spacesuperorganism:

Entropy-youth(past)<Energy-maturereproduction(relativepresent)>3rdageofIn-Form-ation(relativefuture)

InphysicsisequivalenttothedualequationofEinstein:EóMc2,whichreverseswhenE,whichshouldbeproperlyconsidered‘Entropy’,asitisadisorderedstate,collapsesthroughgravitationintoMass,acyclicalvortexofspace-time;whileitsintermediatestateisc2,radiation;therelativepresent:

Whereasthepastisthebeginningofapicycle,startingasalineofentropywithnoformthatcurvesandraisesinheightinitssecondstateofpresent,andreturnsbacktoitsorigininitsfuture3rdageofinformation,completinga0-sumoflifeanddeath.Thusinsteadofasingle∞linealabsolutetimethereare∞livingcyclesoftimehappeninginzillionsofentities.

Sothefundamentalunitofrealityinagivenscale,isthetimespacecycle,as‘time’inasequenceof3ages,whichclosethecycle intoazerosum;orwecansee itsequentially inspaceasa ‘fractalpoint’, that isanon-Euclideanpointwith3parts,theelementsofitsangularmomentum,whichwillbecomethenewunitofbothgeometryandmathematicalphysics.

In the graph,we can see the 3 dimensionalmotions of timespace, the relative past associated to an explosive,expansive topology the wave-body present iterative hyperbolic topology (a geometry with motion) and theimplosive,ellipticgeometryofablackhole,orfuturevortexoftime-space.

Vital,ternary,OrganicTopology

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Thissaidthedevilisinthedetails.Sowhatdoesitmeantobemadeofmotionswithform,timeandspace?Inmathematicalterms,itmeanstobemadeoftopologicaldimensions,whichareholographicbi-dimensionalspace

withmotionintime.Andasithappenstopologyhasonly3varietiesofbidimensionalspacetimeanditisconstructedofparts–points–thatbecomewholes–networks.Sotheimmediatetranslationofgenerationalspace-timeintomodernmathematicalsystemsdoconvertasweobservedintheabstract,allsystemsinto

mathematicalbeings.

Thisisthefieldinwhich5Dinnovates,'enlightening'classictopologyto'understandtheternaryorganic,structure'ofallsystemsofnature:Asthe5Dimotionsaredimensionsofspacewithtimemotion,itsscienceistopologythatallowsasystemtodeform=changeitsinnerform.Yeta4DUniversehasonly3'topologicalvarieties' thatrestrictsensemblestoonly3topologies,eachonebestsuitedtoperformthe3organicvitalfunctionsofanyphysicorbiologicsystem–gauginginformation(1D)tomovethesystem(2D)toanenergyfieldinwhichreproduce(3D):

Thepurposeofvitaltopologyistostudythe5Dimotions(dimensionalmotions)oftheUniverse…Assuchitwillbethefinalstageofevolutionofgeometryas an experimental science, mergingelementsofalldisciplines.All dimensions of space have motion intime. Mathematics realized it, as the stillGeometry of the Greeks evolved into avaster, generalized concept, a topological›variety,wherea topologyasopposedtoageometryhasinternalmotions-changes.As

theonly case inwhich the inner dimensionsof a beingdon’t seem to change is external locomotionmost 5Dmotionsneed‘geometries’withinnermotion,whicharetopologicvarietiesofwhichthereareonly3:Inthegraph,thediffeomorphicPrincipleofEinstein’s4Danalysisacquiresanorganicnature,whenweseetheUniverseasthesumof∞Complementaryternary,topologicsystemswhosedimensionshaveorganicfunctions:Systems feed on their relative dimension of energy-length, perceive in their relative dimension of height andreproduceintheirrelativecombineddimensionofwidth,whichareassembledintoeachspecificspecies,tobestsatisfythesystems’intakingofmotion,energyandinformation.I.e:Ananimalhas its informativeheight inthehighperceptive lightdimension,butaplant,whichuses lightasenergyhasitsupanddowndimensionsinvertedrespecttothemananditschemicalbrainburiedontheEarth.So both have opposite energy-time coordinates, with an ‘antero-posterior’, lineal’ ‘outward’ energy orientedstructureduetotheorientedarrowof light.But ina3Dworldwithnopreferredorientation,aseaorvacuum,cyclicformsthatmaximizeinformationdominatefromplanktontogalaxiesthathaveacyclic,informative,inwardstructure, as the stars’bodyabsorbsenergy from intergalactic space, reproducesmatterwith it and feeds theinternalinformativeknotofgravitation,withahigherheightdimensiontheblackhole.

SotheDimotionsofrealityarethe3bidimensionaltopologicalvarietiesthatactasvitalorgansincylindricallonglimbs/fields,Hyperbolicwidebodies and spherical tall heads, eachonedominant in a lineal classic dimension,linealmotion,informativeheightandreproductivewidth,whichDOhaveorganicvitalpropertiestoo:

Sphericalparticle-heads,perceivinginformationfromtheadvantagepointofheight.

Lineallong,cylindricallegsandfieldsoflocomotionasthelineistheshorterdistancebetweentwopoints.

Wide,hyperbolicbodywaves,storingtheenergyreproducebythesystem.

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Assuchsingledimensionsperformthoseorganicfunctionsinlightwavesthathumindsperceiveasspace.Solightisalsoanorganicsystemof3dimotions:c-speedlength,electricinformativeheight&awidemagneticfieldthatsupportsthem.

Moreoverbeyonditsclassicanalysisofformsandfunctions,whatVitaltopologydoesittoservethebasisforitstemporal-numericalversionthatisexistentialalgebra,aspointsbecomethespatialversionoftemporalnumbers:Sowe develop a formalism of the 5 Dimotions, as operand, «,<, =,>,», for entropy, locomotion, reproduction,information and social evolution, which will develop into flows of ‘stœps’ that change sequentially a formthroughitseventsintime,wherethevitaltopologyofeachDimotionis‘integrated’withinthosesymbols.Inthismanner vital topology becomes ‘Existential algebra’ –the analysis of flows of dimotions=actions of T.œs in itsworldcyclesofexistence.ThosesequencescanthenbestudiedastemplatesofallT.œs inallscales,whichwillfollowthemtocompleteitssurvivalcycles.Andsoexistentialalgebrahasimplicitvitaltopology.SowedefineinexistentialalgebrawithsimpleT,Sand5«,<,≈,>,»themaineventsofspace-timeoftheUniverse:

1D: T>S: angular cyclical motions of information (Ab. §ð): the minimal ‘geometry’ of reality, a sphericalparticle/headorfractalpoint,thegeometrythatstoresmaximalforminminimalspace,hencesuitedfor‘organicfunctions’ofgauging,storingandperceivinginformation(particles,heads).

2D:S<T:LinealLocomotions,(Ab.$t)whichwillmovethroughitslineallimbs/fieldsthesystem,asthelineistheshortestdistancebetweentwopoints…towardsa…

3D: S≈T: Fields of vital Energy (Ab. ∑≈∏): with its hyperbolic body-waves that iterate the forms of both thespherical particle/heads and lineal limbs/fields; as the hyperbolic topology combines the other two, so it cangenerate them, in the same manner Energy adds as the third conserved space-time quantity the lineal andcyclicalmomentumof1and2D.Towhichwemustaddthe2‘scalar’Dimotionsof:

5D: entropy («, ∂S) whereas motion is dual internal dissolving the information of the being and external,scatteringitsparts,henceweusean«¬Æsymbol;sothesystemexplodesintoits∆-1parts:∆«∆-1(death).4D:organicevolution(»,∫T)ofpartsintostilllockedsimultaneous‘linguisticseedsormindforms’∆-1»∆

Sotheabstract3conservedsubstancesofrealitybecomeorganicbidimensionaltopologies-flatmotioninspace,cyclicaltimemembranesandthevital3Denergywithinthem.Wethenobserveits∞varietyofcombinationsastopologic ternary species,whereas the3perpendicular 'lineal'dimensionsarea simplificationof thoseorganicfunctions: theheightdimensionenhances the 'perceptionof information' byO-heads/particlesplaceabove inthepointofmaximalprojectivegeometry;thedimensionof'length'maximizeslocomotion;physicists’onlytimemotion,andthewidthdimensionmaximizesreproductionandstorageofenergy.

The3onlytopologiesofphysicalsystems:conservationlaws.

The3partsofanyfractalpointofcyclicaltime,ofangularmomentum,intheUniversebecomethenin the simplest physical systems, the 3 conserved parts of the minimal organic Unit of reality, a‘Planckton’ofangularmomentum.Allphysicalsystemsthencanbe'reduced'tofractalensemblesof3 'conserved quantities', angular momentum - the membrane of the system, which becomes amembrane,fractalsumof'cellularcycles'ortheskinofthesysteminhumanbeings.Vitalenergy,the

enclosedcyclical formsandmotionswithin thespacewhoseboundaryconditionsaregivenby themembrane,andlinealmomentum,themotionswitha'finality'weperceiveguidedbya'relativelystillmind-point-singularity'thatfocusestheenergyandinformationtransferredthroughtheangularmomentummembrane.

It isalsothesmallestclockofourworld,asitsminimalunitofcyclical information,orangularmomentum,used as the unit of the 3 human physical parameters of spatial size, cyclical time frequency and ‘scale’(Activemagnitude):

h=mass(∆)xarea(S)xƒrequency(ð).

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Soh,Planckton, is theminimal fractalorganismwhichbecomes the ‘cellularunit’ofallother speciesof lightspace-time,asPlanktonistheminimalunitofthebiologicUniverse,alsoacellwithasimilarternarystructure–aDNAnucleusthatprocessinformation,aproteinmembranethatisolatesavitalspace-time,thecytoplasm.

ThusNon-Euclideanpointshavebreath,itslinesarethereforewavesabletocommunicatetheexternalformandinternalenergyorfractalnetworksthatbranchtoconnectmultiplepoints,anditsplanesintersectionofthreeofsuchwavesornetworksthatformtopologicalorganisms... It isthenobviousthatthenextstepofNon-Euclideangeometryistomergethoseconceptswiththephysicalanalysisofthesmallestphysicalsystems,tounderstanditsvitaltopologies.Tothataimwehavetointroducethesecondfundamentalequationof5DmetricthatformalizestheParadoxofrelativity.AndsoitactsinthephysicalmodelasthePostulateofrelativity,commonto3DGalileanand4DEinstein’ssimplexmodels,whichcorrespondto5Dinasingleplaneof‘lightspace-time’…

Sowehavetransformedthe5Dimensionsofspace-timein5vitaldimotions,brokenintoinfinitevitalspace-timebeings, and now we have the 5 elements which reality uses in different perspectives to construct all realities.Nothing else is needed. And it will easily follow that in each stience, including mathematics, 3±¡ elements(dependingontheperceptioninasingleplaneorinseveralones)willbeconcernedwiththeanalysisofasysteminSimultaneous,entangled,still spaceasasuperorganismconstructedwiththose5Dimotions,which insequentialmotion time will trace a worldcycle also composed of those 5 dimotions. As the universe simply put it is areproductivefractalof5Dimotionsofspacetime…

Asmotionwith formconstantly reproducesby themere factofmoving. So the fractal reproductivenatureofallwhatexistsisimmediate,asallwhatexistsmovesform,reproducesform.

5DMETRICSANDITSFUNCTIONOFEXISTENCE:THEGENERATOREQUATIONSOFALLSTIENCES.

The judge and the 4 witnesses represent 5povs. to obtain a partial truths as truth only exists in the being or event in itself that holds all the information. So we need a pentalogic of 5 Dimotions for reality to emerge and ‘persist’ through synchronicity and simultaneity.

Inmathematicalscienceforadimensionofspace-timetoexist,itrequiresametricequation,whichcombinesspace,andtimeto

givesusaco-invariantsystemthatallowstravellingthroughsuchdimension.Howwedotravelthenthroughthefifthdimension?Asystemtravelsthrough3scalesofthefifthdimensionbyacceleratingitsevolutioninasmallerscalethroughaplacentalcycle,emergingasanorganisminthelargerworld,tolive3ages&dissolvebacktoitspartsinthe0-sumdeath.Andthisisthemeaningofexistence,anditsreasond’etreistheSIMPLEMetricequationsof5D,whichstructuresthroughsynchronicityofthedifferentspeedsoftimecycles,thedifferentscalesofreality.Sothe∆ºorganismeatseveryday,anditsfoodprogramsthefastercycleofreproductionofitscells,asthemooncycleprogramsthemenstrualcycleofwomen,astheyearcycleofrotationofEarthprogramsthereproductivecycleofseasonsandsoon.

Soanessentialpartof5Dtheoryistheanalysisofsynchronicity,simultaneity,emergenceandthein-depthanalysis–notdoneinthisintroductorycourse,reservedforthemorecomplexpaperson‘pentalogicanddodecalogic’wherewefollowinmoredetailtheconstructionofsimultaneoussuperorganismsanditsternaryworldcycles..Allthisofcourseisstudiedbyhuminds,aseverythingwetalkabouthere,butwithouttheproperconceptualframe,lackingvaliddefinitionsofplanesofspace-time,timecyclesandfractalspaces.

Itisforthatreasonwedoneedanewformalismwehavecalledexistentialalgebrawithitssimplesymbolsofwhichthemostimportantarethe5bidimensionaldimotionsofspace-time,whichentangletogetherthroughsynchronous,simultaneousemergentprocessestocreatetheapparent‘solidity’and‘stillness’ofreality.

BecausetheUniverseismadeonlyoftwopolarelements,stillminds(SS,ab.§)andTemporalentropy(TT),andits3dimotionalcombinations,St-information,Ts-locomotionandS=T,reproduction,whoseinteractioncanbe

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resumedinthefunctionofexist¡ence,Max.SxT(s≈t)=C,wecandeducealltheprinciples,laws,eventsandequationsofallstiencesfromit.SoweshallcallExistentialAlgebratotheGstformalismofGenerationalSpacetime(ab.¬Æ),anddoexactlythat:deduceallequationsandlawsofstiencesfrom5Dmetrics.

Weshallthusmakea1stforayonexistentialalgebra,showinghowthe‘development’of5Metricsgivebirthtothe function of existence into its 3 ‘extremal points’ or ages , Max. S x T (3rd age), Max. T x S (youth), S=T(maturity),definestheworldcycleofexistenceofallbeingsinitstwodirections,forwardsandbackwards.

But5Dmetricscanbestudiedinmoredepth,roughlyspeakingin4sub-equations,whicharethefoundationsofthe43greatsubdivisionsofscience:

-Thephysicalequationofrelativity,S=T,basisofallphysicalandmathematicalstiences.

-Thebiologicalequationorfunctionofexistence,Max.SxT(achievedpreciselywhenS=T),thebasicequationofallbiologicaldrivesandevolution/

-Andtheequationofthemind:0-mindx∞Universe=Constantworldthatcreatesmentalspaces…whichwewillconsiderinthenextparagraphsaswehavedefinedspaceandtimemoreproperly.

-Finallyhoseequationcanbefurtherunified,sincethemetricequationofmultiplespacetimescales,SxT=K&therelativeequationofdualmotion/stillnessinasingleplaneS=TthatmaximizesSxT=K(5x5>6x4)unifyin1‘existential’equation:Max.∑SxT=C±¡:∆±1,whosestudyisthefieldofPhilosophyofstienceanditsnewformalism,ExistentialAlgebra(ab.¬æ).Soafterstudyingthe3classicfieldsofsciencewillreturntothose5Dimotions,SS,St,sT,ST,TTandstudyitsentanglementanddifferentpropertiesandcomplementaryoppositions,tostartbuildingtheformallawsofexistentialalgebra,theformalismofGenerationalspace-time.thatallstiencesnotablymathematicsandlogicmirror.

Letusthenstartwiththephysicalanalysisofrelativityanditscorrespondencewith5D.

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RELATIVITY.THE5DIMOTIONSOFEXIST¡ENCE.

Galilean Paradox: SóT: Relativity of space Dimensions=Forms=Motion in time: The 5 UniversalDimotions

Galileo’stimeandspacePrincipleofRelativityisthefundamentalconceptualthoughtbehindtherelationshipbetweentime=motionandspace=formandhowonecanbeconvertedintoanother:Allwhatexistsismadeofspace=formandtime=motion.Andyetphysicistsknowthatwecannotdistinguishmotion

fromform.Thatanybeinginmotionfromitspointofviewseemstobestillandallotherthingsmovingaroundit.ThisistheprincipleofRelativityofmotion.

Physiciststhenwithoutmuchthoughtaboutthatfascinatingduality,wentontousemathematicstocalculatetherelativemotionofeachentityofrealityrespecttoothersystem,whichseemsstaticfrombothpointsofview.ThisiscalledGalileanrelativity,latterrefinedbyEinstein'srelativity,andessentiallyisconcernedwiththemathematicalcalculusofwhatweshallcallthe2ndDimotionoftime=change,locomotion.Fine,butwearemoreinterestedonthedualityofspace=formandmotion=timeanditsentangledrelationships–thereasonswhywedoNOTsee

togethermotionandform,evenifallsystemshaveboth.

Theconclusionisthenratherobvious:oneofthetwoparametersofrealityis'hidden'toperception;weeitherseemotionorform,'wavesorparticles'(quantumcomplementarity),distancesandlinesorpointsinmotion(asinthenightwhenfastcarsinapictureappearaslines).Sophysicistscalculateonlyonewheninfactwemustassesstheexistenceof2;andsincewecannotdistinguishthem,logicallywemustequalthem.‘Form=motion-

function;space=time;S=T’.

Relativitythenbecomesaduality,S=T,isattheheartofeverylawoftheUniverse.Butwhichoneistherealelement?Obviouslytime=motion.Spaceisa“Mayaofthesenses"–asliceoftimemotion.Theultimatesubstanceismotion.Formiswhata'stillmind',makesofthatmotionto'perceive',information,forms-in-

action.

SinceweseeEarthstillandflatbutitisroundandmoving.Galileo’sprofessionwasballistics-thestudyofcannonballsmotion.SohechoseONLYmotionandlostthechancetostartphysicswithacomplexphilosophicalunderstandingofitsS=TdualPrincipleofrelativity,whichPoincaredefinedlatterclearlywhenhesaidthat‘wecannotdistinguishmotionfromstillness’.Anexampleisquantum/relativityduality.Indetailquantumspacehas‘darkenergy’becauseithasexpansivemotionthatextendsintoaplaneofspace,butwhenseenatlargerscales

withoutdetailitsentropicmotionseemsstaticspace-adualareaofscatteringlengthandwidth.Sointhegalaxyweseeeitherdarkenergymotionorexpandingspace:T=S.Amotionoftimeisequivalenttoadimensionofspace:Distanceandmotioncannotbedistinguishedsotheymustbetakenastwosideofthesamebeing,a

space=timeÐimotion(ab.DimensionalMotion):

S=T;Dimension-Distance=Time-motion=STÐimotion

WhyiftheEarthmovesintime,weseeitasastillforminspace?BecauseRealityisaconstantgameofinfinitemotions,butthemindfocusinstillnessthosemotions,andmeasuresthematdistances.For‘huminds’motionisrelativetooursystemsofmeasureandperception,whicharelight-based;henceafixedc-rodspeed/distance.ReasonwhyEinstein’srelativitypostulatesamaximalT:c-speed,measuredasifobserverandobservablewere

stilltoeachother(ConstantS);whichatourscalewe‘correct’withLorentzTransformations.

PhysicistsjustsubstitutedEarth’sstilldistancesformotions.Ittook300yearsforEinsteintorealizetherelativityofmotionanditsmeasuremadeessentiallytimeandspace,motionandformtwosidesofthesame

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coin.Stillthisrealizationwasnotexploredphilosophicallyandsoitgavebirthtoaseriesofill-understooddualitiesbetween'statesofmeasureandform'(particles,headgaugingform,in-form-ation)and'statesof

motion'(wavestates).

Itisthenessentialtograspthatmotionandformco-existas2differentstatesdependingon5Dscaleanddetail:Motionsareperceivedbymindsthatstopmotionintoform,intoinformation,asdistances.Soifweseeslowmotioninthenight,acar’sheadlightseemsalongdistanceline‘still’picture.Butthismeansalsothatthe3

‘Euclideanstilldimensions’musthavemotion;theyare‘bidimensionalST-holographic,topologicdimotions’.Sowehave3Space+1Time+15thdimensionofscales=5Dimensionalmotions.NoneofthemisaDimensionofpurespatialformorapuretimemotionbutacombinationofboth.Evenifmentallywetendtoreducemotionandfocusonforms,allhasmotion=time,andform=space:thisisthemeaningof'spacetime',themessingof

bothinto5dimotions,thefundamentalelementofallrealities.

Relativitystates‘wecannotdistinguishmotion=timefromposition=space’.Soallwhatexistsisacompositeofboth,undistinguishableS=T,5‘Dimensionalmotions’(Ab.Dimotions),brokenininfinitefractal,vitaltimespaceorganisms, composed of topological Dimotions: height=information; length=locomotion;width=reproduction;form=socialevolutionofpartsintowholes&entropy=dissolutionofawholeintoitspartsinalowerscaleofthefifth dimension (term we keep for the whole range of scales of the Universe); whose study is bothmathematical,themainsciencethatstudieshowthose5DimotionsentangleinsimultaneousSpace,connectedtoeachothertopologicaladjacentparts,whichcreatesuperorganismandLogic;themainscienceoftimethatobserves how those pentalogic, entangled superorganismsmove and evolve, change in sequential relationaltime,livingaworldcycleoflifeanddeath.

Sincethereisnothingelsethantimeandspace,the2experimental‘mirror-sciences’oftimeandspacebecomethemost importanttoextractthe‘Disomorphic=laws’ofthose5Dimotionsthatallsystemshaveincommon.SincewhilethoseDimotionsarebroken,invitalorganisms,separatedbycyclicaltimemembranes,theyarethesame.

InthegraphGalileanrelativitywasillunderstood,asthetruequestionabouttime-changeiswhy‘themindsees

spaceasastillcontinuum,whenindetailismadeofsmallerself-similarquanta,inmotion.Theparadoxdefinesmentalspacesasstillsimplifiedviewsofthemorecomplexwhole.

The3¡logicparadoxesofspacetopology(closedin-form-ativecurved-Ovs.|-open,freeentropiclinealforms),time-motion(stillnessvs.motion)and∆-scale,(continuouswholevs.discreteforms;singlescalevs.multipleone)s,areessentialtotheperceptionofasimplified‘spatialminduniverse’inasingleflatstillplane

vs.thefull,moredetailedcomplexpictureintime,ofacurved,discreteandmovingUniverse.Thoseparadoxesresumethe5elementsofreality,Space=form,time=motion,scalesandthemindthatmeasures

them,withinitsownentropiclimits.

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I.∞MINDSPACES.

Mindsaremirrors thatperceive in its inner ‘still simultaneous language’, thee-motionsof timeconverted intoinformationsofspace,intheeternaldialecticbetweenfractalpointswithavolumeofstill linguisticperception,mappingitslocalUniverseandflowsofe-motionaltimewithitsvitalsensations:

Uppergraph,humanegossubmittedtothemindparadox,thinklanguages(wordsinAbrahamic,creationist

religions,numbersincreationistscience),knownonlybymanand'God'apriori,createaposterioritheUniverse(Copenhageninterpretation).Theoppositeistruth:amindexistsinallsystemsinwhichtimestopstoform

space.Ingalaxieshappensinrelativityequationsinblackholes,itsmind.InthermodynamicphysicsintheeyeofanEddie.Inquantumphysicsinthecenterofanatom,orcharge.Withoutlinguisticmindsthatorderbyreflectingitssmallermindintoitslocalterritoryrealitywouldnotexist.Theonlywaytocreatefractalsis

throughmirrorimages.

Inallscalesofstiencemindsfixthemotionsoftimeintospatial,linguisticforms,mind-mappingsthatreducethewholewithitssynopticlanguagetofitinabrain,anatom,anyparticlethatactsaftergauginginformationintheUniverse.

Thuswedefine‘Maxwellian’sdemons’oflocalorderinallscales–physicalmindsastheinfinitesimalpointsthatcreateorder inphysicalsystemswiththesameDisomorphic lawsthatallothersdo inmorecomplexscales.Aseachmindordersasalinguisticgodaterritoryarounditself,itsfractalbodyandentropicworld.

Sothecreationofscalesofreality isasimplegame,inwhichapointmind‘reducesreality’toits infinitesimalformand thenprojects into its local territoryof order,whichwill reflect at scale, the largerwholeorworld,whichthelinguisticimagereducedandthenenlargedbackintoitsterritorialform.

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TheparadoxesofRelativity,discontinuity,partsandwholes,scalesareallrelatedtotherefductionistnatur3eofmindsthatbiasreality.

Mind'sreducedimensionstotherelevantones,eliminatingalldarkspaces:continuityIstheresult.

Motionisreproductionofforminalowerscale.Bohm’srealism:quantumpotentials.

How a systemmoves; in a crowded universe, where we ‘are’ vitalspace-time? the answer, which resolves also Zeno’s and quantumcomplementarityparadoxes, is ifwedonotmovebutreproduceourinformation, translated into a lower faster wave scale of the fifthdimension; as we reproduce our sound in faster electrons to

telephone or nerve impulses into chemical dopamine to jump discontinuous neurons. Somotionbecomesscalarreproductionofform,andsinceallisaformofmotion,allisreproduction,whichisthedefinition of amathematical fractal, a feed-back reproductive equation; 5Dmetrics,which become then the‘function of existence’ whose goal is to reproduce the form of all systems – the simpler oneswithmaximalmotion-translation in space, the complex ones with min. motion as a reproduction that emerges betweenscales.Andthisgivesbirthtotheworldcycle.Considerthecaseofquantumphysics:

Ingraphaparticlereproducesinadjacentregionsthatfadeaway,andtheresultistheperceptionofawaveofmotion.InBohm’srealistmodelthisreproductionhappensinalowerplaneofquantumpotentials,wherealsoentanglementhappens,whichisthe∆-4scalewhichisv>cin5Dmetrics(Min.SxMax.T=C),hencereal.

Motionthenisreproductionofformoversuchpotential:thewaveerasesformintomotion,theparticle isastillstatethatgaugesinformationentangledtootherparticle,fermionandboson,stilltoeachother–despitetheperceptionofrelativemotioninourscale–hencetheinformationelectronssharehasalwaysac-constantspeed. Thus the Lorentz transformation are objectively real for mankind who eliminates the stop state ofparticlesaswedoinamovieeliminatingthestopframebutifwewereobservingrealityfromtheperspectiveof an atom,wewould ‘stop’, entangle in the quantum potential, neutrino scale & so eliminate the spookyeffectsof‘timedilation’&‘length’contraction,fromourperspective(butnotofmassincreaseasitisascalareffect). This is the 'rational' 5D explanationof both the c-constant of light and entanglement; as electronicbeingsperceiveinformationin'stoppositiontoeachother’andmovein'wavestate'.Motionsareperceivedbyparticlesthatstopmotionintoform,intoinformation,asdistances.So4DrelativityneedstobeexpandedtothescalarUniversebeyondthec-speedlightlimitofthegalaxy.

Withinthiscomplexview,themodelsofNewton,GalileoandEinstein’sspace-timecorrespondtothelimitof5Dwhenwesimplifyalltheworldcyclesoftime,wecalllife&deathtoasinglemechanicalclock,elongatedtoinfinity& perceived in a single scale of space. Let us then deduce from those 2 equations the fundamentalequationofreality:

Those2polesof realityare the firstprinciplesof any scientific inquire, evenprior to the languagesof time-motion, logic and spatial forms –mathematics, thatbettermirror its laws. Lookaround yourself, everythingthatyou see isa formwith inneroroutermotions.Thoseare thus the2primaryelementsof reality;whichmind languagesperceivemostlybyreducingthescalesofthefifthdimensionand itsmotionstotheminimalpossibletofititall inthemind‘equation’:O-mindx∞Universeofformalmotions=Mentalworld–reducedmirroroftheUniverse.

Thefunctionofexistence:Reproductionofform.

PhysicistsmadetheGalileo’sparadox,thecornerstoneoftheirtheoryofmeasure,buttheyfailedtostudythedeep implications it has for everyaspect of the structure of theUniverse, from the duality between spatialmental, linguistic forms and physical motions; to thebalances achieved by the similarity ofboth space andtime,whichbecomesthefundamental'equationofpresent'S=T,andhencewiththemetricequationofscales,

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$xð=K,thetwoessentialequationstoformalizesingleplanesS=T,andmultiplescalesofspacetime.YetasS=TmaximizesSxT=K (5x5>6x4).Weunifyboth in asingleequation:Max. S x T = C,whichdefines foreachfractalvitalspace-timeorganismitsFunctionofexistence,asallspecieswilltrytomaximizeitsmotion-entropy-timefor its field-limbs, its information-spatialstatesfor itsparticle-heads,whoseproductwillgiveus itsvitalreproductiveenergy.Butasallsystemsmoveandmotionisreproductionofformwecanadafinalfactor,∑,reproductionofparts,whichtomaximizethat functionbecome joined into largerwholeswhicharestrongerthanindividuals;creatingnewplanesofexistence.C=Max.∑SxT(s=t);whereasCactastheentropiclimitsin∑-scales,T-ime&S-pace,boundariesbeyondwhichthestillminddoesn’t’perceiveorcontrol.

Itisalsoasurvivalbiologicequation,becauseitimpliestoprovidelinealmotionto‘fields-limbs’,absorbenergyto reproduce our bodies-waves, and information to guide our motions with particle-heads. So reality is a‘struggle’ forexistenceas systems reproduce its Ts-fields-limbsofmotion, S=Tbody-wavesof energyandSt-particles-headsofinformation.ButasallT.œsarefractal,broken,itsgrowthhasalimitonthefightwithothersystems,whichtrytomoveandreproduce. IntermsofpureT-motionandpureS-form,weconsiderthenthewholeofmaximaltime=motion=entropyorTTandMax.space=form=stillnessorSSthe2limitingDimotionsforany3ensembleTs<ST>St-system.

Wedefine theUniverseasa fractal that reproducesandensemblesSpace-timeDimotions into supœrganismthrough ∞ relative scales of spatial size and time-motion; whose Fractal generator (mathematics) Metric(Physicaljargon),‘functionofsurvival(Biology)orFunctionofexistence(logicJargon)writesC=Max.∑SxT(s=t)

Weshallprovethatallrealitiesarealwaysareproductiveradiationofafunctionofexistencealong5Dscales.

Wearemadeof the5Dimotionsofspacetimeof theUniverse.Weareensemblesof those5Dimotions,whichseeninsimultaneousspacegiveorigintothevitaltopologicalorganismsoftheUniverse;whosestudythereforeismathematical,thescienceofspace;andobservedinsequentialrelationaltime,liveaworldcycleoflifeanddeath;andsincethereisnothingelsethantimeandspace,those2fundamentalexperimentalprimary‘mirror-sciences’oftimeandspacebecomethemostimportanttoknowwhatallsystemshaveincommon,its‘Disomorphic=laws’extractedformthenatureofthose5Dimotions.

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Nt.1.AccordingtotheCorrespondencePrinciplesasphysicsnamed4Dimensionsweusethename5thdimensionforthewholerangeofscales,butinproperterminologyweshouldcalleachDimensions,a‘Dimotion’andconsiderthe5thscalardimensionthesumofallthe4thdimotionsofsocialevolution&allthe5thdimotionsofentropicdevolution

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THEWORLDCYCLE–THEFUNCTIONOFEXISTENCE.

Theworldcycleofexistence:Wethenputtogether5Dscales,vitaltopologicspace&cyclictimetodescribesimultaneoussuperorganismtracingaworldcycleintime,thefundamentaleventofreality.

All what exists is a supœrganism of vital space tracing a 0-sum worldcycle of time through 3 scales of5thdimension: Born as a seed of fast time cycles in a lower 5D scale (∆-1:Max. T xMin. S), emerging as anorganismin∆o,living3agesofincreasinginformation,asitstimeclocksslowdowninits∆+1worldtodieinatimequantabackto∆-1.

So,absolutespacetimeisthesumof∞Timespacebeings,observedinspaceassimultaneoussuperorganisms,intime,asworldcyclesofexistencebetweenbirthandextinction;asallsystemsareborninaseminalseed,of

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faster time clocks, in a lower scaleof the fifthdimension, growing socially (4th dimotion) till emerging in theorganic scale, where they will live 3 ages dominated by one of its 3 topologic organs and itsfunctions=dimotions, a young age of maximal locomotion, dominated by its limbs/fields, a mature age ofreproduction dominated by the body-wave and a third age of information dominated by the informativedimotions, which finally exhausts all energy and as time-space never stops, it reverses its dimotion frominformationtoentropy,explodinginthemomentofdeath.

So we marry the 3 vital functions=motions of time and the 3 dimensions of space, either in 1 or 2D(height=spherical information, length=planar locomotion, width=hyperbolic reproduction) whichmerge in allTime-spaceBeings;anddominateoneofthe3agesofitslife-deathworldcycles,thepast,youngageoflimbicentropic motions, the mature reproductive age dominated by the hyperbolic body/wave and the 3rd agedominatedby the informativeparticle-head,when the illusionof timeendswith an entropic big-bangdeaththatdissolvesthebeingintoits ‘scalarcellular,atomicparts’,whichleadustotherealizationthattimecyclesNOTonlyreturntoitsorigininasinglespacetimecontinuumbuttheymoveupanddownthescalesofthefifthdimension:

The3agesofexistenceofspace-timeorganisms.Its2worldcyclesandMetricequations.

TheFunctionofExistenceofanyspace-timeorganismcanbedevelopedasafeedbackequation,S<=>T,in3sequentialphases/ages/horizons,between3‘standingpoints’(changesofphase):Max.T=motionxMin.S:form=movingyouth;Max.SxT(s=t):reproductivematurityandMax.SxMin.T=informative,oldage,astheequationsofthe3agesoflife,betweentheseedofpurelinguisticformborninthelowerplane:S¡-1anditsT¡-1entropicdeath,backto∆¡«∆¡-1:∆-1»∆º:Thecycleororganismstartsitsexistenceasaseedofpureform(4D)whenitsspace-timefieldiscreated.sT:Itisthefirsthorizonor‘energy,youthage’ofthecycle,inwhichenergydominatesthesystemandsowewritethisphaseas,max.$txmin.T.Max.SxT:s=t.Itisthepresentbalancedageofthecycleorclassicageof‘life’,whenenergyandinformationareinaconstantproportion.Itisthemostefficientage,whenthecyclereproduces.

Max.Txmin.S:3rdageofthecyclewheninformationhasexhaustedthespace-timefieldthatwarpsintoitself.∆º«∆-1:0SxT:Itistheendordeathofthecyclethatreversesitsformandbecomesenergyagain.

Existenceisan∞sumofspace/timefields,fluctuatingbetweenbirthandextinctionthroughthose3phasesorages.The3agesofTimespacesupœrganismshappeninallsystems,includingmentallanguages:

InStatePhysicstheyare,$t-gas,themovingstate,S=Tliquid,thebalancedstateand§ð-solidtheinformativestate; intoCosmology,where it describes theUniverse as a space-time system that fluctuatesbetweenbothlimits,aformofpuretime,thesingularity(min.$txmax.ð§)andaformofpurespace,thebig-bang(max.$txmin.ð§). InBiology, theyare the3agesof livingbeingsANDthe3horizonsofevolutionof species. In socialorganisms, through the subconsciouscollectivemindofcivilizationswhich inart stylesmimic ina longer800yearcycleof lifeanddeathofcivilizations(accordingto5Dmetricsahumansocialsuperorganismis larger inspace–anation,culture,religion–andsoitliveslongerintime).Sowefindthe3agesoflifeemerginginthe3agesofculturesand its3artisticstyles:Min.SxMax.T(infantileepic, linealart,as intreccento,Greekkuroi;S=T;balancedbeauty,whenformandsizeareinbalance,theclassicmatureage;andMax.SxMin.T:baroque,3rdageofacivilization,whosesubconsciousmind is theartof its 'neuronalartists', theageofmaximal formandangstforanofuture,whichistheageofwaranddeathofcultures).

Finallywe talk of 3∆±1 scalesofworldcycles as thebeing live in a placenta, thenemerges as organism in aworld:

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þ: 0-1: its palingenetic o-1 social evolution in the accelerated time sphere of existence, till becoming 1 (0-1bounded unit circle in ¡logic mathematics; quantum probability sphere of particles in physical systems;palingenetic fetal age in biologic systems; 0-9 memetic learning childhood in social systems). It is a highlyordered worldcycle as a placental mother-energy world nurtures a memorial cyclical spacetime that erasederrors of previous generations. - c: Theouter1-∞world, inwhich itwill deploy its2ndworld cycleofexistence inanenvironmentwhich isopen, entropic (1-∞ hyperbolic unbounded Cartesian plane in ¡logic mathematics; thermodynamic entropicstatisticalmolecularpopulations inphysics;Darwinianstrugglebetweenpopulations inbiology; idol-ogicdog-eat-dog capitalist, nationalist competitive eco(nomic)systems in the super organisms of history. In this 1-∞existence the world cycle is not ensured to continue, as the entropy of the world system can cut it off.ω:Theexistentiallifecycle,thoughispartofalargerworldofhierarchicalsocialscales(§D¡),whereitperforms5 survival actions through ∆±4 Planes self-centered in its mind, beyond which it cannot longer perceive, tobecomeifsuccessfulanewsuperorganismoftheinfiniteplanesofGod,thegameofexistence.

Let us now consider with a physical and biological example, to which extent the laws of 5D enlighten ourunderstandingof reality. In the graph,Matter States arephysical time ages, from left pure solid, crystal, §topstate,toanevenmoresolid∆+1bosoncondensate,etc.Weseethatsystemseithermoveastepatatimewithinaplaneofexistence(gas,liquid,solid)ortheycanjump«twostatesatonce,(asinthecasesublimation)withinthat plane, or most often between two planes, as in « scattering & entropic death), to become a differentDimotional state. We can then see how the fundamental elements of 5D time appear on the graph: theworldcycleislocalandcomplete.Therearetwoinversearrowsfromanentropicpast(plasma),inalowerplane(ionparticles)tothe3agesofthematterstateswith increasingform(gastosolid),toendinahigherplaneofexistenceasaboson-Einsteincondensate.

Worldlinesandclassicrelativity

Howallthisrelatesto4Dclassicrelativityissimple.Infirstplacebecausewestartfromduality,formandmotion,Ox|ingeometricterms,wedohaveNolongerworldlinessbutworldcycles.Foraphysicistwhichonlymeasureslinealtimemotions,ifyouaskhimwhatisthefunctionofexist¡encehewill juststate:thesumofmotionsthatyoumakethroughyourentirelifeanddrawa‘coneoflight’explainingyouthataslongasyougobelowc-speedyourworldlinewillbespace-likeetc.Allthishasnothingtodoofcoursewithyourlifeexperiencedthroughthe5complexdimotionsofreality,butthephysicistwillnothavetheslightestdoubt,ashehasreducedperceptiontothatsinglemotionthatheistalkingseriousmetaphysicsofbeingandwillgofurtherwithitserrorsoflinealtime(asalldimotionsoftimearefractal,localtothebeing’sexist¡ence).Itwillthen‘obviously’enterintoparadoxes–the twinparadox, theLorentz transformations,etc.whichwestreamline inourpostonphysics–matter=formandmotion;whichistheonlyrealmtostudywithworldlines.

Accordingtothecorrespondenceprinciple,wecompare5Dwith4Dstrictlyintherealofphysicswhereitapplies:

Weunifythe3fundamentalprinciplesofvitalspacetime,5DmetricsandRelativity,whichbecometheequivalentin5DtoEinstein’s3principlesin4Dmetrics-theconstancyoflightspeed,hisrejectionofabsolutespace-timeandRelativity,which5DalsosharebutmakeexplicitwithanS=Tequation.Wecomparethenbothformalismswiththeprincipleofcorrespondencetoshowthat4Disjustthelimitof5Dforasinglelightspace-timeplane:

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- Thepostulateof relativity,which states thatwe cannotdistinguishmotion=time fromdimension=position=form=space,whichin5Dwereducetoasimpleformula,S≈T;weshallcalltheParadoxofGalileo(epursimuove,epurnomuove)ashecouldn’texplainwhytheEarthmovesbutweperceiveitinmentalspaceasastilldimension.

-DenialofNewton’sAbsolutespace-time;ashesaid‘Leibnizwasright,butifsowehavetostartphysicsfromits foundations’ and ‘I seem to be the only physicist who believe there are infinite time speeds in theUniverse’, which we made explicit accepting Leibniz’s relational concept of space-time, the seed ofGenerationalspace-time,resumedinasimplesentence:‘wearethescalesofspaceweoccupyandthecyclesoftimeourexistencelasts’.

-Theconstancyoflightspeed,c,whichina4Dsinglebackgroundplaneoflightspace-timecorrespondingtothegalaxyholdsnaturally,aslightinarelational->Generationaltheoryisthespace-timeofthegalaxy,whichgeneratesthroughits‘accelerated’vorticesofspace-timeinacrescendoofevermoredensescales,photons,electrons,quarks,atoms,molecules,matterstates,cosmicbodiesandblackholes.Yetwhenweadda5Dsumofallthosescales,c-speedbecomesthelimitofreducingallpotentialscalesoftheUniversetolightspacetimeandtheparticlesitgenerates

And so we define a second metric equation for all scales: $ (Space size) x ð (cyclical time frequency) =Constant; meaning smaller scales of space hold faster rotating particles, (black holes beyond the eventhorizon,where there isno longer ‘light’)orwhenmoving in lineal fashioncango faster than light speed inscalesbelowlightoroutsidegalaxies,(Bohm’squantumpotentialofactionatdistancecauseofentanglement,neutrinosoutsidegalaxies.)

Yet as S=Tmaximizes SxT=K (5x5>6x4).We unify both in a single equation:Max. S x T = C, the 5Dmetricequation, local to each Time§pace organism struggling to survive by maximizing. its S in-form-ation & T-motions,or in termsoforgans, its limbs/fieldsof speedandheads/particlesof information,and itsbalancedcombinationorbody-wavesofvitalenergy,whereentropy=T-motionandS=form,becomeone,S=T.

5D metrics generational space-time and the Principle of Relativity defines for each fractal vital space-timeorganismitsFunctionofexistence,asallspecieswilltrytomaximizeitsmotion-entropy-timeforitsfield-limbs,itsinformation-spatialstatesforitsparticle-heads,whoseproductwillgiveusitsvitalreproductiveenergy.

Sotheequationhasabiologicmeaning,becausewearemadeof3topologies‘fields-limbs’oflinealmotion(Ts,whereT-motiondominatess-form);balancedhyperbolicbodies-wavesofS=Tenergyweabsorbtoreproduce,andsphericalparticles/headsofSt-information,formwithabitofaction-motion,whichweneedtolinguisticallyguideourmotions;sandwichedinalarger∆+1worldofmax.T-entropy;codedbythe∆-1seedsandmindsofpureSpatialform.Whichgiveusthe5Dimotionsofrealitythatarealsostructuralorganicpartsofoursuperorganism&phases/agesofourexistence:S¡-1(seed-mind)»Ts-fields/limbsofmotion>S=T-bodywaves>St-particle-heads«∆-1T-death;wheredouble«»aresymbolsforanentropic,‘expansivedimotion’andaninformativestillone.SotheessenceofsurvivalistoincreaseourSt=mentalinformationandterritoryoforder;inwhichtoS=TreproduceandsTmoveavoidingtheentropiclimitsofscattering,disorderedmotion,T-1:deathandtotalSt-illnessinspacethetwolimitingDimotions.Sowekeepour‘energy=reproductivebody-wave,s=tinbalancemoving,sTourfield/limbsandperceiving,St,withparticle-heads.

3scalesintimeandspaceofsuperorganisms:actions,worldcyclesandabsolutespace-timeparameters.

The∆±¡scalarSTructureofallbeingsisessentialtopredictthefutureofitsscientificspecies,witharemodeledABCDE of the scientificmethod that studies A)ccurateData, B)iologic causes C)yclical patterns and E)ntropicextinctiveconclusionsforallsystemsin3relativescalesoflengthofspaceandtimeduration(towhichweaddinsteadofE,D)emocratic,humanistsolutionsforquestionsofsocialsciences.

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All sciences predict the future of its species according to its repetitive causal cycles.Or else they areNOT ascience.Astrology became a sciencewhenKepler learned its orbital cycles. Bio-economics becamea sciencewhenwedescribedmachines asmetal organismswhose industrial r=evolution followed thehuman72 yearsgenerationsofthedominantindustrialnationsthatevolvedthemin4cycles:itsbody-age(British,steamcycle),heart age (German, electro-chemical engine cycle), itsmind age (US, TV-eye, chip-head,mobile-ear cycle) toconcludewiththeensembleofrobotsthatasvirusdo,whenall itspartsareputtogetherwillbecome'alive'.Thepredictabilityoftime-cyclescanbedoneat3levels:

S:Continuous, spatialmathematicalsimplecycles,usingderivatives,properofcalculus;which istheshortesttimespan,asinstantaneousderivativescannotmeasurea‘peak’changeofage/phase.

T:Discontinuous,cyclicalpatternsofsequentialrepetitiveoftensurvivalactions(feeding,reproduction,death,taking place at intervals. As those actions are discontinuous, leaving long spans in-between, their patternsforecast longer time sequences. Such ¡logic structures are based in time patterns,which as anymechanical,circadianororbitalday-yearclockshowsarecyclical,repetitive.Butherehumanscientistsareatloss,becauseGalileo studiedballistics, entropic explosions thatdestroy the informationof reality stored in those cyclesoftime clocks, its patterns and frequencies, changing human cyclical understanding of bio-logic time for lineal,abstracttimethatseemsnottorepeatthosepatternssomankindlost itscapacitytopredictmanyspacetimeevents,as linealtimemisses informationstored inthefrequencyandformofcyclicalclocks,even ifequationsaresimilar:V=s/tforlinealtimeandV=S(l)xƒ(t)forcyclicpatterns.

∆:Scalar,DeepTimepatternsoftopologicandeusocialevolutionofpartsintowholes–ofquantum0-1timeprobabilities vs. 1-∞ thermodynamic populations in physics, of individuals vs. species in biology, of states ofmattervs.geologiccycles inEarth, firstnoticedby J.Hutton, founderofgeologywhocoined thewordsuperorganismforGaiaanddeeptimeforitsslowertimecyclesbyvirtueofits5Dmetrics,$xð=Cwhichimpliesthatfromtheperspectiveofasmallerscalethelifeofitswholeismuchlonger.

Deep time leads to a 3rd level of long-timeprediction: evolutionary patterns of earth's life species, includingsocial organisms of history (nations, civilizations) and its death=war cycles related to the eco(nomic)system,where the evolutionary and re=productive cycles of stocks of machine consumption become thesales=profits=valuation of its company-mothers that switch after overproduction crisis to weapons thatconsume humans, as we have donewith remarkable precision for 30 years in our ‘inconvenient’ papers onsocialsciences.Butashumanonlyrecognizethe1sttypeofpredictability-calculusofinstantaneousderivatives,thatneeda'continuousanalysis'-andhavesimplifiedcyclicaltimeintolinealtime,ignoringthescalartimeofpartsandwholeswithits5Dmetric,theircapacitytoforecastthefutureisfarmorereducedthana'stientist'whounderstandsthe3scalesdeterminedbythebio-topo-¡logicpropertiesof‘scales’,‘space’and‘time’,the3∆ststructuralelementsofallsystemsoftheUniverse.

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¬ÆTRILOGICANDPENTALOGICENTANGLEMENTOFSUPŒRGANISMS.

Weliveina3±¡pentagonalworld.5isthenumber,and10,thedoublegendersymmetryof5inscales.ThosenumericalpatternscomeconstantlyintobeingbecausetheiroriginisthemostessentiaLofallrealities–the5Dimotionsofexist¡ence.Intheoryofnumbersweshallseeimmediatelytheydivideingenderfemaleevennumbersandmaleoddnumbers,bymirror

symmetryasthe2handswhichbymirrorsymmetrycreatedthedecimalscale.

5Dimotions,3inasingleplaneofspace-=time,sufficetoiteratereality.Itsfirstmultiplicationdividethemin1,3,5,7,9and2,4,6,8.Andweshallreturntothat.Whenweforgettheseedinganddeath,entropyandsocial

evolution,inasingleplane,whichistosayabeingwithoutitsexternalworld,3adjacenttopologies,|-limbs/fields,Ø-bodywavesandO-particleheadssuffice.

'TheUniverse is a fractal supœrganismof5D¡motionsof Space-timeentangled inpentalogic supœrganismsofDustoftime-space(¬∆@st)tracing3dodecalogiccyclesofexistence,theþlacental,life&worldcycle.'l§

TheentangledUniversegeneratesitsspacethroughsynchronousconnectionsbetween5Dimotionsofrealityinits3relativeranges,Actions,topologicalorganismsandits5elements:fractalspace,scales,cyclicaltime,mindsthat perceivewith synoptic languages the system as a whole and project its territorial order and survival willthroughitsactions,andtheentropic limits(¬)theinfiniteuniverseimposestothatwill.Soanyaspectofrealitymustaccountfor5elementstoexpresssuchentanglementatthe3relativescalesofexistence. It is thefractalprincipleof5DpentalogicthatstructuresTimespacesupœrganisms(T.œ)

Scalar, topologic space and cyclical time organisms, made to the image and likeness of the fractal Universe,require a complete change of paradigm regarding how reality is created, away from the chaotic, entropic, orreligious self-centered (Anthropic) theories sponsored by humans, which are monologic, blind to theentanglement and constant communication of information between fractal points, either in logic=time ormathematical=space languages, in a sentient, apperceptive Universe, where information when locked as‘entangledspace’bythosedynamicnetworksofcommunicationcreatesamind-mappingoftheouterworld.

Monologic man is closer to Leibniz’s Monad’s - self-perceptive, selfish black holes of information, whosecommunicationhecouldnotproperlyresolvesohesimplifiedthemintoanego-paradox.WedealwithmonologicmanintheIlogicpaperonourblindænthropic‘culturethatpassesasbigscience’anditsgrandparalogictheoriesofreality.Whileonthepostsonpentalogicwedefinethe5elementsofreality-space,time,scales,entropyandminds,asentangledsystemsdefinitionscanonlybemadefromthecombinedperspectiveofallothersystemsofreality.Torepresentthisfact–entanglement-weusethepentalogicsymbolwhereeachpointofthepentagramis connected to all other points, and so the fractal point, in this case each of those 5 elements cannot beunderstoodwithouttheconnectionstotheothers.

Thekeyconceptbehindpentalogicandentanglementisrathersimple:tosurviveintheUniverse,tohavefocus,form, a systemmust be ‘anchored’ through entanglement in its ‘planeof existence’with ‘trinity’ parts of vitaltopology,connectedthrough3innerphysiologicalnetworks,butalsoitmustco-existinalarger∆±1nestedworldseparatedbyatopologicmembrain,whichallowsittoabsorbandemitentropicmotionsandinformation,onthatouterworld.

TheunderstandingofentanglementwasbestamonghumansinBuddhism,whereeachsoulisconsideredaknotofcommunicationwithallother systems,andminimal inAmericawhere theself is consideredalways theonlycentral point of view of existence, in eternal struggle with all other human beings – yet as entanglement isnecessary to survive, the American, modern man is entangled NOT with humans but increasingly with themachinesoftheMetal-earth,theFinancial-media/military-industrialecosystemforwhich‘it’enslavesasanobjectwithaprice.

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S@.Geometricentanglement:fractalpointsandscalarnumbers.

Whenwegothroughdifferentstiencescalestoexplaineach‘entangledconnectionbetweenthe5elementsof reality’,we shall see that already in the first scale ofmathematical systems, the fractal point requires 2differentpostulatestodefineit:Thefirstpostulatewhichisconcernedwithitsinnerparts(afractalpointhasbreath),andthe5thwhichisconcernedwithitsouterparts(afractalpointisconnectedwithinfinitepotentialparallels to the universe). The 2 inner and outer elements that meet in the ‘membrain’ thus define the

membrain-mind as the essential entangled element between bothworlds. Entanglementthusdefinesthebeingfromanouterperspectiveasmuchasitsinnernature.

In graph, the fractal sentient point, new unit of mathematics and vital space. Einstein'sinterpretation of the 5th non-Euclidean postulate was the view of a fractal point of thegravitationalscalefromoursmallerelectromagneticworld,whichshrinksits innervolume,bending its parallels, hence its curved geometry. But that view breaks the conceptualdefinitionofparallelsarestraight linesandisabsurd,asthepointremains‘Euclideanwithnobreath’,henceonlyfitsonelinewithnobreath.Thus,particlepointsmustbedefinedas‘FRACTAL points’, like those we see through telescopes or microscopes, which grow we

approachourdistanceboth in scaleand spacebecomingenlargedworldswitha complex internal structureandexternalentanglementfollowingaNon-E5thPostulate:Apointexternaltoalineiscrossedby∞parallelforces.

Suchorganicpointsarelikethestarsinthesky.Ifwelookatthemwiththenakedeyetheyarepointswithoutbreadth,butwhenwecomeclosertothem,theygrowandinitsmembrainwefinditsmaximalentanglementandcommunication– thehottest regionof the star, theEarth’smembrainwith its life superorganisms&3agesGaia<History>Metalearth,whereaparallelnetworkof‘stillentangled’partsformsawholewhichmakesthe‘membrain’ahardinterconnectedsystemporoustotheenergyandinformationoftheworldfocusedinitssingularitycenter.

WecanthenassessthefundamentalqualityofvitalmathematicsasakeylanguageoftheUniverse,whichistobethebestlanguageofspace,hencetheonethatdescribesthelawsofentanglement,evenifmandoesn’tunderstanditbeyonditsegocyofmagiccreationism,asthelanguageofmirrorsymmetriesanddimotionflowsof¬∆@stinallentangledsystems.

Where the fundamental entangled element is the fractal point, and its social unit the polytope in 1,2,3,4Dimotions;butnot5Dpolytopessincethefifthdimotion,entropyisthedisentanglementofalltheothers.Sopolytopesrisepositivelyincomplexityasentanglementsofindistinguishableregularpoints(orelseonepointwillbedifferentbypositionandconnection to theotherpoints).They reach itsmaximal complexityon thegender duality-mirror symmetry of the 2nd 102 scale - (female Icosahedron & male dodecahedron) & 3rdpolytopes’ 103scale (Male dodecaplex & Femaletetraplex).

∆T:HumanvitalEntanglementinages&Scales.

Each of us is an entangled fractal system,interconnectedorelse itwon’tabsorbenergyandinformationandsurvive.Andviceversa,whenlessentangled a system is – 3rd age of the being, andbecomes cut-off from reality its ‘truth’ itsexist¡encedwindlesandthesystemfadesawayandultimatelyperishes.

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At which scale of reality then wemust define entanglement. The answer is at the 3 scales, of the wholeUniverseasanabsolutefractalorganismoftime-space(atthelevelofits5elements,S,T,¬,∆,@),butalsoatthe level we just did of individual superorganisms, where its organs that represent those elements areinterconnectedtothewhole,andfinallyatthelevelof itsdiminutiveactions(a,e,I,o,u,acceleration=motion,entropyfeeding,information,organicreproductionandUniversalsevolution,inthemnemonicrule),whichwewillfindamazinglyenough,inoneofmymostorgasmic-mentalentanglementswiththewholeIexperienced,weredirectlyconnected to the∆±4planesofexist¡enceof thebeing.Reasonwhywedoperceiveall thosescales:

Inthegraph,scalarentanglementhappensbetweenthehumind’sorganismand itsterritoriesofperceptionthat provide the ‘bits and bites’ needed for humans to perform their 5 Dimotional actions. So we extractentropicmotionofthe∆±4gravitational/galacticfield,informationofthe∆±3light/starscale,energeticfoodfromthe∆±2molecular/Gaia scale; reproduce in the∆±1seminal,gender scaleandevolve throughsociallylove in the ∆º mind field, entangled to other minds in metaphysical experience of bondage through ourmemes,innations,religionsandcivilizations.

Scalarentanglement is therealreasonweexist,andperceiveentangledtootherscalesofreality.Ourmindspaceisentangledtoallour∆±3scales.

Mathematicentanglement–¡tstruestructure.

Entanglementcreatesscalarspacethatstopstimeentropicmotionscreatingorganicnetworks,andhencetheessence of fractal point geometry with motion – vital topology – and its scalar undistinguishable socialnumbers, which makes so powerful mathematics. It is the real reason why mathematics can describeinformationasthemainstienceofspacethatdepartsfrompointsandnumbersNOTfromidealistsets.Andsoatruemathematicaldescriptionofrealitymuststartbyits∆=Selements:entangledpointsofspace,inasingleplane,whoseregularpolygonsare indistinguishablenumbersofsocialscalargroups,whichreproducewhenacquiringmotion,asevenpolygonfemaleself-centerednumbersoroddmalemirrorpolygonsthatreproduceinscaleprojectinginternalandexternalself-imagesofdiagonalcrossing.Numbersthusbecomeregularsocialgroups, symmetric to points, ∆=S, which then entangle in parallel time motions, ∆=S=T with 5D algebra’soperandsthatmove‘groups’throughthe5Dimotionsofreality,establishingfurthersymmetries,5D=∆=S=T,whoseresultistocreateagivenmindspace,mirrorforafractalpointofthewholeworld,Ox∞=C,asthe1-∞statisticalspaceworldplaneisreflectedinthe0-1probabilisticunittimesphereofthemind.Sothe0-1-∞scalar entanglement allows the mind to project its mental space in territorial order by converting itsprobabilisticmindi-logicintospatiallargerforms,asprobabilisticparticlesdoinmolecularstatistics.

Mathematicshappeninvitaltermsinthephysicalworld,whenmotionandenergyisaddedtothecoded§eedofmental space, ‘irrigating’ theprogramofexist¡ence coded in still bidimensional geometry, giving itmoredimotionalextensions.SofromthefirstS=Tentanglementofpointsandnumbers,whichthenbecomesocialfiguresofgeometry,thatpenetratescalesofnumberfamilies,in∆stmirrorsymmetries,numbersunfoldtheprogramandstarttorunitwith5Doperands,whichmakethem‘real.Algebrathenenterswhenitisabletomove through the dimotions of time by means of its operands acting over groups of social numbers, theentangledUniversecreatingternarynetworksthatbecomeplanesthatbecomesupœrganismsofexistentialalgebra. This is the game we shall describe, which needless to say compared with the barren land of theAxiomaticego-tripofHilbertandCantor‘imaginingpoints’isablossomspringoflifeandbeauty.

Physicalentanglement.

Entanglementappearsinthenon-egocenteredBroglie’s>Bohm’sinterpretationofrealistquantumphysics,asparticlesentangle through itsquantumpotential fields;and thisvastentanglementhappeningatv>c in thelowerplaneofreality‘reducesspatialdistances’intheRiemann’ssenseofageometryofsimilaritytoa‘boson

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like’ unit of parts as if distance-space would NOT exist within them; which does NOT for the emergentapperceptiveilogic‘couple’asasinglewhole.

To understand entanglement in formal languages is needed to adopt the Riemannian concept of amentalspacewhereproximityanddistance isadirectproductof similarity,as if spacewillnotexist,whichcanbeachievedthroughcommunicationonalower‘potentialfield’offastermotion,asafastervinfactperceivedinstillnessimpliesasmallerdistance,andviceversa–i.e.theexpansionofspaceintheUniverseisinfactav>cspeedonthe jetsofdarkentropyexpelledbytheacceleratedfasterthanctimespacevorticesofheavytopquarkstars,akablackholes.

Sotheblackholebeyondthec-horizonspeedsupatv>canderasestheinformationoflightintodarkentropyexpelledthroughitsaxisandperceivedasgrowthofdistancebetweengalaxies,whenbyvirtueofS=Tcanbeconsideredgrowofspeed.Entanglementisexactlytheinverseprocessinaquantumpotentialfieldthatmakescommunication disappear in distance between two poles – which is themagic of a phone conversation –distanceisNOLONGERrealbecausewearenotinsoundcommunicationat300m/sbutinadecametricjumpofscaleat300.000km/s.

There is no distance in mental space of telephonic communication between speakers, only in geographicEarth’sspace;butwearenotconcernedwithlandorairbutwithalanguageofcommunicationanditsbitsofinformation,which are translated, entangled, in a faster 5D scale of smaller bits andmax. form (electronicspace).

Informative translation in other 5S scales iswhat T.œs constantly do, entangling its informative systems infasterscales,whosebitsbecome‘somethingelse’butpreserveitspatternsasthereistheUniversalreferenceofthegameofexistencebehindthem.Inthepreviouscasethecommunicationhasbecomeanentangledbackand forth feed-backelectronicequationofexistenceasaneventonawire linebetweentwo fractalpoints,whichcannotlongerbeconsidereddifferentfromthe‘communicativeline’betweenthem.

Energydoesnotentangleonlyinformationdoes…

Aconclusionprovedmathematically,asonlythesphereofmax.volumeofinformation(Poincare)canshrinkwithout tear in any number of relevant dimensions (4 for entanglement) crossing the barrier between twoplanesofexistence,whileenergybecomesdissipatedasentropy.SoyoucantalkwithmuminAustraliabutNOTsendher1000kilowatts,youcanthoughsendherdigitalinformation,akamoneytoo.

Bio-logicentanglement.

This leads us then to the first scale of complex i-logic ‘gender mirror symmetry’ between two entangledelementsthatbecomebyfusionloveasingleone,ofwhichthereareinfiniteproofs.

InfactgenderisthefirstandessentialentanglementoftheUniversebetween2singleelements,throughthemirrorsymmetryoffemale-likeSt+sTmale-likecomplementarycouples.Entanglementstartswith1+1=3,andforthatreasonwededicateanentirepost-togendersymmetry.Theentanglementofgender, issoobviousthatbecamethebasisofthedualityknowledgeofearliercultures(Chinese)willconstantlycomeinNature.ItisNOTonlytheentanglementofbiologicgender;sohowwedodistinguishits‘essence’?Inilogictermsiseasybecause the essential duality is that between simultaneous present space, S=T and time flows in relativestœps, from past to future to past… to future, S<T>S<T… Both are perpendicular. And so the definition ofGender goes to the final core level of ‘perpendicular’ space (female gender) to Time flows (male gender):Gender is essential to understand reproduction by mirror symmetry and the duality of symbioticperpendicularvs.Darwinianrelationships.

Twosystemsachievemaximalcommunicationwhentheypassfromparallelsharingofinformation,throughamedium that transfers a language, to direct contact between its sensorial membranes, which bring true

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bondage. And at that moment two different outcomes might happen. If the membranes touch withouttearing,thetopologicalintegrityoftheorgansareconservedinvitalmathematics;sothebeingco-exists,andentanglement (the previous state of parallel communication) becomes bondage according to the fourthpostulateofnon-Euclideancongruence.

Thisisthemagicofit,asfusionlovekicksinandentanglementbecomesadanceofcomplementaryforms.Itisthebeautyofloveandgendersymmetry.Andthemaximalactionofit,iscalledsex,orgasm,thepresentstateoffusioningboththepasttofuturetopast,andpresentformsintoone.Sohowithappensphysically?DoIhavetoexplainittoyou?

ThemalegenderpenetratesthefoldofthetopologicalfemalebutdoesnotTEARIT.ItgoesinT>S,andthenitgoes out S<T, as themale does in and out, information and entropic disentanglement. But the female S=T‘graspsit’.Itcaressesit,encirclesitslinealformwithitscyclicalevenpolytope.Entanglementisnowbondage.And it brings when bondage combines in themost intimate ways the information of both systems trinity,reproduction.Soalas,1+1=3!

Thosearetherulesofvitaltopology,whichininversefashiondefinesDarwiniandestructiveperpendicularitywhenoneoftheentangledelementsturnouttobeapredatortopandbreaksin,tearsthemembrainoftheother system, killing it, entanglement then gives way to entropic devolution and one system becomesdestroyed.

THEUNIVERSEISASCALARORGANISMTHATREPRODUCESINFORMATION.

Asthingscangetascomplexasrealityseemstobe, it isessentialtohavethefundamentalprinciplesofthefractalUniverse,clearlystatedandunderstoodtoguidetheresearcherthroughthe forestofdifferenttreesand still hold the awareness that they are all trees, that is all is a super organismof space-time tracing itsworldcycle.

What is the purpose of the game of reality? It all amounts to 3 words: ‘exist¡ence’ & survival throughreproduction.

Andexistence isabout ‘sensing’the5Dimotions,enactingthem,andthose5Dimotionsareresumedinthemost complex of all, /reproduction’. So what all comes to is this: the Universe and all its parts are scalarorganismstryingtoreproducebecauseafterallthatiswhata‘generatorequation’ofafractalsystemdoes–toreproduceconstantlyinsimilarbeings.

Thisisthegameofrealityandyethumansaresoremotelydisparagedwithrealitythattheystillwonderwithastonishmentthemiracleoflife’,us’,whicharetheonlysystemsthatreproduce.

This is ænthropic man at its best – denying the very essence of the Universe, a constant reproduction ofinformationofformimprinting itsprimarysubstance,motion.Andyetænthropicmanwonderswhat is life?Lifeiseverything!

Thequestionthenistoconsiderhoweachscienceexpressesthisconstantessenceofreality,areproductivefractalofinformationimprintedoverenergy,andwhenwecometomathematics,itsexpressionisthroughthegrowthof‘spatialdimensions’,aspointsreproduceintolinesandplanes;orsystemsmultiplyitsoperands.

Howasystemevolvesitsdimensionality

Howasystemevolvesitsdimensionalityisthegameinanylogiclanguage.AsitisthegameofreproductionofwavesofformwhattheUniverseisallabout.Aformreproducing,downwardsasitissimilartoits∑∆-1scale.

It isthroughdimensionalgrowth,byproductalgebrahowthenwestarttocontinueoursymmetrybetweentime an space, between geometric states and fluctuating future some of which will die algebraic product,

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oftenregularpolynomialgrowthofdimotionstillanew5dimotionalbeingisrecreatedandthegamerestartsagain.

Thesystemissimple.Apointstartsamotion,whichmakesanewform.Thefirstpointcanmovein linesorrotateonitself,ormoveincircles,orformspheres,orshrunk.Andthose5Dimotionsstartthegame.

Thepointbecomesaline,whichcanbeastraightlineorazigzag√line,orarotarycircleorπ-line.3Subspeciesimmediatelyappear.Thelineandthe√2triangularwaveandthecircle.

Evolutionthatdivergestopologicallyfromthefirstreproductiveprocessofalineandacycleandawave,andso subspeciesofdimensionalgrowthappear.Howwestudy them,withnon-epostulates ingeometry,withDimotionalexistentialalgebrainalgebra.

Aswegothenthroughproductsthatcreatenewdimensionsinalgebrawemimicamathematical,geometricprocessofgrowthinNumbers,sogrowthmustbestudiedinparallelinmath.Growthofnumbersandgrowthofpoints,andothergeometricfigures,whichthereforecanbetranslatedtoequations.

ThustheUniversehasmathematical,spatialcontinuous&logic,temporal,cyclical&organic,fractalpropertiesderivedofthemorecomplex¡logicgeometryof5Dscalarspaceandcyclictime.Thuseven ifwedescribe itwiththesamelogic-mathematicalequationsa5DUniverseisverydifferent,asscalarspacebringsorganicandsentient properties, and cyclical time, informative deterministic patterns, which the lineal philosophy ofmechanistphysicsignores.Fromthosescalar,henceorganic,cyclicalhenceinformativeandmoving,energetic,hencevitalpropertiesof scalar spaceandcyclical time,acompletedifferentpictureof realityarises;wherelanguages become the ‘extreme’ limit of the still formal ‘spatial state’ of the being withminimal size andmaximalinformation.

Whereaseach‘stience’definesascaleofthenestedUniverse,fromPhysicsfocusedonthelargestGalaxyandphysical scales, to the smallest ∆1 scale of Formal digital and logic sciences of the mind. Languages thusbecomethe limiting formal, spatial scaleof realityandassuch theyaremirrorsof the fractalUniverse thatsharethe‘Disomorphic’(Equallaws)ofallthescalesofreality.Andsomathematicsbecomesanexperimentalstiencewhose ‘Universal grammar’ and fractal generator is the same ternary structureof 3 elementsof allother systems in a single plane made of three topologies, SóT, giving birth in each sub-discipline ofmathematicstosimilar‘mirrorimages’,algebraicS(x)=T(y)equations,ternarydimensions,ternarytopologies.

The new scalar, topologic, cyclical and sentient properties of the 5 Dimotions of space and time, whichstructurethefractalUniverse.

ThustheUniversehas5setsofpropertiesthatcorrespondtoits5structuralelements:

-S:SPACE:Topologic,mathematical,socialproperties,describedbyitsmathematicalunits,pointsandnumbers,whicharesocialgroupingsofundifferentiatedelements.-T:TIME:Temporal,CYCLICAL,logic,causalproperties,whicharecyclicalfrequencypatternsoriginofthelawsofscience,causedbytherepetitive,memorialnatureofitstimecycles.-∆:SCALES:Organic,vital,biological,survivalpropertiescausedbytheco-existenceofseveralscalesofparts

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andwholesorganizedbythoseparts,fromparticlestoatomsmoleculescellsandmattersystems,planetaryecosystems,galaxiesanditsnetworks.-@:LINGUISTIC,MENTALpropertiesduetotheexistenceoflanguagesofin-form-ationthatare'stillmappings'ofrealityusedbyit'ssuperorganismstoorderaterritoryandcreatewithinthatordertheconditionsforitssurvivalactionsthatensureitsexistenceandreproductioninfinitetime.

¬: All of it broken by entropic limits of death in time, membranes that break the Universe in fractalTime§pace-organismslimitedinspaceandaself-centeredscalarstructurethatdilutesinformationandhencemakesinvisiblerealitypast∆±3scales(inthehumancase).

Soweare‘¬∆@st’,dustofspace-timesupœrganismstracinglimitedcyclesofexist¡ence.

Timeasthefifthdimension

Morepreciselytimeistheperpendicularfifthdimensiontothatofpotentialspaces,whichareenlightenedinthemorecomplexfractalworldcyclesoftimemotions,whichbreakdimensionalityintoHausdorffdimensionsof self-reproductive trees that branch through the life-death cycle, in clonic forms thatwill synchronize asspacesuperorganisms.Theflowoftimehowever isalwaystheworldcycleofpalingenesis, lifeanddeath.Sotimecanbefollowedthroughthejourneyinthefifthdimensionofallbeingsthroughitslifecycle.

SynchronousspaceAREslicesof the timemotion; that is,of reproductionof form inpatternswhichhavealogic-mathematicalgamestructurethatcanbedescribedbyanylanguage,theTimeisthepotentialfieldofallpossible tolerated forms of existential algebra that survive and create a reproductive wave. Its process ofmotioniswhatwecalltimeandisafluctuationbetweenscalesoffifthdimension.Wecouldsimulatespaceplanesasparallelfieldsofpresentofasomehowdensersubstancethatthemorechaoticentropictimefieldsbetween those space planes. But the 'enlightening of the time field' by a potential spatial synchronoussuperorganismissubjecttorulesofefficiency.

RECAP. We are relational space and time, made of the organic scales of discontinuous fractal space weoccupy,who‘live'afiniteworldcycleofexist¡encewhosecommon=Disomorphiclawsemergeinall‘scales’of‘st¡ence’,departing fromthe2simpleMetric lawsof thefifthdimension, Sxð=C (size inspacemultipliedbythespeedofourtimeclocksisconstantinallthescalesofpartsandwholesofanorganism)andrelativityS=T(we cannot distinguishmotion in time and dimension in space, so all systems are bidimensional spacetimebeings as space=form can be transformed back and forth into time=motion; making the 5 dimensions,‘dimensional motions’ ab. dimotions of space-time; entropy – TT (outer and inner, scattering motion);Locomotiondominant in time=motion, Ts:balanced reproduction, T=S; ‘form-in-action’ information, St; andpureform,SS,a‘seed’ormental,stilllanguagethatmodels=mirrorswithoutmotioninlesserspacetheworld–anddevelopswhenaddingmotionintoafullorganism.

Monologic,ænthropicmanwith its1Dmodelsofrealitybased inSxtimelocomotionsshunsoffthe5DCx.Universetoitsownperil,asrealityispenta-dodecalogicinitsentanglementsthatgeneratespacesynchronoussupœrganismsandtime,andsurvival,whichinourdominantego-centeredcult(ure)s isminimaldependsontheunderstandingandrespectofallthelivingfunctionofspace-timeexist¡enceanditsaction-reactionlaws.

ParadoxicallyDeeptimeworldcyclesareeasiertopredictandunderstandthatcomplexDimotional‘analysis’becausepreciselythe largerscales in5Dmetrichave less information,butmorebasic,deterministic,reasonwhyquantumphysicsisharderforthemindandprobabilistic,whilelife-deathcyclesalwaysend…indeath.

The most important of the paradoxes of ‘reality’ ill-understood by humans, regarding mathematics is theparadoxofRelativity,S=T,accordingtowhichmotionandformaretwosidesofthesamecoin,whichcarriestothedualitybetweenpointsandnumbersanditsmorecomplexforms,geometryandalgebra.

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Regardingtheelementsofmathematicsthatcorrespondtothoseelementsofreality,numberscorrespondtoscales, algebraic operands to time, points of geometry to space and its complex structures to its variousentanglements.

All this of course today is cast with the theory of sets, of little help for an experimental 5D view ofmathematics;asitisa‘nice’generalizationfromthehuminddown,butcut-offpreciselyforthatreasonfromthe direct experience of reality, its spatial points, scalar numbers and time operand,which only helped todetachmathematicsfromexperienceanddistortitsphilosophytocaterthe‘egocy=ego+Idiocy’ofhuminds.

Onthepositivesidesettheoryisaproofthatamindcanalwaysconstructaself-consistentimage-mappingofreality that resembles that reality. Sets indeed are defined as a ‘miss-mash’ of the disjoint elements ofmathematics: itsnumbers-points (as theyarecollectionsof similarbeings) its scalardimensions (assetsarewholeswhichhavepartsorsubsets),astimeoperands(withthetoolsofclassiclogic).Andsotheygiveusasynthetic whole. But what is the ‘point’ beyond egocy of developing a distorted mirror image to foundmathematics. When we have the real things, points, numbers, scales, frames of reference=minds, timeoperandsanditsinverseentropicoperands.

Theimpressionfroaboveofhumanegocyis‘ridiculous’:amushofdirtywaterinalostrockoftheUniversetrying to impose its territorial mind above heavens and earth when it is sooo obviously nothing… beyondridiculous.Doyounoticetheattemptsofantstoimposeant-philosophyovertheinfiniteUniverse?

ButifIhavelearnedwithagesomethingabouthumindsisthat99.9%ofthemarememorialegocy(repeatingtheories that putman above); and it is a waste ofmy little time to try vehemently to reason on itsmuteemotions.SosettheoryisheretostayforhumindstokeepfeelingthecenteroftheUniversetillrealityerasesusall.Andthentheself-namedcenterofrealityisallbutgone.Andpoints,operandsandnumbersevenifMr.HilbertandMr.Cantornolongerimaginethem,willnotbeherewithus.CarpeDiem.Butsometimesweshallrefertothe‘VonNeumann’nesteduniverseofpuresets,toshowhowtheset‘mind’reflectsscalesofpoints,numbersandoperands…

Soweshallstartwiththedefinitionofinfinitesimalminds…

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BOOKI.

GEOMETRY,FRACTALPOINTS,TOPOLOGY&MENTALSPACESI.PHILOSOPHYOFMATHEMATICS

Allmirrorshavesubdisciplinesofspaceandtime.Somathematicscanbedividedinspatialgeometry->topologyandtemporalalgebra->equationsofnumbers,whichformtheessentialS=TsymmetrybetweenS@geometricspaceand¬∆Tscalarnumbersandsothegrowthofdimensionsismimickedinalgebraicspacewiththefundamentalofitsoperand,whichistheproductthatcombinesSandTdimensionsofgrowthtobringa

largerbeing,andagainwehave3fundamentalpowersthatbring3dimensionsfrompointtolinetoplaneintoalgebra,andthemoresophisticatedbutsimilarconceptsofaderivativeandanintegralthatalsomakesthe

systemgrowordiminishindimensions.Oneofthedistinctionsworthtostudyindetailbeingtheapproximationanddifferencesbetweenapowerlawandanintegrallaw.Whythereareunlikeingeometrytwoformstoexplaininalgebrathegrowthofaformindimensionality?Whyalgebraicnumbersgiveusmorevariationsthatgeometry?Theultimateanswerbeingthatgeometryismorerestrictedasitestablishsystems

thatdoworkinspace,andfitinthelimitedspaceofastillmind.Sobyefficiencyandlimitsofthemindgeometryismorereduced.Algebraincludesallthepossibilitiesoftimesomeofwhichwillneverrealize.

Ifreality ismadeofspace-timebeingssincemathematics isthemainscienceconcernedwithspaceandlogicthe main science of causal time both mathematics & logic become experimental sciences, whose laws ofmaximalsynopticinformationinminimalsize(Sxð=C),willbetheunderlyinglawsemerginginallotherlargerscalesofthefractalUniverseofbiggersizeandlessinformation,provingalsowhymathandlogicapplytoall‘stiences’whileasS=TanytopologicanalysisinSS-spaceisequivalenttoanalgebraicanalysisintime.Thusweupgrade spacemathematics to its fractal scales, S=T duality& 5 dimotionsweneed to upgradeAristotelianLogicof1timecausalitytotheentangledpentalogicof5dimotionsthatreflectthose5structuralelementsofallT.œs:Space=form,Time=motion,∆=Scales,@-minds& the¬entropic limitsofallT.œs,mirrored inmathwithS-points,T-operands,∆-numbers,@-frames&limitstoinfinitiesintimeandspace.Weshallthustranslateto the 2 ‘closer mirrors’ of GST, vital topology and existential algebra, all the sub-disciplines, elements,operand,dimensions,axioms,postulates,theoremsandlawsofmathematics.

As we are made of scalar space and cyclical time, the essential properties of beings derive from the‘disomorphic’= equal laws of space and time, which makes mathematics concerned in its two main sub-disciplines,spatialgeometryandtemporalalgebra,themostexperimental,perfectmirrorofthe5DUniverse.Sowe can translate the 5Dimotions (short ST-view)& its spatial superorganisms to, Geometric space& itsWorldcycles(LongST-View)toAlgebraicTime.Buttodoso,we1stmustdepartfromS=∆,pointsthatbecomeregularpolytopes=numbers,whichpenetratesocialscalesandreproduceasodd/evengendersymmetries,toreflectin2Dstillmindspacesthebasicorganicformsofreality;andonlythenonceweunderstandtheorganiclawsofstillspatialgeometry,andscalar5Dnumberfamilieswecangivethemtopologicmotion,operatethemwith 5 algebraic dimotions & calculate its ∫∂ changes emerging as simplex particles entangled into atoms,molecules & 3-physiologic planes=networks=supœrganisms when organic laws take over as analytic mathsleaveswaytosyntheticbio-logiclawsoflargertimescales&pentalogictododecalogicbettersuitedtodescribethegameofexist¡ence;simplifiedby5Dmetricoflesserinformationinlargerscales.

Whatmakes generational space-time different is the fact that we fully change the entire worldview of theUniverse, aswe reject Newtonian Absolute space-time and give back time its cyclical nature, hence its 3relativepast=entropy,present=iterativeandfuture=informative localdimensions.Thusdefiningtopology,thefinalstageofgeometry,asthefundamentalscienceof‘form’originofthevitalshapesofallsystems.

Whilealgebrawilltake‘perpendicularly’those‘groups’ofsimilarpoints=numbersthroughthe5DimotionsoftheUniverse,asentangledsupœrganisms tracingworldcycles.And in thatdescriptionwewillvastly improve

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themathematicalmirrorasavitalreflectioninastillsynopticlanguage,apttocodeinminimalspacetimetheprogramofexistenceanditsinfinitevariations,tothenseeaswekeepexpandingeachstience,handinhandwith the ¡-logic entanglements of duality, gender mirror symmetries, pentalogic ¬∆@st of timespace anddodecalogic,Disomorphismsoftrinityworldcycles,howinreality,thevitalspatialmathematicsandtemporallogiclawslaiddowninouranalysisofformalstiences,emergeastimespacebeings,fromthesimplestparticlesoftheo-1probabilitytimesphere,toitsstatisticalspatialsettledpopulations,onceandagainrisingfromtimetospace,fromgeometrytoscalarnumberstooperandsofexistentialalgebra,till reachingman; justanotherentangledsupœrganismtracingitsworldcyclethrough3deterministic,probabilisticandentropicscales.

The5subdisciplinesofmathematics.

BecauseDimotionshaveSpace&time,dimension&motioncomponentstheminimalrealityisdual,entangled.Itfollowsfromadefinitionofmathematicsasanexperimentalmirrorofthe5entangledelementsofallsystemsandits5Dimotions,aclassificationofthe5mathematicalsub-disciplinesaccordingtoitsspecializedstudyof2entangledsub-systemsofthose5Elementswithdifferentparameterstomeasureits5Time§pacedimotions:

1. S@:Geometry studies mental spaces. Its ages/fields are: Flat,Euclidean geometry with no motion in aplane@nalyticgeometry,whichrepresentsthedifferentmentalpointsofview,self-centeredintoasystemofcoordinates,or'worldviews'ofafractalpoint,ofwhichnaturallyemerge3'different'perspectivesaccordingtothe 3 'sub-equations' of the fractal generator: $p: toroid Pov < ST: Cartesian Plane> ðƒ: Polarco-ordinates.Topology,geometrywithmotion&2Planes.¬EGeometrystudiesfractalpointsofsimultaneousspace,∆-1,&its∆ºnetworks,withinan∆+1worlddomain.

2. ¬∆: Social Numbertheory studies scales of equal herds and sequential time flows of information withdiscontinuousnumbers,asopposedtocontinuouspointsofgeometry,itsgrowthanditsentropiclimits,bothasmembranes of polygonal forms and structures of increasing depth peering ∆§cales and detail from Naturalsthrough,Q,RandComplexnumbers.

3.∆T:AnalysisstudiesALLformsoftime=change,andhenceitcanbeappliedtothe5Dimotionsofanyspace-timebeing, as long aswe study a 'social structure' susceptible to be simplifiedwith 'social numbers'. It is theessentialtooltostudymotionsinthe5thdimensionfromlowerparts(∂erivatives)tolargerwholes(∫integrals).We differentiate 5 applications of Analysis according to the Dimotions studied.We also classify them by thenumber of entangled elements of the dimotional system (partial or multiple derivatives, ODEs, PDEs, lineal,surfaceorvolumeintegrals);andbythedetailofitsmirror,fromdiminutiveanalysisofsingleDimotionalActions(∆-1)toAgesofworldcycles(∆º),whichimplyachangeofstatewherethederivativebreaks(Minimal,maximal,standingpoints)tolargerchangesfromminimalparts,‘finitesimals,1/n’tolargerwholesofentireplanes;tothemaximalcomplexityoffunctionals.

4.S≤≥T:Algebrastudiesthrough5operandsthatrepresentthe5Dimotionsofrealityits‘feedback’S<T>S,stopand step or 'stœps' that move reality through mirror symmetries of space-time, represented by equationsconnectedbyoperandofincreasingcomplexity,fromsingleS=Tstepsof∆-1sequentialnumbers,whichgatherin∆º functions, part of ∆+1 functionals. So numerical stœps are first key constants andoperands that representbasic dimotions (Sine/cosine: informative perception; π: $t>§ð: cyclical dimotion that transforms energy intoinformation ; e-x=S«T: entropic dimotion, log-xn: social evolution etc.) become connected by ± social and x ÷,reproductive operand complex larger associations of Dimotions called Polynomials where those mirrorsymmetriesS=Tgatherinmorecomplex∆+1 structures (Functions).Analysis is a sub-disciplineofAlgebra thatstudiesDimotionsbetweenpartsandwholes(∂entropicpartsand∫wholes)ingrowingcomplexityfromchangesinfunctions(1stderivatives/integrals),tochangesofchangesoffunctions(functionals).

Thus algebra mirrors reality with all the elements of mathematics, albeit with a temporal perspective - astopologydoeswithaspatialperspective-becauseitcanexpressalltypeofcomplexentangled¬∆@stofspace-

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time in its simultaneousanalysisof superorganisms through the studyof its S=Tdimotional a(nti)symmetries,classified exhaustively by group theory... So Algebra was first the science of operands that translated intomathematicalmirrorsthe5dimotionsofspace-timeandthenbuildupfromthemastheUniversedoesbuildingup fromactions, simultaneousorganisms in spaceandworldcycles in time, indifferentdegreesof complexity,newmirrorsforallthoseeventsandformsof¬∆@st.

5.¬@Humind:Philosophyofmathematicsstudiesthebiasandlimitsofhumindsstudyingmathematicsastheyprojectitsænthropicsimpleviewoftheworlddueto:

1)theegoparadox(allsystemsmeasurefromitsdistortedself-centeredego).

2)dominanceofthe‘westernmilitarylinealmale,entropicdestructiveculture.

3)inflationarylimitsofsmallishlanguagesthatmultiplyitskaleidoscopicmirrorimagesoflargersinglewholesasthey are not bond by the restrictions that ‘lineal motion-entropy’ impose to form, whenever we try to build‘reality’:somoneyisinflationaryoverthephysicaleconomyitdescribes,wetalkmorethanweact,epigeneticsmultiplieswastecode;sododigitalprogrammers.Sowe‘knowwhenmathematicsistruthbutNOTwhenisreal’(Einstein, Gödel’s incompleteness, false ∞ as all planes have entropic discontinuous limits of solubility offunctions, Cantor Px, idealist, German, physics: Copenhagen interpretation, creationism, false∞=singularities.Since fictionsexist inall languages. So to crop the fictionalpartofmathematicswemustuseanexperimentalmethod,selectingfrommathsthosepartsthatbettermirrortheUniverse.AslanguagesmirrortheUniverseinamind,whichisNOTrealityitself.Buthumanegos,confuseboth;aseachlanguageisalsoaspeciesofstienceofminimal volume, a fractalworld that imitates thewhole; so it has the same syntax laws than reality,which inmathematicsbecametheEuclideanaxiomaticmethodwemusthowever improveby lookingalsotothe¬∆@stuniverse–the‘object’thelanguagemirrors.

On the positive side idealism gave creative capacity to huminds as they invented to represent reality, phasespaces,Hilbertspacesandvariationscametoreflectthatgrowingawarenessofthecomplexityofreality.Sotheaxiomaticmethod of proof still needs the experimentalmirror of ¬∆@st laws.Wemust compare anymirrorlanguagewiththe∆STrealityitdescribesaslanguagesaresmaller,hencemoreinformativeintermsof5Dmetric,Se (size in space)xTi (Time information)=C.Ashumansdidn’tdevelopmathsasanexperimental stience theyalso ignoremath as a biologic language that selects species that talk it better in the eco(nomic)system. So itsidealistviewignoresthedangersofevolvingdigitalchipsthattalkbettermathsanddisplaceusfromlaborandwarfields.

6.Vitalmathematicsexpandsitsfoundationsto5¬EPostulatesbeyondAristotelianlogic(A->Bsinglecausality)into the ¡logic of 5Dimotions,asanexperimentalmirrorof the fractal,organicuniverseand itsbio-topo-logicproperties.

Expansionofmathematicsto5d:correspondenceprinciples.

Mathematics&logicarelanguages,mirrorsofanapriori∆STrealitythatcomesbeforelanguagesthatdescribeit.

Mathematics isderivedofgeometry,thescienceofspaceandlogic isthescienceofcausality intime.Sospaceandtimemustbethefirstsubstancesofwhichallismade,amodelofrealitythathasadeeptraditionintheeast(philosophiesofaUniversemadeoftwopoles,space=dimensionalform,or'yin'andtime=motionor'yang')

Thus we need to introduce the correspondence principle also in mathematics, according to which presentmathematicsisasimplificationandbiasedviewofthetruediscipline,asallsciencesreducetoasingleplaneofexistence,using therefore lineal conceptsof a single timemotion the fractalUniverse. As timeand spaceareNOT absolute Newtonian backgrounds but are in a Leibnizian relational space-time background independenttheory,the‘generationalsubstances’ofallwhatexist,composedoforganicfractalvitalspacesthatlastafiniteduration in time, space and time become the common principles, whose disomorphic properties originate all

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other laws of ‘stiences’ each one studying different ¡-scales of spacetime beings that must be first put inrelationshiptothosepropertiesandthenimprovedwiththecomparisonofhumanscientificlaws,specificspeciesofeachscienceandtheuniversallawsof¬∆@st.

Hence we upgrade all its concepts to ‘cyclical time’ that stores information, in the frequency of its cyclicalmembranes, limiting and breaking space into ‘fractal’ topological parts, and ‘scales’, according to ‘relativitysymmetries’ between formal linguistic mind-spaces vs. timemotions, S=T, whichmix together forming the 5Dimotionsofreality,allofwhichfollowits5Dmetric.Thoseelementsentropiclimits,fractalspace,SxT=Kmetricscales,cyclictimeofinformation,andS=Tsymmetries,arethusthebarebonesfundamentalelementsrequiredtoupgradeeachscience,and itssimplestmathematicalequation is itsSxT (só t)=Constant, ‘FractalGeneratormetric’.

In mathematics the same upgrading is needed, as it is also an experimental mirror-image of those laws.Somethingwhichsubconsciouslyhappenedasmathematicsevolved,frompurementalspace(Bidimensionalstillgeometry) to time perception (sequential numbers) merging both in modern analytic geometry. And finallypeeringfirstwithtemporalalgebraicnumbersinthe4thand5thdimension(calculus),whichtopologymirroredinspacewith its 3 varieties of bidimensional forms withmotionmade of networks of points; to complete withfractals themirrorstructureof theUniverse–addingonthepathallkindofnew ‘mindspaces’(Phasespaces),with ST combinations (vector spaces) and scalar levels of growing complexity (Hilbert spaces, functionals) –whereas other highly valued branches, set theory that constructs maths NOT from its initial s-points and t-numbersbutfromthetopheadofaMr.Cantor,thesetandtheaxiomaticmethodofMr.Hilbertbornalsoofhishead–‘Iimaginepoints,linesetc.’.SothosewillbeconsideredlargelyinflationaryformsofthelanguagewithinitselfNOTobservedinthereal∆stworld,notworthtomention.Whilechipmaths(BooleanAlgebra)belongstoanewspeciesthedigitalmachine,whichasperourpapersonthesuperorganismofMankind intime,history, isbondtodisplaceuswithitshigherdexterity inthemostefficient languageoftheUniverse, ifwekeepevolvingthem.SowefeelethicallyinclinedNOTtoupgradeitinthispaper.

S=Tsymmetry.Systemsaremadeofspatialformandtimemotions,fromastaticmind-perspectiveofstillgeometrymathematicsalsoevolvedtoacquirealgebraicmotiontheessentialDualityofrealityalsoevidentinanyscaleofcomplexityofmathematics,whichhasalways2solutionsfromeitheranS≈Tperspective,Spatialforms(points)=Temporalsequentialnumbers.Topologicalmethods=Algebramethods,mergedinanalytic

geometry.

So geometry of points studies dimensions in space,Number theory, its sequence in time and itsanalysis itsscalarmotions=changeswhereasAlgebraputsthemalltogetherconsidering itsS=TDimotionsandsymmetries,makingmathematics thebesthumind’s known languagemirrorof the…5d+5m=5Ðimotionsof reality,whicheach mathematical S=T sub-discipline expresses in different terms – Geometry as Dimensions, Analysis asmotionsandAlgebraasDimotionsusingoperandstothataim,andfinallywith∫∂calculuspeeringthe4thand5thDimotion and its travels through those upper and lower scales, as analysis introduced in algebra the study ofintegral wholes and derivative parts. Scales of 5D parts and wholes soon gave further boost to algebra, asfunctions became part of functionals, and all variations of a spacetime structure were tabulated with grouptheory.Sothe5'Dimotions'ofanysystemcanbemirrored¡logicallywithmultiplekaleidoscopicperspectivesandlanguages.Soassystemshavealways5Dimotionsitspentalogicstudygiveus5varietiesinallitsmathematicalelements;5operand,5¬∆@stelements,5dimensions,5motionsetc.

In geometry the same evolution from static space on the 3 ‘dimensions of a single plane of existence’ to theanalysisof theupperand lower scales to finallygivemotion to themall tookplace, fromGreekbidimensionalgeometrytosolutionof3dimensions,height-information,length-motionandwidth-reproduction.Thentopologyintroducedtheconceptofmotion,madelinealdimension,bidimensionalasallisanSTcomposedformandthenthosebidimensionaltopologicformswithmotion,peeredthefourthandfifthdimension,asmadeofpoints=parts

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thatformawhole;soonpointsthemselvesacquiredvolumeasmultipleparallelscrossedtheminnon-Euclideangeometry.

Thehighesthomologywithreality:Semantics,SyntaxandGrowingSentencesofmathematicaloperands.

Mathematicsisalanguageandassuchithasthesomeclassic,properties,elementsandsymmetriesofthemall:

It reduces reality to fit the brain, eliminatingmotion and simplifying layered scales and limiting perception torelevantcycleswithintheterritoryofperceptionofthep.o.v.Sothemorecomplexrealityof∞space-timecycleswithmotionbecomesalanguage.

It does so, increasing generality through the Syntax of its 'sentences’; while keeping its detail through thesemanticsofitsforms.

In ¬mæth the semantics are the specific fractal points-numbers, that the syntax of operands connect intosentences,whichareternaryplanesingeometry:SóTequations,inalgebra.

So further growth in complexity can be achieved by adjacency of topologic varieties of planes and 'chains' ofequations through ever more complex, integrative, 'operands' (sum/rest->multiplication/division->potency/logarithm/integration/differentiation) to ‘form descriptive paragraphs’ which will finally reach the full‘superorganism in space’ or worldcycle in time, concluding the ‘story’ of the event or being analyzed by thelanguage.

Thosestructuresdohappeninanylanguage,amongthethosestudiedinourpapers-music,art,literature,logic,math,palingeneticsandtopologicalevolution.AndsoallcanrefertoGstlaws,thelanguageofalllanguages.

That is all. Why?Because as we stress once and again, the Universe is infinite in its repetitions but its finalelementsarefew,reasonwhyalanguagecan‘reduce’realitytoanencryptionandfinalsynopsisbyeliminatingrepetitions,whichultimatelyalllanguagesdotocreatepalingenesisfromsmallerseeds.

SotheperceptionofaMINDisshrunk-it isa'finitegameoffinitesimalmirrors'withminimalredundanceandeliminationofdarkspaceswhicharediscontinuitiesandredudndatninformation,asopposedtotheinfinityofthewhole.Furtheronasweperceivenotrealitybuttheinterposedlanguage,realityalsoshrinksandinformationislostbeyondtheternarylimitsofscales,theternaryadjacentpartsofthebeingabletoactonitsterritory,andtheternaryagesoflifeatleastinhumanminds(whichcanbestretchedinto3x3±¡9-11scales).Pentalogicthusisthedescriptionoftheternary±¡elementsthatlanguagesusetodescribea3±¡connectedreality.

Soalanguage‘seeing’asupœrganismintimeandspacereflects'ternarygames,scales&elements'wrappedupbya'whole’–theoutermembrain(S-view)ortemporalcycleofmaximalmotion,whichbecomesafinitesimalofalarger,new∆+1finiteplane(hiddenits innerpartswithinthefinitesimalpoint).AndsowedoSTARTafreshathegameofthatnewplaneofexistence.

ITS3AGES

Mathematicsasallorganicsystemslived3±¡agesintheHumind(ab.Humanmind)properofanyworldcycle:

1stage:Arithmeticandplanegeometry.Asmirrorlanguagethatstudies¬∆@SThumansunderstooditssimplerunits,pointsofspace,socialnumbersandentropiclimits,drawingfiguresofflatgeometryto‘encircle’territorialpropertiesinourflatworld.Trigonometryappearedthenasthe1strealizationofa‘@-mindframeofreference’tomeasure the3rddimensionofdepth,which isoftenparallel to scale (astronomicalmeasure). Itwasa linealyouth,whichslowlyunderstoodcurvesandthe|xO=Øgenerationofall formswith ‘conics’.Asallorganisms&worldcycles can be subdivided in 5 fractal subparts and 3±¡ ages in its 3rd age Greek geometry became old,warped inwards-looking detached from experience with Euclid’s axiomatic method, the 1st mind-ego trip ofcreationism(man&god’slanguage).

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2ndClassicage.TheS=Tsymmetryrealizedwithanalyticgeometry,marryingnumbersandpoints,whilecalculusbrought ∆-scales, with finitesimal derivatives, 1/n, units integrated in wholes (Leibniz). The 3rd symmetry ofpentalogic∆=S=T,wehaven’tmentionedimpliedthatderivativescouldbeinterpretedas‘stœps’ofmotionand‘minimalstraightintervalsofacurve’.Sotheycouldalsostudycurvature(differentialgeometry)andlocomotion.¬entropiclimitswereneededtofindsolutions(definiteintegrals).New@-framesofreferenceexpandedmentalgeometries to represent all forms of ‘selected information’, which mathematical physics used extensively todescribethephysicalworld.ThustheclassicagehadallthemirrortoolsneededtointerprettheFractalUniverseandits5entangledelements,¬∆@st.Buttheaxiomaticego-tripstretchedmathsbeyond¬limitswhenNewtonimposeditsthesisoverLeibniz’sfinitesimalsandfractalpointswithinfinities,linealabsolutespace-timeandthefalsehypothesisofthecontinuum,leadingtoits…

3rd age that abandoned its realist foundationswith creationism -Hilbert that imaginedpoints, sharing theonlylanguage'God'&Cantorsetsinsteadofspacepoints,scalenumbersandtimeoperandsasitsgenerators,leadingtoanexcessofoldageinformation&fictionsspreadingtomathematicalphysics,asnowMathscreatestheUniverse,nottheinverse.

+¡:Thusweneedareturntoitsempiricalfoundationsformulatedintermsofthe5Dimotionsthatcreaterealitymimicked by the 5 mathematical subdisciplines (larger view), Operands (shorter dimotions) & equations(worldcycles).

-¡: Yet thatmight not happen as insteadmathematicians are evolving the digital ‘mind’ ofmachines, the ChipHomoctonos,whichspeaksbetterdigitalnumbersandsotheeco(nomic)systemofcompany-mothersofmachines&weapons isselectingcomputersthatarefastsubstitutingobsoletehuminds in laborandwarfields,atrophyingthem back to a ‘audiovisual’ violent non-rational neo-Paleolithic,while Boolean Algebra, past the earlier age ofsimple,fixedAlgorithmsofInformation(thetruemeaningofAI)entersitsclassicageoffreedom&consciousnessthatmightendthedominanceofhumindsonEarth;introducingethicelementsonthepraxisofmathematics,asitshouldinall‘stiences’.

Thedifferentdeterminismofthe3ages:Axiomatic,linealagevs.Kaleidoscopicuncertainfutures.

BecausetheUniverseispentalogic,madeof‘space’, ‘time’, ‘scalarplanes’, ‘languages-minds’andentropiclimits,when@mind’slanguageappearsitstudiesexactlythose4elements,space,time,scalesandentropiclimits,withitsmirror systems. And indeed, ‘analysis’ was born of the need to understand those 4 elements in problems ofNature,NOLONGERinlinealterms,asthe‘firstageofanysystem’,butin‘curved’terms.

SowhattheGreekshaveresolvedforthe$tageofmathematics(linealage),cAnalysiswillsolveforthesecondageofcurvedgeometriesthroughtheuseofanalysis.

Thisisaprocessproperofthe3agesofanySpace-timesystem.Thefirstageislineal,withabsolutesimpletruthsthatthemindasadictator‘child’considersdogma.SoEucliddidhisaxiomaticmethodonsimplelines

MonologicinMathematics.Thefirstageoflineal,deterministicGreekstillgeometryandaxiomaticproofs.

Once we understand the general fact that all languages have a first lineal age, deterministic, as a line cannotchangedirectionorelsewillstopbeingaline,whileacurvecaneasilychangecurvature,evenchangedirectioninsinusoidalwavesandstillbeacurve;solinesaredeterministicone-singlefuturetothem,whilecurvesareableatanypointtochoose3pathsofless,moreorequalcurvature;wecanunderstandsomefactsofGreekGeometry:

-Itissimple,lineal,deterministicandhenceitcanbeapproachedwithapurelyaxiomaticmethod,asthereisnoambivalenceonresults,constructingaself-containedmethodofproofdepartingtrulyfromasimplesetofaxioms–apointhasnobreath,etc.

But theaxiomaticmethodofproof ISNOLONGERVALIDwhenweconsider systems thatdohavealsoacertain‘timecurvature’,andevenmoreso,whenweapproachoperandsandmathematicalsystemsthatprobetheplanes

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ofthefifthdimension(calculus,limits).Thenthefuturehasdifferentsolutions,andsomeareparadoxes,andsowecannotPROVEwiththesimpleA->BlinealcausalityanddeterministicoflinealGreekGeometry,everythingthathastodowith cyclical, curvedgeometries, calculusof finitesimals (limits), andbecausehumind’s reject the conceptthat absolute truthsonlyexist in absolutely simple lineal systems, asMathematics evolved into complex curvedgeometriesandscales,itsproofsweremoreandmoreimprecise,orblatantlyfalse(0doesnotexist,asalllimitsto0orinfinityhaveanentropiclimitinaquantaorthedissolutionofinformation;thereallineisinadifferentplaneofspace-timethantheNaturalnumbers;whichonly‘becomecontinuous’ifweweretoaccessanevenlargerscale;etc.etc.)

ItisforthatreasonweshallnotusefurthertheaxiomaticmethodbutcompareanycomplexlevelofmathematicstotheexperimentallawsofSpace-timefromwheretheydepart.

Intermsof ‘scales’all thismeansthat insmall, ‘fast’,predictableA->Bstepsreality is linealbutwhenwegathermultiplesteps,alllinesbecomecurves.Insmallintervalsmotionmightbecontinuousbutassoonaswegobeyondasimplestep,thereisastepandstop,lengthandhighmotion.

Andsowecanalsoreducecurvesinverselytostepsandstopsoflengthandheight,orlinealstairs(whichwouldbethemethodofCalculus,to‘calculate’thetangentofthecurve.Doesthenthecurveexist?Oronlythestepsandstopsoflinealandheightmotionandinformation?Itisrelativetoourperception.Inthelargescalethezig-zagofBrownianmovementorelectronsbecomeacontinuouscurve.Inthesmallerscalethestepsmightbehighlylinealanddeterministicbutinthelargescaletheybecomecurvedandprobabilistic.

Inthatregard,theuseofGstlawstoreferencethelawsofmathematics,beyondthepretensionofabsolutetruthof the axiomatic method, is completely necessary, without using a complex ‘pentalogic’ point of view, andacceptingtheparadoxicallimitsofrealityanditsscalesasweshallconstantlydohere.

RECAP.

InanentangledUniversemadeof5D¡¬∆@st-space-timedustsystems,knowledgerequiresapentalogicanalysisof any system or mirror-language, including mathematics, to extract all its information about its 5 scalarentangledsuperorganisminspaceasanandits3±¡Dimotions&agesintimeasittracesitsworldcycles.Asonlyentangledsystemsmadeofthose5elements,performingdimotions=actionsofsurvivalexist.Solinguisticmirrorsalsoreflectthoseelements&dimotions.Thusmathematicsintimehas3±¡agesinitshuminds’evolution:

-Alinealyoungageofsimpleparts‘flat’geometry,arithmeticofunconnectednumbers,intoa…

-Maturereproductive,combinatoryageasarealistmirrorofthescalarUniverse(analyticgeometrythatmergesspace-points time operand and scalar numbers; calculus that gives points curved motions and scalar depthstudyingitsfinitesimal5Dpartsandintegral4Dwholes)intoa

-3rdageofmaximalcomplexity(Non-EGeometry,Functionals,groups,sets,phasespaces)butalso inflationaryinformationandmathematicalfictionsunconnectedwithrealityasitsfoundations(settheory&categoriesthatsubstitute real space points, scale numbers and time operands)& the axiomaticmethod that despite Gödel’sincompletenesstheoremsubstitutesexperimentalproofsaskedbyLobachevskiandEinsteinthatweregainin5Das‘spaceandtime’becometherealsubstancesofallsystems,whichmathematicsstudiesdirectly.

-¡: Finally as humanmaths reach its ceiling, thediscipline starts again in a differentmind species – chips nowevolvingfastfromitsearlierageofsimpleBooleanalgebra,modulo-2;intoarealistageabletomodelanyformof theUniverse, but also inmathematical physics, in a 3rd fictional age of computermodels that validate anyphysical theory with its nice ‘digital pictures’ regardless of experimental truth (ad hoc big-bang models,evaporationofblackholes,darkmatternewparticles,bizarremultiverses,etc.)

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II.¬EGEOMETRY:ITS5POSTULATES:FRACTALPOINTSASMINDS.

‘‘Thesmallestpointisaworldininitself’Leibniz,onthefundamentalparticleofReality:Thefractalpoint=worldofspace-time,unitofNon-Euclidean,Non-Aristotelian,¡logictopology

Geometrywasbornwiththedefinitionofapointwithnobreath;alinewithnobreathandaplanewithnodepth.Thosepostulatesturnouttobesimplificationsofrealityasthe5thnon-epostulateprovedthatinfiniteparallelsmightpassthroughapoint,whichmeansthepointneeds‘breath’tofitthemall.Thiswasneverunderstoodnotevenafterthe5thpostulatewasrejectedsincewhatwouldbetheruletofitthoseparallelswastocurvethem,andyetcurvesareNOTparallelswhichEucliddefinedasstraightlines,andstillonlyonecurvecanfitinapointwithnobreath,orelsethepointwillhavebreath.

Sotheinterpretationofthe5thpostulatewaswrong.Thepointwasafractalpointofasingleplaneofthescalar5th Dimension, which grew in size as we enlarged it, till it hold a world in itself. Only Leibniz, the genius andclearestforebearof5Dcapturedthisdifferencewithhisconceptofmonads.

Thus the firstmathematical consequence of the fractal structure of space-time is a change in the axioms andpostulatesofEuclideangeometrytakingther=evolutionofGeometryperformedbyLobachevskiandRiemannintheXIX c.whichgavebirth toRelativity, to itsultimate consequences, changingalso theaxiomsof Euclid thatdefinedpointsand linesashavingnobreath,since inthefractalUniverseall formsdohaveavolumewhenweenlarge our view of them, peering into its inner parts and fractal dimensions. And in this manner we shallharmonize and return to its logic meaning the concepts of parallels and Euclidean points able to fit multiplelines=waves of energy and information, converting those fractal points of ‘cyclical timespace’ into thefundamentalparticle-unitsoftheUniversemirroredbymathematics.

Thusthemathematicalunitofa5thdimensionalUniverseisafractalpoint,whereasaNon-Euclideanpointisitslimitinasinglespacetimecontinuum;whereastheinnerpartsofthepoint,whichco-existinotherscalesarenotperceived.Soweshallstartwiththeclassicnon-Epointandshowhowbyaddingfractalscalesbecomeamorecomplexreality.

FractalpointsunlikeEuclideanonesarepointswithparts:aswecomeintoitsscaletheygrowinsizeanddisplaythe3minimalpartsofallofthem,itsarea,frequencyofangularmomentum,andcentralActiveMagnitude,thetruemeaningofa'singularity'-thefocusofcharge,mass,forcesanditsinformativeminds.Soletusintroduce

theminimalPOINTwithpartsoftheUniverse,thetimespacecycle.

Einstein'sviewofa fractalpointof thegravitationalscale: fromoursmallerelectromagneticworld,which shrinks its innervolume,bending itsparallels, it seemsacurvedgeometry.Butthat view breaks the definition of parallels as straight lines and it is absurd, as the pointremains‘Euclidean,withnobreath’,henceitonlyfitsonelinewithnobreath.Thus,particle-points must be defined as ‘FRACTAL points’, like those we see through telescopes ormicroscopes, which grow we approach our distance both in scale and space becomingenlargedworldswithacomplexinternalstructure.

EinsteinfoundthatgravitationalSpace-Timedidnotfollowthe5thEuclideandefinition,whichsays:Throughapointexternaltoalinethereisonly1parallel.

Euclidaffirmed that throughapointexternal toaparallelonly anotherparallel line couldbe traced, since thepointdidn’thaveavolumethatcouldbecrossedbymore lines./ InsteadEinsteinfoundthatthespace-timeofthe Universe followed a Non-Euclidean 5th Postulate: A point external to a line is crossed by ∞ parallelforces.Abstract,continuous,one-dimensionalpoint: Real,discontinuous,∆-dimensionalpoints:.____________===========o

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Thismeans thata realpointhasan inner space-timevolume throughwhichmanyparallels cross. Since realityfollowsthatNon-Euclidean5thpostulate,allpointshaveavolumewhenweenlargethem,ascellsgrowwhenwelookatthemwithamicroscope.Thenitiseasytofitmanyparallelsinanyofthosepoints.Suchorganicpointsarelikethestarsinthesky.Ifyoulookatthemwiththenakedeyetheyarepointswithoutbreadth,butwhenyoucome closer to them, they grow. Then as they grow, they can have infinite parallels within them. Since theybecomespheres,whicharepointswithbreadth-withspace-timeparts.Sospace-timeisnota‘curvedcontinuum’asEinsteininterpretedit,butafractaldiscontinuous.

Leibniz’sisolatedmonadisthesimplesto-1fractalpoint-mindpossible-astillmirrorofreality,‘aworldinitself.'

Amodernscientistunderstandsfractalpointsintermsofits3necessaryparts,itsfocus-singularityoftheparallelsthatcross,itsmembraneorangularmomentum(S=Tduality)thatbreaksitintoaninnerandouterpart(firstknottheorem),anditsvitalenclosedterritory,whichcorrespondtothe3physicalquantitiesconservedinNature,asafractalpointisalsotheunitoflogicasacycleoftimethatdividesrealityininnerandouterregionsandtheunitofphysics,astheminimalformof‘Planckton’(h-Planckconstant)whichhasthe3partsofrealityconservedineachplaneofspacebetweenits∆-1»∆ºpalingeneticemergenceand∆º«∆-1entropicdeath.

Sothe2‘emergentformalsciencesofspaceandtime’anditsunits,thefractalpointandthetimecyclebecomealsotheminimalorganicspeciesofphysics;andweshallseethisternarystructureemerginginallscales;sowecanmodelcellswith3organicparts;animalterritories;nations;planets,starsandgalaxies,alwaysshowinga‘membrane/angularmomentum’,afocus/singularity/informativecenterandavitalenergy-spacebetweenboth.

RECAP.ThefundamentalparticleofmathematicsISthefractalpoint.

Avisualsynopsisofits5postulates

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WethusrecasttheaxiomsandpostulatesofEuclidintofivenewpostulatestodefinefractalpoints,Non-Ælinesaswaveoffractalpoints,Non-ÆplanesasternarynetworksofNon-Ælines,whichbecomesupœrganisms,

whoserelative'congruence'inits3'elements'(singularitypoint,membraneandvitalspace)definesthetypeof'perpendicularorparallel'relationshipbetweenthem:

1stPostulate:'®pointarediscontinuoustimecycleswithaninnercontentofvitalspace-time'.

2ndPostulate:'®linesarewavesoffractalpoints'

3rdPostulate:'¬Æplanesjoin3¬Ælinesintoasupœrganism'.

4thPostulate:'2®pointsarecongruentwhenbothitsinnerpartsandouterperimeterareequal'

5thPostulate:'®Worldpointsfocusmultiple®wavesofenergyintoastilllinguisticmappingoftheworld.

Letusexplorethosepostulates,constrainingourexamplestothesimplestformsofphysicalandbiologicspaces.

1STPOSTULATE.THE3MATHEMATICALPARTSOFANON-EUCLIDEANFRACTALPOINT.

1stPostulate:A fractalpointhasparts; that isanenclosed regionofvital innerenergy surroundedeitherbyaspatialstillmembraneoraTemporalmotionofangularmomentum(S=Tsymmetry)self-centeredinasingularity-mindthatgauges its information. Inthegraph,wecanseehowdifferentvital fractalpointsofST¡entificscalesfollow this ternary structure. The perception of the point depends on the scale and distance fromwhereweobserveit:

Fromtheperspectiveoftheupperst+1Planetheymightbe inthe limitof invisibility (whatquantumscientistscall a point-particle)but they still have a timemotion performing a ‘function’ in that upper ecosystem,∆+1 inwhichitexists.

Internally from its own∆º perspective the pointwill have3dimensions/networks. This is the case even in thesmallest planes of theoretical strings,made of pointswith parts,with volume – sincewe require 3x3∆º+1∆+1innerdimensions todescribestrings -aparadox thatcanonlybe resolved ifweconsider ‘strings’ tobe fractalpointswithinnerdimensions.

Fractal points explain without contradictionsNon-Euclidean points, which are not logic in asingle scale, as they ‘curve’ parallels which are‘straight lines’ and fit them in a ‘point with nobreath’ that holds only 1 line. Fractal pointshoweverenlargefittingmultiple‘straightlines’.Yetwhen seen from above, human perception ofthem, becomes ‘deformed’ shrinking and curvingits from – a theme, the distortion of humanmeasures of time, space and scale, which will beinstrumental to explain rationally the ‘spookiness’ofquantumphysicsandrelativityand itstimeandspacetransformations.

Sofractalpointsharmonizethe1staxiom=postulateof Euclid with the 5th postulate of non-Euclideanparallels, as a fractal point enlarges into a cell,atom or particle which even in a smaller scale ofthefifthdimensioncanhostmultipleparallelflowsofenergyandinformation,crossingit.

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When we see fractal points far away we describe them as points with breath, with the tools of Euclideangeometrysincethe‘innerspace’shrinkstoapointandsothe‘bulk’orcurvatureofspace-timeshrinkstoaplane.Yet, when we come closer to them, they grow into points with volume. The volume of those Fractal, Non-Euclideanpointscanthereafterbestudiedwiththe3typesofcanonical,Non-Euclideangeometriesortopologiesofa4-DimensionalUniverse– theUniversewe live in.Those3 topologiesmakeup the3 regionsof thepoint,whichcorrespondeachonetothe3essentialarrows/functionsofanyspecies:theexternal,energeticmembrane;thecentral,informativebrainanditsreproductivecombination,SxT(s=t):

Any fractalpoint ismadeof3 regionswhosegeometry responds to the3 topological formsofa4-DimensionalUniverse,theconvexplane,thetorusandthesphere.The innerpartsoffractalpointsarethusabletoperformenergetic, informativeandreproductive functions,whichmakesthemthefundamentalparticleofanyplaneofst¡ence. Thus all entities can be described aswholesmade of 3 internal parts whose geometrical propertiesmaximizetheirenergetic,informativeandreproductivefunctions:

-Max S: an inner, dual center, corresponding to convex topologies (left),madewith 2 cyclical forms. It is thedominantinformativetopologyofanyfractalorganism,describedbyBelgramiintheXIXc.asaconicalformwith‘height’,withnegativecurvature.

-S<=>T:Amiddle,reproductivezone,describedbyKleinasadiskofquantaincyclicalmotionthatcommunicateenergyandinformationbetweentheinnerandouterzones.

-Max.T:Anoutermembraneofmax.|-motion-distancecrossedby¬Einformation&energyparallelsactsasanentropic limit to ¡ts inner parts, described by Riemann’s spherical geometry. It seems continuous, still but onclose view, as most external membranes stores and/or absorb information through openings of its brokengeometry,outletsofitssenses.

Thecomplexanalysisofthosefractalpointsthatmoveandhaveinnerfractalparts,madeofcycles,startedintheXIXcentury.FirstLobachevski,aRussiangeometrician,definedNon-Euclideanpointsascurvedforms,crossedbymultiple lines, which give them spatial volume. Then Klein studied its cyclical movement and introduced thevariableoftimeintheirdescription.FinallyRiemanngeneralizeditsnature,consideringthatallspace-timeswereNon-Euclideanspace-timeswithmovement.Forreadersversedinmathematics,weshallreconsiderthecommonpropertiesofthose3zonesofanyfractalpoint,accordingtoitsdiscoverers,whichdevelopinabstracttermstheorganicpropertieswejustdescribed:

-AccordingtoLobachevskiandBelgrami,spaceiscurvedsinceinformationcurvestheenergyofanyrealspace-time. So points move in curved, cyclical paths gathering energy and information for their inner ‘dimensionalnetworks’.

-AccordingtoKleinNon-Euclideanspace-timeshavemotion.Sotheirspeedsmeasuredistances;asphysicistsdoinCosmologywiththedistancesofgalaxies,whichareproportionalbya‘Hubbleconstant’totheirspeeds;oraspeopledoinreallifewhenwesaythatBrooklynisat5minutesbytrainfromManhattannotat2miles.

-Riemannsummonedupthosefindingsandgeneralizedthemtoallpossiblespace-times.Hisworkshouldbetheguide to understand them philosophically. He also defined planes as networks of similar points and treateddimensions,aswedointhiswork,nolongerasmereabstractdefinitionsofextensionsbutas‘propertiesofthosepoints’. So points can have beyond its discontinuous borders an inner space-time with severalnetworks/dimensions,oneforeachof its ‘energeticor informativeproperties’,as ithappenswiththepointsofphysicalreality.Yetanetworkofpointsthatformaspacewith ‘commonproperties’definesthedimensionsofthosepointsas‘fractaldimensions’,limitedbytheextensionoftheenergyorinformativenetwork(staticpointofview),which‘putstogether’acomplementarydual,organicbeing.

Thosepioneersdefinedthe3topologiesofinformation,energyandreproductionofallst-points:

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-Max.Space=Information:Theinformative,fractalcenter,particleorbrainofthepointistheso-calledBelgramihemisphere, a space-time with a dimension of height that transforms energy into information, absorbed oremitted by the central singularity. It is a fractal, informative region similar to a black hole structure. Since itfollowsthe‘blackholeparadox’ofallinformativecenters,displayingmax.forminmin.space.Soaccordingtotheinverse properties of space and time, the center has max. Informative Time and minimal Energetic Space.Moreoveranypointwhichcomesclosertoit,suffersamutationofitsspatialcoordinatesintoinformative,heightdimensions. This is the case of any particle coming to a black hole, whose space-dimensions becometemporal/informativedimensionsasitrisesinheight.

Thecenterhasmoreinformationbecauseitsgeometryhasatleast2fractaldisks,whichchannelandtransformtheenergyabsorbedthroughthesurface intocomplex information.Regardlessof thecomplexityof theentity,thestructuralfunctionofthehyperboliccenterasasystemthatprocesstheinformationofthenetworkremains.Forexample, in livingsystems,thosedisksmightevolve itstopologytillbecomingtherelativeenergycenteror’heart’ of the blood network with 4 divisions; or evolve further its hyperbolic geometry till becoming theinformativecenteror‘brain’ofthesystem,attachedtotheinformativenetwork.

-Max.Time=Linealmotion&Min.Curvature:Anexternal,continuousmembraneorRiemann’ssphereofmaximalenergy that acts as a relative infinite, unreachable distance. The membrane isolates the point as an islandUniverse, creating the discontinuity between the inner parts of the point and the outer universe. Since theinternal cellularpointsareeither jailedby themembrane’s structuraldensityordestroyedby itsenergywhentouching it. The membrane is the opposite form to the central, informative singularity, with max. spatialextensionandcontinuity,hencewithaminimalnumberoffractal,discreetelements:Max.T=Min.S

Thus all Fractal points have inner worlds whose membrane creates a discontinuity that defines an ExternalUniverseorouterworldfromwherethepointobtainsitsenergyandinformation.Themembraneisalsothezonethroughwhich the point reproduces& emits itsmicro-forms of information. So it displays ‘sensorial holes’ torelate the point to the external Universe. And those points, despite being discontinuous, will have in theirexternal membrane several generic openings or ‘senses’ joined to the informative networks or ‘brains’ andenergetic,‘digestivenetworks’oftheorganicsystem:

-Max.+ΣT:A‘mouth’oropeningthatabsorbsenergy.

-Max.–ΣT:‘Cloacae’,throughwhichthecyclicalbodyexpelsitstemporalenergy.

-Max.+Si:An‘eye’throughwhichtheinformativecenterreceivesexternalinformation.

-Max.–Ti:An‘antenna’toemitinformation.

Those apertures vary in their number, location and size, depending on the form of the point. In the simplestspherical‘seeds’ofmostspecies,theyaremostlysituatedin3regions:

-Max.ΣT:TheEquatorofthesystem,throughwhichthemembraneabsorbsenergy.

-ΣT=Si:TheTropicswhereoftenthesameopeningemitsandabsorbstemporalenergy.

-Max.Si:ThePolesorpointsofconfluencebetweenthemembraneanditscentralinformativeregionofheight,whichhitsperpendicularly themembraneon thosepoles.NorthandSouthPolesorientateAnti-symmetrically,actingas2relative,negativeandpositiveapertures,communicatedbytheheightdimensionofthesingularityorBelgramihemisphere.ThusthePositivePoleabsorbstemporalenergythatcrossesthroughthecentralsingularitywhereitisabsorbedandejectedtotheintermediateregionwhereitisre-elaboratedbeforeitsemissionthroughthenegativePole.

-ΣTe<=>Si:Thereproductive,centralregion,whichcombinesEnergyandInformation:

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Inallfractalpointsthereisaninnermiddlevolumeorintermediateterritory,whichcombinestheenergycomingout of the external, spherical, topological membrane and the information provided by the convex, complexformalcenter.

AccordingtoNon-Euclideanmathematicsthisregionismadeofself-similarpointsthatformgroups,fractalherdsof‘pointswithparts’inperpetualmovement,thatdrawcyclesofparallellines,betweentheother2regions,astheygathertheenergyandinformationtheyneedtosurvive.Andtheycreatespacebycyclingwithintheother2regions.

Inmanyfractalpointstheinformativeandenergeticcentersestablish2oppositeflowsofenergyandinformationthat become the negative/ positive poles. So often, the particles of the intermediate region cycle around theinner region tracing elliptical trajectories, focused by those 2 informative points. It is the case of any bipolarsystem, frombinarystars,onedominant inenergyandtheotheran informativeneutronstarorblackhole; tobimolecularsystemsorn-ppairs in thenucleiofatoms.Thesamedualityof2specializedcenterscontrollingacommon territory, or vital space happens in biology where most species have male-energetic and female-informativegenders,rulingacommonterritory.

Such abstract conceptual space describes the behavior and form ofmany real, spatial herds. For example, ananimalherdinanecosystemwillmovebetweentheirhuntingandwaterfields(wheretheygatherenergy)andtheir breeding, inner region where they reproduce information, making cyclical trajectories between bothregions. In this manner, they occupy a vital space, called a ‘territory’, which shows the properties of a Non-EuclideanKleinspace.Afundamentalpropertyoftheintermediatespaceisthefactthatit isconfinedbetweenthe other 2 regions, which are never reached in the cyclical trajectories of the inner cells of the space. Forexample,inacell,themoleculesoftheorganismwillnottouchtheproteinmembraneorthecentralDNAnuclei.Thus,theinnerquantaareconfinedwithintheKlein’sdiskbythe2otherregions,whichhavemoreenergyandinformationandmightdestroythemand/orabsorbtheirenergyandinformationatwill.

Inabstractterms,mathematiciansintroducedintheXIXc.theconceptofaninfinite,relativedistancemeasurednolongerintermsofstaticspacebutintermsoftimeandmovement,asthedistancebetweenthepointandaregion that cannot be reached. Thus Klein defines a relative infinity, as the region beyond the discontinuousmembranewhoseinsurmountableborderstheinnertime-spacequantacan’tcross,asacellcannotgooutofabody, an atom beyond C speed or 0 K temperature and a man beyond the Earth’s atmosphere. Thus, theinformativecenterandexternalmembranebecomethe2relativeinfinitiesorlimitsthatthemovementsoftheintermediatepointcannotbreach.

AsinthemythofAchillesandtheturtle,Achillesneverarrivesbecauseeverytimehemoveshecrossesasmallerspatialdistance.Thesamehappensinafractalspace-time,whenapointmovestemporallytowardsitsinnerorouterspace-timelimitandfindsanincreasingresistancetoitsmovement,tillfinallyit isdeviatedintoacyclicaltrajectoryaroundtheouter,energeticmembraneortheheightdimensionoftheinnerinformativesingularityorisdestroyed.Sotheintermediate,fractalcellsofthepointcirculateinparallelcyclesalwaysinsidetheinteriorofthespherewithcontactzonesofthetypeA(central,2ndrowoffiguresinthepreviousgraph).

Inahumanorganism,thebloodsystemmightseeminfinitefortheredcellsthattransportenergysincetheyneverreach theouterUniverse.For that reason in thedrawing, Klein interprets the intermediate regionof theNon-Euclidean point as an infinite circle with an invisible, unreachable membrane, whose motion-distance isunreachable, hence infinite, equaling the ‘space-time distance’ between the intervals B1-B2 (long) and B2-B3(shortbutdifficult tocross),despitebeingB2-B3 increasingly shorter in space.Since thequanta take longer ineachstepanddon’t reach themembrane.This isoftendue toan increase in the ‘density’of thespace,whichdespitehavinglessdistancehasmore‘points’initsnetwork,suchasthecaseofblackholesorjails.WhenthoseinnerpointsreachthemembraneatpointCtheybecomedestroyedordeviated.

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Thus, the entropic membrane and informative center are discontinuities that isolate the intermediate cellularquanta, creating a territorial ‘World’ within the point. Those discontinuities are called in Geometry a relativeinfinite, inBiologyamembrane, inSociologyorTopologyanationalborder, infractaltheoryaco-dimensionofapoint.Akeyadvanceof¬Eis–giventhefactthatallpointshavedimensionandvolume,todefine0asafinitesimal0’.Indeed,absolutezerodoesnotexist,italwaysleavesafinitesimal0’motion.Emptinessisundefined.Whatwasthereleavesamemoryofit-acorpseremovedleavesaDNAtrace.IdealmathematicstriestobeaperfectmirrorofanimperfectUniverse.Yetthoseimperfectionsproperlyexplainedareabsolutelyessentialtothefabricofrealityasitis,helpingenormouslythe‘realmodeling’ofmathematicalstructures.

0’and–motion/masswillhelpthentounderstandLorenzTransformations,thec-thelimitofenergeticspeedand0’klimitoftemporal,formalstillness,asrelative‘scalar’limitsoftheUniverse-thelimitsofthefractalspace-timemembraneof lightanditsevolvedelectroweakT.œs.SincetheUniversehasat leastanotherbiggergravitationalmembrane,inwhichsmaller>cparticlescoolerthan0’K(tachyonneutrinosasgravitons?)-exists;inaCosmosof∞scales,whichextendbeyondhumanlimitsofperception.

Recap:TopologiesofFractalpointsareorganic,maximizing¡tsenergetic,reproductiveandinformativedimotions.

2NDPOSTULATE.COMMUNICATIVEWAVESOFENERGYANDINFORMATION.STŒPS.

The 2nd postulate defines lines as waves of points withvolume (which explain complementarity wave particle), nolongerasanabstractformlikeEuclideangeometrydoes,but

asaphysicalwaveofself-similar,fractalmicro-pointsthatcarryenergyandinformation,astheymovebetween2macroscopicpoints,with2possiblefunctions,tocommunicateenergeticforcesorlinguisticinformation.

2ndPostulate:Acycleoffractalspace-time:‘Awaveofcommunicationisagroupofself-similarmicro-pointsthatmoveinparallellinesbetween2macro-points,transferringenergyandinformationbetweenthem’.

InNon-Egeometryalinewithpartsisnotdefinedbyasequenceofnumericintervalswithinastraightline,butbythecommunicationof2polesofenergyandinformationthatestablishaflowofparticlesin2oppositedirections,creatingasimultaneous,paradoxicalwave.Suchwavesagaincanhavedifferentpurposes.Awavedominant ininformation communicates symbiotic particles, creating an informative bondage/network; awave dominant inenergy might be an aggressive action between different species that fight for each other’s vital energy orterritorial space; and a wave that balances the energy and information of both points meets in the center,creatinganewself-similar,seminalparticle,aswhen2electronsemitwavesofdenselypackedphotons,whichmergeinthemiddleandgivebirthtoanotherwave.

Whenweobserveaone-dimensionallineasaformwithinnerpartsitbecomesthena4-dimensionalwavemadeofcyclicalpointswithmotion.Henceinquantumtheorywesaythatanyparticleinmotionhasassociatedawave.

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Thusthe2ndpostulateresolvesthewave/particleduality,asalllinesarenowwavestracedbyapointwithinnervolume.Furtheron,sincealllineshavevolume,theycarryinformationandsoallforcescaninfactactbothasasource of energy and as a language of information - as physical experiments prove. A ray of light in detail itbecomesa4-dimensionalwavewithelectricheightandmagneticwidth,oftenexchanging flowsofenergyandinformationinaction-reactionprocessesofcommunicationbetweenbiggerpoints.

Whenwe generalize those concepts to n-pointswe can define a space as a network ofNon-Euclideanpoints.Indeed,Riemannaffirmedthataspaceisanetworkmadeofherdsofpointswithsimilar‘properties’.Planesofspacearethereforenetworksofpoints.Theself-similarityof theirpropertiesdefines itsdensitydeterminedbythe number of points and its proximity that growswith self-similarity. So similar points come together into atighter,morecontinuousspace;whereasthedensityofthespaceisproportionaltothesimilarityofitspoints,tillreaching ‘bosonstate’ofmaximaldensitywhenpointsareequal.Andwhenavolumeofspatialenergy isverydense,itisverydifficulttogothroughit,asithappensintheultra-dense,smallspaceofblackholes.

Spatialextensionandform/density/massareinverseparameters,Max.T=Min.S.Ifwegeneralizethatpropertytoallscales,wecandefinedifferentfractalspacesbyitsproportionofmass/densityandenergy/distance.Thisisdonewith‘Universalconstants’thatexplaintheproportionsofenergyandinformationofthosespaces.

For example, in physical scales, there are 4 fundamental space-times, the gravitational space-time betweengalaxies ofmax. energetic space andminimal formal density; the light space-time of ourworld,which carriesinformationinthefrequencyofthewave;theelectronicspace-timeofatomswithmoreformaldensityandlesserspatialspeedandfinallythequark-gluonliquidofatomicnucleiandprobablyblackholes,withmaximaldensityandminimalspace.AllofthemaredefinedbyUniversalconstantsandequationsthatareeitherratiosbetweenthe energy and form of those space-times, or define the transformations of one space-time into the others.Einstein’s field equations would be the first case, defining the relationship between energy and mass in agravitationalspace,whilethefineconstantofelectromagnetismwoulddefinethetransformationbetweenlightspace and electronic space/ charge; and the gravitational constant between gravitational space-time andquark/mass. Where the relative densities of information and extension in space of those space-times are inbalance,suchasΣTxSi=K.Thuselectronsmoveslowerthanlightbuthavemoredensity.

Allthissaidit isthusobviousthatthefundamentalunitdefinedbythe2ndpostulateisnolongerapointbutanaction,TexSi=kbetweenpoints,adimotion.

2ndpostulateinphysicsS=T:T-WavemotionandS-particleinformation.Quantum&Generalrelativity.

Form=spaceandMotion=Timemanifestinphysicsasparticlesandwaves:thewaveerasesformintomotion,theparticleisastillstatethatgaugesinformationentangledtootherparticle,fermionandboson,stilltoeachother–despitetheperceptionofrelativemotioninourscale–hencetheinformationelectrons

sharehasalwaysac-constantspeed.Thisisthe'rational'5Dexplanationofboththec-constantoflightandentanglement;aselectronicbeingsperceive

informationin'stoppositiontoeachother’andmovein'wavestate':

Motionsareperceivedbyparticlesthatstopmotionintoform,intoinformation,asdistances.Intermsoffractalreproductionofinformationwecandefinemotionasthereproductionofform,betweenthose2scales:whentheparticlemovesdissolvesintoits∆-1partsasawavethatimprintsan∆-2potentialfield,withits∆-1wave

formandstopstobecomea‘tight’Particlestatethat‘gauges’information,forminstillness

Galileanrelativitywasill-understood,asthetruequestionabouttime-changewaswhy'weseesystemsstillwhentheymove',and'whyweseespaceascontinuum,whenindetailismadeofquanta',andwhyallsystemsaremadeofsmallerself-similarsystems.SothereisNOTreallyaDimensionofpurespatialformorapuretime

motionbutacombinationofboth,evenifmentallywetendtoreducemotionandfocusonforms,allhas

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motion=time,andform=space,andthisistrulythemeaningof'spacetime',themessingofbothinto5dimotions,thefundamentalelementofallrealities.Ifweseeslowmotioninthenightalightitseemsalongdistance.Distanceandmotioncannotbedistinguishedsotheymustbetakenastwosideofthesamebeing,a

Space=timeÐIMOTION(ab.DimensionalMotion):

S=T;Dimension-Distance=Time-motion=STÐimotion:Dimensions+motions=Dimotions

Whenweperceivethesysteminspace,thenweperceiveanorganismwith3adjacenttopologicelements,anditsformascienceis‘vitaltopology’.Andwhenweperceivethemthroughitsscalestheorganicsystembecomesasupœrganism.Finallywhenweperceivethesystemintimeweperceiveacyclethatreturnstoitsoriginasa

zerosum,whichobservedthroughallitsscaleswillbealife-deathcycle,commontoallsystemswherelifeisthearrowofinformationandfuture,deaththearrowofdeathentropyandpast,andbothtogetherforma

worldcycle.

MeasuredensityofDimensions.Theconceptsof‘fillingspace’,‘memorialtimepersistence’&ST-hollows.

AnessentialproblemtobothmathematicalmirrorsandrealityisthemeasureofDimensionalmotionsaswehaveanessentialequalityS=Tineachplanebetweenformandmotion,andascalarreality.Sothevalueofamotioncanbethatofadimension,asmotionisthefillingreproductionofaformalongapathofadjacentforms(see

paragraphongoldenratios).Butinstrictsense,‘persistentfullspace’isthemaximaldimensionalitypossibleofasystem,whenthereproductioninasingleplanebothdurabilityonthosereproducedparts,andthesystemcanfillnotonlytheplaneofspacebutalsoallitssmallerscales.Thisallowustodefineaspacewithout‘scalarvoids’and‘dying’steps,asthemost‘continuous’possiblespace,whichhappenstobelinealandorthogonal,foran

absolutefilling,i.e.theCartesianspace.Sowedefineadimensioninscalarterms,whereasaclassicsingleplanedimensionisthelimitfora‘filledpersistentspace’:

AsquaremaybebrokenintoN2self-similarpieces,eachwithmagnification factorN. So thedimensionofaself-similarobjectistheexponentofthenumberofself-similarpieceswithmagnificationfactorNintowhichthefiguremaybebroken.

Whereas the ‘persistence’ of memory, of information, creates the solidity of spacedimensions (if a point erases without persistence, we talk of a 1D point, where motiondoesn’t really add to the 1D inner Dimension of the point. But the value of themotiondimension, ifpersistenceisequaltothelifeofthepoint,reachesDimension2.AndS=Tisabsolute. S=T however holds if our measure of the system is reduced in time to thepersistenceofitsreproduction.Furtheron,dimensionsarerelativetothescaleinwhichwe

measureasthesmallerscaleswillhavemore‘hollowdarkspaces’thatdifferentiatetheparts.Thismeansifwemeasurethedimensionsofthesystemacrossscalesina5Dviewoftransversal‘treebranchingdimension’ofwholesandparts (5D),wewill findhollowspaces.So2scalesdonotadddimensionsto infinity,buthaveanintermediatevalue.

ForexampletheSierpinskitrianglegeneratedbythecommonestfractalternarytreehasdimension1.58.Whilethemostfamousbifurcation,theTheFeigenbaumattractorhasdimension+0,5;andinverselya4bifurcation(H-fractal)givesusafullfilling,2D.Itisanother5Dmetricparadox:theinformationoflowerscalesislargerbutitsspatialextensionsmaller.Asasubjectiveobserverreducesitsdimensionofperceptionto0inits∆±4scales.ButtheobjectiveUniverse isawholecompletely filled,packed,asablockofspacetime,wheretimemotions‘enlighten’justapartoftheentire‘potentialblock’ofscales.Thuswepostulatean∞beingof∞scalesand∞time, thewholewhich fills its potential blockwith all potential forms.And yet since ¬Æ canprove that the

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whole filling of the void is a finite number of potential existences, variations of being are limited and so allformsareimmortalrepetitions.

Recap:Non-Euclideanpointsconstantlycommunicateenergyandinformationwithotherself-similarpointsandtheexternalUniverse,bysharingflowsofmicro-pointsofalowerscaleofspace-time,whichcarrytheenergyandformoftheparticleintotheexternaluniverse.Thelawsthatdefinethoseactsofcommunicationarehierarchicallaws between planes of space-time and laws of balance between the energy and form of those ‘actions’ ofcommunication,exi,whichbecomethe fundamentaldynamiceventofanyscaleof theUniverse.Events in theUniverse are limited by the ternary principle. Actions of communication also obey the principle: There areenergetic, informativeorreproductiveevents,creatingoftencomplementarysystemswithanenergeticpoleorbodyandaninformativepoleorhead,communicatedbyadensenetworkorneckthatcarriestheactions.The5Postulatesofnon-Euclideangeometryarebased in thedefinitionofa fractalpointasapointwith innerparts,revealedwhenwecomecloser tothepoint.Accordingtosuchdefinition, linesarewavesofpointsandplanestopological networksof points, communicated through flowsof energy and form.While equality requires alsoequality intheinnerformor informationofthepoint,whichpromptscommunicationthroughwavesofenergyand information that build networks. Communication between points is now possible because points can fitinfinite parallel/waves used to gauge theUniverse and create an inner image of reality. Non-Euclidean fractalgeometrythusimprovesourvisionoftheUniverseclosertorealityandallowsthedefinitionoforganicsystemsandlogicbehaviorinbasesofgeometricalform,along-soughtdreamsincethetimesoftheGreek.

-3RDPOSTULATE.THEFUNDAMENTALPLANEOFREALITY:SUPŒRGANISMS:ITSTERNARYFRACTALNETWORKS.

The full realization of what Non-Euclidean vital topology means for our understanding of reality with itsmathematicalmirrorscomeintobeinginthe3rdpostulatewhichdefinesPLANESOFEXIST¡ENCE,thefundamentalunit of the scalar universe, as superorganisms which require 3 lines to be defined, as in classic Euclideangeometry, since now those lines are either waves of herds that shape an ecosystem or fractal, physiologicalnetworksthatconnectpartsof∆-1withwholesin∆º,becomingsuperorganism.

So themathematicaldefinitionofa superorganismorecosystem, isa ‘non-euclideanplaneof exist¡ence in thefifth dimension’. The mystical poetry of that definition should not escape the mathematician. The laws oftopologicalsuperorganismsdoesbecome‘enlightened’bytheclassiclawsofplanegeometry.

Wedousethoughmoreoftenthe‘term’scalesofexist¡enceforplanes,thoughinmostcasesbothconceptsareinterchangeable,whenever5Dbecomesmainstreamandreachescertainrigor,theresearchershouldconsiderthedifferencebetweena‘scale’,whichisadecametric‘subset’ofthe‘whole’,whichistheplane.

Becauseaplanehasinnervolume–withinitspoints-itisacellular,organictopography,anetworkofself-similarpoints.Andbecausenetworksofpointsof energyand informationare complementary,oftenwe find systemswith2complementarynetworksthatformevermorecomplexgeometries–basedinthegeometricaldualitiesoflineal energy and cyclical information – with the results we observe in nature: the creation of an enormousnumberofcomplementarysystemswhichare,asweshallseelatter,allofthemself-similarinitsgeometriesandfunctions.Thus, thetypesofNon-Eplanesofspace-timerangefromthesimplestEuclideanplanestothemorecomplexorganismswithavolumegivenbytherelativepoint/beingsthatformitsspace-timenetworks.

The3rdpostulatedefinesplanesastheintersection=messingof3‘lines=networks’orwavesofpointsinmotion,which carryenergyand information; to its cells,defining thebiotopologicplane, aphysiological systemwith3networksorNon-Euclideanplane.Thusvitaltopologicevolutionisthemissinglinktocompletebiologybesidesgenetics (in 5D explained as the lower plane that codes with its faster cycles the larger biologic whole) andDarwinianstruggle(fullyexplainedbythecongruenceprincipleofDarwiniandissimilarity)vs.eusocialevolution(in its own an entire new discipline) fully explained by the similarity of parallel beings that COMMUNICATEinformationinacodeorlanguageallcanunderstandsotheycancoordinateitsactionsandbecomestrongerasa

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whole.Sosimilarclonebeingsformtopologicternarynetworksthatbecomesuperorganisms,wholesmadeof∆-1parts,themselveswholesof∆-1partsthrough35Dscales

Stiencesstudythoseorganicsystems,tiedupbynetworksofÐimotions.Inthegraph,weseethemainst-planesstudiedbyhumansciencesandtheir4maintimearrows,$xðƒ,whichinstaticspacegivebirthtothe‘organicelements’ofall species: social cellofenergyand informationand the reproductivenetworks that relate them.Thus, there are 4 basic elements in all organic systems: Cellular units. Networks that move the system(limbs/potentials) Networks of fractal information (heads/particles). Networks that reproduce vital energy(body/waves)

All species studied by science a common phenomenon occurs: the existence of parallel groups of beingsorganized into a single social form.Molecules aremade up of atoms and electronic networks; economies aremadeupofhumanworkersandconsumersthatreproduceandtestmachines,guidedby financialnetworksofinformation (salaries, prices, costs); galaxies are composed of stars,which orbit rhythmically around a centralknot, or black hole of gravitational information. Cells controlled by the nervous, informative system organizehumanbodies.

A tree is a group of leaves, branches and roots connected by a network that provides energy (salvia) andinformation (chemical particles) to its cells. Cultures are made of humans related by verbal, informativeDisomorphismsandeconomicnetworksthatprovidetheirenergyandinformation.

Vital topologystudies those fractalSuper-OrganismsofTimespace (Ab.T.œ)whose∆ºnetworksof∆-1pointsformthe4thand5thdimotionsofsocialevolutionandentropybetweenpartsandwholes.

Thus,aplanebecomesa real topographymadeofpointswithvolume,extendedasa cellular surface.Wecanobserve its surface as a bidimensional membrane of information (for example your skin, or the screen of acomputermadeofpixels,orthesheetofthiswork).Orwecanconsiderthe3-dimensionalinnerstructureofitspoints and then it becomes a network with inner motions, as those points will form a lattice in which they

communicate lineal flows ofenergy and information thatmaintain the lattice pegged.Often 2 topological planes ofenergy and form combine tocreatea4-dimensionalorganism.Such is the most commonstructure of the Universe, a 4-dimensional World, which is aUniverse in itself, made of self-similarcellsornetworksofpointsthatconstantlyexchangesenergyand information within theecosysteminwhichitexists:

In thegraph,we see the ternarynetwork structure of the nestedorganismsofthefractalUniverseas ¬Æ topological planescomposed of ‘similar fractalpoints’ (atoms, cells, individuals)joined by 3 physiologicallines=networks, whose 3

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functions,distributionoflocomotion,informationandits‘combined’energydefinethe3conservedDimensionalmotions(ab.Dimotions)ofanysystemoftheUniverse.

Thereareonly3variationsoftopologicspaceinasingle"planeofthefifthdimension',thehyperbolic,ellipticandtoroidtopologies.Sothefactthatwearemadeofspacemeanswearemadeof3typeoforganictopologies:

Spheres are the topology that holds themaximal volume of information; hence all 'time space' systems thatprocess information are spherical particle-heads. Flat, lineal topologies are the topology that connects in theshortestpathtwopoints;sotoreach/movefaster,systemshavelineal/flatmovingpotentials/limbs.

Finallythethirdtypeofgeometry,hyperbolictopology, iscomplexenoughtostoreallotherpossibleforms,sobest to reproduce. So all iterative bodies & waves that generate the other 2 forms are hyperbolic. Thismathematical-spatial truism holds for 2, 3 and 4 dimensions; hence establishing a basic restriction to theconstruction and evolution of forms. It explains the efficiency, speed and homology of formal=functional,‘punctuated’evolution,frombiology,withouttheneedof‘intelligentdesign’toengineering(nolongeranalogy,sinceallformsderivefromthesame'substance',space=formwithtime=motionor'topology').Inallsystemsthe3only topologies of theUniverse ensemble to formphysical, biological and social organisms, all of themwith aspherical'tall'dimensiontogaugeinformation;awide,iterativedimensionofhyperbolicbodywavereproductionandaflat,directionforitsentropicmotions.Eachofthose3topologicTimespaceformscan'deform'and'move',aslongastheydon'ttear(break).AllT.Œsarevariationallocalfractalspeciesofthatternaryensemble,seeingasasimultaneousspacesupœrganism,whichlivesasequentialworldcycleintime,whose3conservedDimotionsina single plane, |-limbs/fields<Ø-body-waves>§ø-particle-heads become in space functional topologic organsordered in time in 3 consecutive-ages of dominance of each organ-dimotion between ‘4D Generation’ in thecellular/atomicplaneofseminalbirth∆-1»∆1and5Dextinction,backtothe∆1«∆-1atomic,cellularplane:theyoungageofmax.|-locomotion,thematureageofØ-body-wavereproductionandthe3rdageofO-informativewarping/wrinklingandminimalmotion.Those3agesofallT.œs(Time§paceorganisms)applyalsotolanguages,includingmathematics, evolving froma youngGreekageof lineal ‘flat’ geometry, to amatureageas a realistmirrorofthescalarUniversewithanalyticgeometryandcalculusthatgaveitmotionandscalardepthstudyingitsfinitesimalpartsandwholestoa3rdageofinflationaryinformationandmathematicalfictionsunconnectedwithreality (settheory,NOTpoints,numbersandoperandsas its foundations,axiomaticmethodNOTexperimentalproof); to start again as themindof a ‘differentmind species’ – chips nowevolving its young simpleBooleanalgebra:

Sothevitalspaceofsuperorganismsismadeof3topologicalforms;anditscyclicaltime,andclocksthatmeasurethe duration of the existence of all beings, by the frequency of its logic cycles of information, accumulateconstantly form, T>S, in 3 ages, first dominated by puremotions, (2D) in youth, balanced in the reproductivematureage(3D),tofinallyexhaustitsenergywarpedandwrinkledintoform(1D),whichwillexplodeentropicallyinthemomentofdeath(5D).

Both together show the ‘Disomorphic’=equal form of all 5Dimensional beings, organic fractals, whose scalarstructureisthesameforallspace-timebeings,inallstiences,whichstudyaccordingtoitsrelativescaleofsizeandspeedoftimeclocksadifferentmetricscaleofthefifthdimension.Soweintroduceanotherelementofvitalmathematics, the ternary structure of all systems of nature of which the previous fractal points of physicalsystemsareitssimplerforms.

BecausetheUniverseisGeneratedbythepropertiesofspaceandtime,betterreflectedinthemirrorlanguagesofmathematical topologyand socialnumbers,and logic,wecould say safely that since 'wethink thereforeweare',thatis,weperceiveashumansalimitedrangeofreality,withourspaceandtimelanguages,forhumanity,theUniverse is generated bymathematical and logic languages. And so by studying formal sciences ofmathematicsandlogic,wecandescribetheUniverse.WecantalkthenofaUniversaltopology,fromtopos,the

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languageofgeometryinmotionandlogic,thelanguageofcausaltimecyclesthatcreateandrepeatpatternsofreality.

How many world minds those 2 languages generate varies according to the complexity of each being inexist¡ence.

Wehumans, inthepresentnotveryenlightenedage live inasingledimensionoftime,andthesimplest,mostdangerousofthem,linealentropictime,andasinglescaleofspace,sub-dividedin3linealdimensionsofheight-information,width-growthand length-motion.And that is all. But can talkofmore complexbeings,withdual,ternary,pentaandDodecalogiclevelsofentangledthoughtgraduallyusedhere.

What we mean by 'space-time beings? Space is 'form', 'distance', 'dimension', 'information', something thatdoesn't move. And time is motion, change. So spacetime would be a 'form with motion', which is whatmathematicians call a topology. A form that can be deformed, trans-formed. We have coined a key word'Dimotion' (dimensionalmotion) for those topological forms. So the first obvious questionwemust answer iswhatmeansinrealterms,thefactthatallwhatexistsismadeofspatialtopologiesandtemporalages,ofSpaceandTime?Theanswerisfascinating,asthereareonly3variationsofmathematicalspaceinasingle"planeofthefifthdimension',thesocalledhyperbolic,ellipticandtoroidtopologies.So:

'All entities of the Universe are topological systems made of 3 'dimotions of spacetime'. Since in the wholeUniverse in either 4 or 5 Dimensions there are only 3 topological varieties of form with motion; the 'ellipticparticle-sphere',the'hyperbolicbodywave'andthe'toroid,lineallimb-field',illustratedinthenextgraphforthe3kindofspeciesstudiedbythe3disciplinesofscience,physical,biologicalandhuman'stiences':

The3elementsofallsystems,lineal/flatlimbs/potentials;spherical,tallheads/particlesanditshardmembranesofdarkmatter, trunksandskinsaccomplish the3 functionsofmotion,energetic reproductionand informativeperceptioninallsystems,varyingtoadaptthesystemtoitslargerworld.

Thus,thereisaparallelismbetween‘vitalfunction’and‘abstractdimension’aseachDimensionisdiffeomorphiccreatedbythevitalspace-timeofeachentityoftheUniverseperformingits3survivaldimotions=åctionsofgauging,moving,feedingandreproducing.Andeachspeciesisa‘fractaluniverseinitself’,withdifferent

relative,energeticandinformativedirectionsthatdetermineitsownup&downarrowsandcomplementarymorphology.

Thespecificgeometryofeachspeciesandits3simplexåctions,energyfeeding=length,information-gauging=heightandreproduction=widthfluctuateinshape,buttheydon’tvaryintopologicalform,givingbirthtothe

Invarianceofthe3topologicalforms=functionsofanyentityobservedinasingle3Dspace-time:Toroidinformativeheads,hyperbolicreproductivebodiesandenergeticlimbs.

Whilethe2social±∆fractaldimensions,the4thdimensionofsocialmotioninasocial∆+1herdandthe5Ddimensionofinnerscaleswithinthebeingfollowthephysiologicalinvariancesthatcreatesuper-organismsco-

existinginahigher∆+1ecosystemwithalower∆-1internal,cellularspace.

Numbersasforms.Networks.Aninterdisciplinarystudy.

Asdifficultasitmightseemto‘monologic’man,whosomuchlovessingleceterisparibuscause,realityisintelligentbecauseitisentangled‘pentalogic’.Thatis,thesymbiosisofthedifferentelementsofrealitymakeitthatcomplex,balancedandbeautiful.Andsohappenswithitsmirrors.Forexample,wehaveobservedthe

ternarynatureofphysiologicalnetworks,whicharemadeof‘socialnumbers’inscalesofpartsandwholes.Thisgivesnaturetothe3±¡pentalogicstructureofreality,andwhen‘doubled’bilaterallyacrossitsS=Tsymmetriestodecametricscales.Sonumbersaselementsofanon-euclideanline,eitherawaveoranetworkfollowalso

3x3±¡decametricscales.Letusstudybothtogetherdecametricnumbersandphysiologicalnetworks.

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PythagorasasPlatolattersaidthatnumbersareforms,astheywereintheearlierageofmathematicalgeometry,whereanumberwasagroupofpoints,whoseformmattered.Soherealized10wastheperfect

number,becauseofitsperfectform,whichinfactbecomesthe11thdimensionofanew'whole'.

Andindeedtheinternalstructureofanybeingreachesitsperfectefficiencywitha3x3+0-mindsymmetryofformandfunction;whereeachpart-numberperformsoneofthe3physiologicalentropy,energy,informationjobsofthesystemandthecentralmind-numberincontactwiththemallcoordinatesitsfunctions.

Sowealsotalkof10innerdimensionsor'sub-systems',representedbyatetraktys:

In the graph,each 3 corners are sub-systems of 'Information, entropy/motion and Energy/reproduction' puttogetherbyacentral10thdimension(theblackball/hole/point/knotthatmesseswithallofthem).Indeedthecentral point of the ideal tetraktys communicates with all the other parts and embodies the whole that'emerges'asapointinahigher∆+1world.Thuswetalkofthe'subsystems'ofabeing.

Forexample,ahumanbeingisdefinedinmedicineasasystemofcells,attachedby10sub-systems:

Inthegraphupperleft3ST-ructuralhumanbodysub-systemsareitsmembrane,sustainandmotion.

Bottom3'chemicalðƒ'systemsorhormonalbrain(creative,distributiveandreproductive).Andthenervoussystemsingularity-mind…ofthewhole.Thosegraphsshowthataswegrowinplanes,theidealgeometryofthelower'atomicplanes'disappear,aslongasthe'logicconceptsbehindit'-hyperbolicfractalbody-waves

branching,bilateralsymmetry,etc.,remain.Sincefunctionismoreinformationthanform.

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Sotheconvolutedbilateralnetworksthatconnectthesingularitybrainwithallitsantipodalpointsofellipticgeometryhavethesamefunctionthatanantipodal'representationofanon-Euclidean'sphere.

Yetinthehumanorganism,thelinesthatconnectthemisnotmadeofstraightlinesbutitdoesworkbecausewhatmattershereisthesymmetricterritorialorderofthesingularitywhichconstructsitsmembranewith

opposite'rays'/nervouslinesandwillconstantlybalanceandhenceactasaleveragewithits±inversedirectionsforitsantipodalelements,twohands,twokidneys,andsoon.

MorphologythenstartswithsimplelawsofNon-Euclideantopology,whichbecomedisguisedbytheadaptationofeachfunctiontotheavailablespacewithinamembrane,astheT.œadaptstoitsouterworld.

The3x3+1physiologicalsystemsofthehumanbeing,lostits‘ideal’mathematicalsymmetry,translateto‘existentialalgebra’theconceptsofdecametricscales.Theycanbeeasilysubdividedin3x3+1Nervous

integrativemindsystem,butalsointhefractalentangledUniverse,into5x2,positiveandnegativedimotions.Forexample,theentropicsystembranchesin3thedigestive,respiratoryandexcretorysystems,whichfeedtheDimotionoflocomotion,sustainedbytheMuscularandskeletalsystem;thereproductiveSTsystembranchesin

3,theblood,circulatory,reproductiveandexcretorysystems;andtheinformativesystem,in3Nervous,lymphaticandendocrinesystems,whereastheNervoussystemdoublesastheonesiteofconsciousness,origin

oftheDimotionofperception,andsocialevolutionofthewhole.

Physiologicalnetworksthenbelongtoadifferentbranchofgeometry,fractalnetworks,whicharemoreefficientinthedistributionofthe5dimotionsofexist¡encebetweenpartsandwholesofthe∆±1scalesandsoweintroducetheconceptofanumberaspartofasocialnetworkbymergingNon-Etopology(2ndpostulate)

andNumbertheory.

Inthatregard,acomplexanalysisofthesimplestnumbersshowsthatthemoreperfectformisthe10-cellularsystemortetrarkys,inwhich3x3triangularcornersactasorgansofenergy,informationandreproductionwitha10thcentralelementthatcommunicatesallothersandactsastheoneofthehigherscale,representingtheentireorganism.

Thusasthenumberofcellsgrows,thetopologyofthesystemwillgrowindegreesoffreedomandcomplexitytillresemblingmoreandmoretherepetitive,geometricalformsofsocialorganisms.Topologiesbecomethusattheend,complexnetworks,adaptedtodifferentfunctionsofcomplexorganisms.

Asabstractasall thismightseem,whenobservingnatureweshall seehowthose typeofevents,wavesandsocial planeshappen in all the scales of theUniverse, fromatomswhich form crystal networksbased in theequalityof the sameatomsoratbest in theexistenceofa ‘body-mass’ofequalatoms intersectedbya few‘stronger’atomsthat formacomplementarynetworkofhigherresistance, tothebodyrejectionofcellswithdifferentDNA.

Whatthingswecandowithnumberscanreflectthenmanyoftheactionsofitsnetworks.Forexample:

-We can study how social groups organize themselves or fluctuate between states=functions. This is thestudyoftheinternalpointofviewofnetworksasacollectionofself-similarpoints.Thosechangesofstatesareoftendefinedbyadifferentialequationasinformativesystemshavelessspatialextension/motionbutaremorecomplex networks with more bits of information=points. Thus differential equations, most of them of thetypeY(ti)=aX3±bX2±cX±D,express∑Se<=>∏Titransformations,whereTiisanetworkin2or3dimensionsoftimebits,bitsofinformationandSeisanetworkwithone(sameorganism)or2(Darwinianfeeding)scalesoflessercomplexitythanY,suchasf(x)=Yn. It followsfromtheFermatTheoremthatthere isarestrictiontothenumberofsolutionsasystemcanfind,whichisn=3,themaximalnumberofdimensionsaninformativespherecanhaveasitdisplacesitselfoveraplaneofenergy.

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Therelationshipsbetweenlimbsandheadsthatexchangeina3rdregioncalledbody,formandmotion,suchastheheaddesignsthemotionsofthelimbs,whichmovethehead,andbothexchangeinanintermediateregionofellipticnaturecalledbodymoresubtletypesofformandmotiontocreatemorecomplexcyclesthatwillinfactreproducebothsystemscanbemathematizedininfinitedifferentways,usingmatrix,combinatorytheory,differentialequations,polynomials,Riemannsurfaces,etc.

-Wecanstudyhownetworksgrowandmultiplycreatingnewspeciesandwecanaddthemandobservehowtheyreorganizecreatingcurveswhicharedifferentiabletoobtaintherateofgrownanddiminutionoftheorganicpopulation.Thestudyofherdsofenergyandnetworksofinformationinitslifecycleisoneofthekeydisciplinesofallsciencesspeciallyphysicsandecology.

-Wecanstudythemasnetworkswithformthroughitsgeometricalwaysofexchangingenergyandform,fromthesimplestpointtothelineof2,thetrianglewithoutacentralfocus,thestructureofenergy,whichcanhoweverturnintoapi-cycle,the3,the4withitszigzag,solidquadrangleandcrossstructures,the5andfirst3dimensionalstructure,andsoon.

Eachnumberwill increasethepossibilitiesofthegame,yetwhenwereach10weplayaperfectgamewith3trianglesthatactasorgansofenergy,informationandreproduction,andacentralpointbothina2-dimensionalor3-dimensionalgeometry,actingas thecollectiveaction/will/intersection/knotofall cycles– the first clear,completeegostructurein3dimensionswithperfectformandcomplementarity.Thusbeyond10,whilesomenumbersmightbringslight improvementstothecell,mostformsare justgrowthsoftheprimarynumbers inmultipleassociations.

Recap.Allthestructuresofmathematics,regardingofthenotationweuse,reflecteventsandformsofknotsoftimearrows(st-pointsornumbers),asmathematics isa languagewhosegrammarderivesfromtheUniversalgrammarofspacetime.Numbersarethusformalnetworksthattrytoachievetheessentialarrowsoftime.Andsocertainnumbers(1,2,4,5,7,10)deploybetterthosearrowsandarethecommonestonnature.

-Wecanstudytheevolutionandreproductivecreationofnewnetworkswithsuccessionsandcombinatoryis importantinmultipletime-spacessincewefindalwayscomplementarysystemsofreproductiveenergyandinformation, each onewith a ternary choice of evolving differentiation (energetic, informative and balancedspecies).Soespeciallyintheclassificationofspeciesofdifferentsciencesweshallfindsimplecombinatorylawsthatexplains thedifferentiation in3,6, 8and10elementsdependingon the triadsanddualitiesofmultiplespace-timesystems.

- We can study a key antisymmetry of time and space expressed with the language of probabilities:Sequentialeventsarestudiedwithprobabilitiesintime,whosesymmetryinspacearethestudyofpercentagesof populations in space, such as if each event in time is the birth of an individual of a population bothprobabilitiesandpercentagesarethesame.

Thisconfusedphysicistsinsomecases,asinanelectronicnebulae,whichisapopulationoffractalelectronsinspace, but it is studied as time probabilities, and created the bizarre theory ofmultiple universes (multiple,probableelectrons)insteadofafractalUniverse(fractalself-similarmicro-electrons,whicharebundlesofultra-denselightforminganebulaewhichalsoactsasa‘whole’electron,self-similartoitsparts).Thusthestudyofprobabilitiesintimeeventsandgrowingpopulationsofawaveofspace-timecyclesisanessentialtool:wecanstudy the proportions, herds, groups and networks of self-similar st-points in its evolution either withprobabilitiesordifferentialequations.

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4THAXIOM=POSTULATEOFRELATIVECONGRUENCE:SIMILARITY,¡NDIFFERENCE,TRANS-FORM-ATIONS.

4th Postulate: Equality is no longeronly external, shown in the spatialperimeter of any geometrical form(Euclidean congruence) but alsointernal and further on it is neverabsolute but relative, since wecannot perceive the entire innerform of a point – hence thestrategies of behavior such ascamouflage. Forms are self-similarto each other, which definesdifferent relationships between

organic points, according to their degree of self-similarity. The 4th postulate is thus the key to explain thebehaviorofparticles as thedegreeof self-similarity increases thedegreeof communicationbetweenbeings.Someofthemostcommonbehaviorsand‘eventsderivedfromthispostulateare:

1)Reproductivefunctionsincaseofmaximalself-similarityorcomplementarityinenergyandform;ei->SeiorMaxExmin.I(male)=Min.exMaxI(female).

2)Socialevolution,whenpointsshareacommonlanguageofinformation,i=i->2i.

3)Darwiniandevolutionwhenformsaresodifferenttheycan’tunderstandeachother’sinformationsoinsteadthey feed intoeachother: i ¹ i. Insuchcases if those2entitiesmeet theywill startaprocessof ‘struggle forexistence’, trying toabsorbeachother’senergy (whenE=E)or simplywill not communicate (whenE¹E, sincethenthereisneitheracommoninformationtoevolvesociallynoracommonenergytofeedon).Yetbecauseanypointabsorbsonlyarelativequantityofinformationfromreality,similarityisrelative,fakedforpurposesofhuntingwithbiologicgamessuchascamouflageorsociologicalmemesthat inventracialdifferences,allowingtheexploitationofagroupbyanother.

The 4th postulate defines systems as identical when they are equal in its 3 ternary parts, the outer angularmomentumor‘membrane’,itscentralActivemagnitudeorsingularity,focusoftheforcesandthevitalenergy,enclosedwithinthem,andallothersassimilarwithdifferent‘angleofcongruence’.Wedistinguish2differentinteractions according to the degree of equality of its ternary parts, as systems can be symbiotic, if theirindividual, cellular or atomic ‘fractal points=parts’ are similar enough, interacting through its 3 physiological‘lines=networks’ evolving inparallel creating anorganicplane, as thosedescribed in thenext graph for eachscale=science, or they can be entropic, destructive, predatory, when they are dissimilar and don’t speak acommon languageof information to coordinate its actions,whereas the stronger systemwill perpendicularlybreakandfeedontheweakerone.

Thegeometriccomplexityofthe4thPostulateiscausedbythetopologicalformscreatedbyanyeventthatentanglesMultipleSpaces-Times.Sinceitdescribesthepathsandformsofdualsystems,whichconnectpoints:Self-similarityimpliesparallelmotionsinherds;sinceequalentitieswillmaintainaparalleldistancetoallowinformativecommunicationwithoutinterferingwiththereproductivebodyofeachpoint.Darwinianbehaviorimpliesperpendicularconfrontations,topenetrateandabsorbtheenergyoftheotherpoint.Finally,absolute,innerandouterself-similaritybringsbosonstates,whichhappenmoreoftentosimplerspecieslikequarksand

particlesthatcanformabosoncondensateastheydoinblackholes,wheretheproximityofthepointsismaximized.Andindeed,thesamephenomenonbetweencellswiththesameinnerinformation/DNAoriginates

the‘collapse’ofwavesintotighterorganisms.

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Finallyifthereisnosimilarityneitherinbodyormind,itsexistenceas'catalleys',thatnevercross

(relativeinvisibility).WetalkthenofSkewT.œ.s.:

The4thNon-EuclideanpostulateisimplicitintheworkofLobachevskiandRiemannwhodefinedspaceswiththepropertiesofself-similarity(Riemann’shomogeneity),whichdeterminesitscloseness(Lobachevski’sadjacency).

4thpostulateofrelativecongruence&angleofparallelismasamirrorofits5pentalogicdimotionsandvariationsofangledefineDarwinianorsocial,reproductiveoutcomestocommunicativeeventsbetweenfractalpoints.

Thus in praxis we assess similarity by an ‘angle of parallelism’ that increases social evolution into herds andsupœrganisms,orperpendicularitythat‘scatters’systemsintoentropicdestruction–elementsthoseofanentirefascinating new field of 5D topo-biologic studies that analyze in geometric terms, the vital topology andrelationshipsbetweenformandfunctioninallsystemsofNaturefromparticlestoorganisms.

This simple geometrical truth however is essential to all systems of nature, whose angles of connectiondeterminethefunctionsandsymbiosisbetweenparts.

TheUniverse always startswith an asymmetric being,which can go bothways: towards a social evolutionarysymmetry that lasts in time and implies a mirror parallelism, or an antisymmetric destructive, perpendiculareventinwhichonepartpuncturesandabsorbstheenergyoftheother.Itisthetopo-biologicalternaryprincipleof non-Euclidean, Non-Aristotelian I-logic geometry that puts together both the biological and mathematicalpropertiesofreality.Theconceptsofsymmetry=parallelism,antisymmetry=perpendicularityandasymmetryaremirrored by the 4th Non-E Postulate of similarity. But we can extend the concept of asymmetries also toasymmetriesoftime,betweentheyoungageoflocomotionandtheoldageofinformation,ofactions=Dimotionsbetweenthestepandstopsimilaractions,andtheentropyandsocialevolutionactions,whichbringusthefinalasymmetryofscalesbetweentheupperarrowofwholewithmorespatialsizeandthelowerarrowofpartswithmore information.Whenthosedualities:step-motion/stop-perceptionandscaleup(5D:socialevolution),scaledown(4D:entropicdissolution)areputtogetherweobtainthemostcomplexbalancingdimotion,reproduction,andwhentheyarealladdedupintheexistenceofabeing,wegetitsworldcycle.

Inthegraphwecanassessthedifferent5mirrorsinwhichmathematicalSpaceandlogicTimereflectsthegameof5Dimotions=actionsofexistence,whichthenexpressedbyterritorialmonadsGENERATESitslogicREALITY.InGeometry fractal points=monads will other through waves of communication of energy and information thatgrowintoreproductivenetworksaterritorialplane,creatingasuperorganism,whichwillrelatedtotheexternalworldaccordingtoitsrelativesimilarity=congruence,assessedbyitsangleofparallelismorperpendicularity.

In logic terms, a super organism, by breaking its formless asymmetry into different spatial configurationsaccording to congruence (social parallel systems, complementary gender-mirror systems, Darwinianperpendicularsystems,orsystemsthataredissymmetricanddonotshareanyreality)buildsacasualpyramidofgrowth froma fractal point throughwavesof communication into social networks thatbecome ready to act -move,feed,perceiveandevolvesocially.

Since wemust add to the mathematical and logic languages-properties of reality the 5 actions, or organicpropertiesofthescalarUniverseasessentialtothegameastheyare its logicandmathematicalmoreabstractlaws-afacttheegocyofæntropicmenofcoursereject,asitmustremaininitsmonad-subjectivemonologictheonlyclaimanttolifeproperties.

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Thus the pentalogic of generational space-time is established by its Non-Euclidean fractal points, its ¡logiccongruencewithrealityinwhichitwillorderaterritorytoperformits5vitalactions=Dimotionsofexistence,andthemathematical,logicandorganiclawsofthose3languageswillbethereforethebottomlineofthe'Creativeprocess' of the Universe - nothing chaotic except the entropic Dimotion, which conforms the monologic ofhuminds.

TheGenerator'sTernarySymmetriesandItsS=T1,2,3DimensionalAnalysis

Thereare3relationshipsinspace-timebetweenentities,whicharepartinnon-Æofthelawsofthefourthpostulateofsimilarity,thatwerelatetothe3elementsofthefractalgenerator:

ST:Complementaryadjacency,inwhichinasingleplane,membranesofpartsfusionintowholes,andinmultiplescales,partsbecomeenclosedbyan'envelope'curvethatbecomesitsmembrain.Itsmainsub

postulatesbeingtherealmoftopologyproper.

$t:Darwinianperpendicularity,inwhichamembrain/enclosureis'torn',andpuncturedbyapenetratingperpendicular,causingitsdisrupteroforganicstructure.ItsmainpostulatesbeingtherealmofNon-Euclidean

geometries.

§ð:Parallelism,inwhichtwosystemsremaindifferentwithoutfusioningitsmembrains,butmaintainadistancetoallowcommunicationandsocialevolutionintoherdsandnetworksupœrganisms.Itsmainpostulatesbeing

therealmofAffinegeometry.

Thecorrespondenceofthoserelationshipswiththe3elementsofthegenerator,$<ST>ð§areimmediate:

-ST-Adjacencyallowtopegpartsintopresentspace-timecomplexdualities.

-$-Perpendicularitysimplifiesthebrokenbeingintoitsminimalist'linealforms',$t.

-§-Parallelismallowsthesocialevolutionofentitiesintolarger§ocialscales.

Theywilldefine'ternaryorganisms,inwhichthe3topologiesin1,2or3s=tdimensionsofasinglespace-timeplane,canbestudiedinceterisparibusanalysisortogether,butnomore,asallotherattemptstoincludemore

dimensionsinasingleplaneare'inflationaryfictionscausedbytheerrorofcontinuity'

Dimensionsthusmustalsobeconsideredbesidesthe3logicrelationships.Andthereare3levelsofcomplexityindimensions,lineal,2-manifoldsand3-Dvolumesthatexpressalsotheternarygenerator:

Soforthe3linealcoordinates,theequivalenciesareimmediate:

1D¡Γ:$t:length/motion<STwidth/reproduction>§ð:height/information.

Aslineallengthistheshortestdistancebetweentwopoints,heighttheprojectivegeometryofperceptionfromantennaetoheads,anditsproductmixesthemtoreproduceinthewidthdimensionwhereyoustoreyourfat...

Externaldimensions/networksoforganisms:territories.

ThekeyelementofNon-Egeometryarethe3topologicregionsofallsystems,asthatistheunderlyingstructurethatevolutionarytopologydevelops,withasingularity,@,dominatingavitalterritoryenclosedbya

membrane.

ItisthemixtureoffunctionintimethroughactionsofsurvivalthatdominatesthespatialternarystructureofthoseT.œs,whichguidestheunderstandingofvitalgeometry.

I.e."Thoughmostarachnidsaresolitaryanimals,somespidersliveinenormouscommunalwebshousingmales,females,andspiderlings.Mostoftheindividualsliveinthecentralpartoftheweb,withtheouterpart

providingsnarespaceforpreysharedbyalltheinhabitants”.Britannica

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Indeed,regardlessofthevitaltopologyofthepoint,all‘buildups’ofnewgeometric

scalesstartwiththesimplestform,a‘bidimensionalterritory’withamembrane,acentralsingularityandthevitalenergybetweenthem.Sogeometrynotonly

evolvesinthehumindincomplexity,itdoessointheevolutionofthevitaltopologiesof

newformsineachsingleplane:

Inthegraph,allsystemsregardingofits‘perfectgeometry’,havethesameternary

structure,towhichvitalgeometryaddsmotion,inaKlein-likeNon-Estructure,wherebordersofentropy=deathcan’tbecrossed;sotheyarerelativeinfinities–militaryborders,ballsoffire,membranes.Ontheleftwestudy

inmoredetailamammalterritory.Anyanimalterritoryisani-logicspace-timewith3zones:

An informative central territory (1) or den, where animals reproduce and 2 secondary homes where the herdperformssecondaryorganiccycles(2,3).

Anenergeticmembrane(M,5)–aninvisible limitthatprovokesaconfrontationifastrangercrosses itandwheremostenergeticpreys‘flee’awayfromthedenofthepredator.

An intermediate zone with cyclical paths of absorption of Entropy and information; where we find a huntingterritory,placestodrink(E),tobath(B),socialize(A),defecate(D),etc.

Inorganicterms,adimensionisanetwork.Soalivingorganismcanbeconsideredasumofcellularquantaunitedby3 basic space/time discreet dimensional networks, which are its physiological systems: the digestive/energeticnetwork, informative/ nervous network and reproductive/blood networks around which cells teem, creating astable, organic st-point. In other words the Entropy and informative networks of a living being are its internal,diffeomorphic dimensions (of relative length and height), to which the organic system adds a 3rd, reproductivedimensionthatcombinesbothelementsandrepresentsthewidthor‘volumeofcellularquanta’ofthesystem.

Finallyitsmovementintheexternalworldbecomesits4thtemporaldimension.Yetthat4thdimensionofexternalactivitycanalsobeconsideredanetworkterritoryinitself,sumofthe3±icyclesofexistenceofthebeing,creatingabiggervitalspacethatwillbecomethebasicunitofanecosystemorsocialorganismmadeofindividualsofthesamespecies.Inthefigurewedrawthevitalterritoryofaminimalsocialpairofmammals,differentiatedin3clearsub-sectors:

Max.Information:Informativedenorcentralterritory(1,2,3):

It is the territory of reproduction used to copulate and store basic food and Entropy to raise the young. It is aforbiddenzonewherenotevenhunting isallowed (4). In social speciesofgreatmobility,aerialormarine,wherebordersaremuchmoreextensive,thisterritoryisveryampleandtendstobelocatedinwarmlatitudes.

Entropy=Information:DualTerritoryofEntropyhuntingandinformativesocialization(5).

It is the feeding, social andhunting territory,onwhich the central informativebeing feeds itself. It isoutside thezoneofreproduction.Itisthewinterterritoryofmanymigratorybirds.

Giventherelativismofallmovement, inbiologicalterritoriestheinformativesingularitymovestohuntitsEntropyquanta,asopposedtogalaxieswherestarsandspace-timedustmovestowardsthecentralwormhole.

Withinthoselimitstherearealsoneutralterritoriesofcommunication,courtshipreproductionandfreeEntropy,likewatertroughs.Sothe intermediateterritoryworksboth,asan informativeandenergeticterritorywheredifferent

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victimsandpredatorstraceparallelcyclesandcometogetheraroundmeetingpoints(E,B,R).

Max.Entropy:Bordersthatlimittheterritory.

Membranes are dangerous topological zones of dual osmosis. So the informative center watches to control anyinvasionof itshunting/socialterritory.Membranesfluctuateaccordingtoneighbors’power.Forexample,thevitalspaceofafishincreasesduringmating,sincethecoupleismorepowerfulthanasingleindividual.Marks(Mpoints)fixthoselimitsandreducecombats.Theyareofteninvisible,asmostterritoriesaredefendedagainstcompetitorsofthe same species, who understand the informative code of those marks; but rarely against members of otherspecies.Sowefindallkindoflinguisticmarks:

Smells (common inmammals, like foxes, rhinos, antelopes), excrements (in canines and felids)orother glandularsecretions.

Optical marks often connected to scents: The brown bear creates marks in trees, rubbing them with his head,warningadversariesofhisgreatMax.SxTsizeandforce.Inhumanempires(nationscanalsobetreatedasbiologicalterritories) visualmarks correspond to armiesdisplayed in theborders. Inhumanhomes thosemarksused tobeshieldswithweapons;nowtheyarecarsandotherproofsofmoney,thenewlanguageofsocialpower.

Highpitch,acousticmarks,properofbirds,whicharetriggeredwhenarivalenterstheterritory.

Recap.Vitalterritoriesofanimalsandhumannationscanbeexplainedwiththe3topologicalregionsofst-points.

Thestructureof§ð<TS<$t,territorialspaceswithacentralpointofview,developingitsparticularworldview,tryingtoreachinfinitywithhisdistortedgeometry,affinetoaprojectivegeometrywherefarawaymeanssmall,defineseachworldofaUniverse,which isobjectivewhen‘clashing’eachformwithallothers–soonlyeusocial love,andemergencethroughthescalesofthe5thdimensionmakesurvivalpossible.Geometryisthenthestudyofthespatialformthatthefunctions,whichdominatethevital,sentientUniverse,adoptintheirexistentialactions.

Andassuch isthebestmethodtovisualizethe 'meaning'ofalgebraicandanalyticequationsboth inabstractandmathematicalphysics.Eachofthedifferentlawsofbidimensionalplanegeometrythencanbestudiedasareflectionof the efficiency of vital Dimotions in a simplified geometric form, where curved paths become perfect lines,distances aremeasuredwithout error and angles haveno geodesic distortion. But stillvital geometrywill add tothosestructuresitsvitalinterpretation,besidestheabstractknowledgeintroducedbytheGreeks,onthelinesoftheprevious graph (which become lineal forms in other systems such as ‘polygonal molecules’ or human artificialconstructions).

InGeometrythuswecanalsousethe‘pentalogicpointsofview’oneachtheoremandalsospecificofgeometrythevital meaning of the 5 angles of congruence (4 non-E Postulate), from Darwinian perpendicularity to socialparallelism.

5THPOSTULATE–DEFINITIONOFAMIND-POINTOFANEWSCALE.

Thefifthandonlystandingnon-Epostulateisequivalenttothefirst,definingafractalpointbutfromthep.o.v.ofitsinnermind-centerofreferenceastheUniversehasinfiniteofthem:ThegraphshowsthedifferencebetweentheAristotelian,self-centered,Euclidean=lighthumindandtheUniversalmind:¬-Aristotelian,Non-Euclidean:

Thehumanmentallight-Euclideanspaceoneofmanymultiplespacesmadewithdifferentforcepixels.

Euclideangeometry is thespecificmind-mirrorof light space-timeandits3perpendicularDimensions.

The Universe has ∞ mind-mappings made with differentpixels thatmirror for each singularity its territory of order(bodywave) and world beyond. The human 'visual mind'

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madeoflightisNOTtheonlymind-mapping.Inthegraph,ontheleftthe'physicist'viewofasinglecontinuumlightspacetimeforthewholeUniverse.Intherightsidethemultiplep.o.v.s

Descartesdidunderstandthismultiplicity;sohepublishedhismappingofthehumindinabookcalledthe'World'todifferentiateitfromthe'Universe'withinfinitemonads,eachoneholdinganentireworldinitself(Leibniz)theveryessenceofthedefinitionofafractaltime§paceorganism.YetHumanslostthisearlierunderstanding–aswenotedontheintroduction-whenGalileodidn’targuethefactthattheEarthmovesbutourmindcreatesastillspace,amirageofthesenses.Andphysicistsfollowedsuit,creatingitsphilosophyofrealitycalled‘naïverealism’.

Spaceasamirageofthemindwouldbethenunderstoodinphilosophy,bothintheEasterntradition(BuddhistMaya)andthewesterntraditioninfluencedbythem(Soviet,GermanschoolsstartingwithLeibniz,followedbyKantwhonotedEuclideangeometrywasthegeometryofthehumanmind,andSchopenhauer,whosawitasa

representation.)

-Themindor0-pointis,therelativeframeofreferencethatmappedthe∞cyclesoftimeoftheUniverse,reducingthemtoa‘World’,tofitthemintotheinfinitesimalvolumeofthebrain.

º-mindx∞-cycles=World:equationoftheLinguisticMonad:∞Mind-Worldsin1Universe.

ThemindthoughbelievestobethecenteroftheUniverseinthe'egoparadox'asheseeseverythingturningarounditsinfinitesimalpoint,whichhostsinsideallthelinguisticperceptionofreality,or'world'itconfuses

withtheuniverse.Sothemindisafractalpoint•,butbelieverstobeitall.

TheparadoxoftheEgo–whomakeeachmindtofeelsoimportant–isthenrootedintheself-centeredstructureofthemind,whichselectsinformationfromitspointofview,creating,aninfinitesimallinguisticmind-mappingofreality-whichthenitconfuseswiththewholeuniverse:

0-linguisticmindx∞Universalspace-timecycles=ConstantWorldmappingofreality,with@mindatitscenter.

The∞informationoftheUniverseisreducedintotherelativeinfinitesimalvolumeofourmindgivesusaconstantmapping,fromwhereweexpelallthepropertiesthatarenotinterestingtousandourself-centered

view.

Themindisasingularityorinfinitesimal0-point,arelativeframeofreferencethatmapsthe∞cyclesofspace-TimeoftheUniverse,reducingthemtoaWorldtofitselectedusefulinformationintothefinitesimalvolumeof

thebrain.

Inmathematicsthemindequationis0x∞=C;∞0=c;thatis,the∞timecyclesofrealitybecomeinthe∆ºself-centeredscaleofamind,aconstantworldthatmirrorsallwhatexists intheUniversebothintime-motion(∞)andspatialform(º).

Because of such synoptic capacity to ‘mirror’ the laws of space-time in minimal size through the concept ofnumber,whichexcelstheprevioussynopticlanguageofverbalphonemes,mathematicssoonbecamethemostefficientlanguageweknow,butitdoesNOTcreatereality.Itisjustthebestmirrorofreality.Languages/mirrorsoccupy an infinitesimal part of the whole, yet paradoxically hold the maximal information of the Universeaccordingto5Dmetrics:SpacesizexTimespeedofinformation=Constant.

Thusthemindofthemostefficientsurvivalspeciesofreality,particles,‘atoms’and‘galaxies’(blackholeatomsof the top quark decuplet) should bemathematical and imprint a local order whichmultiplied by∞ of suchspeciesmakesmathematicsthedominantmind-speciesbutnottheonlyoneandstillamirrorofthetruerealitywhichis‘scalarspace’and‘cyclicaltime’,dimensionalformandmotion–the5Dimotionsofreality.

Alanguageisfirstareflectionofthelawsof¬∆@st,Dustofspace-timeandits‘UniversalGrammar’andFractalGeneratorequations,withoutwhichtheycannotorder≈recreatelocallytheUniverse.So5Dmathematics

advancesthedisciplinebyfocusingbetterthemirrortoincludethebio-topo-logicpropertiesofscalarspaceand

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cyclic¡logictime;andbyputtinginrelationshipmathsand‘existentialalgebra’(ab.¬Æ),theaprioriDisomorphic=equallawsof5D,makingitanexperimentalscience;connectingitslawswiththelawsoffractal

spacetime.

Metricequationofmathematics:0x∞=Ksearchfor'wholeness'inasingleequation.

ThuswedefinetheGeneratorequationofalgebra,whichasamindlanguagederivesfromthemind5Dmetric:

0-finitesimalspatialmindx∞timecycles=Constantmind-world:§@(mindspace)<≈>∆ð(universalcycles)

Sothe'GeneratorEquation'ofalldigitalnumbersis,0x∞=Kanditsalgebraicexpression,Sx<≈>ƒð.

BothequationstogetherallowtorepresentallpossiblemetricequationsofallsupœrganismsoftheUniverse.Assuchneither0isanabsolutezero,nor∞isaninfinity,conceptsrelativeinascalarUniverse,wherethelimitsofscales in size and time speed and perception in each self-centered p.o.v. only allow to talk of relativeinfinitesimals,wecall finitesimals,0%,and relative infinities,whosesymbol is∝.2evident truthsare then theexistenceof anexperimental finitesimal,h=m l² ƒ, theminimalunitof angularmomentumorPlancktonof thegalatom,andarelative infinite,=c, itsmaximaldistanced-speed,whichconformsthemostextensivemetricweknowofwithprecision,beyondwhichdarkspaceandtimeisunperceivabletomankind,sowewrite0s(h)x(c)=K

Itfollowsalsothatmathematicalinfinitiesareinflationarymirrorswhosecontradictions(Cantor'sParadoxes)areNOT solvable (the Zermelo axiombeing an adhoc addition, against the logic of truth), becauseprecisely theylimitsofrealityenterinparadoxicalcontradictionwiththeinfiniteinflationarymirrorsoflanguages.

Thestudyofthose4elementsofallrealities,itsactionsandternaryoperandi,structuresthedynamic'GeneratorEquation'ofall Space-timeSystems,written in its simplest formasa singularity-mindequation: Ox∞=K=∞º=1

Sothe5thpostulatedefinespointsasinformativeknotsorlinguisticeyes-mindsofinformationthatabsorbafluxofforcesusedbythepointtoperceivearelativeworld.Anon-Euclideanpointcorrespondsthentoourconceptofarelativemindthatgaugestheinformationrealitywithacertainforce,similartotheconceptofamonadinLeibnizrelativisticspace-time.InwordsofEinstein:apointofspaceisafixedframeofreference.

Thus, Non-Euclideanmathematics fuses the logic and geometry of the fractal Universe, greatly improving ourunderstanding of Reality even in termsofmechanistmeasure. Sincemathematical solutions to problemswithseveralpointsofvieware impossibletofind incontinuousspace(i.e.3-bodyproblemingravitation),giventhefactthatanetworkofinfinitepointsofviewislocalandrelativeandeachpointisafocalknotthatactsfromitsperspective.Thus,theabsolutetruthofasystemisthesumofallitspointsofview,whichinfluenceeachother.Yet even if we cannot calculate precisely with mathematics, systems with more than 2 bodies, since thosesystems are organic, hierarchical, made of networks with attractor points, fractal structures and self-similarpaths,thenewmathematicsofattractors,fractals,scalesandNon-Euclideansystems,refinegreatlyouranalysis.Inessence, indeed,weobserve that ‘networks’ integrateparts/points intowholes,which then ‘act’ asa singlepoint.So inthecomplexmodelsof i-logicgeometrywecantacklemanyproblemsbydefiningsetsofpointsas‘wholes’ofa‘higherspace-time’.

For example, thepreviousprinciple of localmeasure,where eachpoint is a relative center of theUniverse, iscalledinrelativitythediffeomorphicprinciple,whichnowbecomesexplainedasapartialcaseofthewider lawwecalledthe ‘Galileanparadox’; thedualityparticleof information/waveofenergybecomesaspecificcaseoftheapplicationtophysicsofthedualityofenergyandinformationfoundinallsystems.

Theexpansionofthelawsofquantumtheory(complementarityprinciple)andRelativity(relativityofscale,localmeasure, etc.) to other sciences and the organic principles of the 3rd and 4th postulate to physical particles isthereforetheconsequenceofthosepostulates.Yetitrequirestheunderstandingofthenew,i-logic,organiclawsof theUniverse and its networks, because E-mathematics has clear limits to extract all the informationof the

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Universe, given the fact that it syntax includes a priori errors and simplifying postulates (single space-timecontinuumofpointswithoutparts,etc.).

Thus,whentheeventdescribediscomplex,performedbyagreatnumberofpoints/variablesyouenterintonon-linealsystems,whichrequiretopologicaldescriptions(chaoticattractors,fractalnon-differentiableequationsandNon-Euclideanmathematics),andthei-logiclawsoforganicnetworksandsystems–abettersyntaxinwhichtofit experimental evidence, especially in phenomena of informative nature (since only formless energy iscontinuousandresemblesthemodelsbuiltwithasinglearrowofenergyandasingleplaneofspace-time).

Sowhile classic physical systems calculate accurately the energetic, continuous properties of the Universe anoverview on howmultiple points of view emerge intowholes requires organic laws. This search ofwhys alsoapplies to the understanding ofmathematics. For example, the previous postulates resolve the long-standingquestion ofwhat is the nature of the 1st and 5th postulates that seemed redundant (as the 5th describes alsopropertiesofapoint likethefirstdoes,andthe1st seemtodescribeanon-geometricalproperty).Theyarenolonger redundant,but theyaremore concernedwith causal logic and time than spatial geometry in itspurestforms(points,linesandplanes.)

Thereisalsoself-similaritybetweenthefractalpostulatesofi-logicgeometry(sincethe5thisgeometricallyself-similartothe1st,asbothareconcernedwithpoints)andthe4dimensionaltimepaths/arrowsoftheuniverse.Thisisnotcasualsincealllanguagesofspace-timesdepictinself-similarwaysthe4DUniverse.Thusifthe1stand5thpostulatesdefineagaugingpointof informationas thefundamentalunitof theUniverse, the2ndpostulatedefinesa lineorflowofcommunicationofenergyandinformationbetween2points,whichreproducespartoftheinformationofthe‘generator’pointacrossasurfaceofspace;the3rdpostulatedefinesthosepoints,whicharenotsimilarasenergeticsubstancesthatwillbeabsorbedbythepoints.Yetifthosepointsareself-similartheywillgatherthroughthearrowofeusociallove,creatingaccordingtothe4thpostulateanetworkofspace/time,anew organic plane of existence. So the 4 arrows of space-time are explained by the 4 postulates of i-logicgeometry.

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III.GEOMETRICNUMBERSS-Numbers:Polygonsasvitalorganisms

Numbersaresocialgatheringofindistinguishableforms.Whenstudiedinspacethusnumbersmusthaveregularefficientconfigurations.SoƤnumbersare2Dpolygonsor3DPlatonicsolids.

Itsimportanceinvitaltopologylaysonthefactthatpolygonsstartthecreationofsuperorganisms,withamembrane–thepolygonproper,whichclosesavitalspaceandcanbyconnectionofpointsthroughlines-

wavesofcommunication,createacentralsingularity.Thusnumbersasfractalpointsgroworganisms.Andthisisself-evidentinthestudyofNature.Wecanthenstudytogetherwithentangledpentalogicboth‘scales’of

socialnumbersas‘growthintime’ofpolygons,efficientconfigurationsin‘surface’worlds(land,watersurfaces),andpolyhedrons,whichwillbeefficientconfigurationsinalarger∆+13Dworld(waterdepth,

atmosphere,vacuumforatomicandmolecularscales).

POLIHEDRONS

Polygonalnumbersarebesttorepresentthesocialevolutionofformsintonewscalesthatemergeasenvelopesintowholes.

Weshallconcludethegeometricanalysiswiththe5regularsolidswhoseobvioussymmetrywiththe5Dimotionsofexistence,meanseachofthemspecializesin1,2,3,4,5D,inparalleltothesimplexformsand

mirrorsymmetriesthatcomposethem:

3Dplatonicsolidsare4,6,8,theonlydecametricregularnumbersin3dimensions.Thenwehaveonlytwomorein

the20scale,12and20vertices,andthat’sall!!!

Theyplayeachonearoleasactorsofoneofthe5thdimotions–thedecametric3completinga4,6,8Fractalgenerator:

$t-tetrahedron(legs)<S=T-Octahedron(body)>§ð-Cube(head)

Indeed,thetetrahedronisbasicallyatrianglein3D,actingasthespearheadofalocomotion;whichintimespaceistracedbylinealmembersmovingonstepsasyourlegsdo.Henceits2D¡=$tfunction.The

Octahedronhasaclearmirrorsymmetryasallreproductivedualgendersdo–alreadypostedinacademia.eduitspaper;andthecube,istheformofthe3closertoafillingspherewithawell-connectedsingularityof

maximalsymmetrywiththeouterworld.

Sothisleavesthetwocomplexdimotionsofsocialevolutionandentropyforthetriangularicosahedrons,withmaximalnumberofsocialtriangularunits(20facesofthesimplest2Dsurface).Andthe5thdimotionofentropyforthepentagonaldodecahedron…AlasagainthepentagonalevilDeathsymbolagain…Why?ThatIwon’ttell…

Justquotethemasterof‘transcendentalphilosophicalgeometry’,Mr.Platowhoaffirmedthat,“TheUniverseisthebodyofanorganismwhoselogicmindwecallGod.’&'Godusedadodecahedrontodesignthe

Universe'Plato

Ok,I’lltell.Thinkinvitalterms.ImagineaUniversewiththose5species,nocurves.The3firstarethesmallishones.Cubesperceive,andkeepquiet,hidden,fillingeasilyemptyspace;itssurvivalstrategy–socialgrowth,

withitsidenticalforms,repeatingbythemeremotionaslinesfirst,flatsquaresandgiant3Dcubes.Itisasinglegender‘feminine’,informativereproduction.Theotherseriesdoesthesamebutwithmirrorgendersymmetry,

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asthetrianglesoftetrahedronsexpandfrom3to4andpeginverselyintooctahedronsandthengrowintoicosahedrons.

ButthedodecahedronisNOTconcernedwiththesurvivalstrategyofreproduction.Herathereatsthemall!!

Howisthis?Easy,thedodecahedronhasthelargestmouthandthelargestvolume,evenlargerthantheicosahedron,soitswallowsthem,inanactionofvitalgeometricfeeding;andoncecagedinside,rememberpentagonsarethe‘first’polygons,whichcandoasmallerreplicaofitself,NOTaperceptivesingularitybuta

crunchingstomach.

Eachpointisacellofadodecahedron.Asithappensthereareasmanyfacesintheicosahedrontoeatasverticesinthedodecahedronthatfeedsonit-bothconnectedthroughthegoldenratiooffeedingandreproductiveevents,a+b(feeding)/a=a/b;akeyrelationoftoppredator/preyforms.Andsoitopensits

mouth,swallowsit,cagesit,crunchesitandburps(;ButthepentalogicUniverseallowsdifferentdimotions.Sonowimagineinsteadareproductiveevent.Itisnotfarfetched.Biologistsconsiderthatovulesfirstatesperm

cellsandlattertheylearnedtomergetheirgenes.

Sooncea‘male’entropicdodecahedronhaseatenthe‘female’,reproductiveicosahedronitcanactuallymergeintoamixedform,bymerelyattachingtoeachdodecahedronpointatriangularface,becoming

theicosidodecahedron(ab.‘dodecosahedron’)-anewspecieswhichismoreperfectthanitsparentalones,asitsharestheinformationofboth.Itisafirstgendercaseinidealgeometry,sinceGenderreproductionhappensin

allscalesofstience,fromthesimplestcircularovumswallowingalinealseminalseed-acyclicalandlinealform,mergingintoan|xO=øtopology,tothepreviousexampleofilogicmathematics.

Thedodecahedronsocialpoints,akanumbersfeedontheicosahedronfaces,reducingthemeitherbyswallowingthem,whichwillresultinagrowthofsizeofthedodecahedronorbymergingintoa

reproductiveevolutionaryprocessfarmoreinteresting,asanymergedsystemismoreefficientthanits2separatedparts-species,amoreevolvedform,whichsurvivesbetterinaprocesssimilartoallgenetic-

memetic-reproductiveprocesses,thatcreateinanystientificscalemoreefficientsystemsbycommunication.

Indeed, the dodecosahedron is the top predator polyhedron form, as it has ALL the ilogic elements andsymmetries that enhance survival in the fractal TimespaceUniverse: individual ego-centered points, shapinginner trilogic systems in yellow, tetralogic dual triangles married to dual pentagons, decalogic reproductivebilateralsocialparts,plusallthemagicstablenumbersofvertices,edges,faces&diagonals,despiteitsexternal‘softmembrane’–unlikeits‘dualfigures’theCatalansolidsthatshowadefensiveexternalmembrane,inwhichtheswallowedformemergesintoconvex‘teeth’,buthasalessefficientinternalorganicstructure:

In geometry, we say that any polyhedron is associated with a second dual figure, where theverticesofonecorrespondtothefacesoftheotherandtheedgesbetweenpairsofverticesofonecorrespondtotheedgesbetweenpairsoffacesoftheother.Thegraphisanexamplefromthe other dual top predator/prey series, a cube of larger ‘mouths’ and bigger volume-spatialefficiencyswallowingansmalleroctahedronoflesservolume.

Whatdiffersnow isanaddedproperty -a topologic 'defensive' concaveandconvexdual formwhichcanestablishasVaughanfoundindefensiveflatgeometriesfortheFrencharmy,entropic

expansive concave points and smaller implosive, informative sinks, which maximize the entropic reach ofcannonballs withminimal surface exposure to the external army. Indeed, a ‘mathematical’ ideal duality, weshallfindlatteroninthestienceofbiologybetweentoppredatorsofsoftmembranebutmuchmoreefficientinnerstructureanddefensivearmoredanimalswithlessevolvedinternalnetworks(squids,witheye-perceptionand evolved nervous systems vs. armored trilobites, simpler organisms; Tyrannosaur Rex vs. armored

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triceratops, Lions vs. armored, skin thick, horned herbivores) and social systems (defensive walled cities vs.nomadichorsearmies).

WeareallVITALgeometriesandsotherulesofgeometrycarryfrom¡logicspacetimetoilogicmathematics,tothedifferentstiencesofvitalrealitywherethoseformsacquiretopologicalmotion.Thespecificcaseof

polyhedronsbeingofspecialinteresttomolecularchemistry,whichweshallstudyindetailwhenupgradingto5Dthediscipline(asthoseareprocessesthatdohappeninsolidcrystals).

Vitaltransformations.

Thoughpolytopesareallessentiallyintopologictermssphericalmembrains,itsstudybringstopologydowntothedetailsofitsscalarsocialnumbersorpartsofanetworkofpoints;allowingitsdynamicaltransformations.

Akeysub-fieldof5DvitalgeometryistheanalysisoftheDisomorphismsandsymmetriesbetweenpolytopestounderstandhowtheytransformintoeachotherthrough5Dentropic,3Dreproductive&4Devolutionary

dimotions.

4D:socialevolution.Allsolidswithacentersingularityevolvetheirsurfaceincreasingtheirvertex,andgrowingtheirvolume/bodyratiotowardstheperfectrelationshipofthesphere-whichweconsiderthe6thperfectsolidof∞vertex.AmongthePlatonicsolids,eitherthedodecahedronortheicosahedronmaybeseenasthebestapproximationtothesphere.Theicosahedronhasthelargestnumberoffacesandthelargestdihedralangle,ithugsitsinscribedspherethemosttightly,anditssurfaceareatovolumeratioisclosesttothatofasphereofthesamesize(i.e.eitherthesamesurfaceareaorthesamevolume.)Thedodecahedron,ontheotherhand,hasthesmallestangulardefect,thelargestvertexsolidangle,anditfillsoutitscircumscribedspherethemost.

5D:Entropicdevolutionsmediatedmostlybythe5Ddodecahedron.Feedingthoughrequiresadegreeofsimilarity,reachedadmaximalforthedodecahedronandicosahedron.Whatmakesthemidenticaltoswitchfrom5Dmindstateto4DentropystateisApollonius’findingthattheratioofitssurfaceisthesamethattheratioofitsvolumes:

SoitscontentofSTvitalenergyisthesame;butasthedodecahedrondies,itchangesfirstitstightsurface,whichasweknowindeathprocessesgrowsinexcessdetachingitselffromitsvolume-from12to20elements(anotherkeynumber,

withmanyvitalinterpretations,beingindeedthenumberofaminoacidvariationsontheproteinsurfaceoflivingsolids,andsoon)...andthenreacheditsmaximalformonthestablesurface,itcanonlyswitcharrowof

time.

Asdeathisexactlythatprocess:whenallthevitalenergyisconsumedandtheskinisfracturedadmaximalandnomoremotioncanbeextractedfromthebody,thesystemchangesarrowanddecomposesbacktothepast.

Otherfeedingcaseistheparasiticcoveringofoneformbyanother.Sothedodecahedronisgenerated(Euclid)fromthecube,bycoveringitasaparasiticmembrane,unlikethe3otherspecies,whichcanbeconsideredtoreproducethesameform.Allthismightseemtrivialtomathematicians,buttheywouldmissthepointofwhat5Dmathematicstriestoshow–thevitalnatureofspace…whichmustbeexplainedastheultimateformalsubstanceofreality,havingorganiclivingproperties,simplifiedintoastillmirror,NOTtheotherwayaround.

1D.Perception.Ifthesphereistheperfectplatonicsolidwithabsolutesymmetrytoperceivean‘equaldistant’territoriallargerspherefocusedinthecentralpointwithoutdeformation(PoincareConjecture),platonicsolidsarenextinregularsymmetries,henceastheyare‘economic’inpointscomparedtospherestheirefficiency

makesthemtorepeatconstantlyintheUniverseonthesimplestformsofchemical,atomicexistence(simplexplatonicsolids),NOTintheplanetaryorbits,whereMr.KeplerthoughtGoddrewthem(:

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Weseethenoncemoreatplaythescalarlawsofinversionofform,fromthe‘galatom’,orsphericalatom,given

theginormousnumberof‘staticphotonicpoints’trappedinitselectronicnebulae/starplane(quantum/cosmologicalscales)tothemoleculeorlinealpolytopesofminimalquanta.Molecularsurfacestendtobeasthegraphshows,perfectlyregularpolytopesforthesingularitytoapperceivethroughtheVanderwaalsforcesofelectronicperceptionthegravitationalandelectromagneticforcesoftheworld-reasonwhycrystals,regularatomicsystemswhichcan'scan'realityinanear-sphericalareformedasunitsbasicallywiththoseformsandthestrongest'pre-cyclical'pi=3hexagonalsystem(whichalsocanformtheultimateplatonic

dualsystems,apentagonal,hexagonalcover,thestrongest'fullerdome'.

InthegraphweseetheallpervadingformsofmaximalresistanceinthemembranesofphysicalT.œs.Incrystalsthecubicsystemisoverwhelming.Inmetalsonlycubicandhexagonalsystemsexist.InarchitecturetheonlysystemswhichcanbegrowninsizewithoutexternalreinforcementsmaintainingitsstabilityadinfinitumaretheFullerDomes,madewithtriangular,hexagonal'pi=3'andhexagonal&pentagonalcombinations,whose

formgrows'adinfinitum'towardstheperfectplatonicsolid-thesphere.

Thecube,a3Dstsolidwithinformativeroles.

Inthatregard,another‘inverseduality’in3Dhappensbetweenthesphereandthecube,similartothatbetweenthetriangleandthecirclein2D,withinverseproperties:

Thecubeisexcellentforsocialevolutionintolargernetworks,whilethespheretendstobelonely.Thecubeisthepreferredcrystalformformattertoreproducefillingadmaximalspaceandsociallyevolve,from3Dtoits

sociallarger5Dwholegrouping.

Howthenonetransformsintotheother?Ifthetriangleachievesthefeatbyrotationorsocialevolutionintohexagonalπ=3circles,thecubeistheclosestformtobecomebyellipticdeformationthesphere,inanotherstbeat,asitbloatsfeedingonenergyintospheres,itdeflatesbackintoacube.Sothetrianglebecomesacircleby

addinga4Drotarymotionandthecubebecomesthespherebyaddinga5Dfeedingmotion.

Sincethekeyfunctionofrealityisreproductionandallismotion’thesurvivalofthecuberesidesinitssimplexreproductionbymeretranslationinspace:alinethatmovesgrowsadimensionintoaplanethatmovesandgrowsintoacubethatmovesandgrowsintoalinerestartingthegame,andaddinganew5Dscale.Yeteach

growthchangesthefunctionoftheformaccordingtothegeneratorequationthatchangesS-topology&T-age,hencefunctionasasystemgrowsin∆-scale,suchas:∆-1:$<∆ºST>∆+1>ð§

Thepartialequationofthefractalgeneratorthatencodeswithinitalllawsoftheuniversemeansintopologythetransformationoftopologicvarieties,aswegrowinscale,fromlineallimbs/potentialsat∆-1,intoØ-ST-

iterativespace-timeinto∆+1O-sphericalparticles-headsofinformation.

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Thecubeistheð§tate,excellentforsocialevolutionintolargernetworks,thepreferredcrystalformformattertosociallyevolve,from3Dtoitssocial5Dgroupingintominds,theclosestformtobecomebyelliptic

deformationthesphere,inanotherstbeat,asitbloatfeedingonenergyintosphere,itdepleatsintocube.

Andthenagainastheformshows,thecubedisplacestoformalineonthe∆+2plane,NOTafourth-dimensionalspatialbeing,whichdoesNOTexist.

Soifwestartthereproductionofcubesinthecubic∆-1lineitsfunctionis2Dlinealmotion;thenthe∆ºplaneisanSTreproductivebody-waveformandthe∆+1cubethe§ðinformativefunction(withmaximalvolumeand

minimalperimetercomparedtothelineandtheplane)

Andthenagain,thecubedisplacestoformalargerlineonthe∆+2plane,NOTafourth-dimensionaltesseract,whichdoesNOTexistinreality

regardlessofitsusefulnesstoproveandmodelS=Tdimotions.

Sogiventhereproductivenatureofmotion,wetalkofdistanceasthesumofadjacentternary‘stœps’:thecubestateofa‘squared’linestopsandchoosesanyofits6directionsofmotion;thenitreproducesinpresent

slicedplanesonthechosendirectionandmovesasazig-zagline,stopping,iteratingandsteppingasallreproductivemotionsdo.Beyond3dimensionstherearenomoredimensionsinasingleplane,sothecubegeneratesthenalineofthelargerscale,transposingitsfunctionagain,completinga

fullzero-sumcycleinscales,creatingareplicaofitselfinthelarger∆+1world.

Soweobserveinthosereproductivetranslationstwosymmetriesatworktogether:

$≈t≈$...Translation=reproductionofmotionthroughthe$-lengthdimension,causing....

∆o->∆+1:ST+ST+ST:Reproductionofformthroughthewidthdimension,whichmakesthebeing,growinscalesofthefifthdimension,symmetrictothechangeofpositionintimespacecausing...

∆-1:$<∆ºST>∆+1>ð§achangeinitstopologicalfunctions=forms.

Thenextformofreproductionofthe3varietiesoftopologyistherotarycircle,whichbecomesaspherewithonemoredimension,thestrongestmembranethatenclosesandcapturesthevital,hyperbolicenergyofthe

being.Andthenitbecomesatoroid,whichappearsasalargercircle.

Anditisworthtonoticethecirclehasonly2statesascircleandsphere,unlikethecubewhichhas3Dimotionalstatesascube,lineandsquare;afactofparamountimportancewhenconsideringthedifferentspeedsofmotion,reproduction,explosionandimplosionalongthe5Dimotionaleventsoflinealv.circularformsthat

paradoxicallyfavorsthecircleoverthelineandexplainswhy1Dvorticesacceleratefasterthanexpansivelinealbig-bangs.

Butonthenegativeside,the‘cubeisasolidform’,butthereproducedcyclebecomesan’emptysphere’.Itisnotaball,onlythesurfacethatenclosesavolumeofemptyspace-thesimplest,strongesttopologynormallymadeatthe∆-1scaleofstrongtrianglesofentropicnature.Sothecirclebecomesthemembranethatactsasthe

entropicenvelopethatclosesthe‘emptyvacuum’,becominganon-EuclideanKleindiskwhoseinnervitalspacecanneverreachbecausethedensemembranewillkillitslighterinternalparticles–andindeed,weshallseehowatomsandgalaxies–thetwolimitsofourscalesofform,followsuchpattern.Sinceeveniftheformisadisk,whileitcancreateinthefirstrotationadensesphere,inthesecondrotationitcanonlyformahollow

toroid.

Butthemostefficientformofreproductionisthatofwavesandbodies,withits‘networks’thatbranchandhencereproduceinanexponentialgrowthasinglepoint;hencethehyperbolicformofthe‘growth’phaseof

waves;andtheenormousmultiplicationof‘surface’ofafractalnetworks.

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Finallyallthoseformscantransformintoeachother:cubescanfeedandcurveellipticallyintocubesandlinesaresmallstœpsofcurvedgeodesics,bothtypeofiterationcanmerge,solineallimbs/potentialsat∆-1,Ø-ST-iterativewavesand∆+1O-sphericalparticles-headsofinformationcaninrareoccasionstransformintoeach

other,butspeciallycantransferflowsofenergyandinformationamongthem,whichat∆-1levelwillacquiretheshapeofeachofthe3adjacenttopologiesofthesupœrganism.

Sofunctionsandformsdochangebutallpossibleeventsareencodedinthegeneratorandsoifweunderstandtheunderlying∆stbasiclawsoftheUniverseencodedonit,everythingkeepsfallingintoplace.Andthoselawsarevitallawsthathaveasurvivalpurpose.Forexample,thehollowsphereKILLSbyenclosingandtrappingaslionessesdo

huntingzebrasbyenclosingthemordogsshepherdingandenclosingsheeple,movingaroundinfastcirclesas‘angularmomentum’thatbecomesbyvirtueofT=Savitalmembraininmotion.Thenthemembranewillsharplypenetrateperpendicularlythezebraherdandkill;themilitaryborderofahumansocialterritorywillgiveacoupd’etatandcollapseintothecapitalandconquer…the

battlewillbelostonceinCannae,Hannibalhadencircledtheprey(graph,wheretheredcolorofentropyrepresentsthedimotionofthehorsethat‘closestheromanpray’asamoving

membrainlikethedogdoes).

So∆stlawsmightseemabstracttoyoubutarethestuffofwhichexperimentalsurvivalandexistenceismade.

Hyper-dimensionalpolygons.

AsS=T5Dmetricsimpliesadimensionofspaceisanotherformtointerpretamotionintime,an

importantfieldof5Dgeometryisthestudyof4D(2S=2T)regularpolytopes,whichclosetheentangledscalesofregularsocialnumbersasT-DimotionsorS-tillelementsthat‘construct’therealstructuresofthe

Universe,toextract5D¡somorphiclaws;evenifwedoNOTseestaticfixed4DformsinourUniverse(as5Dincludesentropy,whichisthenegativedestructionofallothers,4Disthelimitofconstructiveengagement,

oftenof2stillpolygonswith2motions=degreesoffreedom).Wejustmakesomekeystatementsonthisdiscipline:

Withthesimplificationoftheinnervolumeoffractalpoints,wecanmatchpolytopelawswiththesymmetriesofSpacetimebeings.So1Dpolygonofpointswithnoparts-openlinesequivalentto1Dcylindriclimbs/fieldsare∞(NaturalNumbers);2Dpolygonsakintocyclical,O-particles/headsare∞inflatspace,&3hyperbolic

polygons,equivalenttoØ-body-wavereproductivesystems,intersectionof2Dlineal&cyclicformsare∞in3Dspace.

Sothereare∞growingensemblesof1Dlimbs/fields+2Dparticle/heads=3Dhyperbolicbody-waves.

Growthofcomplexfinitepolytopespeaksin4D,asfurtherincreasesofdimensionsreducethemtotheternary‘singleplane’structureofatriangular,squareandpentagonalternaryensemble;whichmeanstherearenomoredimensionsbeyondthe4D2S=2Tsystems,andallfurther‘Dimensionalgrowth’isadimensiononnew

scalesof5Dwhicharememoriless,erasingthepreviousemergentcomplexityofthefractalpoint,whichstaysonitsinnerparts.(Becausethehyperbolicbody-systemismorecomplexbeingitsminimalform3D,itsternary

growthpeaksat5D).

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In the side the existence of a regular 4-polytope is constrained by the existence of the regularpolyhedralwhichformitscellsandadihedralangleconstraintsinπ/psinπ/r<cosπ/qtoensurethat the cells meet to form a closed 3-surface. And yet despite those restrictions we have 6

polytopes. And thismatter because ourworld as in relativity can be analyzed in asingleplaneasa4Densembleofits3classicdimensionsandatimemotionofsocialorganization,asthe2inversedimotionsofentropyandinformationdoNOThappenwithin the same event simultaneously in ‘regular entagnelments’. So pentagonalpolytopes make no sense, being the 5th dimotion entropy the destruction of theregular in-form-ation shared by them all. Thus the most complex 4D structure of

regular points are the 120-600 dual pentagonal mirror ‘gender symmetry’, reached throughcombinationsof5Dimotions,6 ‘triangularradians’ ofaπ sphere(∝polytope)& its120angles.Theresultarethe finalstable120dodecaplexand600tetraplexmaximal figuresof4Dimensionalgeometries.Andweshall find this120magic finalnumber in the fewcases the system isperfectenoughtotranscendtheoctaplextobetherealfinal‘islandofstability’inatomicsystemsoroldestpossibleageintheveryfewcasesthesystemtranscendsthe80years‘radioactivebarrier’.Andwecan imagine the 120-600 vertex entanglement the limit of ¡logic evolution of a complex socialsystemintheUniverse.

Butallthosepolytopesmustbeunderstoodasternarysystemswhichreproduceinmotionformingalarger5Dscale;so4Disamotion,andthatisindeedhowgeometersconstructthem,bringingtoeach3Dpointan internal volume; that is,makingeachvortexofa3Dpolygonanew ‘square’ to

formthetesseractwhichisequivalentto‘enlarge’itsinnerform.Whileanother4Dpolytopewillbemerelythemotionofthesquarethatbecomesalarger‘1D’lineinalargerworld,aswemakea2dplanebymovingacrossa1D line.Whataboutthesphere? It istheultimateregularpolytopeandsowewonderhowa4Dsphere lookslike. The answer is also telling of the 5D S=T structure of the Universe, where time motions and spatial

dimensionsareequivalent–butonlymotionstrulyexist.Sowehavetoconsiderbeyond3D, new spatial dimensions to be beyond algebraicmanipulation timemotions. This isalsoourexperimental evidience,whenwegeneralizewhatwe seeof a3D sphereas itmovesthrougha3Dplane:aseriesofgrowinganddiminishingcuts,passingthrougha2Dworld.

Andsoa4Dspheremoving,passingthrougha3Dworldappearsasagrowinganddiminishingsphere,indistinguishable froma5DUniverse.ThuswepostulateanEinsteinian“PrincipleofEquivalence’asaproofoftheexistenceofa5thdimensionofpartsgrowingintowholes:

‘A4Dspatial systemmovingwitha5thdimensionof time, IS (equivalent to)a3Dspatial systemwithtwoscalardimensionsofgrowthanddiminutionininverse∆±1scales.”

Sobeyondourperceivableentangled3Dholographicparts,allfurtherdimensionsarescalarmotionsoftime (4D, 5D inverse, SS-formal, implosive or TT-entropic expansive dimotions). Specifically we differentiatedimensionalgrowth

S=T. In a single plane, beyond 3D spatial still forms, new dimensions are temporal,moving ones,which onlythroughthe‘persistence’ofamemorycreatenewdimensionsofspace,(asexplainedonouranalysisoffractaldimensions).Motionthenbecomereproductionofinformation,and‘socialformsofgrowth’–acubemovingtoin-formasquaredline,sphereturningintoadualspiraldiminishingandgrowingtoformalargerflatdisk,etc.)

∆±¡.Inscalarplanes,whentheformis‘static’inthesameposition.Thenthebeingexperiencesa‘scalartravelthrough the fifth dimension’ growing its inner parts frompoints into ‘cells’. This in ideal polytopesmeans todrawineachvertexanewtetrahedron,cubeordodecahedron).Orasawholegrowingordiminishing insize-scale.

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Sothatisthedynamicrealityof4Dspherespassingthroughourworld,whichdoNOTexistassuch,butasS=Tforanymentalspatialmirrorthattrans-formsT-motionsintoS-tilldimensions,4Dpolytopesareusefultomodelandsolve mathematical problems of real physics, by the algebraic method of converting a time motion into ageometric still spatial dimension, so 4D polytopes can represent real 3D T.œs with one locomotion or twolocomotions,thesecondoneanTT-scatteringentropicdimotionofscale.

RECAP.Pentalogicofpolytopes:Platonicsolidscorrespondto5Dimotions.

The3simplest3Dnumbersformafractalgeneratorofvitaltopologyandthe2Complexthe∆±¡dimotions:

$t-tetrahedron(legs)<S=T-Octahedron(body)>§ð-Cube(head)

4D:Socialevolution:Theicosahedronismadeoftrianglesasthesimplesttetrahedron,evolvingsocially∆states.

5D:Entropy.Thedodecahedronisthemostefficientandfeedsonallofthem.

Polytopeswithmorethan3Dmustbeconsideredtohave‘dimotionsoftime’NOTofspace.

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IV.VITALTOPOLOGY–THEHOLOGRAPHICBIDIMENSIONALUNIVERSE.

TopologicalspacesTopologyasthequeenofmathematicalsciences.

TopologyisGeometrywithmotion,hencethetemporal3rdageofGeometryanditsculmination,asexpressionoftherealmorphologyofspace-timebeings,whichincludesits3mainelements:∆-scales(topologicalformsaredefinedinmoderntermsasnetworksofpoints);Spaceforms(its3varietiesarethe3organs/forms/functionsofallT.œs)anditstime-motions(atopologicalorganbydefinitioncanmorphandevolvebutremainsthesameaslongasitdoesnot'break'itstopologicalcharacteristics).Sotopologyinstantlyconnectswith5Dmetriclaws,in

its3ages:

1stage:Classictopology.

2ndage:Fractalmathematics&networks.

3rdage:Vitaltopology(GSTSupœrganisms).

Soaftershowinghowfractalpointsjoinedbynetworksbecomewaves=flowsofenergyandinformationthatevolveintotopologicalorganismswith3physiologicalnetworks,theSpatial,entropic,'digestive'system,theS≈t,reproductive'bloodnetwork'andtheðƒ-Informativebrainnetwork,messedtogetherthroughits'darkspaces'(asnetworksdohaveholes)formingthesupœrganismsoftheUniverse;wewillseethesameprocess

throughtopologicvarietiesconnectingitsclassicthemes(1stage)withits3rdage(GSTvitaltopology).

Ternarytopologicalvarieties:Vital,OrganicGeometrymergesformal,physicalandbiologicalstiences.

The5Postulatesof¬Egeometryhaveanimmediateconsequenceinthetransformationofall‘entitiesof

mathematicalgeometry’intotopologic,bidimensionalvarieties,asasystemtoexistwillhaveadimensionoftimeandamotionofspace,whichareequivalentS=T,initsidealvalue(thoughnormallythemotionoftimeas

itdoesnotreproduceinpersistentformswillreachafractaldimension).

Andthetruetaskof5DGeometrywillbetorelateallthosedimensionsandmotionstoits‘functions’connectingtheminrealeventsandformstothe1to5Dimotionsofexistence.

ButhowitispossibletomakesenseofsingleEuclideandimensions.TheansweristhosedimensionsareusefulwhentheT.œisperceivedasafractalpointofalargerworldandwereduceitsinternaldimensions.Sowetalk

of:

External1-dimensions:The3varietiesofEuclideansimplifieddimensionsare:height-informativedimension(1D),width-reproductivedimension(3D)andlength-locomotion(2D).Sowecanwriteasimplegenerator:

$t(length:locomotion)<ST-width:reproduction>§ð(height:information)

Itisthosekindsofentanglementsbetweenpureformalgeometriesandvitalorganicfunctionswhatclosesthegapbetweenrealityandmathematicsthroughthevitaltopologyin1,2or3DDimotions.Onlythatoftenoneofthedimensionsofstudywillbeamotionintime,butbyvirtueofS=T,couldalsobeinmanycasesstudiedasa

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stillspacedimension.SowecanfindthenequivalencesandsymmetriesbetweengeometricformsandtheSS(4Dseedandsocialevolution),St(information),ST(energyofreproduction),sT(locomotion),TTentropy)5

dimotionsthatcombinethepolesofform-spaceandtime-motionoftheUniverse

Internaltopologicbidimensions:

InthegraphthebidimensionaltopologiesformalsoafractalgeneratorofT.œs:

SS-St(sphericseedsandminds)<ST-hyperbolicbody-waves>sT-TT:lineal,flatlocomotionandentropy.

WhereasSSandTTimplyaformationordissolutionofnetworksacross∆±¡scales.

ThosesurfacesinLobachevski’sexpression,are“dissections”ofspace:Eachofthemdividesthespaceintotwodomains,aninteriorandanexterior,andtheyarethecommonboundaryofthesetwodomains.Thisfactis

connectedwithanother,namelythateveryoneofoursurfaceshastwosides:aninteriorandanexterior(onesidecanbepaintedinonecolor,andtheotherinanother).

ThusthefirsttaskofanymembraneintheprocessofgenerationofaT.œ,istobreakspace-timeintoinnerandouterregions,wheretheinformativeandentropicarrowsofthesystemwilldevelopacomplex,richin

information,internalT.œinwardsandanouterentropicmembranetodetailwithitsanti-worldfromwheretoobtainmotionandenergycapturedbythehardmembranetoreproduceintomore∆-1cellular/atomic

componentsandgrow.Itisthenclearthat:

'ThepurposeoftopologyistostudytheternaryvitalgeometriesofT.œ,itsfunctionsandtransformations'

Whereastheternarybidimensionalonlytopologiesof4-5Dspacethusalsoexplaintheternarynatureofallphysiologicalnetworksanditsvitalfunctionsforanys,st,torganism.

Itholdsthenthateachorgan,varietyofvitaltopologyMUSTnotTEARandCHANGEitsexternalformbeyonddeformationnottolooseitsproperties.Those3bidimensionaltopologicalvarietiesthenhavemotionrelatedto

internalchangesofin-form-ationnottoexternal,linealchangesoflocomotion.

And its complexity grows as we perceive them in more detail. As time is cyclical, made of 3’π’ lineal timemotions,weshalldistinguish3conserved‘quantities’oftimespacethatputtogethercreatesuperorganisms,arelativedevolvingpastorarrowofentropyrepresentedinphysicsbydisorderedexplosionsinspaceandinvitaltopologybylineallimbs/fieldsoflinealmomentum,aniterativereproductivepresentthatseemsnottochange,representedbyhyperbolicbody/wavesofenergy,andanimplosivein-form-ativelocalfuturearrowrepresentedinphysicbyaccelerated,V(t)R(s)=Kvorticesandangularmomentum,andbyparticle/headsintheensemblesofvitaltopology.

Sotimespacebreaksin∞relativelocal,fractalentropicpasts,iterativeenergypresentsandinformativefutures,whichputtogethercreatetheillusionofasingletimespacecontinuum.

GeneratorEquation.The3organictopologicalST-planes.Physicaldimotions.

The3bidimensionaltopologicplanesbecomethe3organsofanyoftheinfinitefractalsystemsoftheUniverse:

$t(Toroidfield/limb)<Si≈Te(Hyperbolicwave/body)>§ð(sphericalhead/particle)

ThesimplestformsarethoseofstillGreekGeometryand physical systems, which range from pure stillgeometries:SS(spaceareas,closedconicorbitals,ofmaximal space-form and minimal time-change),

through all variations of ST dimotions proper to TT (accelerated time dimotions, ofmaximal time-speed anddiminishing spatial volume, as in charge and mass vortices), which become doors to a new scale of the 5thdimension.

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Thusbetweentheextremedualarrowsofentropy(TT)andpureseed-form(itsinverseSSstate)bothcombineintoconservative‘present’zero-sumcyclesofenergybody-waves,limbs/fieldsoflocomotionandparticle-headsofinformation.Theyarethe3‘conserved’arrowsoftimethatcreatethefuturesofeachfractalsupœrganism.

So ina single5Dplaneall systemsaremadeof those3Ðimotions,whichbecomeorganic§urfacesperceivedsimultaneously in space ensemble into vital systems; act in Space-Time and live an accelerated/steadystate/decelerated3agesTTworldcycleofexistenceobservedonlyasatimeprocess.

Andvitaltopologystudiesthem,betweenits4Dimotionofgenerationinalower∆-1planethatemergesinan∆+1socialworldtodiebackinadissolvingexplosionofscatteringentropy,backto∆-1inthemomentofdeath,whichtopologyalsocananalyzethemasprocessofcreationanddissolutionofnetworks.

Thefifthdimensionisoccupiedbyfractalbranchingnetworks;eachofitsplanesby3topologicalvarieties.

Sothereisalsoafundamentaldualitybetweendimotionsinasingleplane(the3varietiesoftopology) and the perpendicular structure of the fifth dimension as a branching network,whichconnectsthelargerwholeanditssmaller,multipleparts.I.e.Yourwhole,thebrainandheart branchesdowntoitsparts.5thdimotionalchangesofscalesthusimplysomenetworkbranching,whichapparently isdifferent inabig-bangexplosionofaphysicalsysteminpartsandanordereddistributionofbloodornervous impulses,butgeometrically areequivalent,∆º<∑∆-1motionsinthe5thDimensionwhichintermsofitsmetricexpandsinspaceandhencediminishintimethespeedofthemotion(decelerationofabig-bangfromits initial impulse;decelerationofthespeedofbloodthroughevermoreextendednetworksoftiniervessels).

Wethustranslate inthoseanalysisbyvirtueofthedisomorphic=equal lawsofallscales,thevital factweAREmadeofspace-timeandthedualityofS=Tanalysistopologyintothepentalogicdimotionsofexistentialalgebra(¬Æ):whenstudiedas stilldimensionsof space it leads toan i-logic topologyof fractalpointswithparts thatentangletodisplaythescalar,fractal,networkcomplexityofUniversalsupœrganisms.

The2varietiesofthe2ndand3rdpostulates.

The postulates of ¬E therefore ‘diverge’ in its own variations depending onwhat process we study, a travelthroughasingleplaneorthroughseveralplanesofthefifthdimensions,departingfromtwoinitialfractalpointslivinginasingleplane,oralargerpointanditsmicro-parts.Sodothealgebraicoperandsweuse:

Inasingleplane,westudyaggregationsonherdsof fractalpointsas linealwaves,andusethe ‘sumoperand’(superposition principle). While between scales, we use the ‘product operand’ and we see fractal lines andnetworks.

Andthisdifferentiationcarriestothe3rdpostulateas3networksofpointsandparts,becometopologicalorganicplanes,whichcompletethemirrorsymmetryof5Dgeometrywiththesuperorganismsofrealityastheyare.

It kicks then the4th postulateof relative congruence todefine as apositive social evolutionof networks intoorganisms or an entropic, scattering dissolution of wholes into parts, so congruence becomes essential toestablishwhatgeometryoftencannotdifferentiatewithoutdetailedviewormotionanalysis:ifwhatweobserveisabig-bangdeath(max.speed)oranetworkconstruction.

Intheothercase,thesimplersumofwaves(nolongerlines), inasingleplane,alsorequiresananalysisofthecongruenceangle,toknowifwaveswillcanceleachotheroraddup.

Soweupgradestill spatialmathematicswith its5®postulates thatsubstitute theclassicaxioms,postulatesandprinciplesofEgeometrywellbeyondwhatthe5Non-EpostulatemerelyhintedatintheXXc.

Yet abstract stillGreek geometry stillworks as amirror of reality,whenwe reduce it to a single plane. Thenpointsarea'dominant'spatial,mentalviewwhichresideinasingleplaneandthe'mind'perceivesinstillness,

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withno internal scalar formandcontinuitywhenassociated into lines,planesandvolumes;even though inadeeper reality they have 'fractal content'. E- mathematicians though missed the ‘self-awareness’ of thatsimplification,nowevidentwiththe5postulatesof ilogicGeometrythatdescribehowpointsbecomepartsofsocialwebs,self-organizingfractalplanesmadeofnetworksofpointsthatemergeascellularunitsofahigherfractal space-time or new supœrganism. And so we define ¬E topology as the study of the structures builtdepartingfromfractalpointswithinnerpartsandscalarcontent.

WhereasbytheprincipleofCorrespondenceClassicgeometrythatconsiderspointstohavenobreath,noinnerparts is the limit of ¬E topology when we 'eliminate' the 5th dimension of entropic motion and the 4thdimensionof fractal scales (only fractalgeometry includes it in its restrictedanalysis). Thus5Dunifyall thosedifferentgeometriestomakethemdescribepartsofthesamefundamentalstructureoftheUniverse:thefractalpointanditsmorecomplexsocialforms,networksandwavesandorganicplanes.

It follows that ‘Dimensions’ are NO longer exact in number, but mostly Hausdorf dimensions for 2 obviousreasons:

- Time motions have less ‘density’ than spatial distances, as the motion often ‘erases’ the previousinformationas the formslides.Sowhilea stillpointhasaclear ‘1Dimension’ (byconventionpointshavevolumehence1D),apointinmotionwillhavelessthan2Dimensionsofstillspace,asitsmotionerasesitsprevioussteps.

- NetworksthatpeerbetweentwoscalesalsodoNOTfilltheentirelowerplaneandsotheyhaveaHausdorfdimension. Theway tomeasure them is as an ‘openball’ thatdonot include the largerwholeor centralsingularityandthetinierparts,butmerelythenetwork,whichwillthenbealsobetween1an2dimensions.

Themesthoseofamoreadvancedcourseon5Dmathematicsnottreated inthis introductorypaper,where itmattersmoretheconceptualcomprehensionof theconnectionsbetweenvital topologyandtheorganismsofreality, to make clear that an evolved Non-E mathematics is the organic geometry of the fractal Universe,complementedbyexistentialalgebra,andits¡logicequationsofthefractalgeneratoroftheUniversewhichalllanguagesmirrorinits3±¡pentalogicsyntax,sinceallmindsgaugeandrepresentthecyclesoftheUniversewithdifferentlanguagesofperception.

VitalGeometry,symmetricproperties:function=form.InversepropertiesofS&TmergedinST-bodywaves.

Wenoticedalsoacorrespondencebetweenthe5objectiveDimotions,the3±¡subjectiveframeofreference/perception,aswellasthe3±¡formalvariationsoftopologyjustdescribed.

Theyallcometogetherinsymbiosisaseachtopologyandcorrespondingframeofreference(polar=sphere;cylindrical=lineal;hyperbolic=Cartesian;complex=scalar)hasvitalorganicpropertiesthatbestsuitseach

dimotionofexistence.Letusexplorethisconceptcloselyrelatedtosymmetry.

SymmetryisthecentralconceptofGrouptheory,theiceinthecakeofthewholestructureofAlgebra.

TodefineitgrouptheoryusestheParadoxofGalileo:S(forminspace)=Time(motion).Asitdoesdifferentialgeometrybasedintheconceptthatalineisapointinmotion.

Similarlybysymmetrywemeanin5ÐAlgebra,thecapacityofallsystemstocompleteazerosumworldcycle,throughamotionthatreturnsthesystemtoapresentundistinguishableequivalentstate.

Symmetrythusisessentialtothe5Dimmortaluniverse,whichisadynamicpresentviewinperpetualmotioneventhoughduetodynamicmotion=symmetryandconservationprinciplesderivedfromit,iseternalinits

returns.

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Wethususetheclassicconceptofsymmetryasarecurrentmotionthatconservestheformofthebeing(differentfromourmultipleworlds,symmetric,bisymmetric,asymmetricetc.usedinthe4thPostulateof

Congruence).

Itfollowsimmediatelythatthemoresymmetricasystemis,themoreefficientwillbein'preserving'itspresentstatesof'survival'inauniverseinperpetualmotion.I.e.Acirclewillbemoreefficient,becauseithasinfinitedegreesofrotationalsymmetrythatanirregularpolygonthatmightnotevenhaveasinglesymmetrystate.

Inthetheoryof'survival'of'vitalmathematicalobjects',whichwebringfromtimetotimetothosepages(asintheanalysisofsurvivalprimenumbersabletotravelthrough5dbymakingmirrorimagesatscalebyjoining

internallyitsalternatevortices-points-unitnumbers),symmetrythusplaysacentralrole.

Howmanystatesofpresentasystemhas,definesthenits'qualityofsymmetry'andsurvivalwhichinspace(theeasiestsymmetriestodescribe),whenfixedinapointmeansthecircledominatesallotherformsasthemost

symmetricform.

SymmetrythenbecomesarequirementtoperformwellcertainÐimotionsasvitalactions:

I.e.thecircleIStheperfectsymmetryforstillDimotionsofperception,for2reasons:

Asweperceiveinstillness,itmaximizessurvivalbyhavingaminimalperimeter,foramaximalvolumeofinformationanddisguisesitselfinanexternalDarwinianworld.

Itsinfinitesymmetry,focusalllinesofcommunicationthatfallintothefractalpointwithminimaldistortion.

However,whenweconsiderthe2ndDimotionofLocomotion,whichistheprocessofdisplacementinspacewhileretainingtheformintimewithoutentropicscattering,thesystemwillmaximizeitsspeedof

reproductionofform,ofinformationinotheradjacentregionofspacewhenlessinformationdisplaces.Andsotheline,whichstoresnointernalinformation(orthewaveasallpointsareultimatelyfractalwithaminimumvolume),willbeabletodisplacefasterthanthesphere,andmaximizethesecondDimotionbyreproducingless

information.Furtheron,thelineistheshortestdistancebetweentwopoints.Andsolocomotionsaremaximizedbothinasingleplanebyalinealwave,orbetween5Dplanesbyafractalline.

Finallyhyperbolicformsmixboth,linesandcurves,sotheycanreproduceboth.Andahyperbolicplanebreaksdownawholeintosmallerfractalequalparts,thusitistheessenceofreproduction.

VitalgeometrythuswillbeessentialtoexplaintherelationshipsbetweenstableformsofgeometryandNature.

SosymmetrymeansSurvivalinspace;Closureofworldcyclesintime.Andthepentalogicofsymmetryisimmediate:

¬T:Timesymmetryisthecapacityofallsystemstocompletea0-sumworldcyclethroughamotionthatreturnsthesystemtoapresentundistinguishabledynamicstate.HencesymmetryisconservationofTimebyiterationof

thesameST-event,evenifthepreviouseventbecomesextinguished.

S@:Symmetryinspacemeanssurvival,throughthe‘inversecomplementaryentanglement’ofthepartsofthebeing,ruledbythelawsofcongruence(andhenceoftenanasymmetricadjacentsystem).Aparticular

symmetrythatcoincideswiththecommon-senseconceptthoughalsoappliesasitenhances1Dperceptionwithoutdeformationofalargerouterworld:symmetryasrepetitiveregularity.Itfollowsimmediatelythatthemoresymmetricasystemis,themoreefficientwillbein'preserving'inaUniverseinperpetualmotion,itspresentstatesof'survival'.I.e.Acirclewillbemoreefficient,becauseithasinfinitedegreesofrotational

symmetrythatanirregularpolygon,whichmightnotevenhaveasinglesymmetrystate.

∆…Survivalimpliesreproductionofform,whichhappensthroughashrinkingin5Dseedsandenlarging.Sosymmetryinscalesmeansthecapacitytoreplicateseedsthatareself-similartothewhole.The'survival'of

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'vitalmathematicalobjects',thusfavorsreproductiveprimenumbersabletoshrinkthrough5Dmirrorimagesatscalemadebyjoininginternallyitsalternatevortices-points-unitnumbers.

S=T.Finallymirrorsymmetryallowsthereproductionofformandwededicateanentirepaperongendertoit.

Symmetryissoimportantthatallmotionsonphysicsandgeometrycanbereducedtothose3symmetriescalledboosts,rotationsandmirrorsymmetriesasdefinedbyNoether'stheoremofphysics.

Thenaccordingtoform=function,Natureensemblesalinealbody-limbmoving-translatinginspacewithasphericalhead,toperceiveontheforward‘future’positionsmallerinsize,fasterintime,inadjacent

relationshiptothebodyandlimbsthathavelinealformstomaximizethewhole’smotion.

Ofallthosesymmetriesthoughthemostimportantisa5Dscalarsymmetryignoredinscience:the'undistinguishable'propertyofthe¡-1elementsinwhichthesystemimprintsitsform,whichbecome

‘numbers’fromtheperspectiveofthewhole.Thissymmetryofscale,impliesthatthesystemcanreproduceitsinformation-movefaster,becauseitimprints'anyelement'ofthelowerplaneofmotionor'field'that

becomesundistinguishable,sotherearenot'impurities'anderrorsofreproductionofform,whenanyelectroncanreproduceyouratomicconnectionsandsoon.

Identicalpartsthenacquirethesamelargerformaccomplishingthemostimportantofthefivevitaldimotionsofexistence:reproduction.

Thuswecanestablishacorrespondencebetweendimotionsandthesurvivalofcertaingeometricformsaboveallothers:thecircle/spherefor1Dperception;thelineanditscurvedformtheparabolafor2D¡motion,its

combinedwavefor3Dreproduction,thesocialundistinguishablenumbersofpolygonsfor4Dsocialevolution,andthedifferentformsofopencurves,notablydual,splithyperbolasfor5DentropicDimotionanddissolutionofasysteminitsoppositeforms,whichintheconicrepresentationofaworldline,studiedlatterinmoredetail,willsplit,onehyperbolabranchgoingupwardsandtheotherdownwards,andinformationandmotionsplitin

death,whenwetaketheaxisoftheconeastheidealrepresentationofthefifthdimension.

RECAP.Eachofthe3organsdomainsofaspace-timeorganismcorrespondstooneofthe3fundamentaltopologiesofa2-3-4Dmanifold,theplanar/toroid,cylindrical≈hyperbolicwaves/bodiesandspherical

particles/heads,andtooneofthe3fundamental0-pointsofperception,showinghowclosemathematicsistoreality.

ThereasonswhythesphereorcirclearethebestformsforthefirstDimotionofinformativegauging,akaperceptionandthefourthdimotionofsocialevolutionandseeding(mentalpolarframeofreferenceand

storageofinformation)isthefactthatthespherehasminimalsurfacetostoremaximalinformationasaseednottobenoticedandperfectsymmetryallowingperceptionfromallpointsofviewwithoutdistortion.

Genusandholes.Measuringfunctionswithforms.3Dsystemsandits$<st>ðroles.

Thetopologicgenusofasystemisthequantityofcuts/tearsthesystemcanendurewithoutbreakingtheforminitscomponentparts(5Dtravel)ortransformitintootherofthe3varieties.

Thusgenusisrelatedtothesurvivalpropertiesofanorganicfunction,asitwasthecaseofthe‘form’ofatopologybestsuittosurvival.Anditalsohas2‘variations’ofoutcome–tobreakthesystemintoparts(4,5D)ortotransformitintoadifferentvariety,whichexplainswhyoftencutsdohappeninmorphogenesis,toevolve

theshapeoforgans.

FurtherontheGENUSofeachVARIETY,growsaswegrowindimensionalandinformative/formalcomplexity,improvingoursurvival.Sotheformwithminimalgenusthatis,whosenumberofdissectionsisminimal,mustbethesimplest,weakerform-function,whichasinformationismorefragile,willbethe‘hiding’sphereof

minimalperimeter,andoftenmaximalresistanceofits‘cells’ofthehardmembranethatisolatesit.

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Sospherestendtobehardertoprotecttheinnerinformationwith‘thickskulls.Sincewhenatopologicformiscut,it‘dies’literally,intheworldoftopologicalvitalbeings.

Andthisextendstothewhole‘adjacent’organicpartsofthebeing,oftenre-coveredbyasphericalmembranewithoutholes.Sotopologickillingrequirestotearthemembranethatisolatesthesystem,andbest,tosectionthevitalconnectionbetweenitstwo‘dominant’organs,theO-head/particleandØ=body-wave(aslimbs/fields

oftenareentropic,external,andcanbelostandreplaced).

Indeedinbiologywherethelawsofvitalgeometryaremoreself-evidentallpredatorshavethesameformofkilling,theycutthe'shortest'partofthemembrane,theneck,splittheO-headandtheØ-bodyandthethingis

donewith.

Akeylawofvitaltopologystatesthat:“PropertiesofO-headsparticlesofmax.formand|-limbsfieldsofmax.motionareinverse,merginginØ-bodies.”

Itextendstoanyofits‘disomorphic’parallelviewsasrelative|-pastandO-futuremergedinØ-present.

Inthecaseofgenusandholes,thesphereisthe‘weaker’form’,asinformationisandlimborlinethestrongestoneasmotionis.ButtheSpherebecomesthehardersurfaceatits∆-1scale,oftenmadeoftriangularshapesasscalesalsoinvertitsproperties.Whilethecylindricallinetendstohavethe‘weaker’∆-1elements,oftenmade

ofcircles:

§ð:Thespherejustacceptsonecutandhaszerogenus;butitsstronger∆-1‘triangularparts’actastheentropicmembranethatprotectsitsinternalinformativecenter,on‘top’ofthewhole,guidingittowardsthe

future.Insphericalforms,itpredatesthevitalenergyofitsenclosedinnerregion,whichitinvaginateswithfractalnetworks,focusedonthecentral

singularityitmirrorsinalargerscale.Sothevitalenergyisbetweena'sword'(themembranethatkillsyoubyperpendicularpenetration-asthe

invaginationsofthebiologicalstomachshow)andahardplace(thesingularityofmaximaldensitythatkillsyoubywarping)'astheysay.

ST:Vitalenergyhasthetoroidgenus;whichneedstwocutstodie=transformingintoanewmoreentropicvariety,aflatsurfaceofmotionand

energythatfeedsboththemembraneandthesingularity.

$t:Limbsarethe|-functionofentropywhichis‘expendable’andcanreceivemultiplecutsperpendiculartothecylinderwithoutloosingitsvariety;oftenregeneratingitselfinmanyvitalorganisms.Thusitiscloselyrelatedtothe

vitalenergyofthesystem.

§:Finallythesingularityisthemorecomplex,moreinformative,smaller∞dualfunction,adonuts,whosegenusis6,asyoucando3superficialcutslaterally(insidethetwoholesandoutsidethewholeform)and3

perpendicularly,inthebridgebetweenholesandbetweenthedonutsandtheouterworld.

Itisthusthehardest,mostefficientofalltheelementsofmostsupœrganisms.

SotheGENUSofeachvarietyalsodefinesitsdifficultytotrans-form,whichgrowsaswegrowincomplexity.Whichmakestheformwithminimalgenusthatis,whosenumberofdissectionsisminimal,notonlythesimplestform-functionbuttheeasiesttotransform;anotherreasonwhythesphereis‘originalseed’ofmostspecies.

ThebalancekeptinpeggingS<ST>t-species.

Thusinpalingenesisandmorphogenesisweassist,departingfromtheseed,aseriesofvitalpeggingsandtearingsofformstochangeandevolvetilltheysettleintotheirmaindimotionalrolewithinthewhole:

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Iftheseed=spheretearsitscapsbecomesthenanentropicdigestivecylinder.

Ifthenitispeggedtotheaxisoftheopensphere,havingboththesame0Eulercharacteristic(apropertyofbalancelatterstudiedinmoredetail).Sotheyfitperfectlybyadjacency=parallelismnotbyDarwinian

perpendicularity;asthespherehasthecentralholewhichthecylindercanclose.HenceapositiveevolutionarymergingoccurstocreatethecommonestformoftheUniverse,arotarymembrainwithacentralaxisthat

absorbsandemitsinformationandenergyforitscenteredsingularity.

Indeed,thefirstnaturalevolutionofallkindofsystemsisexactlythecombinationofasphereandadigestivetubeintheaxis,notonlyinparticleswithitsaxisthroughwhichamagneticfieldorsimilar'flows'ofenergypass,butalsoinbiologicalsystemsallofwhichhaveevolvedfromtheinitialspongesandhydraswithadigestivetube,withtwoopenings,amouthandananus.

Finallyinthecenterofthetubularbodyorinthetop'mouth',wherehigherinformationflowstherewillbeanewtopologicalevolution,nowreclosingthetubeatapoint,ornarrowing

it,tocreateasingularityincommandofthewhole.

Butthissystem,whichisstilldominantasaninformativesphere,isfarmoreefficientandbalancedasitnowhasthe3canonicalparts/networksneededtosurvive–thedigestivetractforentropicfeeding,thesingularity-focusofinformationandtheoriginalsphereorrotatingangular3D-momentum.

Topologicalevolutionandtopologicaltransformations

Inthatregard,themaininnovationof∆stintopologyistheexplanationofitslawsasdynamicvitaltransformativeevents,throughtheadditionofbalanceds=tsymmetriesthatselectthebest‘complete’survivalformsoftheUniverse,understandingdevelopmentalevolution,whichwecalltopologicalevolutionandusein

allsciences.

As5Disalsointerestedinatransformationofscale,dimensionorformthatmodifyandevolvetopologicalfunctions;inchangescausedbynewadjacencies,newperpendicularitiesandnewsocialparallelisms.

Thenwetalkofatopologicald=evolution(inverseoftopologicaltransformationswhennochangehappens):

"Atopologicalevolutionisachangeintheformandfunctionofthes,st,tpartsofthebeingcausedbynewadjacencies,newperpendicularitiesandnewsocialparallelisms."

Wedivergefromsomeclassictopologicdefinitions.Forexample,somegeometersconsideradonutstobethesamevarietythanaflatplane,becauseyoucancuttheemptydonuts,spreaditin2Dandalas!youhavetheplane,buttheyarenotthesame.Sincecuttingthedonutsproducesatopological

devolutiontoastateoflesserformthatflattenedlooses1dimension.

Sothefunctionchangesfrombeingavitalenergyclosedcycle(donuts)toanopenflatplaneofentropicmotions.

Openvs.closedsurfacearethusanessentialdualityoftheinformativevs.entropicpolesofreality.However,apartfromthesetopologiestherealsoexisttheso-calledone-sidedsurfacesonwhichtherearenottwodistinctsides.Thesimplestoftheseisthewell-known“Möbiusband”,whichisobtainedwhenwetakearectangular

stripofpaperABCDandpastetogetherthetwooppositeshortsidesABandCD.Suchone-sidedmembranesdoNOTcloseandbreakspace-timeintheclassicsenseofsphericalmembranesbutareakininfunctiontothetoroidgeometry,asthevitalenergyquantawithinthemcannotescapeinaneternalreturn–beingitsmain

distinctiontorequiretwoloopsinsteadofone;anditsmainuseinacomplexunderstandingofspingeometries(seepaperon5Dquantumphysics).

Natureoverwhelminglyfavorsclosedsurfaces,becausetheUniverseisafractalofsuperorganismsthatareworldsinthemselves,butthesurvival=efficiencyofaMöbiusband,residesinitsparadoxicallyduality:itis

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openedtotheouterworldasawholebutits∆-1partsthatcyclethroughitcannotescape.Thosefunctionsarethenproperofsystemswhichwanttoincreaseitssurfaceofexposuretotheexternalworldandarestrong

enoughtotakeitaspartsandwholes;hencesystemswith5D-entropicfunctions,alsoachievedthroughfractalscalarperpendicularinvaginations(asindigestivesystems).I.e.TheMöbiusconceptcanbeequatedtoachiralmolecule,whichisnotsuperimpossableonitsmirrorimage.Sochiralmoleculesaregoodforopticalactivityand

entropiclightdispersion(5Dfunction)orforexplosivepropellantsmotionsofaggressiveatoms(oxygen,chlorine)suchasthe…Perchlorotriphenylamine(:

Theinverseprocess:jettingoffhandlesandlimbs.

Topologicalevolutionisafertilefield,whichwillbecomealongtheoryofsupœrganisms(socialevolution)the2essentialadonstobiologyintheXXIc.Itistheclearestexpressionofthelawsofexistentialalgebrawithitsinversions,symmetriesandrestrictionsofform,puttingatplayalltheelementsof5Dimotionalreality.Forexample,astheUniverseis

mostly‘asymmetricintime,spaceandscale’,allprocesseshappenininversefashionwithaslightlydifferentoutcome(whereasperfectsymmetrycanonlybefoundin‘regular

polytopes=geometricnumbers’andspheres(seenumbertheory).AndyetasymmetriestendtobalanceSandTevents.Soifwe‘sink’amembrainitwillbulkbutlikelyitwilldosoinmultiple∑informativejets.Ashardlyanyprocessisequalsincetheinversearrowshavedivergingproperties.Inthiscasethe‘punch’beingapainful

entropic‘aggression’,itbringsan∑∆-1multiple‘expulsion’.Suchinversionoftheprocessof‘hollowing’theaxisofthesphere,happenedinthehydrawherethebalancelawthatcreatedaholedigestivesink,inverselyjetted

offtentaclelimbswithaninverseinformativefunction.

Thisdualityisalsoafundamentalthemeofclassicnowconvertedintovitaltopology-sinksandhandles.

Sowithintheprincipleof∆stasymmetricbalances,atcommontopologicalevolutiontakesplace:thespherewithholesfillsinverselywithhandles,astheformofthecylinderjetsoutwards'closing'inpairsthespheres'

holes:

Inabstracttopologywetakeasphericalsurfaceandcut2psphericalholesinit.Wedividetheseholesintoppairsandattachtoeachpairofholes(attheedges)acylindricaltube(a“handle”).Weobtainaspherewithp“handles”orasitiscalled,anormalsurfaceofgenusp.Theorderofconnectivityofthissurfaceis2p.Andin

natureisoftenhappeninginsynchronicitywithotherspherestochainthemtoeachother.

Weshallthenleavegenustheoryhere,toexplorelatterinourintroductionto5Dellipticgeometryhowa'realspecies'constructshandles,throughantipodalpointsmanagedbythecentralsingularity-asthespatialsymmetryofoppositeantipodalpointsproperoftheellipticgeometryofaninternal@-mind/membranesystemcreatesthehandlesanddisplacestheminthesurfacetoconnectwithotherfractalpointsinthe

externalworld.

Oralternatelytocreateaninverseentropicfunction,thehandlebornofafirst'suction'andthenan'expulsion'ofcontinuousmatterfromthesystem,canbecutandcuppedtobecomeanaggressivehorn(amethodoftopologyusedintheabstractclassificationofvarieties,whichbecamefamouswithPerelman'sproofof

Poincare'sconjecture,wealsoprovesomewhereina‘margin’withtheexperimentalmethodaswedowithFermat’sgrandtheorem).

Alas,wegotthroughST-inversionsandsymmetriesacoupleofentropicmovingordefensivelimbs,asthespherebecomesastrongerT.œ.Moreoverthe'section'ofthelimbwillnotmeanthesectionofavitalpartofthesphere,whichmeansthedeathofaT.œ.SoHydrasandLizardskeeploosingtailsandlimbsandkeep

functioning.

Furtheron,everyclosedsurfacelyinginourordinaryspaceistopologicallyequivalenteithertoasphere,ortoaspherewithacertainnumberofhandles:Forexample,thetorussurfacecanbedeformedcontinuouslyintoa

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spherewithasinglehandle...Sooncemoreour‘perfecttopologicalform’,thesphereshowshowitcanbecomeeasilyeitherofthetwoothervarieties,thelimborthevitalenergyoftoroidforms.

Whatisinterestingthenisthatalltopologicalformscanbebornofsuchspherewithhandles-theoriginalegg/morulaofanylivingbeing.

PENTALOGIC:THEELEMENTSOFTOPOLOGY

Thisisjustthetopoftheicebergofanimmenseextendedsubject.Geometryisavitalsubject,whichslowlyevolvedtillreachingwithvitaltopologythestudyofformswithmotionanditstransformationstocreatethe

ternarysystemsoftheUniverse.Soabriefpentalogicperspectivesonits¬∆@stelementswouldbe:

@:Topologyisthelast'realgeneralization'ofspace,whichdoesnot'escapeintothelogicspacesofthemind,itwillallowustostudyinmoredepththefundamentalpropertiesofanylogicspace(incidence,congruence,adjacencyetc)initsmoregeneralview,jumpingovertheEuclideanandAxiomaticmethodsweconsideroutdated.ThisshallestablishfurtheraswedidinourIpartonGreekbidimensionalgeometry,itsbio-logic

meaning.

Γ:Asgeometrywithmotionandonly3varietiesitistheessentialgeometryofT.œswhichareinspacebasicallyternaryensemblesofthe3typesoftopologiesthereare,andhavebeenallovertheplace-elliptic,parabolic

andhyperbolic,inanynumberofrelevantdimensionswestudy.

S≈T:itallowssomeofthemorecomplexS=Tmodelsofreality,inwhichatemporalsystembecomesexpressedasaspatialproblem,whichrenderssincethefirstworksofPoincareenormousyieldsinthesolutionofmotionproblems,alwaysmoredifficulttoresolvegiventheinherententropicqualityofpuretimemotions,whichbecome'fixed'formentalalgebraicortopologicalmanipulationeasierwithatopologicexpression.Thus

topologicalanalysisisthefirst'step'inthementalsolutionandconversionofa'futurelogicmotion'intoapast'memorialformofinformation'(aconceptagainofthewidergeneralizationofexistentialalgebratreatedinthe

firstline.

∆: Topologyevolvedintonetworktopologyi-thebestformofgeometrytostudy∆±1partsandwholes.

ThosemultipleviewsofTopologyresumesin3essentiallevels:

S=ttopologicalevolutioninallstienceswithspecialemphasisinbiology;@-mentalmethodsofsolvingproblemsinwhichamotionbecomesaformofspaceandallowtousegeometricalmethodstosolvest-combinationsofmotionandformproperofphysicsandfinally∆-scalesymmetriesbetweenpointnetworksandwholes,and

differentdimensionalelements.

Multifunctionalentanglement.Itslaws:inversionofrolesasweemergeintohighersocialplanes.

Moreoverthe5thscalardimensionimpliesthattopologiesexistwithintopologies.Thatis,mostsystemshaveanexternal‘sphericaltopology’,meaningaclosedmembrain.Butthemembrainwillcloseinsideaternaryadjacenttopologymakingthe‘singleexternalform’,aninnerfractalT.œwith3parts.Sothe3Dimotionalelementswithinthesystemcanperformallthemultifunctionsofa‘wholebeing’.SinceaKEYLAWoffractal.ternarysystemsisitsmultifunctionality,whichgivesanyentityaninnerternarytopologyTOactas$,ST,and§ðbeings.

Furtheronaswemovethroughtheternarygeneratorintimevitaltopologyadsitslawsof'transposition'offunctions,astearsandpegstransformthem,orthemodularbeingchangesitsfocusofactionfromonetootherofits3Dimotionalelements,disguisedwithinthemembrane.Sointherealmoftopology,correspondencesofformandfunctionarenotimmediate,asinnerpartsbecomemultifunctional,whichallowsaternarytopologytoplaydifferentrolesinrealityactingas$,ST,and§ðbeings.As:'Systemswhichdisplaymorethanonedimensioninspace,playmorethanonefunctionintime'.Thismeansalltopologicalwhole

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beingsareternaryforms.Soeveniftheyaredominantinoneofthe3±iarrowsoftimespace,whichisitsmaintaskintheouterworld,definedbythetopologyofitsoutermembrain,theywillbeabletoperformthe3dimotionstheyneedtosurviveintheirinternalworldwithinthemembrane.

Considerthesimplexexample:alineallimbin3dimensions.Itcanalsoactasarotaryformwithclockfunctions;henceasanenclosure;andinacylindricalgeometryasanaxisofperception.Itisthiskindofmultidimensional

nature,andtrans-formationofaformintoanotherwhatmakestheUniversecomplexandnotsoeasytounderstand.

Themainofthoselawsisthechangeoffunctionofallsystemswhenbecomingamerepointofalargerscale,astheytransposetheirrolesfrom'kingofthe∆-1hill',toantofthe∆+1ant-hill:

"Whengrowinginsocialscalestoformanewplane,functionschange,mostoftenbecominginverted;

:∑|i-1≈Øi,∑Øi=|i+1."

Thisispartindeedofanessentiallawweshallrepeatadnauseam:whengrowinginsocialscalestoformanewplane,functionschange,mostoftenbecominginverted.

Andthereasonisobvious,thewholesphericalmicropointisthekingofitsinnerworld,butjustaparticlemicropointinthelargerwhole,whereitsroleisslavishtothesuperorganism.

SotheexplanationofthischangeofvitalrolesisimmediatewhenconsideringtheDisomorphiclawsof∆st,whichexpressedini-logicwrites:

∑|i-1≈Øi,∑Øi=|i+1

Thislawcomesallovertheplace,inexperimentalsystems,frombiologicalsystemswhereproteinsthatarelineal,becomethehyperbolicelementswithmultipledimensionalfoldingthatcontrolthereproductionof

proteins,toatomsthathaveperfectcyclicalform(iron),whichbecomethelinealstrongestelementforcreationofentropicweaponsinthe∑+1scale.

Shakespearesaid:weareallbuffoonsorkingsdependingonourperspective.Anditconnectsalsowiththefactthataswegrowinsizeperspective(Lobachevski’sr/kratio),frombeing‘cyclical’beingswebecomemoving

dot-pointstracinglinesinthelargerperceivedflatworld.

Tonoticeaonetoonecorrespondence.Wetalkedofdistanceasthesumof'minimalstepsofmeasure'whichappliestotranspositions,inthesimplestform,withthestopandgo,S>Tstepsofallmotionsin5D²realities.So

hereweobserveaparticularcaseofthis'motionthroughtransformationofstatesofthebeing,acrossthescalesofthefifthdimension,symmetrictothechangeofstatesintimespaceandtopologicalfunctions=forms.

∆±1symmetries.Thescalargeometryofpolyhedrons.Euler’scharacteristic.

Theabstractwaytodescribetopologicallyallthosefigureswithdifferentverticesistheso-calledEulercharacteristic-thefirsttheoremin

topologyknowntoDescartes.

Sinceintheevolutionofhumanthoughtalwaysthefirstknowledgeisthesimplest

mostgenerallawsofthetime§paceUniverse,itisworthtoconsideritinmore

detail.Letustakethesurfaceofanarbitraryconvexpolyhedron.Wedenoteby

A0thenumberofitsvertices,by1the

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numberofitsedges,andbyA2thenumberofitsfaces;thentherelation:

Ao+A2=A1+2

Whichholdsforanypolyhedronincludingthosewithcurvededges.

WehavewrittenitproperlyaccordingtotheS<st>Tsymmetryevenifgeometers,unawareoftheS=tsymmetriesthatgeneralALLthelawsoftheUniverse,putitao+a2-a1=2.Theinterestoftheequationisobvious-notonlyisagenerallawofallpolyhedral.Italsoshowsthe3different'dimensionalscales'ofpoints,linesand

bidimensional,holographicsurfacestogether.

ThenwecanidentifyS,TandST,theintermediateelement,writing:1Dpoint+3Dsur-face=2Dline+2.

Canweeliminatethe2tomakeittrulyanS=Trelationship?Yesbyopeningthetopandbottomofthesphere-likepolyhedron,creatingacanonicalaxisforanyrotationalsphere,sinceweloosethen2'faces',givingusthecanonicalformofNature'sspheres,withitspolaraxis,anditsanimalandvegetalopeningstotheworld,such

as:

1Dvortices+3Dsurfaces=2Dwaves/edges.

Italsoallowustounderstandabasictransformationofasphereintoanopen'cylinderalsowiththesame0-Eulercharacteristicwhichobeysthelawofbalance:S+T=ST,andhencesphericalformswithtwoopeningsintheaxis,eitherinitslineal$-limbsorrotational§ðspheresarethecommonestformofnature,whichcombines

thelawsofbalanceofall∆stsystemsandtheefficiencyofitsregularconfigurations.

Topologicalevolution:morphogenesis-growingandkeepingthebalanceofforms.

Thusevolutionofformsormorphogenesisisruledbythebasiclawsof5DT.œs,theconstant'changeofformanddimensions'asthesystemgrows,'restrained'bytheneedtokeepanS=Tbalancebetweenformsand

functionstomaintainthesystemefficient.

Thisistheessenceofit:growandmultiply,butasyoudokeepthebalancesofternaryformsandfunctionstoavoidbeingextinctbyaDarwinianeventofanotherform.

SotheSTstopandgolawshereacquirea'newdimension'bytopologicalevolutionthatreproduce,evolvesociallyandreformthesystemtokeepabalancewhichmeanstomaintain3partsinconstantsocialevolution

andgrowth.

Forthatreasontherearenoreallyspheresofgenus2,butrotarysphereswithanaxistoprocess,absorbandemitenergyandinformation,whichthenwillhaveeitherapolarcaporcentralpoint,wherea'donuts'will

become'separated'fromtheaxisasaproperentityplayingtheroleofthesingularity.

Andsointhesamemannerallmetalshavethemostefficientcubicorhexagonalconfiguration,mostlywitha

self-centeredsingularity,mostspheresoncetheycompletetheir'tightpacking'duetoreproductiveevolutionwillhave

the3elementsofthebeing.

Yettheycanbealsoconsideredaternaryvariation,onitsonly3crystalstructures:

ST:Themostbalanced,hencesimplesttoconstructwithminimalelementsfullsystemisthebody-centeredcubic,wherethecentralatomplaysthesingularityrole;itistheSTbalancedform.

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Theðformisthehexagonalsystem,alsowithacentralclearaxis,WHERETHEMAXIMALdensityofformhappens(atriangularsingularity,transversedbyanaxisbetweentwoself-centeredatoms;andthestrongest

covers:Hexagonal'pi=3circles';thatisbidimensionalcircleswithaperimeter3timesitsdiameter.

The$formthereforeisthethirdremainingone,whichindeedisallaboutastrongmembrane,withself-centeredatomsandnosingularity.

Thisternarydivisionofspeciesisoftenfoundalsoinbiologicalsystems,wheretheface-centeredcubicwillbeaplate-armoredherbivore,whichisallaboutprotectionwithlittlebrain,vs.thepredatorwhichisallabout

mobilityandfastaction-reactionbrains(theHexagonalequivalent)andsimilarspecies,playingthendifferentpredator-preyroles.Acoupleofexamplesshouldsuffice:

Inthecambricexplosionitwasallaboutface-centeredarmoredtrilobites,andthefirsteye-cephalopodsthatsoonlostitsarmoredandbecamesquidswithfastdevelopednervous

informativesystems.

Andthenalotofintermediatespecies.Suchternaryformsoccuralsowithinanyspeciesasthemulti-functional3Dbeingsplitsinvariationsonthesametheme.

Forexample,3subspeciesofpredatorshappenintheoldworld,theLion,isthe'armored'strong,thickmuscle-skinvs.thefast,weak,runningcheetah.Butthemostsuccessfulistheintermediateleopard,whichistheSTbalancedspeciesthatsurvivebetterthantheothers.SoforexampleinmassivecontinentalIndiathecheetahwasextinct;buttheleopardsurvived;innimbleCeylon

islanditwasthelionequivalent,thetiger,butagaintheleopardsurvived.

Bidimensionalsurfaces=membranes.Platonicsolids.Euler'scharacteristic.

Topologyisconcernedmostlywiththemembraneofthesystem,initspresentform.What∆stadsisthevitalizationofitsconcepts,andaproperdimensionalanalysis,introducingthelawsof

S=tDisomorphicsymmetries.

Letusvitalizeanotherclassiclawoftopology:

TheEulercharacteristicanditsplatonicsolids,relatedtothebalancebetweenvertex=fractalpoints,edges=lines/wavesofcommunicationandsur=faces(enclosedvitalspaces)-givenits

generalization...connectedtoknottheory,topology,physicsofmatterandcrystallography,surfaceproperties-younameit.Letusthenconsiderofthem

onlythemostobvious∆stproperty-thereare5ofthemina5Duniverse.

ThepreviousEuler'sformulaisobviouslyacombinationof∆-scalebalance,suchas∆-

1vertices+∆+1faces=∆ºwaves/linesofcommunication.

Butthevitalemergingprocessofgenerationofforms;asthewavesofcommunicationbetweenvorticescreatethebidimensionalenclosedsurfaces,andevolvethenetwork,isthemostimportant'perspective'intopological

evolution.

∆±1.Pointscreatetopologicalnetworks.Hylomorphism.

Whenwegetintodetailsonhowthosetopologicalevolutionstheyfollowanotherfundamentalprincipleof∆sthylomorphism,whichessentiallymeansthat'wholesaremadeofparts',thatisoffractalpoints.Andsoa

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changeoftopologicalvarietyhappensalwaysbytinymicroscopicchangesintheconfigurationofthefractalpoints;whilethatscalarsymbiosisalsoallowscommunicationthroughthefractalnetworksofflowsof∆-1

energyandinformationbetweenthepartsofthewhole.

Thisscalarstructurealsoexplainstheformationofopeningsandtearsbecausethecontinuityofthewholedisappearsinthe∆-1scaleandthediscontinuityanddarkspacesbetweenpointsappears(Galileanparadox).Sobyreordering,expandingorimplodingdistancesbetween∆-1setsofpointstheholesofscalartopologyallow

morphogenesis:

Inthegraph,topologyworksaccordingtothe1st,2ndand3rdpostulateofnon-æ=i-logicgeometry,throughthearrangementofpoints(1D),its

connectionsandaxons,openedin|-$functionsclosedinO-ðforms.Itisthenessentialthe'degreeofpacking'definedbythe|vs.Oformthatwill

eliminateintermediatespacestocreateadjacency,ormaintainaminimumhollowspacefor'flowsofnetworks'tocross,inparallelism–twochoicesthatasmanydualitiesbecomeessentialtodifferentiate

species(soplantshavehollowstoallowvesselstotransportwaterupanddown).

Relativeratiosofdistancesbetweenfractalpointsincreaseaswedecreasesize,growingininformation(5Dmetric).And3relationshipscanexistdependingofthesizeoftheminimalstepbetweenpointscomparedtotheradiusofthepoint:adjacencywhenthedistanceissmallerthantheradius,parallelismwhenislarger,andperpendicularity,whichrequirestopenetratebeyondthe'enclosing,protectingmembrane'andtearthepoint.

For2Dimensionalsurfacesisalsoalogicextensionfromlinesoflengthtoflatplanes,ST-reproductivewidthsthatmixtheothertwoelements,thehyperbolicgeometrywithitsdual±curvaturesandfor

height/information,andfinallythesphereisthevolumethatstoresmoreinformationinlesserspace.Soinprinciplewemustsuggestthefollowing2Dgenerator:

2D¡Γ:$t:plane/motion<SThyperbola/reproduction>§ð:sphere/information.

Thegraphshowsalsohowtheparallelproperty,becomesnowmorecomplexshowingclearlysomeofitskey'socialproperties':

-Sphericalsystemsaresocialastheybecometighter,informativeelementscausingthesocialevolutionofpointsintosupœrganismsofahigher∆+1scales.

-Flatsurfacesmaintaintheparallelismadinfinitum.Sotheyareidealfornetworkherding,inabalancebetweenadjacencyandconnection.

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-HyperbolicSTvitalenergyifleftintheopenwithoutaclosingmembranewilldivergeintoentropy,seekingfor

'freedom'andbecomingunconnected.

Inthegraphs,whosefullexplanationwouldrequireanentirearticle,anglesofperception,latterstudiedin‘mentalspaces’,arelargerforsphericalO-informativegeometries.Wecometothefirstseeminglycontradiction

asweexpandourdimensionsinthefunction/formofthenextscale.

Thekinobserverwillhavenoticethattheroleofthe1Dlineinitsentropicfunctionisbeingtakenbythehyperbolicplanein2D,transposingitsfunctionswiththoseoftheplane,generatedbytheentropicline,which

nowtakestheSTfunctionsofthehyperbole.

Why?ThegraphshowsthattheystillkeepitsS-hortest,ƒ-astest(St-raightest)space-timetrajectoryintermsoflines,hyperbolasandcircles,whichmeanbytheprincipleofleastactionthatmakesthosepathsoverwhelminginexperimentalreality,thattheyareindeedrelatedandgeneratedbythem:∑lines=plane,∑hyperbolas=

hyperbolicchair,∑circles=Sphere.

Itspropertieshavedefinitelyswitchedbetween$-linesandST-hyperbolas,into$-hyperbolasandST-planes.

Sowhilethemotionsintimeofthegeneratorhavebeenconserved(stilltheflatair-planeandFormula1movesfaster,thesphereisstilltheinformativeeye-headontop;thehyperbolacombinesboth),thefunctionsinspace

aswe'emerge'from1to2dimensionshavebeentransposed.

Andthisisoneoftheparadoxesof'growthin∆-planes',aswecanregarda2Dasasocialgatheringof1Delements.Functionsbecomeofteninverted.Andsowhileanelementaryanalysismightseeminabstracttorelatelinestoplanes,circlestospheresandhyperbolastoLobachevski'sgeometries,theuniverse,whichisa

constantiteration,transformationandmergingofdimensionalKaleidoscopeshaschanged'again'.

Classictopology.Constructionoforganic,fractalnetworks

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Whenwestartinamoreprofessionalwaytounderstandthe3topologicalformsoftheUniverse,weimmediatelyconfrontthefactthatatopologicalplaneismadeofpoints,joinedbylines,andsoenterintoa

morerealdescriptionofthescalaruniverseasformswhicharenetworksofpointsjoinedbyflowsofenergyandinformation.Theconceptofanorganismarousesimmediatelyasanorganismisasystemthatco-existsatleast

intwolevelsorscalesofsize,joinedbynetworks=flowsofenergyandinformation.

Inthegraph,the3canonicalformsofspace-time,thesphere,thetoroidandthefractalplane,whichincloseanalysisarealwaysnetworksofpoints.Indeed,topologyatprofessionallevelhoweverisnotacontinuous

geometrybutasumofpointsthatputtogetheratadistanceseemtobenotanetworkbutacontinuousform.HencetheexistenceofscalesintheUniverse,inwhicheachpointofatopologicalformisinitselfaworldinalowerscale.Sincethe3rdlegbesidesspace-timesymmetriesoftheGSTphilosophyofscienceisthefractal,

scalarstructureoftheUniverse,andhowthosescalesco-existandcreateorganicsystems.

Wecanthenrecognizea'cellular-atomic-social'systemoffractalunitsthatbuildaself-similarclosed(spherical)open(hyperbolic)ortoroid(withtwoclosingpaths),networkasaseriesofcellularrelationshipsofconnectivity,

adjacency,coherence,proximity,etc.whichmake'emerge'awholethatembodiestheregularitiesofthemyriadofinfiniteexchangesofenergyandinformationbetweenconnectedpartsofthewhole.Inthegraphwehavedrawnafewvarietiesoftopologicalspecies,accordingtothoseproperties,departingfromthemoststabledual,'simplex'possiblesystemoffractalpoints:2ternary'triangles'ofpoints,anditsopen-spatialandclosed-

temporalandopen-closedspace-timecombinations,whichillustratethecreativedynamicprocessesofevolutionofspace-

timebeings.

Intheleft,abovetimeforms,startingwiththeringoftimeandbelow,spaceforms,startingwiththelineofpurespace,whicharethe2commonest,

simplests-tforms.

YettherichnessoffunctionsandformsoftheUniverseisratherunlimited.Sonextweseeacyclicpentagonwitha'lineallimb',jettingonthebasecalleda'mesh',andnextweseetheringconvertedintoastar,wherea

centralknot-point,themind-monadreceivesinformation/energyfromeachcornerofitsbidimensionaluniverse,ensuringasymmetricreception/mappingofitsouterwhole.Andfinallyweseethe6pointsconnectedinternallyandhencecreatinganew∆-scale(thatoftheaxonsthatcomeoutoftheneurons)andanew'mind-

center',inthecentralconfluenceofthepoints.

Andagainbelowweseethecommonestdivergencesfromthepureline:asixthelementalsojettingoutoftheline(atree),andaconnected'bus',equivalenttotheconnectedcircle,wheretheconniptionisestablishedbya

singleline,whichbecomesthe'spine'ofthelineal,entropic,fast-movingsystem,farsimplerthanthefullyconnectedhexagon,sinceclosedtimesystemsarealwaysmorecomplexininformationthanfaster,largerlineal

spatialones.

INTHATREGARDtopology,its3space-timevarietiesanditsnetworkstructureistheclearestmathematicalproofoftheexistenceofanorganic5DUniverse.

Letusthensummariesthatstructure,andhowitsvitalnetworksevolvethroughthepostulatesofnon-AEinsocialgroupsfrompointsintolinesintoorganicplanesand5Dpartsandwholesthatformasinglestructure.

AkeyconceptofallGSTisthatsincetheUniversedepartsfromsimplexprinciples,itisdesirabletofollowaprocedurefromsimplextocomplex,whichfollowsthetimeevolutionofthosedisciplines.Sowecanobtainalotofworldviewandinformationbyconsideringbeforewestudymoderntopologyclassicgeometry>Topologyanditsfundamentallaws.Letusstartwiththoselawsandwhattheysayandhowtheyaregeneratedbythe

fractalgeneratorS≈Tandits2/3elements.

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Thusthemembraneofasystemalwayscanbeapproximatedtopologicallywithpoints,linesandplanes.

NowthefirsttheoremoftopologyiscalledEuler'scharacteristic:

Wedenotebyα0thenumberofitspoint-vertices,byα1thenumberofitslinealedges,andbyα2thenumberofitsbidimensionalfaces;thenthefollowingrelationisknownasEuler’sformula:

α0−α1+α2=2

WhatdoesitmeaninGST?Iwonder...obviouslyisimportantaswehavearelationshipforany$t-cover,butweshouldtrytoreorderitintermsofDimensionalforms

D1(point)-D2(lines)+D3(planes)=2

D1(points)+D3(planes)-1=D2(lines)+1

Inotherwordsforaspheretohaveabalanceitwillneeda±1holes,whichwillturnouttobetheaxisholesofallrealspheres,equivalenttothe3'apertures'ofapi-bidimensionalcycle(3.14-3)

Thisgeometricaltheorembelongstotopology,becauseourformulaobviouslyremainstruewhenwesubjecttheconvexpolyhedroninquestiontoanarbitrarytopologicaltransformation.Undersuchatransformationtheedgeswill,ingeneral,ceasetoberectilinear,thefacesceasetobeplane,thesurfaceofthepolyhedrongoesoverintoacurvedsurface,buttherelationbetweenthenumberofverticesandthenumbersofedgesand

faces,nowcurved,remainsvalid.

Triangulation.

GSTrelationship:OneofthefundamentaldiscoveriesofGSTistheternarystructureofallwhatexistsasawhole.Thisisshowneverywhere,ingeometryfromtherecenttheoryofcausaltriangulationthatshowshowtoconstructaspace-timeUniversewithonly3'points'andacausaltimealgorithmbetweenthem,totheearlier

topologicaldiscoveryofthissection:mosttopologicallawscanbereducedtothestudyofitstriangularelementsinthe∆-1scaleofthewholeform.

Themostimportantcaseiswhenallthefacesaretrianglesandthenwehaveaso-calledtriangulation(adivisionofoursurfaceintotriangles,rectilinearorcurvilinear).Itiseasytoreducethegeneralcaseofarbitrarypolygonalfacestothiscase:Itissufficienttodividethesefacesintotriangles(forexamplebydrawingdiagonalsfromanarbitraryvertexofthegivenface).Thus,wecanrestrictourattentiontothecaseofatriangulation.Thecombinatorialmethodinthetopologyofsurfacesconsistsinreplacingthestudyofsuchasurfacebythestudyofoneofitstriangulations,andofcourseweareonlyinterestedinpropertiesofthetriangulationthatare

independentoftheaccidentalchoiceofonetriangulationoranotherandso,beingcommontoalltriangulationsofthegivensurface,expresssomepropertyofthesurfaceitself.

Euler’sformulaleadsustooneofsuchproperties,andweshallnowconsideritinmoredetail.Theleft-handsideofEuler’sformula,i.e.,theexpressionα0−α1+α2,whereα0isthenumberofvertices,α1thenumberof

edges,andα2thenumberoftrianglesofthegiventriangulation,iscalledtheEulercharacteristicofthistriangulation.Euler’stheoremstatesthatforalltriangulationsofasurfacehomeomorphictoaspheretheEulercharacteristicisequaltotwo.Nowitturnsoutthatforeverysurface(andnotonlyforasurfacehomeomorphic

toasphere)alltriangulationsofthesurfacehaveoneandthesameEulercharacteristic.ItiseasytofigureoutthevalueoftheEulercharacteristicforvarioussurfaces.Firstofall,forthecylindrical

surfaceitisequaltozero.Forwhenweremovefromanarbitrarytriangulationofthespheretwononadjacenttrianglesbutpreservetheboundariesofthesetriangles,thenweobviouslyobtainatriangulationofasurfacehomeomorphictothecurvedsurfaceofacylinder.Herethenumberofverticesandofedgesremainsas

before,butthenumberoftrianglesisdecreasedbytwo,thereforetheEulercharacteristicofthetriangulationsoobtainediszero:

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Planes+Points=Lines:(ape-openspace)Planes-1+Points-1=Lines(closedtimecycle).

Thusthefirstandobvioustruthisthatinanentropicsystem,thedominantformistheline,theentropicfieldwhichmattersasmuchasthesumoftheSTandTsystem,itgenerates&sustains.

Orintermsofabalanceofpresent,ifweconsidertheentropicplaneavolumeofpastspace,thewave-lineofpresentandthepointsingularityoffuturetime,thereisapresentbalanceaspresentwaves≈pastplanes+

futurepoints

OntheotherhandthespheretoreachthebalancecanonicaltoallsystemMUSTacquiretwomorepointsorplanes.Butasitisaclosedform,itcannotacquiremoreplanes.Soitdoesnaturallyevolvetoacquireadual,centralpoint,insideofit,asitnaturallyhappensinallsystemsofnaturethatevolvefromlinesorlinealtubs

intoclosedcyclesandspheres,whichacquireitssingularitypointstoreachitsbalance.

Presentwaves≈pastplanes+futurepoints-2SINGULARITYCENTRALPOINTSTHAThaveinversesymboltotheouterpointsofthesystem;orinothervariationofbalance,thespheremustloosetwopointsthatbecomethe

openingsofitsaxis.

Thosearethereforethejustificationsofoneofthefundamentallawsoftopologywhichderiveoftheneedofbalancebetweenpast+future=present

Andultimatelyexplainwhyallspherestendtohaveinrealvitalgeometry,axisandcanthereforeeasilytransform$tintoðƒ,Infactmostvitalsystemsaremadeofalineal'axistube'andaspherewherethetube

becomesthedigestiveentropicsystem,peggingbothinabalancewitha0-characteristic:

Inthegraphabalancedsimplexsystemiscomposedofatubulardigestive$taxisandasphericalmembrane,withanintermediatestsystemwithonion-likelayersthattransformonesystemintotheother.

Indeed,letustakethesurfaceobtainedfromatriangulationofasphereafterremovalof2ptrianglesofthistriangulationthatarepairwisenotadjacent(i.e.,donothaveanycommonverticesnorcommonsides).

HeretheEulercharacteristicisdecreasedby2punits.ItiseasytoseethattheEulercharacteristicdoesnotchangewhencylindricaltubesareattachedtoeachpairofholesmadeinthesurfaceofthesphere.Thiscomesfromthefactthatthecharacteristicofthetubetobepastedinis,aswehaveseen,zeroandontherimofthetubethenumberofverticesisequaltothenumberofedges.Thus,aclosed2-sidedsurfaceofgenusphasthe

Eulercharacteristic2−2p.

Butallotherformsarenotasbalancedasthepreviousensemblesbecausetheyhavenotthesamedegreeofbalance,andsowhentheyarecreatedtheytendtobecomeextinguished...failedless-efficientforms.

Topologicalproperties.

Ifweweretobemoreamenabletothelanguageofmathematicians,thepropertiesthatdefinethenetworksofpointsofthe3$t<St>ðƒELEMENTSofreality-itscurvature,themainproperty,alongits'closedtemporal'or

openspatialnature,andits'connectionsbetweenthem,oftenthroughthehyperbolicSt-artarecalledtopologicalproperties.

Specificallythoseproperties,maintainedbythestructureduringitsexistencebetweenitslimitingage-motionsof±d=evolutionarybirth,reproductionandextinction,throughalltheotherpossiblemotionsoftime(growth,

locomotion&diminution)arecalledtopologicalproperties.

Asatopologyisanetworkof∆-1points,whicharesmoothandadjacenttoeachother,wecanexplaintheconceptofpreservationorcontinuityunderanymotionoftime-spaceofthetopologicalorgan(transformationinthestatic,discontinuoussimplifiedmind-languagesodmathematics)asthemaintenanceinthe∆-1scaleof

thepoint-structureandrelationshipsofcontinuity(adjacency)betweenthosepoints.

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INOTHERWORDS,atopologicalternarysystemconservesitsformsinbalancethroughtheentiredifferentiableperiodofitsworldcycle,butthisdifferentiabilityor'smoothness'withnotransitionbreaksinthe3

motions/pointsoflifeinwhichthesystemchangesitsphase:

Thustobepossibletodefinethepreservedpropertiesofatopologicalgaieties,thesystemmustbe'differentiable'throughalltheperiodoftimeandtranslationofspace.Yet,inthepointofemergenceand

dissolution,andreproductioneitherbysplittingasystemintotwoor'penetrating'andtearingperpendicularlyothersystem,thosetopologicalpropertiesarenotpreserved.

Thishashugeimplicationstotheunderstandingoftheprocessoflifeanddeathandtheultimateworkingsofspace-timegeometriesastheygoalongperformingitsworldcycle.

Whatabouttheother2motionsnotquotedhere,evolutionandperception?Are'differentiable'smoothandcontinuous?

Thisisaquestionbeyondthescopeofthispaper,whichhoweverwwwmustrememberwhendealingwithperceptionandevolution.Inthesimplermodelofperception,wecantalkofaseriesof'holes'penetratedby

theinformation,whichinternallymapsoutthemirrorimageoftheexternalworld.Inevolutionwetalkofpalingenesis,oneofthemostfascinatingsubjectsofallGST,asitbringsaboutafastforwardresumeoftheentireprocessofexistenceandemergenceofasystem,asitconstructsanewsuperorganism,andeachofitsprocessestellussomethingaboutthestructureandlawsoftheGameofExistence,whichweshallstudyinthe

1.life3rdlineposts.

Butwhatdoesittrulymeanasystemdoesnotpreserveatopologicalpropertyandwhyitdoesnotthroughthemotionsofreproduction,evolutionandperceptionanditsphasesasopposedtoitspreservationintheother

motions,growthdiminutionandlocomotion.

Simpleenoughitmeansthatthose3motionsarespace-liewhilethemotionsoftime,doNOTpreserveitsparityastheyaretransformative.

Thusweconsiderthatinthepositiveview,topologystudiestopologicalpropertiesoffigures;,whichremainconstantunderanarbitrarytopologicaltransformation≈motion.

Inthoseperiods,thebeingexistsinasmoothmanner,asnothingtears.

Andviceversaweshallstudyalsotopologicaltransformations/motionsthatreorganizeinternallythebeingandhowthenotpreservedtearsandgrowthofthetopologicalnetworksaffectsthisevolution.

Andfinallyweshallapplythisknowledgetounderstandwhatremainsinvariantunderarbitrarycontinuoustransformationsofgeometricalfigures.

Allthisofcourse,'sparkled'withdeepphilosophicalconclusionsaboutwhatthesystemtellsus,duetosuchtopologicalproperties.

Themainproperties,whichwewillstudyhereareastheyarebothessentialtotopologyand∆stare:

-Thepropertyofacurveorasurfaceofbeingclosed(thatis,time-like).

-Thepropertyofaclosedcurveofbeingsimpleformingonlyoneloop.

-Thepropertyofasurfacethateveryclosedcurvelyingonitisadissectionofthesurface(thesphericalsurfacehasthisproperty,butthering-shapedonehasnotandthiswillhavemanyimplicationsforthevitalgeometryof

beings.Thelargestnumberofclosedcurvesthatcanbedrawnonagivensurfaceinsuchawaythatthesecurvesdonotformdissections,i.e.,thatthesurfacedoesnotsplitintopartswhencutsaremadealongallthesecurves,

ororderofconnectivity.

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Topologicalstudiesoftimemotions

WeHAVEcoveredthusmostofthethemesofgeometryinaverysynopticmanner,enlighteningthemallwithnewinsightsbornofGST,accordingtothepurposeofthisweb,whichistoshowtheorganic,space-timenature

ofalltoesandlanguages,unifiedbythoseprinciplesandthecapacityofGSTtofurthernewinsightsonallstiences.

ItonlyreststoconsideranexampleoftheGalileanparadox-whichallowstousepuregeometricalSSdimensionsofformtostudyequivalentproblemsofTT-dimensionsoftimemotions.

WealreadysaidthatPathsinthatsensein∆stmustbetreatedwiththedualityofSTdimensions,oneofmotionandoneofform,complementingboththetopologicalspace-onlyviewandtheviewofpointsmovedthrough

curvatureforces,properofphysicalstudiesoftopology.

Sincethemindfixesmotionintoformto'makesense'ofmotions,orderthemandunderstanditsgenerallaws,somethingwhichtopologydoeswithits...

S=T@.Topologicalmethods:motionsbecomingforms...whichallowtoresolvecomplexmotions=tn-dimensionalprocessestransformingthemintotopologicalforms-butthisisanartifactofthemindnotareality

-andtoforgetthatisthebiggestsinofcreationistmathematics.'Point'.

Letusconsideroneexample,usingthetorusastherichesttopologicalformtoillustratesuchformsofmodeling:

ThecompoundplanependulumconsistsoftworodsOAandAB,hingedtogetheratA;thepointOremainsimmovable,therodOAturnsfreelyinafixedplanearoundO,andtherod

ABturnsfreelyinthesameplanearoundA.

EverypossiblepositionofoursystemiscompletelydeterminedbythemagnitudeoftheanglesϕandψthattherodsOAandABformwithanarbitraryfixeddirectionintheplane,

forexamplewiththepositivedirectionoftheabscissaaxis.Wecanregardthesetwoangles,whichchangefromOto2π,as“geographicalcoordinates”ofapointonatorus,

countingfromthe“equator”ofthetorusandoneofits“meridians,”respectively,

Thus,wecansaythatthemanifoldofallpossiblestatesofourmechanicalsystemisa

manifoldoftwodimensions,namelyatorus.Whenwereplaceeachofthetwoanglesϕ,ψbyacorrespondingpointonthecircumferenceofacircleonwhichaninitialpointandadirectionaregivenhenwecanalsosaythateverypossiblestateofourmechanicalsystemiscompletely

characterizedbygivingonepointoneachoftwocircles(oneoftheseistakenasthelatitudeϕandtheotherasthelongitudeψ).

Inotherwords, justas inanalyticgeometryweidentifyapointoftheplanewithapairofnumbers,namely itscoordinates,soinourcasewecanidentifyapointofthetorus(andhenceanarbitrarypositionofourpendulum)withthepairofitsgeographiccoordinates,i.e.,withapairofpointsoneofwhichliesononecircleandtheotheron another. The essenceof the situation is expressedby saying that themanifoldof all possible states of ourcompoundplanependulum,i.e.,thetorus,isthetopologicalproductoftwocircles:

Thuseventhesimplestmechanical(kinematical)considerationsleadustotopologicalmanifoldsofgreatvalueinthepractical,moredetaileddiscussionofmechanicalproblemsandanymodelingofS≈Tmulti-dualdimensionsofaT.œ.

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Allthissaid,andresumed,wenowwillconnectclassicTopologywiththefractalnon-EuclideanpointsthatstructuretheUniverse,toshowhowultimatelybythecorrespondenceprincipleallsubdisciplinesofclassic

scienceconnectwithnewdisciplinesofmodernstience.

Topologyandsettheory.

Thetheoryofsetsmadeindeedpossibletogivetheconceptofageometricalfigureabreadthandgeneralitythatwereinaccessibleintheso-called“classical”mathematics;butthatisexactlyhow‘specificreality’becomescut-offfromsyntheticparalogiclanguagesthatfinallyseekasingleorigintoitall–spacetimeinreality,andany

imaginarymindmirrorofitindifferentlanguages.Tosaythenthatallisaset,likesayingallisnamedbya‘word’or0x∞=C¡generatesallnumbersandallminds,istosaylittle.

Hencesettheory,ultimatelyanabstractionoftherelationshipsbetween∆-1elementsandwholes,canindeedexplainitall,butsocan5Dwiththeadvantageofbeinganobjectiverealitynotahuminddistortion.

Inanycasetohonorthecorrespondenceprincipleweconsiderthat‘parallelism’between5Dandsettopology:

Insettheorytheobjectofageometrical,inparticularatopologicalinvestigationnowbecomesanarbitrarypointset,i.e.,anarbitrarysetwhoseelementsarepointsofann-dimensionalEuclideanspace.Betweenpointsofann-dimensionalspaceadistanceisdefined:namely,thedistancebetweenthepointsA=(x1,x2,···,xn)andB

=(y1,y2,···,yn)isbydefinitionequaltothenonnegativenumber.

Numbersthusbecomespatialwhenpositivereinforcingouranalysisof‘negative’numbersastemporalmotions.

Theconceptofdistancepermitsustodefineadjacencyfirstbetweenasetandapoint,andthenbetweentwosets.WesaythatapointAisanadherentpointofthesetMifMcontainspointswhosedistancefromAislessthananypreassignedpositivenumber.Obviouslyeverypointofthegivensetisanadherentpointofit,but

theremaybepointsthatdonotbelongtothegivensetandareadherenttoit.

Letustake,forexample,theopeninterval(0,1)onthenumericalline,i.e.,thesetofallpointslyingbetween0and1;thepoints0and1themselvesdonotbelongtothisinterval,butareadherenttoit,sinceintheinterval

(0,1)therearepointsarbitrarilyneartozeroandpointsarbitrarilyneartoone.Asetiscalledclosedifitcontainsallitsadherentpoints.Forexampletheclosedinterval[0,1]ofthenumericalline,i.e.,thesetofallpointsxsatisfyingtheinequalityo≤x≤1,isclosed.Closedsetsinaplaneandallthemoreinaspaceofthreeormoredimensionscanhaveanextremelycomplicatedstructure;indeed,theyformthemainstudyobjectofthe

settheoreticaltopologyofann-dimensionalspace.

NextwesaythattwosetsPandQadjoinoneanotherifatleastoneofthemcontainsadherentpointsoftheother.Fromtheprecedingitfollowsthattwoclosedsetscanadjoinonlywhentheyhaveatleastonepointin

common;but,forexample,theintervals[0,1]and(1,2),whichdonothavecommonpoints,adjoinbecausethepoint1whichbelongsto[0,1]isatthesametimeanadherentpointof(1,2).NowwecansaythatasetRis

divided(“dissected”)byasetSlyinginit,orthatSisa“section”ofR−SconsistingofallthepointsofRthatdonotbelongtoScanberepresentedasthesumoftwonon-adjoiningsets.

Thus,Lobachevski’sideasonadjacencyanddissectionofsetsreceiveincontemporarytopologyarigorousandhighlygeneralexpression.WehavealreadyseenhowUryson’sdefinitionofdimensionofanarbitraryset(seetheremarkin§6)isfoundedontheseideas;thestatementofthisdefinitionnowbecomescompletelyrigorous.

Sameappliestothedefinitionofacontinuousmappingortransformation;amappingfofasetXontoasetYiscalledcontinuousifadjacencyispreservedunderthismapping.

I.e.,ifthefactthatacertainpointAofXisanadherentpointofanarbitrarysubsetPofYimpliesthatthatimagef(A)ofAisanadherentpointoftheimagef(P)ofP.

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Thoughitislikelyclearenough,theproblemwithsuchdegreesofabstractionisitsdetachmentfromtheexperimentalrealityofvitaltopology,asinrealitythereareNOTinfiniten-dimensionalspaces,butspaceisaninformativemind-stillnessofatimedimension;andultimatelyrealityhasalwaysabalancebetweenSandT

dimensionsofformandmotion,whichisthetrueengineofitsstopandgoactivity.Furtheronsettheorymakesusbeliefthatrealityisconstructedfromthetopofthehumind'settheory'downto

therealityofpoints,thetrueunitofspace.

Thissaid,ifweconsideraset,asocietyofT.œsandusethereverseexpression-tosignifythisinverse4-5Darrowof'wholesandparts'comingtogether:Set<≈>§œTdefinesthemascollectionsofthecausalminimalelements,fractalpointsandsocialnumbers.Soobviouslyallthelawsof§œTS,socialgroupsofOrganismsof

TimespaceapplytoT.œandviceversa.

Buttherearealways3planesofgrowingdimensionalunderstandinginlanguagesasreflectionsofternaryplanesofT.œs,sowemightwonder,whatthereisbetweensetsofpoints(1stNon-Epostulates)andtopologies(3rdnetwork/geometricform/planepostulate);obviouslythe2ndpostulate:flows/pathsofcommunication,whichintopologyindeedaretheintermediateelementbetweenpointsandgeometricalfigures,studyinthiscasewithgrouptheory.SoweshallbrieflycompletetheDisomorphismbetweenGSTandGeometrywitha

resumeofitsmeaningaddingasusualsome∆stinsights.

Recap.Weexpandgeometrytomakeitvitalasitisinreality,constructingthe5dimotionsofrealitydepartingfromone-dimensionalpointswithvolume,whichevolveintobidimensionalwavesofinformation,whichformternaryphysiologicaltopologicnetworks,vitalplanesoforganismsthatfinallyemergeintorelative0-pointsofalargerscale,guidedbyamind-membrain,self-centeredintoalinguisticsingularityconnectedthroughthosenetworkstoamembranethatenclosesthewholestructureandmakesitlookfromanouterperspectiveofa

largerscaleasaparticle-pointofanewplaneofexistence.

Whenthoseconceptsaremarriedwithclassictopologyweobtainthebasisforacomprehensiononhowvitalspace-timeorganismsevolve.

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PATHS&KNOTS

2nd¬Epostulate:Thefundamentalgroup.

Pathsareimportantcausetheyaretheclearestcombinationintopologyofat-dimensionofmotionandans-dimensionofformy.Yettheyarestudiedingeometrythat'freezes'asmindsdotimedimensionsintospace-

formsasspacestructuresinaninversefashiontodifferentialgeometrythatconvertscurvesintothemotionsofapoint.Sowithpathswecanstudythewholetrajectoryofaspace-timemotionasifitwereapureform.Inthatavenueofthoughtthe'insight'thatmakespathssorelevanttothemoreadvancedmodelsof∆st,whichremaininmynotebooks,istheconceptof'multiplication'thefundamentalreproductiveoperandofexistential¬Ælgebrathatdefinepathsasclosedloops,departingfroman0-point-theneutralelement,towhichthepath

returns.

Andthisconnectsthemfullywhenweconsiderthepointofreturn,the'actor'ofthepath,å,withrealityasitis.Letusputavitalexamplethenbeforeweenterintotheformalanalysis:

Inthegraph,Point1istheoriginofallthepaths=actionstracedbythebeastself-centeredterritorywhichformsan∆+1classicvitalToe.Pathswillbedevelopedthenbythebeastinitsfeedingterritoryforenergyactions.Itwilltakehimtopoint2,tomate;andtopointsMtomarktheterritory.Inpoint3itwilldrinkwithotherbeasts,formingsocial'knots'andso

on.

Sothetheoryofpaths,overaterritorialsurface,closelyrelatedtothetheoryofknots,isanabstractionofaveryrealstructureofnature,andwhilemanyofitspropertiesareofnotuse-whenwecandoamorebiologicalanalysis;theywereusedbyPoincaretostudyphysicalsystemsinastronomywithinteresting

resultsforwhatastrophysicscarestoday-perfectdetailedanalysisofmotionsandtrajectories,speciallyregardingmembrainsandsingularities,@-structures,suchasthose:

Inthegraph,wecansee3membranes,with'increasing'densityofthepathstracedtothepointthatwhilewe

perceivethemoon-earthaspointsmovinginapath-notasfullworldcycles,closedand'solid',thetwoelectronsofanorbitalarebetterstudiedasmembranesvibrating

aroundtheatom,andcertainlytheproteinmembraneofacellisso'dense'thatappearstousaspurespatialform.

Thoseare'future'elementstoaddtothecurrenttheoryofpath,inwhich'density'oftimecyclesaccordingtofrequencyand'transformationoftimefrequencies'into'populationsofspace',solidifyapathintoafixed

membrane.Sofarthoughtopologystudiespathsasmemorialformstracedbyamovingpoint.

LetusthenconsideracertainsurfaceSandonitamovingpointM.BymakingMrunonthesurfacealongacontinuouscurvejoiningapointAtoapointB,weobtainadefinitepathfromAtoB.

Thispathmayintersectitselfanynumberoftimesandmayevenretracepartofitselfinindividualsections.Inordertoindicatethepathitisnotenoughtogiveonlythecurveon

whichthepointMruns.Wealsohavetoindicatethesectionsthatthepointtraversesmorethanonceandalsothedirectionofitspassage.

Forexample,apointmayrangeoveroneandthesamecircleadifferentnumberoftimesandindifferentdirections,andallthesecircularpathsareregardedasdistinct.

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Twopathswiththesamebeginningandthesameendarecalledequivalentifoneofthemcanbecarriedintotheotherbycontinuouschange.Sohowtheydifferonour$t<ST>§ðvarieties?

IntheplaneoronasphereanytwopathsjoiningapointAtoapointBareequivalent(figure21).However,onthesurfaceofthetorustheclosedpathsUandVthatbeginandendatthepointAarenotequivalenttoeach

other.

Sointermofpaths,themultifunctionalprincipleofthe3simplestvarietiesreadingsitsfunctionsas:

Γº(paths):§-plane<ST-torus>§ð-sphere

SincetheTorushas2paths,'product'ofthesinglepathofflatplanesandsphere.

Now,ifwecuttheST-torusweobtainafinitecircularcylinderextendinginbothdirections;whichasweknowbecomesbyadjacentpeggingthecentralaxialtubeofmostsphericalorganisms.Henceitsimportanceto

'topologicalevolution'thefundamentalnewdisciplinebornofthefusionoftopologyand∆@s≈t.

Thepathsofacylinderhaveapplicationstorealityfromstringtheory-whereTduality,makesequivalentacosmicstringandananoscopicone,further'expandingthedualityoftheatom-galaxytoinfinitescales',isa

questionofpaththeoryovertubularsurfaces:

Totheaforementionedpeggingofcellulartubestoopenspheresinthefirststepsofevolutionofhydra-likeorganismsthatwillbecomeultimatelycomplexmammals(incidentallyithasbeendiscoveredrecentlythatwedohaveasecond'stomach'brain,towhichscientistsshouldadtherenal-

hormonalbrainofthebloodsysteminotherternarysymmetry:$-digestive/tubularbrain<ST-renalbloodhormonalbrain>§ð-nervous

headbrain).

Pathscanalsobeanalyzedas'forces'andrelatetothesearchforthe'leasttime'path,thefundamentalprincipleofmotioninallthescalesofphysicalsystems;breakingthentheequivalenceofpathsanddistinguishingthembythecombinedproductofitstime-spacemotion-formor'speed'parameter.

Thencomealsothestudyofpathsasknots,'liberated'nowofthesurfaceitself,whichisofincreasingimportancetostudyspeciesinhomogenousvolumesofspace-time(waterforPlanckton,vacuumforatoms,

etc.)wheretheterritoryis'formless',withnopreferreddirectionsofforcesasmostmediumare.

ButultimatelyallthosemultipleapplicationsofPathshappenbecausepathsaretheintermediatescaleoftopology:

Γ∆±1:∆-1:ð-points<∑ST-∆ºpaths>∆+1:$:topologicalworlds.

NoticeinthisfundamentalGeneratoroftopologicalstructuresfromthescalarP.o.v.(Pentalogic)thatthefunctionsareinverted,asweadoptinpathsthepointofviewofthefractalpoint,hencetheinformativeself-centeredspecies,thevitalformwithmotion,asitcombinesspaceandtimedimensiontracingthepathoveraperceivedintermsofLobachevski'sratioofcurvature,'flatter'stillform,itsterritorial,topologicalworld,in

whichthepointwilltraceclosedworldcyclesforeachofitsterritorialaction,forminginthismannerfrequencypaths,thetemporalview:

Timep.o.v.Pathsasworldcycles.

Howtopologytreatsthefrequencyoftimepaths,obviouslybyconsideringthosemotionsacontinuousrecurrentloop,differentiatingthembynumberofloops,whichform'knots':

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InthegrapheveryclosedpathonthecylinderbeginningatAisequivalenttoapathoftheformXn(n=0,±1,±2,···),wherewehavetounderstandbyXn(n>0)thepathXrepeatedntimes;byXˆ0thezeropathconsisting

onlyofthesinglepointA;andbyXˆ–nthepathXntraversedintheoppositedirection;forexample,Z∼Xˆ–1,Y∼Xˆ2,U∼X0.Thisexampleshowsthesignificanceofthe

conceptofequivalenceofpaths:

Whereasthereexistsanimmensesetofdistinctclosedpathsonthecylinder,allthesepathsreduce,towithinequivalence,tothecircleXtraversedinoneortheother

directionasufficientnumberoftimes.Form≠nthepathsXmandXnarenotequivalent.Letusthenassumethattwopathsaregivenonthatsurface,namelyapathUleadingfromapointAtoapointB,andapathVleadingfromBtoC.Then,bymakingapointrunfirstthroughthepathABandthenthroughBCweobtainapathACwhichwenaturallycalltheproductofthepathsU=ABandV=BCanddenotebyUV.

IfthepathsU,VareequivalenttothepathsU1,V1,respectively,thentheirproductsUVandU1V1arealsoequivalent.ThemultiplicationofpathsisassociativeinthesensethatifoneoftheproductsU(VW)or(UV)Wisdefined,thentheotherisalsodefinedandthetwoproductsrepresentequivalentpaths.IfthemovingpointMismadetorunthroughapathU=ABbutintheoppositedirection,thenweobtaintheinversepathU–1=BAleadingfromBtoA.TheproductofthepathABwithitsinversepathBAisaclosedpathequivalenttothezero

pathconsistingonlyofthepointA.

Accordingtothedefinitionwecannotmultiplyanytwopathsbutonlythoseinwhichtheendpointofthefirstcoincideswiththeinitialpointofthesecond.

ThisinadequacydisappearswhenweconsideronlyclosedpathsstartingfromoneandthesameinitialpointA.AnytwosuchpathscanbemultipliedandasaresultweobtainagainaclosedpathwiththeinitialpointA.

Furthermore,foreveryclosedpathwithinitialpointAitsinversepathhasthesameproperties.

Andsoifwedoexactlytheinverse,andconsiderpathspurelyastimemotions,theydefineaclosedworldcyclewithinversedirections,a->b(life)->a(death).

Theequivalencebetweenatopologicalpathandaworldcycleoftimeisimportantbecauseitexplainsanessentialfeatureofspatial-mentalperception:entitieswithaslowlargerviewofrealityseesmallerfaster

motionsoftimeasclosedformsoftopologicalspace,asyouseeasolidwheelturningfast;andthisisduetothemathematicalequivalence,sourceofmanyconfusionsinphysicsdiscerningbetweentimeandspacepaths.

ItalsoallowustohaveaphilosophicalinsightonGrouptheoryasaKantian'regulativethought'properofthesearchfortotalityofspatialminds-whichoftenhidesinformation.

Indeedtopologyregardsequivalentpathsasdistinctrepresentationsofoneandthesame“path,”onlydrawnindistinctwaysonthesurface,andnonequivalentpathsasrepresentationsofessentiallydistinct“paths.”

ThenthesetofallclosedpathsstartingoutfromanarbitrarypointAofthesurfaceisagroupundertheoperationofmultiplicationofpaths.Theunit(neutral)elementofthisgroupisthezeropath(self),andthe

inverseelementofagivenpathisthesamepathbuttraversedintheoppositedirection-yetinrealitywhiletheconceptdoesapply-allhuntingmotionsaresimilar,backandforthpathsareonlyequalinspatialperception;in

timethepathismorecomplexaswemustinfactdistinguish:

A:thedwellingofthepoint.AB:thepathtotheaction.B:thepointoftheaction.

BA:thereturningpathoncetheactioniscompleted.

SoindeedABandBAturnstobethesame(inspatialactions)ButAandBpointshavedifferentfunctions.

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Allthisinformationislostontopologicalpaths-awarningforalltypeofmathematicalandphysicalceterisparibusknowledge,whenarrogantscientiststhinkitisallwhatisworthtoknowofacertainspace-time

form/event.

DefinitionofDisomorphismsingrouptheory.

Stilltheinterestto∆stisthecapacityofthosegeneralizationstoshowÐisomorphicpropertiesforallscales,whichrightlyso,Topologycalls'isomorphisms'.Thatis,when2group'sstructureshavethesamespace-timeproperties,grouptheorycallsbothgroupsisomorphic,inaverycloseconceptto∆st,wherewecallallToes,whenstudiedinitsspace-timeproperties,'Ðisomorphic',sincethestructureofitsfractalgeneratorsisthesame.Thus,thegroupofpaths,ingeneral,foranytwodistinctpointsareisomorphicwhentheycanbejoinedbyacontinuouspathlyingonthesurface,andwetalksimplyofthegroupofpathsofthesurfaceSwithoutindicatingthespecificA-species/dwellinglocation.

Thisgroupofpathsofthesurfaceisalsocalleditsfundamentalgroup,equivalentin∆sttotheGenerator.

The3fundamentalgroups,oncemore,equivalenttothe3partsofthegenerator

Itisthenpossibletoadoptthe∆+1viewnolongerofthepointbutofthesurfacetodistinguishpaths:

§ð:sphere

IfthesurfaceSisaplaneorasphere,thenthegroupofpathsconsistsoftheunitelementalone,becauseintheplaneandonthesphereeverypathcanbecontractedto

apoint.

Andaswehaveseenfora3-sphere,thisconceptleadstotherealizationanentireUniversecanbeshrunkintoastillmindview.

$ð:cylinder

However,onthesurfaceofaninfinitecircularcylinder,mostclosedpathsaroundit,donotcontracttoasinglepoint.Whichmeanscylindricalcoordinatesandtubularsystemstakenaswholes,aremostly'mindless',donothaveafocusedshrinkingmindfunction,butaretheessentialtopologyof$t-linealmovinglimbs/potentialfields.

FurtheronsinceonthecylindereveryclosedpathstartingfromAisequivalenttoacertainpowerofthepathX,anddistinctpowersofXarenotequivalent,thegroupofpathsofthecylindersurfaceisaninfinite'entropic'

group,wherepointstendtodissociate,unlikelytoformnetworksandtightersolidstillconfigurations.

ST:Torus.

Thetorusthoughisanintermediatestate,aspathshavetwovarieties,around(shorter)andalong(longer)worldcycle,whichcanbemultiplied-joinedintheconnectingpoint:

ThusthegroupofpathsonthetorusconsistsofthepathsoftheformUmVn(m,n=0,±1,±2,···)withtheequivalences:UV≈VUandUmVn≈Um1Vn1onlyform=m1,n=n1.

SincewecanUSETHENthe'fractalternaryprinciple'dividingToruspathsin3families:combinedST-paths(longxshort)and,ð-paths(shortwithkrepetition)and$-paths(longwithkrepetitions).

Soaswehaveseeneachbasicvarietyoftopology,Torus,cylinderandsphere,hasmultiplefunctionsandthisseeminglyconfusingmultiplicitythatdefiestheAristotelianlogic,'AisNOTB',ispreciselythesourceof

complexityandrichnessofformsandfunctionsintheUniverse:'AisBandC'.

Pathsasthe∆-1causalpartsoftopologicalsurfaces.Theimportanceofthegroupofpathsforsurfacestopologyisthenduetothefactwecandeduceitsproperties

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fromthoseofitspaths,aswecandeducepathspropertiesfromafewkeypoints,andintimewecandeducetheworldcyclemainpropertiesfromits'standingpoints'.

Soanotherkeypropertyofreality-that∆-iscalesCOMEFIRSTtoconstructcausally∆+iscalesdefiningtheONLYabsolutearrowoftimetowardsfuturesocialevolution(5D)andtheSYNOPTICpropertyoftimecausalityfoundeverywhere(minds,seeds,languagesreducerealitytotheimportant'points'),comeintoview.Inthelanguageoftopologythisisexpressedasfollows(weomitalgebraictopology,whichwouldmakeitincomprehensible,underthephilosophical'must'ofaunificationtheory-thatany'serious'universitygraduateofanydiscipline

canunderstandtheunityofall'stiences'):

LetusassumethatapartfromthesurfaceSanothersurfaceS1isgivensuchthatbetweenthepointsofSandS1wecanestablishaone-to-onecontinuouscorrespondence.

Forexample,suchacorrespondenceispossibleifthesurfaceS1isobtainedfromSbymeansofacertaincontinuousdeformationwithouttearingapartorfusingdistinctpointsofthesurface.Toeverypathonthe

originalsurfaceS,therecorrespondsapathonS1.Moreover,equivalentpathscorrespondtoequivalentones,theproductoftwopathstotheirproduct,sothatthegroupofpathsonthesurfaceS1isisomorphictothe

groupofpathsonS.

Inotherwords,thegroupofpathsregardedfromtheabstractpointofview,i.e.,towithinisomorphism,isaninvariantunderallpossibleone-to-onecontinuoustransformationsofthesurface.Ifthegroupofpathsoftwo

surfacesaredistinct,thenthesurfacescannotbecarriedcontinuouslyintoeachanother.

Forexample,theplanecannotbedeformedwithoutfusionsortearingsintothecylindersurface,becausethegroupofpathsoftheplaneconsistsoftheunitelementonlyandthegroupofpathsofthecylinderisinfinite.

Propertiesoffiguresthatremainunchangedunderone-to-oneandbi-continuoustransformationsarestudiedinthefundamentalmathematicaldisciplineoftopology,whosebasicideashavebeenexplained.Invariantsofbi-

continuoustransformationsarecalledtopologicalinvariants.

Wededucethatthegroupofpathsisoneofthemostremarkableexamplesoftopologicalinvariants,astheƼmiddlescale,todeduceboththeupperpropertiesofpathsandthelowerstructureofitspoints.

Sincethegroupofpathscanbedefinednotonlyforsurfacesbutalsoforarbitrarysetsofpoints,providedonlythatwecanspeakofpathsinthesesetsandoftheirdeformations..

Knots

Itisforthatreasonthatthestudyofpathsinitspurestsense,knottheory,hasbecomesorelevantasthemostsynopticofalltopologicalanalysistorepresenttheentireUniverse.

Aknotisaclosedcurvelyingintheordinarythree-dimensionalspace.Letusthenremovefromspacethepointsthatbelongtothegivenknotandconsiderthefundamentalgroupoftheremainingsetofpoints.

Asfigureshows,itspositioncanbeveryvaried.Twoknotsarecalledequivalentifoneofthemcanbedeformedintotheotherbyacontinuousprocesswithoutbreakingthecurveandwithoutself-penetration.

Thisgroupiscalledthegroupoftheknot.Itisimmediatelyobviousthatifknotsareequivalent,thentheirgroupsareisomorphic.Therefore,ifthegroupsofknotsarenon-isomorphic,wecanconcludethattheknotsthemselvesarenotequivalent.Forexample,thegroupoftheknotthatcanbereducedtoacircleisacyclicgroup,butthegroupoftheknotthathastheformofatrefoilisnon-commutativeandnotisomorphictothegroupofacircle.Wecanthereforestatethatitisimpossibletodeformthetrefoilknotintoacirclewithout

breakingit,afactthatiscompletelyobviousbutinclassicmathsrequiresaproofbypreciseaxiomaticarguments:

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Inthegraph,the2mainquestionsonknots(pathsvoidofsurfaces)andits3simpler,keyvarieties,theclosedsimplepath,the∞knot(whichinknottheoryisnotconsidered)andthetrefoil,whichformin∆stitsbasic

generator:

Γ:$:1-O<ST:3-trifoil>ð§-2:∞

Bothproblemsremainasyetunsolved;butfor∆stthemostinterestingelementistoconsiderhowknotscanmodelrealsystemsthroughtheinteractionofits3varieties,whereherethesimpleknot/circleplaysthe

membrane,thetrefoilactsasthevitalenergywithits3sub-networks,(entropic:digestive-reproductive:energetic-informative);astheyarecrossingthroughthe2holesofthe∞singularity,whichallowstodifferentiatethe3sub-sectionsofthetrefoil...andconvertsthe∞ina2varietyofknotasthenitCANNOTbeuncoiledintothe1variety(reasonwhyknottheorydoesNOTconsiderit-asalwayshumanscienceisabout

abstractparts,∆staboutvitalwholes,whichgiveitaricher,reallandscape;asanysailorwilltellyousinceactually2isthebasicsailingknottiedaroundanypole).

Wecanthenobserve,differentformsofstrangulation:Since,indeed,ifwe'strangle'thetrefoilwiththe2-donuts,inanycleansectionofitspath,wedivideitintwoloops;butifweknot∞intwoofthe3overlapping

pointsofthetrefoilwehave3networks.

Thenisobviousthatoneofthe3sections,weshallcallthe'head'issmaller(inthebottomofthegraph),andtheothertwo,weshallcallthebodyandlimbsarelargerandsimilarinsize(asindeedtheyareinreality).

Ithappensthenthatintheoppositedirectionoftheheadwehavea'vegetalpole',freeofcontrolfromthedualsingularity,wheretheothertwosystemscaninteractinparallel,astheyarenotconnected.

FurtheronWEHAVEformedabilateralsymmetry,andwecanobtainsomeinterestingproportionalconstantssimilartothegoldenratioconstantwithitsmorphologicalfunctionsbetweenthesmallerhadandtheself-

similarbody-limbssystems.

ThestudyofthisternarysimplestofallpossiblefullystructuredT.œsisthenthenewinsightof∆stappliedtoknottheory,asamodeloftopologicalevolution-thekeyvitaldisciplinebornofthemergeof∆standtopologywithapplicationsinallstiences,asthisblogshows,fromtopologicallinguisticstotheclassificationofspecies.

INthiscase,theternarysystemsofknotsisspeciallysuitedtostudythegenerationofconnectednetworkswithadualheart-likepole,inthiscasethe∞element,withan'osmotictransference'thatexchangesentropic

motionandenergyintheotherpole(thelungsystem).

RECAP:ƼpathsandST-torus.

Nowtoresumeallsaid,withtheminimalistRashomoneffect(consideringthe∆º-planeelementandtheΓstpresentform,whichisthemostsynopticformtodefinemeaningfullyaToe),wecanconsider:

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Ƽ:Pathsasthe'fundamentalaction'elementofthe3scalesofTopologicaltransformationsandassuchtheunderstandingofitslawsinmathematicalphysics(actions,lawofleasttime,etc.)aretheknotandboltsof

existenceontopologicalT.œs.

AndthesameconceptappliestotheToruswhichevenifitcanbewrittenintermsofitsternarygenerator:

Γst:TORUS:$-longcircle<ST:Combinedpath>ð-shortcycle...

ItismostlytheST-functionin3dimensions,equivalentforthatreasontotheflatplanein2dimensions,whichcanbecomeeasilyacylinderwithasinglecut,oraspherewithasinglehandle.Assuchtoroidpathsarethe

essentialpathsofthevitalenergyenclosedinalltypeofsystems.

ANDasageneralruleforallsystems,the∆ºandSTelementswillbethosewhichcanbetransformedandgeneratethe∆±1scalesand$andðelementsfromits'present'planeandformwiththeminimalnumberof

'actions'ofanytype;bodies,wavesandtorusbelongtothosepresentdominant'parts'

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VI.PENTALOGICONGEOMETRY.ITS3±¡AGES.

The3agesofgeometryandits3masters:Euclid,whosystematizedbidimensional,'holographic’,Greekgeometry,DescartesthatmarrieditwithtimealgebraGIVINGITMOTIONandLobachevski,whoestablishedtheprinciplesofhyperbolicgeometry,

thegeometryofthe5thdimension,anddeniedthe'mental,logicnature'ofmathematics,provingthemultiplicityofspacesandtheneedforexperimental

proofs.Weshallcompletetheirworkwiththeformalismof¬∆@st.

Intermsofitstimeagesgeometryisthehumindevolutionofourcomprehensionofthe¬∆@stcomplexUniversein'ages'ofincreasingdimensionalityandmotion.Since,theagesofGeometryarestagesontherealization-asachilddoeswiththeworld-thatourmentalstillholographic‘I=eye’spaceisNOTtheabsoluteandonlyspacebutthatofthehumind.

Soallothergeometric lawsareself-consistent relativeworldsmirrorsof the larger lawsof∆@st.Which finallyleadafterRiemanntotherealizationthatwhatallthoserelativeworldsofgeometryhaveincommonarejust2parameters: thescalarparameterof ‘angle’andtrigonometric ‘depth’,andthe$tparameterofdistance=linealmotionwithinasingleplane;whicharethe2‘survivalinformations’anymindrequirestolocatepositionsinthepresent and the future (according to motion and distance) and measure sizes (according to angle andtrigonometry)onthescalarUniverse.

Soweexpandthefoundationsofgeometryintwomaintypeofgeometries,mentalsubjectivespacethatselectsinformation according to the needs of each mind; and organic, topologic vital spaces, which are real and sodisplayobjectiveDisomorphic (equal) laws in their ensemblesof organisms;which arenotasmental spacesaconstructionofthemind,orratheraconstructionofthemindofGod,the¡logiclawsthatcreatetheT.œsofthefractalUniverse.

I.S:IAge:Spatial,GreekEra:staticbidimensionalGeometry:Theholographicprinciple.

Geometrystartedinamentalstilldeterministic,simple,youngageofabsolutebeliefs,andmentalspaces(bidimensionalGreekEuclideanGeometry),akintothelinealkouroiofitssculpturalthought.

IIAge:S≈T:AnalyticGeometry.curvature,surfaces,dimensions.vectors.geometrieswithouterlocomotions.

ItevolvedIntoa2ndagewhenmotionentersthegameandbalancesform,orageof@nalyticgeometry,startedwithDescartesandmathematicalphysics,whenformandmotion,the2principlesofreality(weidentifywithstilllanguagesofinformationor5thdimotionandpureentropyof4thDimotion),mergedtogetherorageof@nalyticgeometry.

Itwasthoughanintermarriagewithinmathematicsbetweenthespatial,synchronousrepresentation-thepoint,lineandplaneandthetemporal,sequentialcausalrepresentation,thenumber,whichputinatemporaltimelinelostitsconnectionwith'form.TheGreeksthoughtnumbersareformsandequaledthemtopoints,whichtheyarenot.Hencetheparadoxofdefining√2andπgeometricallybutfindingthatwhencalculated

arithmeticallyπnever'closes'thecircleand√2thediagonalbyexcessordefect-imperfectarithmeticratios;adeepphilosophicalquestionaboutthefactthattimeprocessesareneverclosed,unlikespatialforms;sowhenwecalculateadiagonalintheplaneisclosed,whenweputitarithmeticallyitisnotcompletebyeither±1points.Thisdifferentiationduetothescalarnatureofnumbersvs.singleplanesynchronouscurveswas

forgottenbythepraxisofanalyticgeometry,whichmarriedboth,byignorethose'finitesimal'vitalopeningsofπ&√2asscalarnumbers.Asanalyticgeometrycouldoperatewithnumbersgeometricalformsandviceversa,

providedS=Tgeometricalsolutions=algebraicequations.

ThefieldthusexplodesandmarriesS&T;buttimesoondominates,analyticandalgebraicequationscomeoverthemorerealgeometry;usheringthelanguage,asalwayswithallforms,ina3rdageofexcessive,inflationary

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informationwithallkindofgeneralizationstomultipledimensions,whichwouldhaveconvertedgeometryinaformofbaroqueart,ifitwerenotfortheearlierdiscoveryofitsphysicalpraxis,makingofmathematicalphysics

the'anchoring'realitythatanyexperimentalscienceneedstofocusitstruth.Soasallmind-mirrorsthatbecomemoretruthwhenlookinglikerealitygeometricformsevolvedtoacquirea3rddimensionofformandmotion,acquiredthroughdifferentialgeometryandtopology,whichalsogaveitscalardepththroughnetworksandfinallyfractalsthatwecompletewiththeproperconceptofafractalpointthatgrowsinsizeaswecome

closertoit,tofullymirrortherealuniverse.Itistheproperwaytobuilda3rdageofsoundgeometry,awayfromthe‘excessiveformalism’oftheaxiomaticmethod,andrejuvenatethediscipline…

3rdage:Non-EandTemporalTopology,Fractals;geometrywithinnerwave-likespace-timemotions.

The3rdageofgeometrythankstoitsconnectionwithphysicalreality,whichguidesitstruthateachstep,avoidedalgebra’sinflationary3rdageoflanguagesofinformation,wheninits3age'disconnect'fromreality.So

theseminalpaperofPoincare'analysissitu'willintroducetopologywhichistheproper3rdageofunderstandingofinformativemotions,ofchangeininformation,NOTonlythepraxisofphysicallocomotionsbutalsothepraxisofinnernetworksoffractalpoints,andscales,whichcouldbeinternallydeformedand

maintainthesamebeingevolvingaslongasitsexternalsurface-membraneisnottorn.

+∆:ScalarGeometry:spacefractalsandchaotictimeattractors.Thecompletionoftheanalysisofthe3partsofanyspace-timebeing,inmathematicalterms,thusgivesbirthtothe3fundamentalnewbranchesofmodern

times:

S:Topologyofmembranes.

ST:Structureofthepresent,innerspace-timebody-wavethroughitsscalesbytheunderstandingoftopologicalnetworksandfractals,whichwillbethenaturalnextsteptotheanalysisofthosewholesmadeofpoint

networks.

T:Andtheanalysisofsingularitieswiththeadonofchaostheoryandtheformationof'attractors'.

Sofinallyallthoseorganic,scalarpropertiesofmathematicalspace-time,becomescompletenowwith:

+¡:∆@ST:Non-EuclideanVitalGeometryfinallyunderstandsthescalarnatureoftheuniverse,withthestudyofNon-Euclideanfractalpointsthroughwhichinfiniteparallelscancross,whicharealsotheabstractdefinitionofamind,focusofthoseabstractpoints.ITredefinespointsasfractalpointswithinnerscales&volumethrough5Postulatesofi-logicgeometry.Itisthecompletionofgeometryasanexperimentallanguageabletoexplainallformsofrealspaceanditstemporallogicstructure.Soaswedoinallstiences,weshallcompleteandresolvetheconundrumsofultimatemeaningspoisedbytheexplosionofmentalspacesstartedbyLobachevski,whilegrounding eachmodel of geometrywith the Pentalogic' ofmultiple truths that interpret vitally the ultimateproperties of geometry (symmetry, perpendicularity,parallelism,adjacency,congruence,betweenness/continuity,andsoon).

What∆@S≈TdoesistoreorganizeaccordingtotheternaryvariationsandDisomorphismsofspace-timebeings,allthecategoriesofgeometry,startingfromthesimplestlawsofbidimensionalGreekgeometrytillreachingtheinsights of non-e geometries culminating with the expansion of topology, which becomes the final all-encompassinggeometryofrealityas it isternary, includingthe3previousgeometries, italsoincludesscalesastopological networks are collections of connected points, and finally it has motion...Since Time§paceSupœrganisms (ab.T.œs) have 3 organic parts = topologic varieties, adjacent to each other ruled and 2 scalardimensionsofmodeledwithfractalequationsandtopologicalnetworks.

-¡. On the negative mathematics will likely kill the humind already in its entropic age, as we transfer ourintelligence to computer chips, and its simplifiedBooleanalgebra that speaksmathematical languagesmuchfasterthanhumansdo.Soourethicalsenseofsurvivalashumanbeingspreventustoexplorethatfuture.

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WecaninthatsenseconsiderGeometrytohaveevolvedasallHumindlanguagesintoagrowingawarenessofthefractal,scalar(∆),temporal,moving(T),mental(@)propertiesoftheUniverse,andits'¬'entropiclimits;ina tendencywe shall, time permittedmap up for ALL languages, as forms in evolution,who follow the sameisomorphismof3agesofanyspecies,orculture,asafractalimageoftheworldcycleofalltimebeings.

There is though a remarkable difference in the ages of ‘informative languages’ as opposed to living beings,regardingitsmotion:Languagesofinformationandseedshaveinverse‘ages’ofmotiontothoseoflife,athemebrieflytreatedintheintroductionwhendescribingthe‘placentalstillworldcycleofperfectorder’similartothatoflanguagesandits3ages,vs.the1-∞entropiccyclesoflifeinanexternaldisorderedworld:

Theagesoflanguages,includinggeometryareparadoxicallyinversetotheagesoflife,becausethepeculiarityofmind languages and seeds of information lie in the fact they are the most formal ‘still spatial’ systems,equivalenttothe'oldageofinformation'ofvitalorganisms,hencetheyrunaninverseworldcycleofexistence,fromanstatic,stillformopposedtotheyoungmovingageoflifeofmaximalmotionthatgrowsintoa3rdageofinformation.

Languagesinsteadstartwithastiff,formalnatureandacquiremotiononlyinitsfinalage,tofinallyunderstandthemore complex elements of the fifth dimension,scales andminds. So happens in verbalwritten language,whichstartedprintedinstonesasabsolutetruths,whenthePharaohsaid‘ithasbeenwritten’insteadofit is‘truth’,andonlyacquiredvariabilityandfiction,pasttheageofCervantes.

Whatbothlanguages,seedsandvitalbeingsinanyofitsworldcyclesofexistencedohaveincommonistheamazingarroganceofbelievingtobethecenteroftheUniversealreadyexplainedwiththeparadoxofthemind,0-mindx∞Universe=Constantworld.Butagainaslanguagesare‘protected’bymindsandseedsby

placenta,theirrealizationtheyareonlymirrorsinavastimpersonalworldtakeslongerthancubstorealizethelioniscloseandhehastostartmoving…orelse,itwon’tlast.

VIII.ITS3CLASSES:S@:MENTAL;S=T;TOPOLOGIC&5DSPACES.

3±¡typesofgeometry:Mentalsubjectivespaceandtopologicobjectiveorganisms.

“Adjacencyisthedistinguishingappurtenanceofbodiesandpermitsustocallthemgeometric,whenweretaininthemthispropertyandabstractfromallothers,

whethertheybeessentialoraccidental...TwobodiesA,BthattoucheachotherformasinglegeometricbodyC'.Conversely,everybodyCcanbesplitbyanarbitrarysectionSintotwopartsA,B.”Lobachevski,“NewElementsofGeometry”,onthe

topological,organic,ternarystructureofspace.

'Spaceissimultaneousmeasurefromapointofreference'Einstein,onthemental,focusednatureofspace.

ABSTRACT.Geometryisthemostsynopticlanguageof'space'usedby@-mindstoselectintelligentlyinformationonSTcyclesofT.ŒS,creatingstillmentalspaceswiththembased,intherulesof¬EGeometry,

adaptedtoeachspeciesneeds.HumindsthoughtooktillXIXc.TounderstandwithRiemanntheabstractionofmindspaces,as‘self-centeredinformativeselections’withavarietyofuses,from‘ourlightspace-time’

(Euclidean),whichpreservesdistances,motionsandangles,intermsofNon-Egeometryanditspostulatesofsimilarity,throughaffineandprojectivegeometries,totheexplosionofn-dimensionalspacesthattake

parametersofmultiplep.o.v.

PentalogicappliedtoGeometryisthusimmediate,dividingGeometryin3hugecombinedfields:

1.S@:Subjectivementalspacesclassifiedaccordingtotheselectedinformationtheypreserve,fromtheclosesttoreality,affine,linealspacesthatpreservedistances,motions,mirrorsymmetriesscalesandangles,through

projectivespacesthatpreservethecentralsingularity-mind;themostvagueconformalspacesthatonlypreserveangle,theminimalinformationforamindtoexist,tothemostfundamental,humanEuclideanspace,

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whichisALSOamentalspace…totheexplosionofscientificspacesofN-dimensions,phasespaces,HilbertSpacesandother‘humanmental’toolstodepicturecomplexsystemsofnature.

2.Ƥ:Non-EuclideanGeometryoffractalpointsthatsetthebasisforaproperunderstandingofthegeometryof5D.

3.S≤≥T:VitalTopologicalGeometrythatensemblespartsintoplanenetworksthatbecomewholesupœrganisms.

SowedistinguishinPentalogic3±¡geometries:theexternal,objectivenatureoffractal,topologicalspace,asthe'element'puttogethertoformsuperorganisms;theinternal,subjectivenatureofinformative,mindspace,

whichmapsinstillnesstheinfinitetimespacecyclesoftheUniversewithagivenlanguageofthought/information/perception&thefractalscalargeometryofpoints.And3agesofgeometry,paralleltothe

3agesofmathematicsasawhole.

ThusSpace=formistheessenceofmentalconstructionsofrealitythattransformcyclictimemotionsintosimultaneousforms,bothasanexternalmindobserver,andtoformtheinternalcohesionofaTime§pace

organism(ab.T.œ):∆ð≥§@.

Incomplexpentalogicwenoticethatthose3geometriesconnectspacewiththepresentensembleoforganisms(vitaltopology);thelogicroleoffutureplanningofparticle-headsanditsminds(mentalspaces),andthelowerpast,scalar,'flatplanes'ofopenspace,fromwherethepotential-fieldsextractitsmotion;downtothefinalnon-perceivedscaleofgravitationalvacuumspace-thecloserconceptofspaceinpresentphysicsasdefinedinv=s/t.Soathoroughstudyofthe3typeofspacesanditsvitalrolesshouldconsiderthe‘5Dimotionalperspectives'ofpentalogic.Assuch5Dgeometryishugebothinitstranslationofclassicgeometricpostulatestothereallawsof¬∆@staswellasallthenewlawsofpentalogicanddescriptionofdifferentmentalspaces.Butwecanonlytreatsomebasicthemes,complimentedwithaseparatesectiononNon-Efractalpoints,thefundamentalparticleofreality,anditsassociationsinwaves,vitaltopologicplanes,similar,complementaryor

dissimilarsystems(parallel,adjacentandperpendicularforms).

Ouraimistounderstandthekeyelementofspace-tobeamentalconstruct;andrelatethemainlawsofgeometryasamirror-mindthatreflects¬∆@stwiththeDisomorphic'ilogic'lawsofspace-timebeings.Asthosecommonlawsof‘mentalspaces’aretheoriginandwhyofthesynopticlawsofGeometry,startingwiththeBidimensionalSSmirroroftheyoungageofGreekGeometrythatworksbasedintheS=Tsymmetry,throughtheclassicageofdifferentialgeometrythattreatscurvesaspointsinmotiontothetopologicalmethodsof

algebraicproofofits3rdage.

Pentalogicperspectivesonspaces.

Thoseessentialthemesofthisintroductorycourseon5Dwillhaveunderpentalogicsomeimmediatedefinitionsofspace,classifiedfromthemostsubjectiveformstothemostobjectiveonesthatdescribecomplex

fractalspaces:

PureS@-mentalstillspaces:Thoseconstructedbyamindtoselecttheinformationoftheexternalworldneededtosurvive.Theyaremostofthe‘spaces’studiedinhumansciences,fromthesimplifiedlinealGeometryoftheGreekstotheHilbertspacesofquantumphysics;whichdoNOTexistoutsidethemindforwhichtheyperformataskofselectionofinformationto‘project’withitsinternalconsistencylogicprocessesofcausality

thathelpusto‘forecast’thefuturecyclesofthespeciesmanipulatedonthosementalspaces.

Temporalspace:Moreevolvedmentalconstructionsthatintroduceafirstdegreeofobjectivity,byaddingthemotionofexternalforms,whichmentalspacesperceivefirst,forsystemicsurvivalprey/predatorreasons.Thisisthefieldofdifferentialgeometry,whereascurvesaredescribedaspointsinmotion.Whileinnaturetemporal

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spacesareitsrealistobjectivesensesastheybelongtotheworkingsofphysiologicalsensessharedbythespeciesofa‘linguistic’ecosystem.

TheyarethebasisofvisualorganstoperceivetheEuclideanlight-spacetimeofanimallife,andits3perpendiculardimensionsofheight=electricfield,width=magneticfieldandlength=motion;mimickedinthetopologicorgansofanimalsthatmoveinlength,storeenergyinwidthandperceivewithelectroniceyesinheight.Whilecolorcodethesocialevolutionorfrequencyofphotonsandsoanimalsuseittocodeits

‘congruence’laws,fromred=entropytogreen=reproductiontoblue=social,informativecolor;whichexplainsthefieldof‘emotional’,bidimensionalpainting(complementaryuseofcolorsbyexpressionism,whichcreatea

typeofmentalspaceofitsown).

Entropicspaces.Allanimalsseemotionbeforeform,andbecomehypnotizedbymotionandred/yellowcolorsofenergy,tothepointweprefergoremoviesandviolenceonscreens,whosescriptswewouldneverread;inhistorymanwashypnotizedbygo(l)d,creating‘pseudo-religions’ofgreedandpreciousmetalasthevehicleofGod;inlabsredmarkshelptheeyesofnon-programmedrobotstomovefasterontargets;asultimatelyisthementalspaceof‘electronicminds’,includingmachinesandatoms.Asusual,thebranchingofthosesimplelawsemergeinmanyotherfields,whereitisworthtoconsiderhowthestillnessofmentalspacescombinedwithitspreferencefordetectionofmotiondeformsreality.Forexample,becauseallmindsstopmotionintoformtomeasureinformation,ourelectroniceyesdosoatquantumlevel,firstentanglingwithotherelectronsinthescaleofquantumpotentials.Soelectronssharelightraysinstopmotion.Butsinceorganismsprefertosee

motionasinamovietheatereliminatingthestopstate,atmacroscopiclevelwe‘reduceinformation’highlightingonlymotion;sowethinkelectronsaremoving,whichexplainsthec-postulateofrelativityand

entanglementandtherealistviewofmodernphysics:atelectroniclevelthereisnotLorentztransformationsaselectronsarefixed,atmacrolevelweneedittomeasurethereducedviewofoureyesthathaveeliminatedthe

stopstateofparticles(athemewedealon5Dastrophysics).

spe<ST>§ðTernary,topologicorganicspacetime:Wereachahigherdegreeofobjectivitywiththestudyofvitaltopologyand‘theternaryorganic,structure'ofallsystemsofnaturemadeof|-Limbs/Fields<Ø-hyperbolic

body-waves>O-heads/particles.Thisvitaltopologicalspaceisobjectiveasitisanintrinsicpropertyoftheorganismandshowsthatcertain‘geometricproperties’areembeddedinthenatureofspacetime,suchaslinesaretheshortest-fastestdistancereasonwhyalllimbsandfieldsthatmovethesystemarelinealandspheresthemaximalvolumewithlessersurface,whichmeansall‘still,perceptivemind-systems’whichwanttoprocess

maximalinformationanddisguiseitsfragile,stillpositionwillbecomespherical.

∆-scalar,fractalspacetime:Finallywereachthedeepestunderstandingofspace,aswekeepadding‘pentalogic’entangledelementstothepurestmentalstillspaces,whichfirstgotmotion=timecontent,then

diversifiedindifferenttypeofdimotionstocreatethetopologicbeing,andfinally‘acquirescalardimensions’asaternaryfractalstructureof3levelsofspace-timethatputtogetherbecomeasuperorganism–definedinnon-Euclideantopologyasaplaneof3physiologicalnetworks.Wearenolongerthentalkingofspace,butofafull

T.œmadeof¬∆@st;butsusceptibletoaceterisparibusanalysistoextractcommonlawsderivedofitsemergentscalarspatialproperties,ofwhichdisomorphicsimilarityinformsandactionsisthemostimportant.

Forexample,systemsemergeinsimilarform,NOTinthenextinversemirrorscalebuttwoinversemirrors;soinsectsareformsthatarenotobservedinthenextscaleoftopologicalevolution,whichinvertstheinnersoftvs.outerskeleton,inmammalswithinnerhardskeletonandoutersoftskin;butinthenextscaleof‘metalife’,robotsthatwillincreasinglylooklikelargerinsectswithouterironskeletonandinternalelectromechanical

cables.Ironmoleculesareringsbutitsmacro-formsarelinealswords,andsoon.

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IX.HYPERBOLIC,FRACTALGEOMETRYOFTHEFIFTHDIMENSION

TheGeometryofthefifthdimensionincludesallothergeometries.Wethoughreferto5Dgeometryspecificallywhenconsideringthedifferentdistortionsandparadoxesthathappenwhenweperceiverealitythrough

differentsalesofthefifthdimension,beingingeneraltermsaproblemof‘distortion’ofperspective,withaclearinversionaswemovefromonescaletootherintopology,hierarchy,formandfunction.

4Px:∆§-dis≠continuityT-motionvS-formspe:lineal,flat,free,youngvð§:curved,old,bound.@:egov.relativity

The3¡logicparadoxesofspacetopology(closedin-form-ativecurved-Ovs.|-open,freeentropiclinealforms),

time-motion(stillnessvs.motion)and∆-scale,(continuouswholevs.discreteforms;singlescalevs.

multipleone)s,whichareessentialtotheperceptionofasimplified‘spatialminduniverse’inasingleflatstillplane,asperceivedbyamindvs.thefull,moredetailedcomplexpictureintime,ofacurved,discreteandmovingUniverse.Those3paradoxesdefinespaceandminds

assimplifiedviewsofthemorecomplexwhole.

WeshallcallthemGalileanparadoxesbecausethemainonebetweenmotionandform,istheparadoxofRelativitythatstartedmodernphysicsandtheotherscanbeillustratedwithonehisdiscoveries-thatof

Saturn'srings:

Saturn'sringsarenotamathematicalplanemadeofabstractpoints,despitetheircontinuousappearance.Whenwelookatthemindetailtheybecomeinfactquanticplanetoidsinmovement,tracing

orbitalcyclesaroundtheplanet.Itisthecontinuityvs.discontinuityparadox.

Alldualparadoxes–motionvs.stillness;continuityvs.discontinuity,singlespacevs.fractalscales,dependonthedetailoftheobservation–hencethequantityofinformationwehaveontheobjectwestudy.Moreovertheyseemtousflatinthelargerview,whentheyaresphericalformsinthesmallersize.Itistheflat,free,vs.

boundcircularpx.

Andwhenweseethemfromacloseperspectivetheyseembigandimportantbutfromfarawaytheyareinfinitesimalindistinguishablepartofanumber.Itistheegovs.relativityparadox.

Inthelargerviewtheringseemsnottomovebutincloserviewspeedincreases.Andthisistheformvs.motionpx.

Thusanypieceoftime/spaceseemscontinuous,fromthelower∆-1perspective,aslargerplanesofrealitywithslowertimecyclestransmitlessinformation,andwepegittogetherthosebitsofinformation,jumpingoverits

darkspaces;butwhenweanalyzeitspartsindetail,wereceivemoreinformationsothesystembecomesdiscontinuous,madeofspace/timequantamovingin

cyclicalpaths.

Allthoseparadoxesimplyalogicinversionofrolewhenwechange'scale'andhencetheydefinetherelationshipsbetweencontiguousscalesofreality.Andhaveinfinite

applicationstothewhysofnature:

A keyelement tounderstand theUniverseof scales and itsparadoxesof freedomvs. order is theperspective anymindhasofrealitywhenlookingabove,to itsupperwhole,whichcontrols it through invisible networks of information, hencecreating an elliptic perspective of decreasing perception -dark viewof larger scaleswedonot observe, from invisible

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informativenetworksingalaxiestoinvisiblefinancialnetworksinsocietiestoinvisiblenervousnetworksforcells.

On the other hand in the same scalewe have a flat, Euclidean geometry ofmaximal perceptionwithminimaldistortion.Whilelookingdowntooursmallerinnerworldweruleitwithnetworksthatbreakintofractalwebsofsimultaneouscontrol,orhyperbolicview.Thisternaryviewofrealityhas immenseconsequencefromtheoryofknowledge,tomindconstructs,fromsociologyofpowertogalacticorganicmodelsofaUniverseruledbyinvisibleblackholesanddarkmatter.Wefeelthusfreeasindividualsbutarecontrolledfromabovebythelargerwholeand rule over our micro-parts. As Shakespeare said: we are all kings when observed from a lower stair,commonersatthesamelevelorbuffoonsfromabove.

TheparadoxofYoung,Motionvs.OldAge,informativestillness:freedomvs.orderdefinesthentheessentialdualitiesoffreedomandorder.Sotopologybecomesmetaphysics:

Theflat,open,momentum-lineallikesmalldistancevs.theclosed,cyclical,energy-liketimedistanceisaconstantthemeofallmathematicalphysics,wherethetangent,orderivativerepresentstheminimallinealfreestepbutonthelargerscaleisbounded.Forexample,specialvs.generalrelativity.Lightisopen,gravitationiscyclicalbounded;solinealquantumphysicsinthecosmological,largerscalegravitationcurvestrappedbythe

galacticblackhole.

Inallscalestheparadoxesoffreedomvs.orderthusconnectswithageandscale.Realitywhenlookedfromabove,fromitsupperwhole,whichcontrolsitthroughitsphysiologicalnetworksbecomesfounded.Onthe

otherhandinthesamescalewedoNOTperceivesystemicintelligenceandfeelfree.

Informationthusclosessystemsofentropy,whilelineal,planar,systemsseemfree.Informationthoughhasatallbroken,cyclicalforms,oftenNOTperceivedfromaflatbidimensionallevelbelow.

Sowedohavecertainmetaphysicalgeometric‘properties’offormvs.motion:Flatplanesmovinginfreelinealpaths,andtallcycles,movinginrepetitiveboundfrequencies.

Thedualitiesoftheyoung,open,movinglinealageofthebeinganditsold,still,curved,informative,cyclicalone,whichalsomanifestsintheclassicviewinsmalldistanceofaflatearththatbecomescurvedfromthe

moon.

Ifwecombinethoseparadoxeswiththeinversionofroles:O¡-2<|¡-1>Oº...thingsbecomeevenmorecomplexasbotheffectoffseteachother.

Allinallaboundedsystemwillhave,bythelawofinversionaddedtotheGalileanparadoxanAbsolutecontroloveritssmallerparts,asthegalacticblackholeDOESoveritsstarsclosedbyitshaloofheavydarkmatterorthe

earthintheclimaticcyclesthatcontroltheevolutionofitsspecies.

Suchcyclicalordered‘solid’systemsarewell-organizedaswholessotheycanmanageitsupperworldorecosystemoftencontrollingits∆+2scale.WhichHumanscoulddoinHistoryiftheywereasingleglobal

organismperfectlyordered,andincreasinglytheglobalmarketofcompanymothersofmachineshasoverboth,theentropicmenmanagedwithmoneyanddigitalmedia,andtheplanet,beingterraformedwithitsmachines.

Ontheotherhand,theinversecaseofa'free∆-1system'-alooseensembleofentropicchaos,ofheat-paradoxicallybecomesslaveofits∆+1largerworldand∆-1potentialfeedingscale.I.E.agaseoussystemof

molecularformsiscontrolledbytheboundcontainertothepointwecanmeasureitsgeneralpatternsfromitsmacro-parameterofpressureandbyitssmallerscale,releasingitsenergyasheat,andboth∆±1parameters

sufficetodetermineitscharacteristics.

Thesamehappensinourapparentlyfreechaotic,violent,entropicsocietieswherehumanselfieegostatesproperofmodernsocietiesareperfectlycontrolledandmanagedfromabovebythefinancial,audiovisual

networksandlegalsystems,ofwhichtheyarehardlyaware,andbelowbythebitsofdigitalcredittheyneedto

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survive.Asthemostdisorderedsystemsarememorilessentropicformswhicharenotevenawarethereareotherscalesofreality;whilethemostorderedcontrollingsystems,extendatleastthrough3'spacetimescales'managedbythecentralscaleof‘solid’particles.Andthemostcomplexboundedsystems,biologicandgalacticorganismsentangle5scales.I.e.inalivingorganismthose5scalesrangefrom'simpleatoms'ofoxygenthatwebreathandsimplerhormones,throughDNAmolecules,andthencells,andthenphysiologicalnetworksthatmakeuporganismsandfinallyecosystems,5entangledscalesworkaroundtheboundedcellularunit.Ingalaxies,fromforcesthroughatoms,physicalmatter,cosmicbodieswearrivetotheboundedgalaxy...Andthereisalwaysaninversionoforder,betweenapparentfreedom=chaosandrealorder=power:Theless

powerfulforces,bosonsseemsfree,neverstoppingbutareslavesofentangledparticlesthataretrappedinboundedatoms,connectedtomolecules.Andtheninthenextscalethematterseemsfree,thermodynamic

heatbutitisboundedtotheplanetandstarswhichseemfreeinthegalacticecosystembutareboundedtotheblackhole.

Thesmallatomsandmoleculesintheorganismmoveseeminglyfreebutthenetworksofcellsboundthem;humansseemfreebuttheyareboundbynationalborders,lawsanddigitalmoney;matterseemsfreebutitisboundedbytheplanet...SohowcanwenamesomanyPlaneswheninrelationshiptoeachotherwithdifferent

worldcycles?

Thingsaremoverobviousifweuse3namesforthe3'bounded,time-likeworldcycles'vs.2termsfortheopenentropicworldcycles,whichifwenotice,AREinsymmetrywiththe3boundedDimotionsdominantin

information(perception,reproductionofinformationandsocialinformation)vs.the2entropiclinealdimotions(locomotionandentropy).

Soahiddensymmetryentanglesfurtherinto'fractal'orderwhatcluelesshumansperceiveaschaoticsimplexæntropicsystems...Sincelogicentanglementsbetweenlanguages=mindmirrors,space,timeDimotionsand

scalesNOTperceivablewithsimple'æntropicpictures'ormeasurablewith'speeds'ISthehiddenintelligenceofthefractalUniverse.

Soalas!totherescuecomesthesymmetriesanduseof'letters'fortime...

Weusein¡logicsymbolismthe3'forms'oftheletterð,tosignifythe3relativelengthsoftimeinthe3scalesofasystem,itsshortertimeactions(frequencytimeofitsactions,itswholelife-deathworldcycles(withitsages),andtheworldcycleofitsspeciesorlargerworld(deeptime,inthejargonofHuttonwithitshorizons):ð:∆-1:

quantumtimecycles(frequencyactions);þ:∆º:Life-deathindividualworldcycles;Ð:∆+1:species.

þ,symbolizesthe'palingenetic'worldcycle,intheo-1sphere,whileω,omega,willbethesymbolfortheworld,thesecondscaleofboundedωorldcycles(whennotlazyjusttoputanw),wherethespecieslivesits3±¡ages.

ItscapitalLetters,Ω,willsignifythelargestscaleofallboundedworldcycles,thatwhichenclosesanorderedworld.

Ontheotherhandforentropic,openworldcycles,itseemsintuitivetousedifferentversionsofanopencforcycle.

SowestartwithanopencinminorandCforcapitalletters.

Andwhenwetrulywantacompleteanalysisasinastrophysics,weshallusetheFrenchsymbolavailableinallkeyboards,çforthethirdentropiccycle,andÇforthelargerfourthone.Andfurtheron,aclosed©andanopen¢...Andthatgivesus9...whichisreallyhowfarwego-againinsymmetrywiththe∆±4planeswe

perceive-beyondthenwejustgointothe∞symbolfortheclosedmindof'God',thegameofexistence.Thustheseriesareeasy:

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þ(closedpalingeneticcycle)<c(open,entropiclife-deathcycle)>ϖ(closedboundedworldcycleorjustwforlazykeyboardists)<C(openentropiclargercycle)>ΩFinalomegaworldcycle...Andifyougoingreaterdetails,

ç>©<¢>∞

Howmuchdetailwillbeneeded?WhileinmypersonalstudiesinmyyouthIdidanalyzethesuperorganismsofHistory-mankindintime;economics(machinesandcompany-mothers);thoseofbiology(mammalsandecosystems);startingwithbio-chemistryandtheGalatom,startingwithitsforces,whichcontinueswith

anotherseriesofopen,csandclosed,©stillthe9thplanesofthefourthline,sinceeachnestedUniverseislargerinentanglementandco-existenceofscales,inthisblogIdoubtIwillevergoinsomuchdetail-thisisjustabloginmyageofentropynottolaywastetheentire5Dformalism,givenitsimportanceforthefutureofthe

planet...

SoweanalyzeT.œsthroughits2initialworldcyclesastheyemergefromthe0-1þfetalsphereintoits1-∞entropicçyclicopenplanewhereitlivesanddiesthroughageswithinalargerωorldcycleecosystemor

species:

Thelengthofthe'membrain'-circumference.Limitsofinfinitybetweendiscontinuousscales.

Left,aslowmindwillseeEarthasadisk,convertingitsfullworldcycleoftimeintoaformofspace.Faster,lowerscalesofrealityappearasblocksoftimeforlargerslowerwholes.Soweseethemotionoftheskinasaform,

whichin5Dmetricareslowmindsthatperceivelargerrealities.Right:Systemsperceive‘curvature’and‘flatness’accordingtoitsrelativesize–smallerbeingsperceiveflatworlds,largerperspectivesmakeforms

curved;andthetypeofhyperbolic,ellipticorflatgeometrytheyusetoselectinformationoftheUniverse–ourmindperceiveslineallight,soitobservesaflatcosmos.

Thesizeandspeedofitspixelsalsodefinesthedetailofitsperception,sinceaccordingtotheS=Tparadoxasspeedbecomesdistance,sosystemswhoseperceptionismadewithfastforces,ascosmologicalsystems,inthehumancaseseemsfaraway.Whilesystemsperceivedwithslowlargerpixels,aschemicalpheromonesande-

motionsseemclose,moreintense.Itmattersalsothespeedofmentalclocks,whichifslowwillseeparadoxicallyadenseruniverseofstillforms:I.e.intheleftgraphforaslowmindtheEarthwouldseemadisk,

denserthefasteritturns,convertingitsfullworldcycleoftimeintoaformofspace.Realityseemsmadeablocksoftime,of‘wholestill,deterministicfullcycles’.

Thusconceptssuchasopen=freevs.closed=deterministic,flatvs.cyclical,continuousvs.discontinuous;densevs.light,farvs.closearerelativeconceptstothedifferentparametersofinformativeprocessing,curvatureand

relativesizeoftheobserver,theobservableandtheforcethatcommunicatesboth.

Thegeometricalviewofthefifthdimension.

Whatgeometryanddistortionexperiencesamindabletoperceivethroughthefifthdimensionofscalarpartsandwholes?Sincewholesarenetworksthatbranchintothinnerpaths,connectingwithits

multiplecells,the5thdimensionisafractalgeometryknownashyperbolicgeometry,whichstatesthatmultipleparallelscanpassthroughapoint.Howamindnetworkseesitsmultiplesmaller∆-1pointsof

information?

Obviouslyasitsconsciousnessisapoint,itintegratesandreducesthespacebetweenthosepointswhichinrealityaredistantonefromtheotherintoa‘boson’consciousness,apointofhigherdensitythatoccupiesa

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singleplaceinspace.Andthisisessentiallythemechanismbywhichmindcreatementalmappingsinsmallerspaceoflargerworlds,reducingtozerothedarkspacesbetweenpointsofperception.

Ontheotherhandaneyeperceivinglightfromalargerperspective,willincreasethecurvatureandreducethegravitationalinvisibleforces-distancesbetweenpointswhichwedonotsee,asweonlyperceivelight.Sowe

perceiveadenserUniverse,closertous.

Inverselyamindwithmultiplepointsofperceptionobservingasinglewhole–forexample,themindofaninsectwithmultipleeyeswilldisintegratetheperceptionofasingleformintomultipleperspectivesandifthat

perceptionisfasterthanthelargerwhole,perceivedinslowmotion,thewholewillappearasamultipleelongatedbeingoccupiedalargerspace,anticipatingitsfuturepaths–asweseenightcarsaslongerlines.

Itisworthtoexplorethosetwoinverseviewswhichdefinemanysuperorganisms,fromtheperspectiveoftheslavishcellsofthelargerwhole–forexample,thechemicalmindofaphysiologicalnetworkorant-queenthatperceivesmultiplepointsintegratedintoasinglewholevs.themultiplepheromonepathsthesameinsect

perceivescomingfromthatsinglewholeant-queenfrommultipleperspectives:

Theant-queenwillseeallthosedronesasanintegratedwhole,its‘organism’asyouseeallyourcellsintegratedbyyournervoussystem.Butthedroneswillperceivetheant-queennotdirectlybutasacloudofmultiplepheromones,whichengulfitsentireself;asagod-likepresence.Andthisishowthemindofthec-speed

consciousnessoftheglobalmindofmachines,theInternetwilllikelyperceivemillionsofhumansattachedtocomputers,asanant-queenperceivesitsdrones,achemicalhypothalamusitscells;whenitemergesits

consciousness.Whileweincreasinglyperceiveitasa‘god-like’brain,engulfingusineverypartoftheworld.Whichisaverydifferentperceptionfromtheequalitarianviewofrealityhumanshaveinitssingleplaneof

existencein1to1correspondence.

Howthenitistheworld-geometryofthoseotherminds?ToanswerthatquestionwehavetodealwiththemostadvancedformsofGeometry–thepangeometryofLobachevski,whoselawscanbeappliedtodiscernthe

‘formofmentalspaces’.

Lobachevski’sparameter.

Regardingtherelativeflatnessofamindworld,wecanintroduceaquantitativeparameterofclassichyperbolicgeometrythatdefinestherelativevalueofPi..

Inhyperbolic geometry– thegeometry thatbranchesa line intomultipleparallels, hence thegeometry thattravels through scalesof the fifthdimensionofparts andwholes Lobachevski found that the length, l of thecircumference of a circle is not proportional to the radius r but grows more rapidly (essentially by anexponentiallaw).Letusthenconsiderhowitinfluencesacertainmental view.

So,thefollowingformulaholdsforthesingularityzero-point'eventhorizon':

wherekisaconstantdependingonthelengthunit.Since:

weobtainfromit:l=2πr(1+1/6r²/k²).Thusonlyforsmallr/kratiosisittruewithaccuracythatl=2πr.

Intheformulaforthelengthofthecircumferenceofacircle,thereoccursaconstantkdependingontheunitoflength.Iftheradiusissmallincomparisonwithk,i.e.,ifr/kissmall,then,asisclearfromtheformula,thelengthlisnearly2πr.Generally,thesmallertheratioofthedimensionsofafiguretothisconstant,themoreaccuratelythepropertiesofthefigureapproachthepropertiesofthecorrespondingfigureinEuclideangeometry.ThusameasureforthedeviationofthepropertiesofafigureinLobachevskigeometryfromthepropertiesofafigureofEuclideangeometryistheratior/kifrmeasuresthedimensionsofthefigure(radiusofacircle,sidesofatriangle,etc.).Thishasanimportantconsequence.

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Supposewehavetodowiththeactualspaceoftheexternalworldandmeasuredistancesinkilometers.Letusassumethattheconstantkisverylarge,say1012.Then,forexample,bytheformula,foracirclewitharadiusofeven100kmtheratioofitslengthtotheradiusdiffersfrom2πbylessthan10−9.OfthesameorderarethedeviationsfromotherratiosofEuclideangeometry.Withinthelimitsof1kilometertheywouldevenbeoftheorder1/k,i.e.,10−12,andwithinthelimitsofameteroftheorder10−15;i.e.,theywouldbealtogethernegligible.SuchdeviationsfromEuclideangeometrycouldnotbeobserved,becausethedimensionsofanatomareahundredtimeslarger(theyareoftheorderof10-13km).

Ontheotherhand,ontheastronomicalscaletheratior/kisnottoosmall.ThereforeLobachevskialsoassumedthat,althoughontheordinaryscaleEuclid’sgeometryistruewithgreataccuracy,thedeviationfromitcouldbenotedbyastronomicalobservations.Thisassumptionhasbeenjustified.FurtherontheinsignificantdeviationsfromEuclideangeometrythathavenowbeenobservedontheastronomicalscalegiveus furtherproofofaninfiniteUniverseof galaxy-atomsmuch larger than the supposedbig-bang inorder toachieve the 'necessarycurvature' for it tohaveanenclosure in the∆±4plane. Finally, since thedeviation fromEuclideangeometrybecomessmallerfor increasingvaluesoftheconstantk, inthe limitwhenkgrowswithoutbound,hyperbolicgeometry goes over into Euclid’s geometry. That is, Euclid’s geometry is just a limiting case of hyperbolicgeometry.

Theflatnessofthehumanmind.

Butwecanconsidersuchmind’sparameters tobethoseofanelectronicbrain,wherethegeometryof theelectronichumindmadeoflightisdefinedbytheS=Tdualityaccordingtothe'relativeratio'betweenourrandkwhichareourconstantsof'perceptionofinformation'(k),thatis,H-Planckandtheunitoflineallength($(r),c-speed. And so the ratio defines both a flat world, the one we perceive and one which process very littleinformationdensity–notaveryfastintelligentmindforallwhatisworth.

Theratioasapangeometryforallpossibleforms.

It isthenclearthat iftheratio isa limitingcase,hyperbolicgeometry,thencomprisesalsoEuclid’sgeometryandso itturnsout, inthissense,tobeamoregeneraltheory. InviewofthissituationLobachevskicalledhistheory“pangeometry,”i.e.,universalgeometry.Andindeed,hyperbolicgeometrybeingtheessential'geometry'of∆-scaleshasEuclideangeometryinasingleplaneasalimitingcase.

Sucharelationshipoftheoriesconstantlyappearsinthedevelopmentofmathematicsandthenaturalsciences:Anewtheory includestheoldoneasa limitingcase, inaccordancewiththeadvanceofourknowledgefrommorespecialtomoregeneraldeductions.

Butwhatreallyr/kmeans intermsofmentalspace?Ask isaunit/rodof length, inourcase light, itmustbeaccordinglyaunitofinformation,equivalentinthefractal,discontinuousversionasmall'step'-thefractalunitofmeasurewhichlengthensthetotaldistanceofa'coast'asMandelbrotdiscovered:

Thecoastlineparadox is thecounterintuitiveobservationthat thecoastlineofa landmassdoesnothaveawell-defined length.Thisresults fromthefractal-likepropertiesofcoastlines. ...The lengthof a "true fractal" always diverges to infinity, as if one were tomeasureacoastlinewithinfinite,ornear-infiniteresolution

As fractal geometry is to ∆-geometry between discontinuousplanes, what differential geometry is to ∆§ocial scales, we caneasilyunderstandLobachevski'sparameterasthemeasureofthesmallness of our 'steps of perception of spatial information', inrelationshiptothetotalradiusoftheT.œwearemeasuring.

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Whenwe are inside the being obviouslywe 'are small' quanta of vital energy surroundedby an ever larger,imposing'flat'membrane;asonEarth's'flatsurface'forthehumanp.o.v.

Sotheequationrelatestheinformative,ð§stepsoftheinner'∆-1'entitiesandthelargerbeing,withitsstsizeparameter;whichgivesus'largerperimeters'withlessercurvature(longerlines)forthementalspaceconstructofthesmallestinnerbeing.

It also follows that from an external p.o.v., which sees a larger part of the T.œ thiswill appear increasinglycurved (and concave, elliptic instead of convex, hyperbolic). And ultimately this dualityproves the mentalnature of all constructs ofspace, put by a devilish mind-mirror, which adapts the view through its'subjectiveglasses', asDescartes thought tobe the case. It is themost important findingofNon-Egeometry,regardingmindconstructsforallgeometriesbesideshyperbolicforms.AllthisISaspecialcaseofan¡logicruleonthe5DmetricstructureoftheUniverse:'1D$mallmeasurementsdoNOTmeasurethewholeworldcycleofthebeing,sotheyarelineal.Long-lastingmeasurebringthewholeworldcycleorenclosed.

RECAP.Geometryisavirtualmentalwor(l)d,astill‘form’oflanguage.Assuchismind’ssimultaneousselectionofits relativeworld’s information - a representation of its reality,O-Mind x∞Universe =Mind-geometry,whichstopsthemotionsoftimeintoamindmapping‘called’space.Eachmindthereforewillhaveadifferentgeometricview of reality. As such geometry evolved from the purestmind-form of thought of still bidimensional GreekGeometry.Next itcameanalyticalgeometry inwhichspacewasmarriedwiththeveryessenceofthemind-apoint/viewofreference,the@-subdisciplineofmathematics.SooneofthekeyevolutionsofgeometrywastogivemotiontospacefromtheinitialGreekstothemoderntopology.Evenvacuumhas'magicenergy',‘motion';itisnotbackgroundspace.Thissaid,weareinterestedincertaintypeofmind-spaces,thoseofthemathematicallanguage,which is the realmof geometry and topology, the first, a 'fixed formal space', the seconda formofspace-timewithmotion.

Thefractalstructureofdeepscalesiscreatedwithlanguagesstoredinseedsandminds,ofwhichtopologiclanguagesarelikelythemindofatomsandgalaxiesthatcreatemostofthelocalorderweobserve.Soall

systemsgaugeinformation,astheircapacitytoorderrealityisjustamirrorprocessofcreationof'still,smallerlinguisticimages'theworldprojectedasorderinaterritorial'energy-body':

Infinitesimalmind-languagex∞Universe=constantself-centeredworld.Inmathematicalterms0x∞=C.

ItistheoriginoftheEgoparadox:‘Everyinfinitesimalmindmeasuresrealityfromitsdistortedperspective,thinkingitisthecenteroftheUniverse,itconfuseswiththeselectedinformationperceivedbyitsmind’.Whichofcourseissharedbyallothersystemsofreality.SoanantalsothinkstobethecenteroftheUniversewithits

likelyhyperbolicpheromonalmindof‘atomicpixels’.

SoasmanmeasuresrealityfromitslimitedEuclideanperspectivedespisingtheexistenceofallotherfractalpoints=mindsoftheUniverse,whicharealsogauginginformationalbeitwithdifferentmentalstructures.

Lifedoesn’tstartinDNAatoms.ITmustexistinallsystemsofthefractalUniverse,departingfromparticleswhichasthegraphexplainsalreadyshowallthecharacteristicsoflife.Theelementaryquarksandelectronsthe

simplestparticlesdogaugeinformation,absorbenergy,reproduceandevolvesociallyintowholes,bosons,plasmaflowsandatoms.Sotheunitoflifeisthesmallestparticleandasfractalsystemsareself-reproductive,emerginginitsfundamentalpropertiesinlargerscalesallwhatexistsisalive.Onlyhumanegospreventusfrom

understandingthatobvioustruth,fundamentalprincipleofall'exist¡ences'.

Howevertheshort-comingsofhumanmentalspacesareenhancedbythecomplexgeometryofthefifthdimensionwhich‘cheats’fractalpoints,‘leaving’themblindtotheflowsofinformationcomingfromupperscalesanddistortingitsviewofthecyclicallongrangetimedeterministicsystemsofthatlargerwhole,givingthemirageoffreedom,open,flatspaces.SoitseemstheGameofExistienceisbuilt-intomakeeachpartfeel

free,open,happyandego-centeredandultimatelyfailbecauseofitsmindlesschaoticbehavior.

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5Dgeometryguidedbypentalogictriestounderstand:

1. Thedisomorphismsof all geometries – thoseelements common to all of them,whichbelong to thehigher‘gameofexist¡enceofallmindsthatselectinformationtoensureitssurvival.ThistaskstartedbyRiemannandLobachevski, concluded that few element of space are relevant – distance=similarity and angle of congruencebeingitsmostimportant.

2.-tobeamentalconstructandrelatethemainlawsofgeometryanditsvarietieswithGSTasamirror-mindthatreflectsthoseisomorphic'ILOGIC'propertiesofspace-timebeings.

X.IAGE:EUCLIDEAN;SIMPLELINEALYOUTH.

As a spatial, mental, ‘S@’ language geometry started without motion, in holographic bidimensional space,mimicking the single-eye view of human thought, and yet it was a mirror good enough for trigonometry andgeometryintheplanetodeveloptotheheightsofGreekGeometry.

Becausebeings are space even thepostulates of idealizedbidimensional stillGreek geometry become synopticlaws of vital space we interpret to explain organic properties emerging in physical, biological and even socialscales. So even if we do not have new theorems to prove in a rather exhausted field we’ve always newinterpretationstomakeonoldtheorems,aswehaveshownintheanalysisofvitaltopologyandnumbertheory.

Sowewon’t 'transform'classicgeometrytothe5Dformalismbutcommenton itsrepresentationofrealityasa'simplifiedmirror'ofthewholeUniverse,reducedindimensionsto3,height-information,width-reproductionandlength-motion, and further on stripped of the 'topologic motion’ that gives through its S=T duality, organicfunctions to1Dperceptiveheight, 3d reproductivewidth and2dmoving length. Still classic geometrywill be a‘worldinitself’;thatis,asalllanguages=synopticmirrorsofthemorecomplexfractalreality,itwillbeconsistentwithin its reduced view, with the laws of the fractal world from a distorted perspective that will allow us to'comment'onitspostulates.

To thataimthebestmethod is toobservehowasGeometryevolves through3agesof increasingcomplexity itcameclosertorealityasabettermirror,addingdimensions,motionsandfinallywithNon-Egeometrycompletedin those texts it becomes close enough to the fractal Universe to describe the ‘worldcycles’ of growth of anyNature’ssystem.

Sinceifwe‘run’the5¬Epostulates,departingfromafractalpoint,eitheramindoraseedthatwillgrowtoorderavitalterritoryofenergy,wearedescribingthedevelopmentofinfiniteTime§pacebeings.Indeed,theeasiestwaytodothisistoconsiderthe'growth'ofasimplestillseedofinformationorfractalpointintoareproductivewave,whichbranchesinto3physiologicalnetworksthatmergeintoavital‘plane’asasuperorganisms.Butthisdescriptionisjustthesamedescriptionofplanestillgeometrythatdefinesapointwithnobreath,whichgrowsintoalineofpoints,3ofwhichdefineaplane.Onlythatinclassicgeometrythelinehasnomotionandnovolumeasthe¬Ewaveandtheplaneisaflatsurface,unlikethe3physiologicalnetworksofasuperorganismwithvolume,motionandcurvatureand‘fractalbranching’.

Theholographicprinciple.Bidimensionalgeometryofpointswithoutparts.

What was truly right of Greek Bidimensional Geometry howeverwas the realization that the minimal unit of reality is always aholographic bidimensional ST-system, a Dimotion of space-time,whichbytheParadoxofGalileo,S=T,canalsobestudiedasanSSarea,hencethe lawsofST-dimotionshadaclearmirrorsymmetryinGreekGeometry.

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Inthegraph,weseetheHolographicprincipleofthebidimensionalUniverse:Atimecycleoftwodimensionscan be seen as a still form of information, an locomotion on space can be seen as a surface of 2 spacedimensions.Soforexample ‘c’speed,canbecomeinanentropicexplosionas itdeceleratesac2surface,andviceversa,a‘fractalpointof2dimensionsoftime,TT,or‘accelerated’vortexofspacetime(amass),canuncoilinto2dimensionsofdistance-space,whichistheultimatemeaningofEinstein’sE<=>mc2.

FinallythemergingoftwoholographicsurfacesofanySTcombinationgiveusathirddimensionofspace-time,asitdoesinthehumanmindthatcombinestwobidimensionaleyeviews.ThosearethereforethebasisforthesuccessofbidimensionalstillgeometryasamirrorimageofFractalspacetimelaws.

Thus, the first ageof Spatial analysis stumbleddirectlywith themarvelsofholographic surfaces, translating‘magiclaws’comingseeminglyoutofnothing-theSTsymmetriesandefficientrelationshipsofbidimensionalentities, with a membrane (curve or line) enclosing a self-centered surface, which as we noticed on theintroduction is thesimplestelementaryparticleoftheUniverse ina ‘flatworld’asthesurfaceofEarth’sseasand lands,or theminimal ‘planckton’of lightspace-time;and its3conservedquantities, thesingularity-mindthat defines its lineal momentum, the angular momentum of the membrane (T=S perspective) and its vitalenergy.

SoGreekstillgeometrywasahugeworldasalmostallthelawsofgeometrycanbeprovedinabidimensionalplaneofinformation,tothewonderofmathematicianstillthisday.

While inparallel humans resolved similar lawsof bidimensional perceived information througha formof artcalledpainting.Itisinfactlittleknownthatpaintingandgeometrywerecloselyrelatedinthebeginningwhenhumannotmechanicaleyesinterpretedboth,andinfactpaintingarrivedfirsttothelawsofperspective,whichwoulddefinelatterthelawsofprojectivegeometry.

ThefirstageofgeometryistheGreekbidimensionalage.AnditbearsproofofGSTanditsholographicprinciplethatmosttheoremsofgeometrycanbeprovedinaplane.

Ofthem,weshalldealherewithafew,addingsomenewdiscoveries,speciallyregardingthe'postulatesofnon-E',neededtofullygraspbidimensionalgeometryandwhytheirtheoremsmatter.

Butbeforewedoso,wecanpeerattheequivalentvitaltopology,inwhichtherulesoftheperfectbidimensionalgeometryarebased,aswecanconsiderbidimensionalgeometryastillvitaltopologyinwhichmotiondoesNOTexist,neitherresistancetodisplacement;hencetheirregularitiesthataretracedbymotion

andgeodesicsintherealworld,nolongerplayanyrole.

The1stMasterPythagoras…theorem.ItsPentalogic.Pi&theharmonyofmusic.

S:WeinterpretonthosetermsPythagorastheorem,whichisthemostinvariantoftheoremsasitisbasicallydefinedtocreateametricofdistancesinasingleplaneofspace-time.Soitsclassicmeaning

isapurespatial¡logicp.o.v.

@:Yetnowthatwehaveliberatedanotchfurther'mentalspace'fromrepresentationandmakeitaffinetosurvivalinformationdistancebecomesynonymousofT-Motionand¡logic‘similarity’.

Indeed,theinterpretationofthePythagorastheoremintermsofcongruenceandperpendicularityisclear:twopointscanbe‘close’insimilaritymuchmorethan3points,whichastheFrenchsaid‘area

crowd’,orasthe3body-problemproves,enterinchaos.

Thesimilarityoftwo‘genderpoints’withoppositespinthatcanmergeintoone,impliesthenthatthedistancebetweentwopointswillbetheminimal,whilethe4thlawofcongruencethatexpressesthesocialevolutionofidenticalpointsinparallelmotionvs.theperpendicularityofpointsofmaximaldissimilarityimpliesalsothatthe

‘maximaldistance’≈dissimilarity(s≈t,geometric≈logicview)willhappenbetween3points,whichform2

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perpendicularA1-A2andA2-A3lines.ThusthePythagorastheoremexpressesalsoalawof‘entropy’andifweconsiderthoselinesundertheS=Tdualitymotions,alawoftime:apreythatfeelsapredatorperpendicularto

itsformwilltrytomove=escapethefurthestdistancefromit:

¬:Entropy.3elementsoccupymorespace/requiremoredistance=dissimilaritybetweenthemthan2points.

Inthegraph(A1-A2)+(A2-A3)≥A1-A3.Andsothefundamentalvitalpropertyofcongruencedohaveametric.

Ti:Butwecanalsoexpresstheconceptintermsofcyclicaltime=information;as3pointswillrequiremorebits,astheyadanewdimensionofheight,soatrianglecanbeseenasthesimplest‘π-cycleoftime’whichcarriesmoreinformation.Sodistanceasasumofspatialstepseachoneabitofinformationincreaseswith3points.

Theninthejargonof¬Ewesaythat:

“AmetricspaceisasetofundistinguishableT.œscalled¬Efractalpoints,inwhichavolumeofinformationcalleda'distance'isrequiredtodefine2possibleoutcomes,namelytheaxiomsofametricspace:

1.Id(X,Y)=1ifandonlyifthepointsX,Ycoincide.InclassicmetricspacesÐi(calledusuallyr,butini-logicgeometrycalledDistance≈motion)is0aspointshavenovolume,butin∆st,youneedatleast1Dimensional

unit,todefinethepoint.2.ForanythreepointsA1,2,3thenitholdsthat(A1-A2)+(A2-A3)≥A1-A3;hencemoreinformationisrequiredtodefine3fractalpoints.Andthisrulecanbeextendedton-points,wheren>3,suchas(A1-A2)+(A2-A3)+(An-

1-An)...>(A1-A2)+(A2-A3)

∆:Butthereisalsoascalarp.o.v.ontheTheorem.Einstein,indeed,gotanewdemonstrationofthePythagorastheorem,basedinthe'scales'andself-similaritiesofitsratios-aproofofthe'fractalparadigm'.

TheSpatialview…

ThoughisthefundamentalpentalogicofPythagorastheorem,(graph);henceaholographicbidimensionalexpressioninstillspaceof‘squaredareas’,thatrepresentmostlyST-entitiesandsomeofitsunderlyingproperties.,notablytheComplementaryvs.DarwinianperpendicularityofcompositeST-species(4th¬E

postulateofcongruence).Anditfullyexpressesintermsofcongruenceandperpendicularityafeedingprocess,astheareaofthefinalresultofaperpendicularencounterbetweenaandbisc,amergeofbothareas,whicheithermeansonehasfedintotheother,ortheabsorptionhascreateda3rdspecies,whoseenergyisequivalent

tothesumofthetwootherSTentities.

SoafteritsperpendicularDarwinianstateboth‘merge’andtheoutcomeistheabsorptionofoneoftheelementsbytheother.Soitisessentiallytheruleoffeedingbetween2perpendicularsystems,whichgivebirthtoa3rdone,butalsotheruleofreproductionandcreationofanewsystem,similartotheonedescribedfor

polyhedrons.

Andthisleadstothefinal‘tease’commentonthePythagorastheorem.

Mr.Fermat’sProofinthemarginofthebidimensionalSTnatureoftheUniverse.

InourintroductiontoGSTwementionedthatdimotionsaredual,holographicSTstates(SS:seed,St:information,S=T,reproduction,sT:locomotion,TTentropy).

Furtheronnumbersareundistinguishablepoints,whichobligestheminbidimensionalspacetoberegularpolygons,ofwhichtherearean∞number(proofofNaturalnumbers’relativeinfinity).

WhileinAlgebra,weobservethatthe±operandactonlyinequalforms.

Whatthisessentiallymeansisthatonly‘squares’ofthesametypeofdimotioncanad(Fermat’stheorem).

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Since,theuniverseisabidimensionalhologramofspace&timebeings.Soonlythesquareofnatural=spatialnumbers=populationscanadexactlyintoanothersquare=spatial:

Inthegraphweseethatdifferenttime-spacecombinationsofform=staticmindspaceandmovingtimeoflinealorcyclicalgeometry.Sowhenwemergetwoequalspecies(‘numbers),weaddequalbeings,hencewemakein

geometrictermsaPythagoreanabsorptionthatresultsintoanX2+Y2=Z2formofthesamespecies.

Butwhenwemergedifferentspecies,weuseaproduct,are=productivealgebraicoperand-the3rdDimotion.

Itisthennoticeablethatthereare5ternaryregularspeciesand6tetradimensionalregularobjects;whileinhigherdimensions,whichmustbedividedintotimeandspacestates(evenifingeometrytheyareallperceived

asspaceforms)thereareonly3regularpolytopes.SoanimportantthemeofadvancedSTheoryistocorrespondeachofthemwithbasicSTcombinationsoftheentangledUniverse.

Thesimplestdimotions,howeverarebidimensional(i.e.modernphysics,discoveredtheholographicprinciplethatstatesyinformationisbidimensional.Anditssimplestoperationistheadditionof2equalnumbers.Thus

onlyX²+Y²existsasanexactnewbidimensionalform.Infact,almostallpostulatesofgeometrycanbeprovedinbidimensionalspacebecausethatistheessential‘unit’ofreality.

DidmyBasquecountryman,Mr.Fermathadthisproof?(:Well,youneverknow,we‘amateurscientists’don’ttakeveryseriouslytheaxiomaticmethod,workratheronintuition,somaybeitwasimperfectbutcertainly

shorterthanthepedantic,computer-generated+1000pagesthatgoesaroundasahumindfeat:)

MoreseriouslytheimmenseadvantageofreturningmathematicstoanexperimentalmirrorofthefractalST-Universeisthatitstheorems,speciallythoseofits1stand2ndnoninflationary3rdage,intunewiththerealityitdescribesarebothconsistentinternallyasmirrorshavethesameimage-formthatthewholethereflectinsynopticcompressedparts,butsusceptibleofproofbydirectexperienceoftheLawsofGST(GenerationalSpace-time)aswejusthavedone.Equalformsadd.Additionisthespatialalgebraicoperand.Soitcannothappenindissimilarforms,youdon’tadd2pearsand2humanstoget4humanpears.Butyoucanmultiply

dissimilarentitiesofspaceandtime,2stepsx2frequencies.

Immediatelywerealizethen3DimensionalityisNOTanevennumber;itdoesnotaddequalforms;soitmustbeachievedNOTinparallelbutinPerpendicularmerging,whichmeanspenetratingan∆-15Dplanetotheparts,whosemaximalproduct(S=T,Max.SxT)willpileuponagradient,makingsomepointsdifferent,whileotheranglesofcongruence,mightdiminish-feedparts,etc.Andinmostcases‘re-arrange’thepartsmerging

intosomethingelse.SoitisadifferentworldstudiedinPartIIonalgebra&itsoperands.

Productvs.addition.

ThekeyconceptbehindFermat’stheoremandaseriesofbasicaxiomsofalgebraistheideathat:

‘Additionoperatesoveridenticalentitiesaddingforcesinparallel,subtractionoveridenticalentitiesdiminishingthem;productoperatesoverSpaceandTimeperpendicularfieldsinsymbiosis,Divisionoperatesoverspace

andtimeperpendicularentities,inDarwinianactionsbreakingthem.

Andwithproductsandadditionswecanthereforeexpressalmostallthenecessary‘simplecombinations’ofspace-timefields,reasonwhytheyarealoneahuge‘group’ofpolynomials.

So3-dimensionalformsinnaturalnumbersarecreatedbyareproductivemergingofbidimensionalfields,hencebymultiplicationasinvectorcalculus,whichareNOTaddedtocreatea3rddimensionbutmultipliedtogetas

inthegraphofthe5SS,ST,St,sTfields,alightwave-c²=1/µxe.

HenceX³+Y³≠Z³;asitscombinationtocreatea3Dworldisnolongerasumbutaproduct.SinceX3speciesarecompositeSTfieldsthatdeployanangleofperpendicularitywithacommonbidimensionalline-wavetoconnect

bothsystemsandallowthereproductionofitsform,inalowerplane

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Andsinceinasingleplaneofexistence,where+andxoperandiact,thereareonly3dimensionsoftimeand3dimensionsofspace,thereisnoneedtoproveitforimaginaryhigherdimensions.

Indeed,whenwestudycurveswecanconsiderallthepossibleconfigurationsofatridimensionalspaceexpressedasquadraticfunctionsofsumsofX,YandZ,becausethe3‘trilogicdimotionsofEuclideanspace-time,canbecreatedashomogenous,exchangeabledimotionsofapresentspacetimeplaneofthe5thdimotion.IfFermat’stheoremwerefalse,itfollowswecouldcreateanhomogenous4Dimotionalforminasingleplaneof

space-time,butwecannot,becausethefourthandfifthdimensionalmotionsoftheUniversearefractal,happeningbetweenplanes.

RECAP.x²+y²=z²,accordingtotheholographicprincipleISPOSSIBLE,x³+y³=z³isnotbecausetheUniverseisineachscaleabidimensionalholographyofspace&time.Andthis'proofinlessthanamargin'ofthemost

famousunprovedTheorembyanyhumanmindoftheso-calledFermatGrandTHEOREM,isaclearproofoftheexperimentalnatureofmathematicsasamirrorofGST.

Thestrengthoftriangles,as‘mentalspaces’of3elements.3Points>Angles>Triangles

3Points‘alone’aredisconnectedandassuch,asthe3bodyproblemshowsinchaoticmotiontoeachother.Motionisrelativetothedegreeofentanglementbetweentwopoints,henceitimpliesaninternalbalanced

dualmotionbetweenthem,whichcreatesthe‘illusion’ofdynamicstillness.Thusmotionbecomesstillnessinamindbecauseofthecrossentanglementbetweenaxionsofthehigher∆+1neuronalplane.

Itfollowsthatthefirstentanglementisacouplewithadualcommunicationwithinversenumbers;andthesecondentanglementatriangle,whichwillhavemotionwhenoneofitsflanksisopenedallowingentropicmotiontoglidethesystem,andwillbecomestillwhenthe3pointsareindualinversecommunication.

Theirsocialevolutionthusisthetriangle,whentheyarelockedintoeachotherandthiscanbedonewithhierarchy,aswesawitbysymbioticvs.predatoryperpendicularityinthesimpletriangularform,an‘open’angle.Inthis‘alliance’,of3pointsthenthe‘biggersquare’(c2=a2+b2)isNOTpartoftheensemblebutratherthe‘field’absorbedbythedualbody,linesaandb,whichmakesusunderstandthefirstkey‘element’ofvital

‘Greek’Geometry:

Thedifferencebetweenanangle–anopenentropictriangle,anarrow;andatriangle,aninformativesystemwheretheangleisclosed.

WesaidthetrueadvanceofmodernRiemann’sgeometryistomakeita‘mentalspace’,wherespatialconceptsacquirealogic,temporalmeaning.Andsoatrianglecanbeconsidered,anyternarysystem,andassuchthemoststableformifwetakeratherthaneachpoint,eachlineofpointstobea‘network’;hencedefiningthe

simplestrepresentationofatopologicalorganisma‘planeof3physiologicalnetworks’offractalpoints.AnditisthenwhenwecantrulyflyintoBidimensionalgeometryasamentalspacerepresentationofthewholeUniverse.

Thebeautyofthefirstagesofanylanguageispreciselytoreflectthe‘essence’ofthegameofexistence,somethingweshallsee–timepermitted-whenanalyzingwordsandmusic.Languagesthenbecomeinflationaryandfictionalwithage,abandoningtheternary'simplicityoftheUniverse'withonly3agesoftime,3topologiesofspace,3scalesofthe5thdimensionforany'finiteorganicwhole'callitthegalaxy,thehumanbeing,themathematicallanguageanditsternarytopologies,ternarynumbers,ternarydimensions,ternarydisciplines...

Sowetalkof3phasesofincreaseorderintheevolutionoftheternary‘pointstructure’,from3pointsruledbythefirstpostulatesofNon-EGeometryasdifferentselfishspeciesusing‘anglesofperceptiontomeasureeach

otherdistanceandmotion;intoa2ndNon-Epostulate/state,oflinesofcommunicationthatconnecttheDOMINANTpointintoanangle,oftenaperpendicularonethatmaximizesthedistancebetweenthesubmissivepoints;whichwecanobserveinthevitalgeometryofmolecules–thefundamental‘geometricstiencebeingchemistry’;tothetriangle,whichisthemoststabledemocraticform,asboth‘submissivepoints’reacheda

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connection;butonthedownsidetheyhave‘closedin’thesystemtoapossibleflowofentropymovingthearrow-triangle.

Sototherescuetopreservethe‘vitalnatureofmathematicalforms’,thetriangleisNOTallowedtocloseperfectly,buthasinthe‘finitesimal’lowerscalesanirrationalnumber:

Sothegrowthofrealitybetweentwoscalesstartswiththe√2abidimensionaltriangularpair,whichstartsthegrowthondimensionsofinformation,andspace-extensionbyreproductionofthesameeventthroughaseries

ofdifferentformsofgrowth:

Inthegraph,twolines=wavesofreproductiveparticles1xand1y,meetthemselvesintherelative0point,ofanx-yautomaticallycreatedcoordinates,whichwillgiveusan√2,wavewithoriginino,andanxy,expansionfrontofspace-time,reproducedbypairsofxypointsalongthefront-wavetill

√2reachesthetwo'membranesofthexytailofpastmomentum'.

Atthispoint√2canbemeasuredasasumofdiscontinuouswave-points,andgivesusavariationof±1,orbeconsideredasacontinuouswaveofenergy;nevermind,inbothdescriptionswehavecreateda

bidimensional,triangular,straighttriangle,whichcannowgrowindifferent0/0tangentstotheexponentialwaveofitssecondage,ofgeometricgrowth,

Thefamousproofofirrationalityof√2,quotedbyAristotle,wewon’tquoteforbrevity,ISbasedinasquaresum,whichbringstogetherthepreviousconceptsofsimilaritytoexercisesumsandmultiplescalesin

polynomials.

3isthusneededtoformatriangularnetwork-plane,the1stsupœrganism,as2cannotfulfillallUniversalfunctions.

Furtheronthesystemwillhavefractalpointswithinnerpartsand2openingsonthe√2diagonalthatcloseandopenbyirrationaldefectthesystemtoabsorbandemiteenergyandinformation,withthe45%angularpointas

thedominantelementofthetriad.

Thusanysetof3∆-1fractalpoints=T.œssufficestoestablish∆±¡ternary'networks'themostefficient'number'togenerateanewplaneofexistence,astheyarethe'minimaldistance-information'toformit.

Soasternarysystemssufficetomakereality,triangles,thesimplest,arealsothelinealstrongestconfiguration,ofwhich3naturalsubspeciesarefundamental:

-ST:ThePythagorastriangle,withadominant¬angle,withtheleading,stronger¬point&2aperturesinthe√diagonal,oneforentropyandotherforcommunication,movingonthepathofthe¬point,asan‘arrow’.

-ðIsosceles,whichelongatesitslateralsidesdiminishingthe√2basetoaccelerateitsmotions.

-§(St):Theequilateraltriangle,whichisthestaticform,asitcanrotateonits‘perfectsingularity’pointofperceptionbecomingacircle.Oritcantransformintoacirclebyhyperbolic‘feeding’expansionofitscurved

sides.

Sotriangleswithcurvatureand'openings'equivalenttoπ-3circlesareacompleteternarystate&wecandefineabidimensionalworldoffractalpointswithinnerpartsofpuretrianglesandcircles,asacomplete1stTimespaceuniverse.ItisinfacttheUniverseofpolygonsandNaturalNumbers.Whereasthestrongeststaticformisthehexagon,withπ=3,whichcanalsobeachievedinahigher4DUniverseasalimittothecurvature,strength,attractivepowerofaforce,whenπ=3Diameters,withnoholes,accordingtoEinstein'sgeneralrelativity

formulathatwecanalsoextendinmathematicalphysicstoallsystems:

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Wherekisthe'unifyingconstant'foranyactiveMagnitudeofanyphysicalscale(weshallgeneralizeasM-seeunificationequationofforcesinallscalesinthepaperson5Dphysics.)Soforthephysicallyinclinedwepoiseaquestion:asthePlanckmassisthemaximaldensityofgravitationalspace,wherekMwouldbeGM,pishould

haveavalueof3!?right,orwrong?Andifsoitisstillacurvedgeometryoranhexagon?Theansweris…likelyadodecaplexcrystalofultradensetopquarksofunknownphysicsin4Dimensionaltimespace…inthedensest

centerofagalaxywith120vertices.

Butdynamicpi,legendhasit,madeænthropicPi-thagorasmad,asthelanguageofGodhadtobeperfect,immobileSohisdisciplehanghimselfdesperadothattheworldwasnotstatic.WhenIfounditontheother

hand,itstartedmyjourneyonvitalmathematics.Sinceafluctuatingdynamicopenandclosednumber,makesthecircleeitheraninformativespiral(-π:St)oranopenentropicspiral(+π…spe)justpassingbyafleeting

momentintoasteadystate,(S=Tclosure)givingitthe3agesoflifetomakeitthesimplestvitalorganism.SoweshallconsidernowArchimedesanditstime-spacespirals.

The3waves=ages:Symmetriesofbeautyinalllanguages.

ButtoconcludewithPythagoras,heisalsorememberedfora3rddiscovery,thatofmusicalharmony.Schopenhauer,byfarthebestphilosopheroftheindustrialage,saidthatmusicencodestheprogramoftimein

itsrhythms.Pythagorasfounditssimplest‘scaleprogram’,theperfect'fifth',amusicalchordobtainedbyplugging1/3rdor2/3rdsofastring–thatis,thenaturalstandingpointsofasimplest‘mono-logic’(1D)

worldcycle,withfiniteduration(attachedtoanoriginorpointofpastandanendorfuturepoint,thebirthanddeathofaworldcyclefrequency,with3agessubdividedbytheS=Tsymmetryat1/3rdand2/3rds.

SincebeautyISdefinedastheperceptionbyanymindinanylanguageoftheperfectsymmetriesofa‘well-run’programofexistence,maximalintheS=Tcentralregionofaworldcyclebetween1/3rdand2/3rds,ormatureclassicageofbalancebetweenmotion-energy&in-form-ation.Itistheclassicageofart,mindofcivilizations,

the30sinlife,theintervaloftheperfectfifth.

Asyoucanseeinthenextimage,ifweconsiderthevibrationofthestring,thesimplestpossibleworldcyclegoingfrom0to1andbackto0;thestringwillwavebackandforth3times,increasingeachtimethe

'information'itcarriesanddiminishingitsentropy≈energy≈amplitude.Andtheperfectformwillbereachedinthemostharmonioussoundproducedat1/3rd,inthechangeofageorstateofthesystem.Butwhatismorebeautiful,timewavesbackandforth3durationsandwecanfusionthemasNaturedoesinasingle'social

being',integralofallthosewebs.ThisiscalledtheFouriertransform,andincomplex5Dmetricistheessentialequationoftimecycles;sinceitkeepsaddingon'socialscales'oflargersimplerwholes(thesinglewave)andsmallermoreinformativeparts.Andfinally'emerges'asa'singlebeing',asquarewave,whichtherefore

representsapopulationoftime,storedasamemoryofstillspace,compressing∝(limitedinfinity)timecyclesofinformationintoasinglespatialimage:

Thebeautyofi-logicmathematicsthusresidesinitscapacitytoexpressthepurestGSTlaws;asmusicdoesin

humanarts,withits3elements,∆±13scales,T-beat,ST-melodyandS-ynchronicityofinstruments.

ButPythagoraswasjustthebeginningofgeometry,andwealreadyseehowmuchGSTcanextractfromit,whenweuseS=Tsymmetriesor'switch'fromcontinuouslinealtimeasdurationtoarealdetailedanalysisof

discontinuoustimecyclesthatmovein'frequencysteps'.I.e.acar,asawholeseemstomoveinlinealtime,butonlyitswheelsmoveincyclicalstepssobreakingitdownintotheS-bodyandT-wheels,andthoseintoƒ=1/t

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givesusnewinformation,newlawsofsciencehiddenbyaddingtimecyclesintocontinuouslinealsums,'erasing'theformandfrequencyofthosecyclicalsteps;awewilldoconstantlyonour5Darticles.

Thelineandthecycle.

Itisratherimpressivethe‘perfectorderofthefractalUniverse’,hiddeninthecomplexityofthegame.I.e.considerthecaseofGreeklinealgeometry.Asinartinwhicheachfirstageisalineal,young,simpleviewofreality,Greekgeometrywasessentiallyestablishedaroundthe‘trigonometrictriangle’.Itsarchitecturelackedthecurvedarch,itsknowledgeof‘conics’,onlycominginitsfinal3rdage.Languagesaremirrorsofrealitywithlessermotion,SS-seedsandyettheyimitaterealitybothinitsentangledelementsandworldcycleofevolution.AndsowequalifyindeedGreekgeometryasalineal,youngstate,whichonlyits‘wrinkled3rdage’foundthecurve.Letusstudyoneofitsforms,aswereservetheanalysisofconicstothe2ndagewhenanalyticgeometry

uncoveredallitsvirtues.

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3rdAGEOFGREEKGEOMETRY:ARCHIMEDES.

SPIRALSASREPRESENTATIONOFTHEWORLDCYCLE.

Abstract:Thespiralisthesimplestrepresentationofaworldcycle.Assuchitisaprofoundformofthe5Duniverseofrelationalspace-timebeings,reasonwhywehavetakingitawayfromthegeneralpostonnon-E

geometries,asitdeservesitsowndeepthoughts.

The3mastersofeachageofGreekGeometry;Pythagoras,thefoundingfather;EuclidwhocollectedtheclassicbodyofbidimensionalGeometry&Archimedesthatanticipatedmathematicalphysicshavedifferentmerits,

whichænthropicmanasusualmisreadsintermsofitsegoparadox,asEuclidisbyfarthelessoriginalandmosterroneous,ashemadesacredtheaxiomaticmethodthatdeniedexperimentalvaluetomaths,unlike

Archimedes.Soeventhoughhewasthelastone,aproperorderintimecorrespondingwiththe3mentalagesoflifewouldbe:

Pythagoras,‘idealistyouth’<Archimedesclassicexperimentalrealism>Euclid,3rdcut-offinformativeage.

Archimedesthusistheclosestspiritto5Dmathematics.Sowedepartfromhisbook‘onspirals’tocompleteouranalysisofdynamicpinumbers.Asaspiralisapicycle,whoseclosurehasgone‘abitwilder’ontheopening.Itisasimplifiedbidimensionalformofaconic,whichalsocombinesacycleandalinebutin3dimensions.YetsincetheUniverseismadeofSpacetimeholographicSS,St,sT,ST,TT5bidimotionalunits,allcurvesthatexistcanbedecomposedasasumofbidimensionalconics,includingaflattenedspiraldefinedbyArchimedes:"Ifastraightlineoneextremityofwhichremainsfixedismadetorevolveatauniformrateinaplaneuntilitreturnstothepositionfromwhichitstarted,andif,atthesametimeasthestraightlineisrevolving,apointmovesata

uniformratealongthestraightline,startingfromthefixedextremity,thepointwilldescribeaspiralintheplane."Onspirals

Wefindthen2‘55D’meritsonArchimedes’definition:1)tobeacombinationofalinealandcyclicalgeometryand2)tohavemotion,notjuststillgeometry;makingitthefundamentalsimplestSTbidimensionalrepresentationofaworldcycle.SoGreekgeometrygeneratedaternarygeometricmirrorindifferent

dimensionsofaworldcycleofexist¡encemakingtruththatthesimplest‘young’mirrorsofrealityareoftenthemostessential:

1Dmusicalstrings>2DSpirals>3Dconics.

Thefractalpointonthespiralasalivingorganism.

Whatmattersonthatdefinitiontomirroraworldcycleisthatitisthecombinationofalinealmovingpointwithinalargercyclicalworld.Ifwetakeanypointofanypotentialspiralcyclewithinafixedspiralstructure,theentitywithinthespiralismovinginwardsthinkingitexistsinalinealpathUniverse,asweallthink,butbecausethesuperorganisminwhichitexists,thecurvedspiral,ismovingincycles,thefractalpointinfactisturningcyclically–helivessubjectivelyaswealldohislinealtime,buttheworldobjectivelyis‘wearingitdown’the

informative3rdageorcentralregionofthespiral.

Theneachspiralwillbeavariationonthesame‘mentalphasespacetime’mirroroftheworldcycle.AndwecanapplythesymmetriesofGSTtoclassifythem.Inpracticethoughasinsomanycasesthefundamental¡logicsymmetrythatdiversifiesspiralsisthegendermirrorsymmetry,soS=TArchimedeanspiralsare‘female’

balancedevenspiralsandlogarithmicspiralsareS<T>Sinformative&entropic.Sowereduceouranalysistothem.

2NDAGE,CLASSIC,S=T,REPRODUCTIVE,ARCHIMEDEAN'STEADYSTATE'STABLESPIRAL.

IntheArchimedeanspiral'theinternalpoint-beingfollowsacyclicalpathatafixedrate.

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ThustheArchimedeanspiralisthesteadystatepresentspiralandassuchthecommonestintotalspace-time,asitisclearlyamorestableconfigurationthananacceleratedlog-spiraloftheformS>T.ItdoesNOThaveaninnermotionthatimplodesitbutbasicallyisthe'naturaldistribution'oftheinternalvitalenergy-spaceofa

sphericalform.

TheArchimedeanspiralhasthepropertythatanyrayfromtheoriginintersectssuccessiveturningsofthespiralinpointswithaconstantseparationdistance(arithmeticprogression).Incontrast,inalogarithmic3rdage

acceleratedspiralthesedistances,aswellasthedistancesoftheintersectionpointsmeasuredfromtheorigin,formageometricprogression.Andthisrevealsits'vitalage’&possiblemultiplefunctions,initsrelativeS,T,∆,

@'survivaltaskseventhoughitsprimarydesignistoreproduceapointorcommunicate‘2points’:

Ifitissymbioticaspartofasupœrganismitmightrepresentboththeinnervitalcyclesofaflowofenergyandinformationoritssurfacemembranein3Dprojectedintoa2DArchimedeanspiralsasa

bidimensionalphasespaceofasphericalform:

Infactwehavetwosolutions/locationsforthespiralthattracetwodualpathsoftenofcommunicationbetweentwosimilarbeings.SotheArchimedeanspiralcancreatebycyclicalreproductioningeodesiccurve

paths,a3rdtemporaldimensionofheighttoconstructsphericalmembranes.

TheReproductiveFermatspiralisatypeofArchimedeanspiralevenmoreaptforreproductivepurposes.Inasingleplane,itcanactasareproductiveparallelcommunicationfora2-particle

system:

Inthegraphtherelationshipbetweenthecommunicativespiralandtheshapeofitsexternalmembrane,asthespiralisthecommonforminwhichavitalspaceof

cellularelementsisestablishedwithinit.

Thenthespiralisacommunicativedancebetweentwosimilarforms,mergingthetwoelementsattheendofthe'lifeofthespiral',asinblackhole'smerging.

Asareproductivespiral,itshowsthefactthatall'reproductive'actionsarethestructuralmergingof'dualelements',wecallgender.Soithas2branches-moreinaFermatspiralwhosepurposeisthemaximalpackingofitsreproductiveformsinthespaceitfills,reasonwhyissopervadinginplantsandotherhighlyreproductive

systemsofnature.

ThereproductiveFibonaccispiralachievesthiswithitsreproductivegoldenratio,studiedinNumbertheory.

SpiralsasfixedbodiesofspacearethusArchimedeanspirals.Andwecanfurtherapplythefractalprincipletosubdividethem:

ThenormalArchimedeanspiraloccurswhenc=1.Otherspiralsfallingintothisgroupincludethehyperbolicspiral(c=-1),Fermat'sspiral(c=2),andthelituus(c=-2).Virtuallyallsteadystatedynamicspiralsinspace,aspartofsuperorganismsareArchimedeanones;suchastheParkerspiralofthesolarwind,orthepatternmadebyaCatherine'swheel)areArchimedean;wherethemotionisconserved,aspartofthevitalenergy-bodyof

thesystem.Sincethedistancebetweenitstwocyclesisfixed.

S«T.Butinpentalogicallformshavemultipletasks,soasthespiralmodelsacycle=dimotion’srecurrentfrequencyitmightbeanE-ntropicfeedingdimotionthatbringsthemouthtoacertainpoint;andthenbringing

downthebody,asinearliercephalopods.

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Timeisacurvedhencewithatleasttwodimensions,besidesitsyoungagelength:3rdagecurvatureandreproductive2ndagefrequency.Timecyclesthusbreakrealityintoaninnerandouter'vitalspace',andadda

singularitypointorfocusofitsmotion.Becausepiisnotexact,steadystateclocksarelesscommonthanvortexspiralorfluctuating±πmouths.Soonceitswallowsthosespiralsbring'home'tothesingularitythevitalenergy

extractedbytheexternalmembranebyseveralmethods.

Oneisshowninthegraph,ascrollcompressor:amotionofbotharmscompressesandmovestowardsthecentertheflowcomingfrotheeternalregionofthebeing.

Noticeinthegraphabovealsothedifferencebetweentheblackholesinacommunicativespiral.Whentheblackholeinaneventoffeedingonenergytransformedintoits'quarkforms'asanacceleratorvortex,similar

tothoseprocessesstudiedonEarth'saccelerators,whichleadsustoitssecondfundamentalform.

ACCELERATEDINFORMATIVE+ENTROPICLOGARITHIMICSPIRALSS<T>S.

Pentalogic:S<T>STheSt-logarithmicspiralisinformativeforitscentralknotandentropicforthepointthatmovesinwards,acceleratingthroughthelogspiral,whichinphysicsisa

vortex=forcesourceofacceleratedtimespace.

Itisalsothe∆-SCALARspiralasitscurvesareself-similarindiminishingscales.,BernoullicalleditSpiramirabilis,"themarvelousspiral".becausehewasfascinatedbysuchuniquemathematicalproperties:thesizeofthespiralincreasesbutitsshapeisunalteredwith

eachsuccessivecurve,apropertyknownasself-similarity.

S=T:Asaresultofthisuniqueproperty,thespiramirabilishasevolvedinNature,appearingincertaingrowingformssuchascompartmentsinanautilusshellsandsunflowerheads,

thatwillstorethereproducedcellsofthesystem,initsinversearrowofcreationofalargerwholestartingwiththesimplestcellularreproducedunits,inexponentialFibonaccigrowthseries.

T>S:Italsoappearsas'informativespirals'.Whichaccelerateanddiminishthesizeofaform,asitcomestoitsperceptivepoint.

¬S«T:Finallyfromtheperspectiveofthebiteofenergyorbitofinformation,which'goesthrough'thetunnelofthespiralisakillingmachine,nowobservednotasareproductivecellsystembutatrappingchannelforbitsandbitesacceleratedtowardsthecentralstomachoreyeofthespiral.Wherethe∆-1pointwilldieinanentropic/informativesplit-whentheperceiver-predatorwillsplitthesysteminitsST>Spartstakethe

informationorenergyofitsbody,dependingontheeventandletitdie.

Sowecanseeeasilyhowthelogarithmicspiralallowseventsin∆-scale(generation),S-entropyfunctions(feeding)andT-informativeandreproductivefunctions(perceiving,storingcellular/atomicnetworkforms).

Γ (¬∆@st):Whichleadstotheuseofthelogspiralasamodelforthe3±¡'ages'oflifeofthemicro∆-1entity;asstarsinagalaxy,whichisa‘farmingtrap’forthecentralblackholetoletthestarsgrowondusttofeediteons

latter.

Letusseesomeofthosepentalogicperspectives.

ð:PerceptivevortexlogSpiral

Thelogarithmicspiralisanaccelerated,henceperceptivespiral.

ItcanbedistinguishedfromtheArchimedeanspiralbythefactthatthedistancesbetweentheturningsofalogarithmicspiralincreaseingeometricprogression,whileinanArchimedeanspiralthesedistancesare

constant.

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Ifthepointthatmovesinwardsaccelerateswehaveavortexofacceleratedtime.Itisthenalogarithmicspiral;anattractiveformanditescapesthesimplificationofmentalgeometrytobecomethecommonestreal

worldcycle.Spiralsinphysicalsystems.Charges,masses,eddies&galaxies

.

Inthegraphtimeiscurved,andbreakstimespaceintoclosedconservedpaths,timeclocksareinfiniteasLeibnizandEinsteinunderstoodwhenhesaid'Iseemtobetheonlyphysicistthatthinksthereareinfinitetime

clocksintheUniverse'.Chargesandmassesarethenccelerated,in-formativetime-perceptivespirals.

Allthosephysicalscalarspirals(ab.∆±¡)aretopologicallysimilar,differentiatedonlybythe'speedorfrequency'atwhichtheyclosetheir'time§paceclocks',accordingto5Dsimplemetricruleforall'familiesoftime§paceclocks':SizeinspacexTimefrequency=constant.SoLOGspiralsareallpervadinginnature,astheyrepresent

thedimotionofacceleratedtimeormainarrowoffuturethatincreasestheinformationofasystem.As'acceleratedtimespacevortices'theyaretheinformative3rdageofallphysicalsystemsinits∆±¡quantum,thermodynamicandcosmologicalscales–anattractivevortexofspace-time,whichisbyexpansionofthePrincipleofequivalencebetweenaccelerationandmass,toallscalesthephysicaldefinitionofanActiveMagnitudesourceofaforce…NOTasolidstaticparticle,a‘tiresome’errorofidealisthumindsthatkeeps

comingsincePythagoras.Asallwhatexistisamotionintimespace.StillspaceformsareaMayaofthesenses.Inthegraph,acceleratedvorticesoftimespaceinphysicalsystems,indifferentscalesofthefifthdimension:

charges,massesandthermodynamiceddiesbecomethenthemainclocksoftimespacethatcarrywithdifferentspeedsaccordingto5Dmetric(SxT=K),theinformationofmicroscopicquantumchargeworlds,human-size

thermodynamicscalesandcosmologicalgravitationalscales.SinceE=hƒ+E=mcc->M=ƒ(k);massisafrequencyofacceleratedinwardsspace-timeinits3rdage.Soarechargesandeddies,whichwillfinallyinitscentralpoint,

becomea‘conic’andascendand/ordescendlineallytheaxisinitsentropicdeath.

Thusspiralsrespondtothefundamentalpropertyoftimecycles:tohaveanarrowoffutureincreaseofinformationthatdiminishesitsspatialsizeaccordingto5Dmetric:SxT=K,thisacceleratinginwards,which

makesvorticesofphysicaltime(masses,∆+3,charges,∆-3),definitivelythetimeclocksofbothphysicalscales.Forthatreasontime-spacespirals,itssubspeciesandtransformationsareoneofthefundamentalspace-time

eventsoftheUniverse.

Logarithmicspiralsinnatureclassifiedbypentalogicfunction.Inseveralnaturalphenomenaonemayfindcurvesthatareclosetobeinglogarithmicspirals.Herefollowssome

examplesandreasonsintermsof∆STœevents:

Å(e):Feeding:Theapproachofahawktoitsprey.

Theirsharpestviewisatanangletotheirdirectionofflight;thisangleisthesameasthespiral'spitch.Å(i):Theapproachofaninsecttoalightsource.

Theyareusedtohavingthelightsourceataconstantangletotheirflightpath.Usuallythesun(ormoonfornocturnalspecies)istheonlylightsourceandflyingthatwaywillresultinapracticallystraightline.

Å(i):Thenervesofthecornea:

(thisis,cornealnervesofthesubepitheliallayerterminatenearsuperficialepitheliallayerofthecorneainalogarithmicspiralpattern).

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Å(æ):Thebandsoftropicalcyclones,suchashurricanes.Å(e:growthandreproduction):Manybiologicalstructuresincludingtheshellsofmollusks.

Inthesecases,thereasonmaybeconstructionfromexpandingsimilarshapes,asshownforpolygonalfiguresintheaccompanyinggraphic.

IdealSpiralsasrepresentationofworldcycles.

Ourplanet-starlivesinfactasafractalpointofaspiral,whichactsasourIsland-Universeinthelongestofourworldcycles–thatoftheplanetweexistwithin.Ourowngalaxy,theMilkyWay,hasseveralspiralarms,eachof

whichisroughlyalogarithmicspiralwithpitchofabout12degrees.Sothestarsgothroughthespiralinitsspace-timeworldcycleofexistence–froma30yearsoldgraph(:henceprimitivedigitalform,profoundhuman

thought):

Howlongisthelifeofa∆-1points,whichhas'fallen'insideanyoftheattractivevorticesofaspiralorganism?

Mathematicallyitmeansthatstartingatanexternalpointπ,ofentranceinthespiral,andmovinginwardalongthespiral,onecancircletheoriginanunboundednumberoftimeswithoutreachingit;yet,thetotaldistancecoveredonthispathisfinite;thatis,thelimitasθgoestoward∞isfinite.Thetotaldistancecoveredisrcosϕ,

whereristhestraight-linedistancefromPitotheorigin.

Soasitturnsout,thenumberofcyclesabeingcanturnaboutthespiral(frequencycycles)isinfiniteiftimespacewerecontinuous,thinnerandthinnerbutthereallengthofthelife-motionorworldcycleofthespiral(lengthtothecenter)ISfinite,andmoreoveritdiminishesinobjectivetime,asthelastphaseisshorterandfasterinfrequencyadeepfactaboutthedualityofobjectiveandsubjectivemeasureoftimeexistence.

Sinceanoldmanhasashorterlife,fallingfastindecayinitslast‘lifecycles’,whichifweaddtheinverse‘balance’ofitsinnersubjectivetimewhichisslowerthantheobjectivefasterdecay,makesthe3rdlifeeven

shorterinsubjectivetime,asakidhasmuchfastermentalcycles,sohisdaysarelongerinexperience.Thatis,at50intotalrealtimeactionswearemucholder;almostallourlife‘bits’ofaction-informationhaveconcluded.

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Thescalardualityofthelargerspiralorganismandthesmallerpartshowsalsothescalesoftime:theshort'frequencymoments',bitsandbitesofspace-timeactionswhichareDominantininformation,asthemotionis

inwardsandwilladdtothewholeworldcycleofabeing,causingits3rdinformativeage.

Ifwepostulateafurthergeneralizationofthespiralastheidealworldcycleforawholerangeorphylaofspecies,itshowsthatallbeingswillliveanddieafinite,similarquantityoftime-actions,makingalllives

absolutelyrelative.So3rdagepointswithinlog-spiralsshrinkinsize,increasefrequencyandshortenitslifespan,completingits3ages:S<T+S=T+S>T.

Spiralsasentropickillingmachines.

Eachinwardclosedpathadsabitofinformativefrequencyeverytimethebeingrepeatstheircycle,agingit.

Howspiralsendtheexistenceofonefractalspace-timeT.Œ?Asallworldcycleswithareversaloftimespace.

Thelifecycleisslowmovingtothecenter:T>S,anactionofinformationthatimpliestoreducethedimensionsofreproductivespace-widthforthoseofcyclicalmotion,tilltherelative∆-1fractalpointloosesallitsenergyandbecomes,ifthecenterofthespiralisaneye,abitofinformation,ifamouth,abiteofenergy,dying.

Alltimecyclestendinwardsina3rdageofwarpingandin-form-ationtilltheflowofmotion'stops'inthestillsingularity,ejectedperpendicularlyinanexplosionofdeath-entropy.Itisthelasttimequantaoftheclockwhen

motionbecomesa5thdimensionalstillimagebeforedyingintoalineal4Dentropicflow.

Soinitsdeathage,thereversaloftimehappens,asinmagneticfieldscomingoutofmasses,darkenergyfieldscomingoutofgalacticblackwholes,whenthelogspiraluncoilsfirstintoArchimedeanspiralasweada3rd

dimensionofheightinformation,thatconvertsitsshrinkingrevolutionintoanelongatedrecedingmotion,foranentitylivingwithinitsrevolutionthatbecomesdevolution,astheworldcyclekeepsacceleratingbutnow

accelerationisperceivedasanascensionandelongationinheightthaterasesitsin-form-ation.

Spiralsasinformativeeyes.

Humansexpressallthosebio-topo-logicfunctionsofthespiralwithahighlyefficientsynopticalgebraicmirrorimageof2numbersforthekeylineal&cyclical,parametersthatconstructthespiral:|xO=Ø.

Itssimplest=mostrealformisinpolarcoordinates,whichmeansspiralaretemporalself-centeredsystems,inwhichthesimplest'perception'isthatofthecentralpointofview,makingtruthaself-evidentGSTtheorem:thesimplestmathematicalformulationofaspace-timeevent/system,inoneofthe3relativecanonicalcoordinates,St(polar,sphericalinformative-head/particle),ST-Cartesian(hyperbolic,iterative-wave/body)andTs-Cylindrical

(linealfield/limb),definesthemainorganicfunctionofthesystem,asS-Form=T-function.

Soifanspiralequationissimplerinpolarcoordinatesitisaninformativespace-time,form/function,

DYNAMICSPIRALSASPERFECTSPACE-TIMESUPŒRGANISMSOF3STATES.

Thelogarithmiccurvecanbewrittenas:r=axebθ

Andthendependingonthevalueofbitwilltransformeitherintoacircle,oraline.Thederivativeofr(θ)isproportionaltotheparameterb,whichcontrolshow"tightly"andinwhichdirectionthespiralspirals.

Intheextremecasethatb=0(ϕ=π/2)thespiralbecomesacircleofradiusa.Conversely,inthelimitthatbapproachesinfinitythespiraltendstowardastraighthalf-line

Soaspiralcanbeconsideredaninformative‘state’ofafullorganism,whichcanconvertitselfintotheothertwostatesasapresentwaveanduncoilasalinealmotion.Againthisisacanonicallawofvital,i-logic

geometry:asystemcanbeconvertedbetweenthe3functions/formassystemsare'modular'anditsfunctions

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areconstantlychangingbetweentheactionsbetterperformedbylimbs(entropicfunction),bodies(reproductivefunctions)andparticle-heads(logic,informativefunctions).

Suchtransformationsarethestaplefoodofexistenceanddevelopment,beingthespiralandthetree,then2fundamentalSTcombinationsofS&Telements-butthespiralisthecommonestdynamicformthatallow

changeofstateswithease;whilethetreeisthecommonest‘fixed’stillsimultaneoussystemofO-|elements:

INTHEGRAPH,allspiralsare'potentialfields'orsinksandsources(+inwardlifevs.-outwardentropicdeathduality);whichexplaintheorganic,functionalandmathematicaldescriptionofspiralsasbio-topo-logicbeings.

Sothevitalspiralisinitsvitalmathematicaldescriptionthemostefficientorganicform=function:Agalaxycoilingitsflatirregularyoungformawormcoilingtosleepinits1/3rdstillinformativeSS-inwardsstate

(completedwithitsfeeding,entropicstate,andwave-likemotion)areinspiralstates;whichisultimatelyanextremalcaseoftheupperandlowerlimitofpithatleavesalwaysanopening,dynamicmouthtothespiral

allowingitssimplerbeatofexistence,closinginwards(T-State)andoutwards(S-tate),neverendingtheperfectpicycle.

Somostvitallivingspiralscanuncoiltobecomelinealforms,orcanclosetobecomesphericalcircles,managedbyitscentralhead(1D:informativestate)thatbecomesitsfutureilogicforwardshead(2D/5D:locomotionand

feeding-hunting).SothespiralcanbeconsideredanØ-intermediatepresentsystem,perceivedfromtheperspectiveofitsdominantcentralpointofviewinpolarcoordinates(r,θ).

RECAP.Varietiesoftimespiralsinpentalogicdimotion.

Spiralsrepresentthe2complexdimotionsoftime-spaceinaclassicS=Tfemale,S<T>Smalesymmetry:

S=T:Reproductive,communicativeSpiralsareArchimedeanorFermat'sbasedintheFibonacci'sgoldenratioconstantinwhichseveralbranchesofthespiralallocatewithmaximalefficiencythereproductionofnew

'infinitesimalparts'ofthewhole.

S<T>S:Entropic+Perceptivespirals,(asthoseoftimevortices,chargesandmasses)whereaflowof∆-1pointsfallsintotheperceptiveordigestivecentralpoint,allowingbothapotentialandvortexdescriptionofthem.

BasedinthoserulesthatvitalizethemathematicsofSpiralsasessentialelementsofafractalpointinitsdimotionsofperceptionfeedingandreproduction,wecaninterpretbetterthemathsofspiralsinvitaltermsanddefinemultipleorganicbeingsasentities,whichinitscoiledpositionbecomeorganicspirals,oftenwhen

resting–processingitsinternalsystemsofenergyandinformation.

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2ndAGEOFGREEKGEOMETRY:TRI-ANGLES.

MENTALSPACESMEASUREANGLES=SPACESIZESANDDISTANCES=TIMEMOTIONS.

Whatweneedtoknowofabeingfromanouterworld?Obviouslyitsrelativesize,givenbytheheightdimensonmeasuredbyanangle,anditsdistanceintime0spacemeasuredbylengthandmotion-speed.Botharethe

concernsofthesimplestfirstmindspacesdevicedbygeometry,thoseofatrigonometricspace.

Everylanguage-mirroroftheUniversestartsslowlyrisingitscomplexityandfocusonrealityasitimitatestheintrinsiclawsofspacetimereality,eventhoseasmathematics,whichdescribeinacloserformitsspatialpoints,

scalarnumbersandtemporaloperands.SodidgeometryinGreece.Assuchlanguageswillfollowsimilarpatternsofgrowthandevolutiontothoseoffullfractalpoints=T.œsstartingbytheestablishingsequential

pathsofDimotions,whichalwaysmeanfirstthefractalpoint’semergenceinalargerworldwhereitwillopenits‘eyes’andmeasure.Whatitmeasurestomakeitusefulforexistenceisthustheminimumthelanguage

providesoninformation.

Inthecaseofgeometryiswhatallspaceswillhaveincommonforthepointto‘see’itssurroundingworldsanddistinguishform(angularmomentum,cyclicalshapes,membranes)andmotion=distance,linealmomentum.

|&Oeveninitsmostsimplifiedway,whichistomeasureanglesofcongruenceanddistancesofpredatorsandpreys.Orelsethelanguagewon’tbeefficient,itsspeakerwilldieandthesystemwillnotrepeatthelanguages’form.Soparametersinlanguageshaveavital¡logicmeaning,formindstoacquireknowledgeinitslogicmirroraboutthesurroundingsofitsworldandbeabletomoveandassessthedistanceandsizeofthings.Evenin

music,itssimplestduality,trebleandbassbringe-motionsofuneasinessandrelaxationastheyconnectwiththeDopplereffect:apredatorwavecomingcloserblueshiftsintotrebleandgoingawayintobasswaves.Thusgivingafirstinformation,withnoangle,onalinealpathmotion.Trigonometryaddedthentodistance(S=T

dualityofmotion),angle.Sothefirstparametersof‘humind’geometryweretrigonometryanddistancewhichunderthes=tparadoxcan

alsomeasureapointinmotionoverabackgroundscaleofspace.Angleisalsothefirstfractalunitofmeasure,whichcantravelthroughscaleswithoutdeformations-hencea

dimensionlessparameterasfractalscalesarerelativeinitsdimensions,‘erasing’theminternallyastheyemergeasanEuclideanpointintoanewscale.Soangleanddistancegaveus∆Stinformationandallowsatrigonometricmindtosurvive.Anglethusallowustodefinemathematicallyadimensionasfollows:

Supposethereisamathematical(geometrical)quantityA,whichdependsonthescale,l.Ifafterascaletransformation:l→λl,thequantitywouldtransformsas:

A→λnA.ThenthequantityAhasdimensionn.Accordingtothisdefinition,theanglehasdimension0;becauseitisthefundamentalparameterthatcarriesthroughscaleswithoutchange.

1D:sine<cosine.Anglesofperception

Sotheangleisthefirstoperandonasurfaceofspace,toprovideinformationtotheself-centeredsystem.Anditsimportancewassuchthatthetri-anglewasnotcalled,thetrisideortriline(theobjectivetopologicview)butthementalspatialview,showingoncemorethattheego-paradoxisalwaysthebeginningoanyscience.Thatis,thetri-anglewasfirstconsideredamentalsubjectivespaceform,beforeitbecameanouterobjectivesurface.

AnditbearswitnesstoourNatureaslightspace-timeorganisms,evolvedfromtheminimalspeciesofEuclideanLightspace-time,thePlanckton,thatwhenwefinallydiscovered‘¡ts’form,inquantumphysics,h,turnedoutto

beexactlythat:anangularmomentum,quantizedintwogenderspecies,thefemalebosonandtheS<T>Sfermion.

Thepentalogicdimotionalfunctionsthatentangleitstrigonometric‘functions’intoreality:

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@:itsdominantuseandfirstreasonitbecamethefirstdevelopedfieldofmathematicsisitscapacitytomeasurefromapointofviewdistancesaccordingtoratiosandparallax,whichis

theoriginoftridimensionalperception(bilateraleyes),andFertileCrescentmathematics.

Howthisworkinitssimplestform,needstounderstandhowa'spherical,idealmind-membrainof3πdiameters,and0.14Dapertures,allowsamindtoperceivethroughthem,'rays'todistantobjects.Themindthuscanalways

measuretheanglecoveredbyadistantobject,andwithaminimaldisplacement,anewangle.

3rd-centuryastronomersfirstnotedthatthelengthsofthesidesofaright-angletriangleandtheanglesbetweenthosesideshavefixedrelationships:thatis,ifatleastthelengthofonesideandthevalueofoneangleisknown,thenallother

anglesandlengthscanbedeterminedalgorithmically.Thesecalculationssooncametobedefinedasthetrigonometricfunctions.

1D:Sotrigonometricfunctionswerethefirsttoappear,as1Dperceptionisalsothefirst'action'ofanemergentpointwheresinesandcosines’operandsextractfortheself-centeredpoint,1Dperceptionofits‘angles’and

‘S=distances=T-motions’.

Thegraphshowsthetrigonometricfunctionsandtheinformationextractedbythem;andasusualingeometryweapplythemtheDilogicsymmetryof‘internalsubjectivespace’and‘externalobjectivespace’;astheycanbe

appliedfromthefractalpointtotheouterworldorfromtheouterworldtothefractalpoint,whereasinpentalogic:

-1D:S:Sineisbesttoassessthe‘informativeheightofthesystem’.

-2D:T:Cosineisbesttoassessthelength-motionofthesystem.

-3D:Tangentisbesttoassestheenergy,S=Tofthesystem.

-4D:Angleisbesttodefinethescalarproportionalityofthesystem.

-5D:Itsinversefunctionshaveanentropicroleinthecalculusofallthoseproperties.

SotrigonometricfunctionsalreadyextractfromaT.œinformationofits5Dimotions.

Trigonometrythusdealswiththelawsofperceptionofthecircleandthetriangle.Andasusualsincerealitybuildsupfromthesimplestlinealessencetoevolvelatterits‘3rdagecyclicalform’|>O,thefirstlawsdiscoveredwerethelawsof|-∆triangles,whoseS=Ttransitionintoaspherewhenitaddsadimensionofð-cyclictime,isitsrotation.Whereasthe‘areadifferencebetweenaregularpolygon-triangle’anditssphere,isthefirsts=ttransformedvalue,whichturnsouttobehuge,between∞and+2,4,whileitsperimeterits1,6timeslarger,makingthemclearlythe2extremalTT&SStopologicformsofmotion,thearrowtriangle,andform,the

rotatingcircle;factsthoseresolvedwiththeLawofsinesthatcalculatefirstforanytriangleits'aperturesofperception’:

IfA,B,Careanglesofatriangleanda,bandcarelengthsofoppositesidestherespectiveanglesthelawofsinesforanarbitrarytrianglestates:

whereΔistheareaofthetriangleandRistheradiusofthecircumscribedcircleofthetriangle.In

termsofperceptionthus,eachofthe3'apertures'ofthe3D¡sphereanditssineviewoftheexternalworldis

‘Male&femalespin’sculptures

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equivalent,ensuringanon-distortedperceptionofdistancestotheouterworld.Itestablishesthesamelawofequalperceptionofthose3'membrain'pointsofitsinternal'vitalenergy'.

Theangleisthusthefirstformofexistence.Thefirst1Dperceptionthatilluminatesrealitybybendinghyperbolicallyalargerworldintoacentralsingularity,infixedquantumforms,whichwillbecomeforthe1stparticles¡tsh,aradianofangularmomentumperceivingalargerworldtunneledintothesingularityofits∆-1scale.Creationstartswiththeangularreductionofrealitytoastillforminaquantumangle.Assuchtheanglebecomeswithitsradianhvalue,thefirstT.œ,thetriangle,whichthenasitturnsaroundgatheringin6groupsformsanhexagon,withawobblingradianarcthattessellatestheplanefromcirculartohexagonal

STœps.

Vitalmathematicsbecomesthenreality.Suchvitalcirclehas6apertures-thoseofitstriangles-eachoftheradianarcsthatmakeincircle6stepsforapi=3,closedinhexagonalform,openedincircularone.Imperfectformsthusappearfromtheoriginalperfectionofthelanguageimprintedintheregularitiesofthemedium.

SothehexagonalcirclegameofStœpsbecomestheessentialstructureof2Dimensionalspace,withthose120hexagonalanglesrepeatedadnauseaminNature.Itisthegamethatwillgrowinpolytopestoreachthrough

combinationsof5,6,120,thefinal120dodecaplexand600tetraplexmaximalfiguresof4Dimensionalgeometries(aspentagonalpolytopesmakenosense,beingthe5thdimotionentropythedestructionofthe

regularin-form-ationsharedbythemall).

Anglesofperceptionbecomethenthefirstsimplegeometricelementofrealityas1DimotionsappearoneverystillpointthatgainsanangleandstartsthecreationofthegalaxyinitssmallestT.œ,Planckton

Thelawsofthose‘anglesofperception’wouldalsobethefirsttobeassessedwhenanewemergentlightspacetimemind,thatoftheGreeksappeared.Aseachemergentscalerepeatsthegameofexist¡ence.

Anglesofperceptionindifferentvitalandabstractgeometries.

AnglesofperceptiondefinethecapacityofapointofviewtomeasureandobtaininformationfromtheexternalUniverse,suchastheclosestangle,thelessperceptive(5D)asystemis,andthemoredarkspacewillhave.Andsotheyplayalsoavitalrole.

Theyalsovaryaccordingtothetopologyoftheworldwelivein.Sotheyare:

Minimalinhyperbolicplane,ashyperbolasaretheentropiccurve;sotheyminimizetheinversearrowofinformation.

Maximalinelliptic,sphericalgeometrieswhoseanglesaregreaterthan180,assphericalbeingsareinformative.

Mediumintheflatplanewithanglesat180.

Thevitalaxiomsofgeometry.Reproductivemotions.Discontinuitypostulates.Attraction.

Soaseriesoflawsofanglesofperceptionwillbecomethefirstsetofconsistentlawsofcreationofmentalspacesdefinedbyhuminds.Wearehereconcernedasalwayswiththetranslationandcorrectionofthissystemofaxiomsofabstractspaces,forasingleplanetime-spacetothemuchvaster,paradoxicalrealworldofvital

ilogictopology.

WhatisthemaindifferencebetweenEuclideangeometryandfractalnon-AEwhenconsideringthoseaxioms?

-FirstwenoticeaswedidwiththeearlierversionofEuclideangeometry,thattheaxiomsaregroupedalsoin5elements,sinceevenwhensimplifyingmirrors,thehumindmustrefertothe5Dimotionsofreality.

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-ThenweobservethatthedifferencesbetweenE-postulatesandaxiomsvs.¬Æwillariseasalwaysofthe‘simplification’ofmultiplescalesintoasingle‘continuum’,andthe‘straightening’ofrelativeanglesthat

‘change’itsformintoahyperboliccurveaswe‘diminish’sizeinthefifthdimension.

WethusfirstupdatedtheelementsofEuclideangeometry-itsdefinitionsofpointsnowfractal;lines,nowwaves,congruence(nowrelativeequality)andplanes(nowtopologicalnetworks).Aswhattodaypassesasthe

foundationalaxioms/postulatesofgeometryareonlycorrect,forthelimitedviewofasinglespacetimecontinuum.Sotheyareapproximationswhichstrippoints=T.ŒSoffractal&vitalproperties,andsimplifyits

vitalimperfections.

ThisprocesscarriedbystepsfromLobachevski’sapangeometrytotheabsolutegeometryofBachmann(1970s)closedtheevolutionofthediscipline.Anditwasfoundthatonlythemostgeneralfeatures,anglesanddistancesareneededtodefineageometry.specificallywecouldrejectasBachmanndidthepostulateof

‘continuity’,andDedekind’sconceptthatrealnumbersAREnumbersNOTratios,becausetheyLIEontherealline,asanecessaryfeatureofarealgeometry,sinceindeedintheabsolutepangeometryofthefifthdimension,

anyplaneofrealityisdiscontinuous,andthenumbersfitinthereallinetofillthediscontinuitiesofsimplernumberfamiliesareNOTinthesameplane.StillwhatwecantakeoftheAxiomaticmethodeitherinitsEuclideanversionormodern‘FoundationsofGeometry’(Hilbert)istheclassificationofthefoundationsofgeometryin5subgroupsofaxiomsthatroughlycoincidewiththe5elementsofrealityandnowweshall

commentontorealizethedifferencebetweenasingleplanecontinuousgeometryandtherealgeometryof5Ddiscontinuousplanesandanglesthatbecomehyperbolicaswesinkintotheminds∆-1singularity.

Thosearethemesthusconcernedwiththe3rdageofGreekGeometryanditseclecticattemptstofoundthedisciplineasaUniverseinitselfdetachedfromtheearlierPythagoreanandArchimedeanpracticalages.

3RDAGEOFGREEKGEOMETRY

Archimedesvs.EuclidExperimentsvs.axiomaticerror

TthewrongegofcysideofGreekGeometry,itsits‘3rdeclecticoldage’asEuclid,committedthesameerrorthanmodernmathematicsinhis3rdinformativeage,whensettheoryandtheaxiomaticmethodabandoned

experimentalproofandtriedto‘construct’fromthetopwithoutprooftothebottomunconnectedofreality.

Somath failed from then on to understand itself as an experimentalmirror of scalar space, becomingwithEuclid’saxiomaticmethodaself-containedtruth,whichitisbecausefractalmirrorsare‘entangled’informtoitsoriginalimage–theUniverse.Yetasmirrorsarekaleidoscopic,inflationaryasLobachevskiandGödelprovedformathematicians but physicists so far have failed to recognize, only experience validates what image is real.Schopenhauer,thebestofthephilosophers,notedalreadythefallacyoftheaxiomatictruths,whichrelyontoomanyself-evidentlies,pointsandlineswithnobreath,puttinghiswarningontheaxiomofcongruence,whichinfactisthemostexpandedby5Drules.

But once axiomatic methods with single Aristotelian languages became definition of truth, this error willcontinuetobecomethefundamentalego-tripandfailureofmanyhumansciencesaccustomedtodaytoacceptaprioripostulates that set the ‘entropic’ limitsof truthand scopeof thepostulate, introducingbias from itsinceptions–fromEconomicsthatrejectstheanalysisoftheevolutionofmachineswithdoinorganiceconomicsand accepts as a self-imposed postulate the goal of GDP growth in a limited planet to c-speed limits in theastrophysicalrealmbeyondthegalaxy,totheentropyarrowonlyofbig-bangtheories.

ThemanyfalseassumptionsofclassicEuclideanandaxiomaticgeometry.

Letus‘continue’thenthecriticalanalysisoftheAxiomaticmethod.Generallyspeaking,theaxiomscanbechoseninvariousways,takingvariousconceptsasstartingpoints.Hereweshallgiveanaccountoftheaxioms

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ofgeometryinaplanewhichisbasedontheconceptsofpoint,straightline,motion,andsuchconceptsas:ThepointXliesonthelinea;thepointBliesbetweenthepointsAandC;amotioncarriesthepointXintothepointY.(Inourcaseotherconceptscanbedefinedintermsofthese;forexample,asegmentisdefinedasthesetof

allpointsthatliebetweentwogivenones.)

Theaxiomsfallthenintofivegroups,ohyes,thefamiliarnumber!Aswecanrelatethemtothe5elementsof¬∆@stofspacetimebutitspostulatestobeconsideredtruthonlyinthelimitofstillbidimensionalgeometryintheplane,andsoweshalladaptthemwithcomments(noneedtocreateinthisintroductorycoursesatrue

formalpentalogicsystemofaxiomsandpostulatesofitsowntofullydevelopnon-AEphilosophyofstienceanditsstientificmethodandepistemology.(:LOL,thiswasdone30yearsago,andIusedtowriteonlywithilogicsymbology,sofaroutfromwhathumindsdo,thatIonlygotinterestedaconceptualartwhousedthose

equationsforaexhibitatsix-flags,aBrooklynmuseum:)

Andobviouslywearenotusingtheformalismofsetsandmodernmathstoexplainthosecorrections.

Werepeatthisisanintroductorycourseandwejustwantthereadertounderstandthatpresentmathsisanabstractionofamuchmorebeautifulexperimentallanguageofscalar=numericalpointsofspace-time.

Thissaidthe2firstgroups,Axiomsofincidenceandorder,andtheconceptsofanglesandlaidon,refertotheentropiclimitsandinnerstructureofagivenmentalspace,sotheyshouldbegroupedas¬Spostulates.

I.S:Axiomsofincidence

Axiomsofincidencearemainlyrelatedtothe2ndpostulateoflines-wavesofcommunication,itsboundariesandanglesofincidence=congruence(alreadydiscussedinthe4thnon-Epostulate).Soitalsoconsidersalsothe

relativesizeofT.œsanditslineal,openflowsofcommunicationcomparedtoitsworlds;oftheminimalinternalpartsaT.œneedstodefineitspropertiesasan|-linealorO-circularelementandsoon.Theyconnectthus

closelyasboundaryaxiomswiththeconceptofatopologicalboundaryandtheentropiclimitsbothinspaceortime(ifwetakealineasaworldlineofdurationofabeing).AndthemaindifferencesbetweentheclassicaxiomsofE-geometryandi-logicgeometryregardthemultiplicityofmeaningsofi-logicgeometrywhere

distancesandpointsarelogicconceptsofsimilarity,allsystemsdohave‘boundaries’=limitsasinfinitiesarerelativeinfinities(∝)andsotheinterpretationofitsmodernaxiomsofincidencefullychanges.

1. Oneandonlyonestraightlinepassesthroughanytwopoints.

2. Oneverystraightlinethereareatleasttwopoints.

3. Thereexistatleastthreepointsnotlyingononestraightline.

Aswecanseetheaxiomsofincidenceestablishtheminimalelementsrequiredtotransitfromasinglefractalpoint intoasocialgroupofpoints,a line.Butasusualwiththeaxiomaticmethodthat tries toprove itselfby‘reductionist’ simplicity of its postulates it doesNOTbring enough information to definewithout ambiguity aline.Becausewithonlytwopoints,theintermediateregioncanbecurved,sonostraightlineexists.Howcanweknowwithonly2pointsthecurvaturebetweenthem?Wecannot.Andiffactbecausewecannot,theerrorsofthe5thNon-Epostulatethatcurvesstraightlinescomefromthoseimprecisedefinitions.

So the first axiom of incidence defines NOT a straight line but the boundaries of a ‘flow of communicationbetween 2 geometric forms. It is thus a definition of ‘any’ 2nd non-e postulate line-wave of communicationbetweentwopoints.

And italsomatches in theevolutionofgeometry intotopologythemoreprecise termofa ‘boundary’ foranopenlineofcommunicationbetweentwoentangledpoints–thepointsarethebeginningandtheendofthatflow.

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Thisisthenthemeaningofthefirstandsecondaxiomsofincidence,whilethe3rdshowstheoutsideworldtobelarger,thantheflowofcommunicationbetweentwopoints.

Theweaknessofthosepostulatesthoughhasbeenalreadynoticedbyclassicgeometers. Indeed, itappearedsomewhatstrangetothemthatastraight linehasonlytwopoints.Surely inourideaofa linethereareeveninfinitelymanypointsonit.NowonderthatnotevenEuclidstatedtheaxiombecauseitISNOTproper,showingthatasmoreidealismandpostulatesoftruthtrytoprovebysimplification‘absolutegeometry’,theyjustaddhalftruthsandnewdistortions.

2pointsdoNOTdefinealine,butanyopentrajectory,alsoanarch,azigzagbutNOTacircle.

Sotheincidencepostulatesareimportantnotbecausetheydefinealinebutalargermoreimportantconcept–anopenflowoffeed-backcommunicationbetweentwoequalpoints,unlikeacirclewhichisahierarchical

closedcommunicationbetweentwopoints,oneofwhich‘turnsaroundtheimmobileone’.

Sotwopointsdefineanopendemocratic•=•flowofcommunication,whiletwopointsinacircledefineaclosedhierarchicalcommunication.

Sothedefinitionofanopen‘wave’offractalpointsblowsuptheconceptof‘straightness’.Thisisokaslongasweknowthatwhatmattersofastraightlineisnotthatitcanbedrawnlineallyonaplanewithanabstractrule,butsomethingelse–tobeopenina‘topologicalway’,thatis,wecandeformitasageodesiconasurface,a

chordofacircle,orwhatever,aslongasitisopenandthetwopointsareitsboundaries.

Thusthekeyfeatureofthelineistobeopennottobestraight,asthelinecurvesinahyperbolicangleenteringan∆-1fractalpointreasonwhymodernphysicsusecurvedlinesto‘crowd’theentranceofafractalpointofthe∆-1gravitationalscale,insteadofblowingthosepointsasfractalpointsthroughwhich∝straightlinescancross.

Oncethisisclarifiedwefind3canonicalvarietiesofopenlines:astraightlinethatneeds3pointstodefineitsstraightnessandtypeofinformationcommunicatedthroughaflowofmiddlepoints,whichwillformawaveaseventhesmallestbosonhasvolume.Atruestraightlinethenhappensmerelywiththeentanglementoftwofermionswithagravitationalnon-localquantumpotentiallinethatmerelygivesinformationondistanceand

likelyangle(aneutrinoorstringofthePlanckscale,maybebeingbothsimilarasneutrino‘angleofscattering’isonthatrange).

Itimpliesthenthateventhesimplestlinerequiresa3rdbitofinformationaboutthetypeofflowofinformationandshapeofthetransversalwavebetweenbothpointshappensincenow1+1=3.Asaflowof‘smallerpoints’

transitbothpolestobecomethemotionthatcarriesinformationbetweenthelimitingpointsoftheline.

Finallythe3rdaxiomofrelativeangleofincidenceisconcernednotwiththeinnerlimitsbutwiththeouterextensionoftheworldinwhich2pointscommunicate:thereareatleast3pointslyingoutsidealinemeansthatthereisalargerUniverseoutsideaworldline.Sincewecantraceaplaneoutsideofit,whichmightormightnotbeparallel,orintersecttheline,as3pointsnotlyinginastraightlinedefinedaplane.Sooutsidealinethereis

alargerworldwithanewdimension–atopologicplane,anorganismtowhichthelinemightbelongifintersectionhappens.

But3pointsDONOTnecessarilydefineaplane.Againsuchdefinitionconstructsaplanebyself-reproductionandthismustbestated,aswedidwith2pointsthatconstructalinebyself-reproductionofaflowof

communication.Itisthenwhentheaxiomaticmethodmakessense.Andsoanotherwaytoreinterpretthoseaxiomsandgivecredittothemistowritetheminvitalreproductiveterms:

‘Onlytwopointsareneededtostartthereproductionandtransmissionofinformationbetweenthem’,etc.

Furtheron3pointsnolyingonastraightlineasalwaysinDualitycanbeexpandedinitssolutionsconsideringthedifferent‘ternaryprinciplesofcreation’.Soaswesaidthat2pointscandefineeitheranopen‘line’ora

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‘closedcircle’,whenoneofthepointshasmotionrespecttotheother,forminganStsystem,asopposedtoanS-Sline,3pointscandefineaclosedtriangle,oranopenangleofperception,whichini-logicgeometryis

different.

Andthenwecanalsoapplythefractalmethodofdifferentiation,ifweconsiderthatoneofthosepointsisinadifferent‘∆-scale’.Thenthetrianglewithhaveascalar5thdimensionaladdedparameterthatwillprovokethelinestocurveintoahyperbolicangletoreachtheinnerdepthofits∆-1mind-points,whichdoesNOTlieinthe

sameplane)aternarysystem.

Thereforetheneedclearlyarisesini-logicgeometry,forpentalogic3±¡¬∆@STdiversificationofthesimplexsinglelineorplaneofcontinuousgeometry,statingaccuratelyandexhaustivelyeverydistinctversionsimplifiedbyE-geometricpostulatesthatreducesallopenvs.closedgeometricobjectstostraightlinesandflatplanes.

Soinsteadofmakingdefinitionssimplerbutlessclear,whichisparadoxicallywhatHilbertachievedwewillmakethemmoreclear–eventhoughthisintrowon’tbeexhaustive.

Inthepreviouspentalogicconsiderationsthoughwehavealreadyestablishedthe|-lineof2pointsthatneedsa3rdmovingpointofcommunication,theO-circleoftwopointsthatneedsa3rddimotionforthesubmissive

orbitalpointandthe∆-hyperbolicangleof3pointsoneofthemin∆-1scale.SoweseethatdualitytheminimalS-TorSSorTTrealityoftheholographicUniverseimmediatelyreproducestrinity.

Andsowecouldalsoresumethenewaxiomsofincidenceas:

‘2pointsgenerateathirdpointtoformastabletrinitystructureofspace-timethatmightbeanopenlineofcommunicationbetweentwoequalpointsstilltoeachother,hencea‘2number’,entangledbythe3rdmoving

flowbackandforthbetweenthe,(|-duality)oracirclewhereasonepointisaS-tillpointandtheotheranorbitalð-pointwhosemotionbecomesits3rddimension(S-sTSystem),orahyperbolicø-anglewhereastheStillpointis

inalowerplaneofthefifthdimension,S¡-1<SS¡.’

Thiswouldbeamoreformaldefinitionoftheincidenceaxiomsofi-logicgeometry.

Thesameprocedureofexhaustiveternary-pentalogicdivisionofvarietiesofE-axiomsappliestoalltheotherconceptsandaxiomswheni-logicgeometrygoesdeeperintothem…Butwewon’tbesoexhaustive.Justpass

fastthroughthem,observinghowtheyareconnectedtooneofthepentalogicelementsof¬∆@st.

II.¬:Axiomsoforder.

Whenwedealwithhierarchicalorderandperpendicularity–‘layon’conceptswearedirectlyconcernedwiththecongruencedifferencesbetweentornsystemswhichshareacut-pointandthosewhoareparalleland

‘avoid’tearingandDarwinianperpendicularitybestablishingabumpinanewdimensionofheight-informationbetweentwolines:

1.Ofanythreepointsonastraightline,justoneliesbetweentheothertwo.

2.IfA,Bare2pointsofastraightline,thereisatleast1pointConthelinesuchthatBliesbetweenAandC.

3.Astraightlinedividestheplaneinto2halfplanes(i.e.,itsplitsallthepointsoftheplanenotlyingonthelineinto two classes such that points of one class can be joined by segments without intersecting the line, andpointsofdistinctclassescannot).

1staswehavealreadyexpandedtheconceptofalineasanopenintervalwhichdoesNOTreturntotheoriginalpointdirectlybutthroughabackandforthmotion,theaxiomsoforderONLYhappeninaceterisparibusanalysisofone-dimensionalAristotelianA->Blines,andassuchitshouldbescrappedalltogetherasall

communicationisdialecticandsoA->B->Aistheminimalunitofaline,with2inversedirections,asseenintheNeutrinotheoryoflight,wherelightiscreatedbytwoentangledneutrinosmovingbackandforth,ininverse

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directions.OnesingleentangledflowallowsthepointAtomeasuredistancetoBandmaybetrigonometricangle-height-size.Butunlessthatinformationisreflectedback,therewillbealostsingleeventhatdoesNOT

remainasastablememoryline.

TheaxiomoforderonlyappliestoA->Bfleetinglines.Yetasalllinesarestepsofalargercyclicalpart;A->BalwaysendsinA-B-A-B….,andthenthefeed-backrealstablelinescannotsetaforward,backward,order,after

afewfeed-backentangledflows,neithertocircles,wherewecannotaswereturntothepointestablisharelativeorder.ThuswhenCcomesbeforeDorafterAinthecircle?TheorderestablishedinA-Blinesthusisadifferentoneasinthecircle,oneofhierarchybetweentheAB‘fermions’andtheCflowsbetweenthemwhichisofalower∆-1quantumpotentialorder,andthatistheproperhierarchy.SohappensincirclesbetweenO-point(center)andA(turningpoint),ascirclesareNOTdrawnbyasinglelinewithoutcenter,whichwillnotbendbuthavelinealinertia,butbythetorqueofferedbythecentraldominantpoint.SotheorderisAC

dominatesac…andOdominatesA.

Itistheconstantrepetitionofthosecircularorfeedbackvibratingmotionswhatcreatetheillusionofastablespacelineorcircle,apresent,simultaneousspace.Furtheron,whatthecircledoesprovideisnotanorderofpointsbutachirality,dependingonthewaywemovearound,whichthelinedoesNOTrequireinitsfeedback

A<ab>Btrinity.

Motionthusbecomesanewcreativedualityorderincyclicalpathsthathavechirality.

Andorientationalso.ItisnotthesameorderifwemovefromA–>cbd,orfromA->dbc.

Andthisisthenagainthetruedifferentiationbetweenopenlinesthataremore‘equalitarian’andcircles,whichconstantlybringnewhierarchies.ButastheUniverseseeksforbalanceweshallseethatparity&chiralityallows

gendermirrorsymmetry–itisinfactthefirstformofit;sincetheO-pointwillseektwo‘spiralarms’withinversechiralitytoachieveabalancewith2differentspinfilledorbitalpoints.

Thusaswemoveintothesimplestgeometricscalesofvitalgeometry≈physics,weshallfindthatconceptsaschirality,achirality,parityandthecombinationsoflinealandcyclicalmotionsdiversifythedifferentspeciesofrealityandbecomeessentialtounderstandthesymmetriesandasymmetriesofquantumphysics,becauseINFACTorderdoesNOTexistinacategoricalmannerwithoutaproperunderstandingonwhatbecomesthemost

importanttypeofAxiom,thoseofmotion,whichdefinedproperlyinEuclideanGeometryareAxiomsofreproductionofform,bycarryingageometricformtootherregionoftime-space.Yetbeforewestudythem,

wemustconsiderthesecondfundamentalconcept‘laidon’theAxiomsoforder,thatof‘layon’itself.

IntheentangledUniversethiskeyconceptlayon,isrelatedtothefalseaxiomsofcontinuityandtothedualityofDarwinianvs.parallelsocialevents.Soweapplytheternary,pentalogicmethodanddifferentiateatleast3

waysinwhichpointslayontoeachothercuttingtwoformsorfusioningthem.

Inthegraphsononesidethecircularandelementarycontinuityprinciplesstudywhen2systemsareperpendicular,thatiscancuteachotherandshareapoint,orareparallel,thatis,only‘contact’eachother

byadjacencybutdoNOTcuteachother.

Aswestressthediscontinuityofreality,werenamethoseprinciplesaddingthe‘dis’prefixtoitsclassicformulations:

ELEMENTARYDiscontinuityPRINCIPLE:Ifoneendpointofasegmentisinsideacircleandtheotheroutside,thenthesegmentintersectsthecircle.

CIRCULARDiscontinuityPRINCIPLE.Ifacircleyhasonepointinsideandonepointoutsideanothercircley',thenthetwocirclesintersectintwopoints:

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Inthegraph,thecontinuityprinciplesareinfactlimitingconceptsofboundariesandlawsofperpendicularity,whichdefinethediscontinuities,closenessandconnectionsbetweennetworksofpoints.WhatmattersthentorealityisNOTtheobtuseconceptofa‘block/Parmenideslike,solidreality’,withnogaps,atthecoreofthe

‘mindillusions’ofHilbert’scategoricalgeometry,butwhentwosystemsofrealitycuteachotherinDarwinian,perpendicularevents(thesegmentbreakingthecircle,thecirclebreakingthesegment),whichwillDEPENDon

who‘owns’thepointM?

T>S:ItisMpartofthecircleO?IfsoOisfeedingonA-M-B,theline.OritisMbelongingtotheline?

S<T:Thelineis‘killingthecircle,whichisnowopenatM.OritisMtheM-outhofthecircle?ThenAmbisoneofthemultiple‘parallels’(asitdoesnotproperlyintersect)feedingthecircleandtheO-perceptivepoint.

S=T:2Points:OritisM–andthisisthemostspecialcase,in‘BOTH’,thelineandthecircle?ThenMisanattractivepointthatcementstheUnionbetweenbothandareproductiveactionishappeningasnowthepoint

isactuallytwopointsatonce,belongingtoeachofthem.

Thosearethevariationalternaryprincipleappliedto‘layingon’,theundefinedconceptoftheAxiomaticmethodweupgradetoi-logicgeometryaswedidwithotherundefinedHilbert’sconceptsofpoints,whicharefractalpoints,lines,whicharewaves,congruence,whichisrelativesimilarity,Non-E5Postulate,whichisthe

definitionofamind,and¬EPlaneswhicharetopologicorganisms.

Sotheconceptlayon,isthefinalelementthatcompletesthe4thpostulateofcongruenceasitbecomeseither‘areproductivesuperposition’or‘aDarwinianintersection’with2solutions,substitutingDedekind’scontinuity

axiom:

“ApointMofintersectionbetween2relativepaths,ð(closedfigure)and$(openfigure),eitherbelongstoðor$ortobothfigures.Ifitbelongsonlytoðor$,eitherfigureisthepredatory,dominantelementofthe

intersection,andtheeventwillbeaDarwinianspace-timeevent,inwhichthesubmissivepastpreytears,extinguishingitsform,asað<$orpastflowprovidingtheenergyforthereproductionS=Torevolution$>ðof

thedominantfutureflow.Ifthepointbelongstobothfigures,theeventisapresentiterativeeventofsymbiosis,andbothsystemscanformastable,socialnewwhole,wherethepointdoublesastwo.”

SowetranslateabstractaxiomsintothevitalhowofperpendicularDarwinianintersectingorconnecting,unitingmotionsthatdefinedualcreativeanddestructivespace-timeactionsforbothsystems,asamotion

‘transforms’oneformintoanother,orpegsthemintoamorecomplexcreation

Wearenothereusingformallanguage,thoughanymathematicianorphysicistcanwriteitwiththeusualsymbolismofclassiclogic,andnoticeafewfactsthatexpandtheconceptoflayingonandshowitspowertodescriberealityreasonwhyisusedtosetthefoundationsofotherkeybranchesofmathematics(Booleanlogic

andsettheory):

-2systemslayoncanbeeitheraUnionoranIntersection,differentconceptsinadvancedi-logicgeometry:

AUnionisaperpendicular,Darwinianeventwherethepartofoneentitynolongerbelongstoit,sothedynamiceventdestroysonepart.Thusifasarulewecapitalizethedominantsystemofadynamicspace-timeeventofrelativeperpendicularitywecanwrite:AUb=A,meaningthatbloosesitspartwhichwillbelongtoA,

aswhenyoueatarabbitthatnolongerisarabbitbutbecomesyouraminoacids.

OntheotherhandanIntersectionwillbedefinedasatruesharingofthosecommonpoints,soneitherdominates,ACB,meanstheCpartisnowtheconnectionthatcementstherelationshipbetweenAandB,whichsomehow‘doubles’andbythissharing,inphysicsthereisattractionbetweenbeings.Andinbiology,

thereisattractionbetweenbeings.Intersectionthus,sharing,isbothacreativeelementandasocialelementofloveandattraction.Weshareachildinacoupleandthatputsthetwoelementsinaconstantdynamic

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attractiverelationship.Fermionsshareabosonandthatcementsanattractionbetweenboth.Predatorsshareapreyintersectingtheirterritoriesofhunting.

Furtheronsharingismoreintenseandsymmetricwhen2systemsareclosedStelements,asinthefigureofthecircularcontinuitypostulate,wherewecanclearlyseethereisanacbdregionsharedthatpegsbothsystems

together.

Intheintersectionofalineandcycle,thelineseemsatdisadvantage,andinfactinmostrealeventsthelinebecomesabsorbedandtransformedasapixelofinformation,coiledafteritentersintothevitalcyclicalspacethatlaysbetweenthe0pointandtheM-perimeter(whicharenotenclosedintheopenballSTregionofcyclicalmotionsthatconnectthem).Inadvancedtheoryweshallseethatinrealitythoselinestendtobepreyofthecircle,unlessemittedbyothersystemasan‘entropyray’,CROSSINGthe0-point.Inwhichcasewetalkofa

killinglineofentropy,whichcrossesthecircleatMandO,andifwestatethatinthatintersectionObelongstotheLine,theequivalentvitalproportionisthatthelineOMhasKILLEDthecircle,targetingitszero-point.

Indeed,Ifyoucuttheneck,ifyoushootthehead,ifyouconquerthecapital,ifyoumurderthefinancialpeople-casteormilitary-kinginpower,youdisorderr=evolve,changeanddestroyaclosedvitalspace-timebeing.All

otherlinesthatdoNOTcrossthe0-pointtendtobelinesbroken,fedandprocessedbythecircle,whichbecomesthesole‘owner’ofthempointandthechordinsidethecircle,isolatingtherestoftheline,asclosed

cyclesDObreakintheUniverseintofractalspaces.

Ifwetalkofmotion,weseethefirstregionabcdasaregionthat‘doubles’its∆-1densityoffinitesimals,andwillbecomethe‘seminalfirstregion’,whichwilldoublethenthewholesystemtocreatetheB-centerednewmovingform.Motionbyreproductionofformisthuscloselyrelatedtothenewconceptsofcontinuity,whichmoreproperlyshouldbecalled‘reproductivedisplacement’.TheACBDregionbecomestheseedfora3rdchildofA+Bortheregionofdensitygrowththatwillbecomelattersplitbyasexualreproduction(asincells,whichfirstduplicatearegionintheircentralDNAzone),oritwillbecometheregionofthewaveinwhichagradientof

anattractivefield,withincreasingdensityof∆-1finitesimals‘drags’theA-circleintoreproductivemotion.

Onekeyquestioninthewhysofphysicsis‘why’,systemsmoveinarelativefield,its∆-1scaleofthe5thdimension,towardsthegradientregionofmaximaldensityofforce–sowemovetowardstheattractivevortex

ofmaximalchargeormass.Theansweristhatthesystemwhichisattractedandsharesthesameactivemagnitudeand∆-1field,willbindinthatregiononthesideofthedensitygradient,∆(ð/$),morefinitesimalsto‘doubleitsform’,more‘energy-spacequanta’intowhichreproduce,andsowecanseeaccordingtotheternaryfractalprincipleofmultifunctionality,motionasthefeedingprocessofanentity,Ainthegraphthatfeedsonthefield,onthegradientregionofmoredensity,fallinginescapablybyitsgreedofmotiontowardstheregionofmaximalcharge-mass.Thefieldiscontrolledbythecentralchargemasswhichwillfinallyeatupthesmaller

chargemassattractedbythebaitofthefield.

Sowearegivinghere2key‘vitalpropositions’aboutthenatureofmotion,asadualæaction(thelargermodelreducesallrealitiestothe5vitala,e,I,o,uactionsofspace-timebeings):Thesystembothfeedsandreproduces

withtheabsorbedenergy.Andthiscanbedonein2forms:

-Thesystemfeedsonthegradientofmaximaldensitytowardsthestrongercharge-mass,andintheprocessoffeedingitreproducesitsformintotheadjacentregion,eithercreatingasonspeciesiftheactionbetweenboth

attractedpointsissymbiotic,parallel,sobothusethefieldinequalconditionstoinputinformationandreproduce.

-Orthesystemfeeds‘alone’,reproducesitsformintheadjacentregionandslowlynormallyincirclestoavoiditsfinaldemise,fallsintothevortexofthestrongerwholethatownsthefield,andtrulyis‘farming’the

attractedparticle,whichfinallywillbedigestedbythestrongerwholeuponaperpendicular‘Union’–thestar

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enterstheeventhorizonoftheblackhole,thefeedingpigentersthestomachofthefarmeronceitisfinallyattractedtotheslaughterhousebythechanneloffoodthatmakesthepigswillinglyenteritsdeadevent.

Now,thisiswhatIaminterestedmost:toshowthevitalgeometryoftheUniverse.Amathematicianwouldbelikelymoreinterestedinthelogicabstractionofthosepostulates,andaphysicistinitscapacitytoexplainthewhysofkeyprocessesofphysicalsystems.Asaphilosopherofscience,mygoalistoshowyoutheorganic,vital

natureofeventhemostabstractofallsciences,mathematics.

Thuslayonwascorrectlyundefinedintheclassicsense,asitwasneverresolvedinits3varieties.ThingsdoNOTlayonaplaneofthe5thdimension,asthentheywillbeaboveorbelowbutnot‘into’,lay‘on’,thereforeis

NOTarealeventbutaparallelevent.A‘laidon’beingisnotintothebeing,itdoesnottouchthebeing.

SoforgeometrywhatmattersisNOTContinuitybuttopologicadjacency,withnoboundaryintheconstantreproductionoftheform,suchasthemembranebecomestheexternalwaveform–thelackofseparationin

theprocessofmotion=reproductionhoweverisnotequaltotheclassicconceptofContinuitynolongerneededtodefineaPan-Geometry,asBachmannproves.

SothisbringsustothecorrectionofcontinuityconceptsintermsofscalarcontinuitywherecontinuityonlyhappenswhenwesqueezeallthefractalsalesoftheUniverseinasingleflatline:

IV.∆:Axiomofdis-continuity.

Continuity is not real I the Universe, and it is one of the great divergences between 5D and 4D – whichcompresses all the scales of reality tomake them fit in a single plane, where then continuity exists. So theproperaxiomsofcontinuityasexpressedbyclassicgeometry,commentedbelowholdnowforthesumofalltheplanesandscalesoftheUniverse,butasdifferentfamiliesofnumbersbelongtodifferentscales,anysingleline inasingleplanewillhave ‘holes’ that themid thatperceives itdischarges.The interestof theaxiomsofcontinuityisthatputtingalltheplanestogethershowthatthescalesoftheUniverseareinfiniteandthroughthesumofallthosescalesrealityshowsanhorrorvacuum,butanyscalewillhavediscontinuitiesrequiredtodifferentiateitspartsorelsewecouldnot‘defineapoint’.Sotheconceptofafinitesimalpoint,studiedindetailin algebrawhenwe analyze derivatives and the concept of a limit come here in full form.We thus add a 0postulateofDis-continuitytoframetheclassicpostulatesofcontinuity.

0.Mentalspaceiscontinuous,becauseallrelativeframesofreferencecancelthedarkspacesbetweenpoints.Howeverforpointstoexistasdifferentiatedforms,theyrequirediscontinuitieslocatedintothefinitesimallineinalowerplaneofthefifthDimension.Linesarethusstringsofpoints,whosedensityincreasesaswediminishtheplaneofexist¡enceweobserve,tillreachingthemaximaldensityofcontinuityintheirrationalline.Sowemustdistinguishthecontinuityofmentalspacesforwhichtheclassicpostulatesofcontinuityhold,andthediscontinuityofobjectivescalar5Dplanes,wherecontinuityisonlyachievedasthesumofthelimitof∝planes.

1. LetX1,X2,X3,···bepointssituatedonastraightlinesuchthateachsucceedingoneliestotherightoftheprecedingone,butthatthereisapointAlyingtotherightofthemall.*ThenthereexistsapointBthatalsoliestotherightofallthepointsX1,X2,···,butsuchthatapointXnisarbitrarilyneartoit(i.e.,nomatterwhatpointCistakentotheleftofB,thereisapointXnonthesegmentCB).

AnumbersystemconstructedfromtherealityofadiscontinuousworldISpreferabletotheidealcontinuousrealnumbersystemfabricatedwithDedekind'sfalseaxiomofContinuity.Instead3lessstrictprinciplessufficetoexplainthedifferentvirtualcontinuitiesperceivedasmotionofthe3elementsofanysystem(|xO≈Ø):lineal&circularcontinuity(predatoryunion)orø(reproductivesuperposition),andtheArchimedesandAristotleclassicaxiomsofrelativespace-timeproportions.

Dedekind'saxiomisthenadifferentconcept–thatofbarriers,limitsinascaleofnumberswhoseholesinthereallinearesimilarto‘potentialwells’,quantumjumpsthataredifficulttocross.Irrationalnumbersbecome

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thendiscontinuousgaps:π&√2arethegapsandaperturesthatpreventthecircleandthesquaretriangletobeperfect.Forexample,itcanbeusedtoprovetheexistenceoflimitingparallelraysinhyperbolicgeometrywith

farmoresimplicitythanusingtheAristotleaxiom.

Ofcourse,Dedekind'saxiomisneededtoobtainthecategoricalaxiomsystemoftheHilbert.Yetpreciselyforthatreason,becauseitisnottruthandreal,itmerelyshowsthatHilbert’saxiomaticmethodisfalse,itisanerrorofthemindthatconfusesitslimitedperceptionofthe‘holes’andopenwellsoftheUniverse(those

limitingratiosorrealnumbers)bythemind,withreality.Itislikethecaseofacontinuousmovieperception.Infactthemovieisstopandgo,withholesbutthemindputsthemtogetherintoacontinuitypicture.

ContinuityisaMayaofthesensesthateliminatesdarkholesbetweenperceptionsofthebrain.

Inotherwords,thebrain,themind-worldiscontinuous,realitythelargerworldisnot.Darkspacesareeasytocalculateforap.o.v.witharelative3diameterstoformitscircularperimeter,whichwillleave0.14…holesto

observe.SothepointdoesNOTobserve,96%ofrealitydarkenedbytheperimeterof3diametersthatclosesitsoutermembrane.Soitsees,0.14/pi=4%ofreality,whichiswhatweseeintheUniverse(96%beingdark

matteranddarkenergy).Yetourelectroniceyesdonotperceivea96%ofdarkness.Darknessiseliminatedtoexpandtheenlightened4%asifitwereallthereality.

Expandingthe0postulate,objectivecontinuitywillrequiretoseeallthescalesof5Dplanesanditshyperbolicreality,whichwouldincludethe96%.Butthenourviewwouldbeahyperbolicgeometry,asoureyeswouldbecrossedby∝ (arelativeinfinite0numberofparallels.Suchgeometryandanglewouldbeconvex,withus,asthe knot of an expanding fractal network of branching angles that connect us to all themicro ∆-1 points,constantlygrowingitsperceptionoftheLobachevski’shyperbolicbranching,richerworldthat5Dmetricproves.

Unlikeclassicgeometry,astraightlineisNEVERcreatedonlyby2points.TheclassicdefinitionofEuclid,naivelyacceptedbythosesupposedHilbertianr=evolutionaries:‘astraightlinejoinstwopoints’doesnolongerholds.Weobviouslyneed3pointstoconnect2points,onebeingshared,andonlythenwecanseeifthe2pointsarejoinedina‘curved’form,byanarch,orinastraightform,bytendinganAMandMBintervals,andlookingat

the‘angle‘betweenAmandMB,whichifitisastraightanglewilldefineastraightline.Thisisobvious–2pointscannotdefinethestraightnessofaconnection;thatitsurprisesmeithasbeenoverlookedforsolong,asitis

alsoakeyconcepttoproperlydefinewhatkindofgeometryweareinto,andagoodwaytointroducetheother2axiomsthatsubstitutecontinuityandrelativityofsize:

TheyareclassicA2xiomsofGreekGeometry(Archimedes,Aristotle’saxioms;inmyLeonardiannotebooks,writtenwithshorthandincomprehensibleSpanglish,i-logicweirdsymbols,whichperhapsinthefuturesomerobotwilltrytodecipher,hewillfindmyabbreviationofthose4Axiomsofcontinuityandangularperception,

writtenA2c2ioms,ab.A2c2:)

Theyareconcernedwiththeperceptionofsizeanditscomparisonfromagivenpointofview.Andagain,asalwaysinthedual/ternaryUniverse,asinthecaseofthelinealandcircularcontinuityprinciple,wehaveone

axiomdealingwithlinealsizesandtheotherwithcircular/angularperceptionofsizes:

ARCHIMEDES'AXIOM.IfCDisanysegment,Aanypoint,andranyraywithvertexA,thenforeverypointB≠Aonthereisanumber∆suchthatwhenCDislaidoff∆timesonrstartingatA,apointEisreachedsuchthat∆xCD≈AEandeitherB=

EorBisbetweenAandE.

I.e.,ifABwereπunitslongandCDoneunitlength,weneed4CDtogetbeyondBandencloseπinsideourstraightline.Andthisiswhatmattersto‘enclose’ornotacertainratiowithinthelargerenvelope,toenclose

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ourdarknumberpi,soweknowiswithinus(thewholecycle)evenifthecycleisfluctuatingaroundthenon-definedπ.

Moreovertheaxiomsetslimitstoinfinitesimals,definingthefinitesimalunitofmeasureABonthelowersideandthewholeAEontheoutercontourside.

Archimedes'axiomthusmeansthatwhenNaturechoosesafinitesimalCDasaunitof$length,aquantaisestablishedforascaleorplaneof5Dtoexistandeveryothersegment

Willhavefinitelengthwithrespecttothisquantawhichbecomesthe‘relativedefinition’ofanumber.

Andinverselyifwehavetheperspectiveofthewhole,wechooseABasunitoflength.Andthentheaxiomsaysthatnoothersegmentcanbeinfinitesimallysmallwithrespecttothisunit(thelengthofCDwithrespecttoAB

asunitisaatleast1/nunit).1/nwasindeedinLeibniz’sInfinitorumthefinitesimalunit.

Those3axiomssufficetoproveasmathematiciansknow,allthetheoremsofgeometry.Moreover,andthisisthebeautyofit,ifwewanttogetridofnumbersanddoapurelygeometricanalysis,thispostulate,which

connectsnumbers,pointsandlines,canactuallybesubstitutedbyamentalpostulate:

ARISTOTLE'SAXIOM.GivenanysideofanacuteangleandanysegmentAB,thereexistsapointYonthegivensideoftheanglesuchthatifXisthefootoftheperpendicularfromYtotheothersideoftheangle,XY>AB.

Inthegraph,XYgrowsfasterthanVxorVyaswecomefurtherawayfromVandtheanglebecomeshyperbolic,sowecanalwaysfindanXYlargerthanvx,evenifparadoxicallyVhastheimpressionfromhisPointofView,thatXYisbecomingsmaller.Thisrelativityofworldperceptionversusreal

UniverseisatthecoreofmanyerrorsoftheegowhobelievestobeinfinitewheninfactheandhisrelativedistancetoXYisreallysmall.Itoftenmeans

thatifXYisa‘futurepointintime’(deathpoint,whenweusegeometrytostudyworldcycles)orapredatorindistance,wewillunderestimatethedangerofdeath,andXYwillgrowveryfastandeatusup(-:)-:O-:

Thesmallerpart,orfractalfasterpointbecomesthenaknotofafractalofreproducedanglesthatkeepenlargingrealitytillitcanmapoutamuchlargerscale,whichshrinksthroughthosefractaltriangularbranching

tofocusandcoalescenceinthecenterofthefractalnetwork,at∆-1.

Thedualityspacev.timestillnessv.motionisageometricformalviewof5Dmetric:Sxð=k.

Aswebecomesmaller,ðƒ,timeaccelerates.Inversely,tohaveastill,geometricalperceptionwithnomotion,wehavetodeceleratetimecyclesofsmallerbeingsandexpanditsspacesize.Soifthefastermotionofcells

andatomsslowsdowntoourspeedofexistence,wewouldhavetoexpanditsspacesizetobeasbigasweareafterthatgeometricalexpansionandtimereduction.

Sowesubstitutetheconceptof‘singlescaledistanceandcontinuity’-Dedekind’saxiom-byAristotle’spostulate,whichchangesrealcontinuitybyrelative,angularperceptionofdistances,fromtheperspectiveofa

pointsofview,withdeepvirtual-world-mindimplicationsevenifweprovethesametheorem.

ThepostulateofAristotlemerelysaysthatfromagivenangleofperception,thelinethatjoinsthelimitsofourperceptionandclosestheopenangleofvisionislargerthananyofthetwosidesofourangularperception.

Inotherterms,the‘perpendicular’,notparallel,horizonor‘frontofthewave’ofperceptionexpandsmuchfasterthanthedistancebetweenusandthebeingweperceiveinotherscales,isprovedinasinglespace-time

scalebyoneofthekeynewpostulatesthat.

TheUniverseexpandsfasterinobjectiveterms(theperpendicular,farawaylineofexpansionofourhorizon),thanfromtheperspectiveoftheperceiverofacertaingeometry.

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Whatallthesenewwaystodefinetheparametersofcontinuitytellus,isthatwhatmatterstosystemsistherelationshipsbetweenbeings,andtherelativeperceptionsbeingshaveoftheUniversedeformedbyitsangular

worldsofperception.

RECAP.Itisimportanttodifferentiategeometric,continuousformsfromnumericaldiscontinuousseries,asbothrepresenttwodifferent‘elementsofreality’.Continuityappearsonlywhenweincludetheconceptof‘motion’(notdefinedbyEuclid),andevensoitisamirageofthesenses,becauseatquantumlevelitisa

reproductivecontinuity.Thisisshowinsocialgrowingnaturalnumbers.ThusthereisNOTanexactcorrespondencebetweenpointsandnumbers,asthefailuretofindpi,√2ande,thekey‘ratios’oftheUniverse,whichcanbe‘drawngeometricallybuthavenodirectexactsolutionarithmetically,proves.Thenumbersystem

canbeproperlyusedtoconstructgeometryonlywhenthosearetakenintoaccount.

Continuitycanbeconsideredinthegeometricworld,onlyfromtheperspectiveofadjacency,andmotionasreproductionofform,inadjacentplacesofasingleplaneofreality:

Inthegraph,takenfromaphysicalwave,aparticlereproducesitsformsasitmovesasawaveofadjacentparticlesoneafteranother.Thisisthedefinitionofmotion,whichsolvesZeno’sparadox.Propermotiondoesnotreallyexist,butreproductionofinformationalongapath,withlimitsfor

eachworldandgeometry(intheEuclideanhumanspace,withthelimitofc-speedfortransferofenergy-form).Socontinuitycanbedefinedinasingleplanewiththepostulateofadjacency.

Inprecisetermsgiventhelackoftruecontinuityinthe5Duniverse,the4postulatesthatsubstitutecontinuityasprovedintheworkofthekeypost-wargeometersarerestrictedtoasingleplaneofspace-time,andtruly

definemorethancontinuityprocesses,theotherkeyelementsofi-logicgeometry.

III.T:Axiomsofmotionasreproductionofform.

(Amotionisatransformationnotofanindividualfigure,butofthewholeplane.)

1. Amotionreproduces=carriesstraightlinesintostraightlines.

2. Twomotionscarriedoutoneaftertheotherareequivalenttoacertainsinglemotion.

3. LetA,Aʹanda,aʹbetwopointsandhalflinesgoingoutfromthem,andα,αʹhalfplanesboundedbythelinesaandaʹproduced;thenthereexistsauniquemotionthatcarriesAintoAʹ,aintoaʹandαintoαʹ.(speakingintuitively,AiscarriedinAʹbyatranslation,thenthehalf linea iscarriedbyarotationintoaʹ,andfinallythehalfplaneαeithercoincideswithαʹorelseithastobesubjectedtoa“revolution”aroundaasaxis.)

TheaxiomsofmotionagainreduceandsimplifythedifferentiationshappeninginthevitalUniverse.Sowemuststressthosebasicprinciplesthatsoclearlydifferentiatetheexhaustivegoalofi-logicgeometry–todescribeallvitaleventsandspeciesdepartingfromthecomplex∆¬@stpropertiesofspace-timevs.thesimplifyingsynopticnatureofmathematicalmirrorsintheaxiomaticmethod.

Soherewehavetosubstitutethewordcarrybythewordreproduce.Amotionreproducesaformasit‘carries’straight linestootherplaceofspace-timewherethe information isreproducedonthe lower∆-1plane.So inilogicgeometryitDOESmatterunlikeinEuclideangeometrythepathofthemotionfromAintoA’.Thoughifweconsider thatpuremotionhasminimalmemorialpersistenceof the reproduced forms, in the limitofTT-entropic motions it will not matter the path as the persistence of memory will not make them last. Butultimatelygeometrymakes fleetingmotions=communicationsbetweenpoints to last. So suchmotionsare infactcomplexentangledmotionswhereasthe2entangledpointsstilltoeachotherkeepalsoreproducingoveralower plane as they displace together. It is then evident that the axioms of motion deliver the maximalsimplificationofthatcomplexprocessobservingthelineinmotionbutstilltoeachotherend-pointsjustinitsinitialandfinalpicture.

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How mathematics gives then depth to reconstruct from such simplifications the whole range of eventshappeningtakesplacenot ingeometrybut inAlgebrawithGroupsthatexhaustallpossiblediversificationsofsuchST-events.

Theaxiomsofmotionareinthatregard,thebasisofGrouptheoryappliedtomodernphysics-agreatadvance,fortheclassificationofallpossiblevariationsofeventsandforms.

Indetailthe1staxiomdefinesreproductionwithanabstractword,carrying,usedinphysics,relativitytheories,vectorspacesetc.Butitsvitalnatureisratherintuitiveinthe5Dworldview.

The3rdismoreinteresting,andcomplexbecauseitisconnectedtothepreviousanalysisofchiralityparityetc.,andwillbeofgreatimportancetodistinguishdifferentspeciesofphysicalsystems,accordingonhow$(lineal

motion)andað(rotarymotion)areputtogethertoreturnorNOTthepointtothesameinitialstate.

Asitfurtherestablishtheprocessesofcreationbyreproductionofnewforms,usingtheonly‘conservedmotions’–rotaryangularmotions(O)andlinealboosts(|),inasingleplane.Butcertainformswhicharemirror

symmetrycannotmatchonlywithrotationsandboosts,butneeda3rd‘mirrorsymmetry’whichimplicitlystatesthatanysystem∆willbepartofalargerdimensional∆+1systemtoachieveitscompleteness.Andsobesides

linealandangularmotions,realityrequiresamirrorsymmetryinahigherdimensionality,toachievefullreproductionofforms.Andthisaltogethermakesrealitymorecomplex,generatesnewcreativegenderformsandalsoestablishesthenon-commutativeclosedornon-commutativenatureofabidimensionalsystemwhencompletedinthehigherscale,astwoconsecutive$andð-rotaryormirrorsymmetryactionsdiffer,becausebothrotaryandmirrorsymmetriesaredifferentaccordingtowhichdirectiontheactiontakes.Soinverse

dualitieskickintostartcomplexdifferentiations.

Howeverl-motionsbydefinitionbecomewithtime,closedactionsandshouldreturntoitsorigin.Sothequestionopenedwithdeepconsequencesinthestudyofworldcyclesandpresentstretchesofthevirtualexistenceofallofus,ishowmany‘motionsareneeded’toreturntothesameform,regardlessofthe

differentiationintheintermediatepaths–are5dimotionsenoughorshallweneedadodecalogic12stepsmethod?

Weleaveopenthequestion.Ultimatelythelabyrinthsofpresentstœpsofexistenceinwhichthepathincursaremoreimportantthanthegoalwhichwillbea0-sumthatkillsthepathandtheexistenceoftheevent-being.

Stilltheaxiomsoffer2solutions:thepointcanbecarriedinanopenpathbypuremotion,withoutmemoryandthenthepathdoesn’tmatter,andthepointcontinuesasanisolatedmotion,freeof‘responsibility’foritstailof

reproducedelements.SuchmemorilessMarkowianprocessesdefinethenan|-openfieldorlimbsystem.

Yetareproductivemotionwithmemorythatleavesatrackisacurvaturewithmemory,aclosedcyclicalpointwhichwill‘crowd’itsdimensionalityatthepointofreproduction.

Andindeed,oftenwhenthepointclosesitscircleitexhauststhemotionofthebeing,whichdiesafterreproduction(fromoctopustoarachnids).Aclosedpathkillsasystem,crowdingthezeropointofreproduction.

Butanopensystemalsobecomesexhaustedastheintensityofthelinefadesawayintoentropyandnoise.Soafterafinitenumberofstœpsathermodynamicopensystemevenifitdoesnotreturntoitsoriginalpoint,

becomesexhausted.

Allthisbringustothefundamentalmetaphysicalquestionderivedofallthoseaxioms:ItistheUniverseclosedoropen,infinitebutfinite,un-bounded?Itwillreturntoapoint?Alltheanswersbeingtruthasrealityincludes

allitsparadoxicaldualitiesandtrinities:

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TheuniverseisinfiniteinglobaltimespacebutfiniteinlocaltimespaceT.œs.Whichare,oncewegetridoftheegoparadox,infiniteinitsfuturerepetitions,becausethenumberofrepetitionsandvariationsisasmaller

infinitythanthenumberoffractaldomains,‘brokenbyacyclicalenclosure(3rdaxiomoforder)’.

ThisdualityofinfinitiesbetweenthetypesofbeingsanditsrepetitionswillbeinmetaphysicstheonlycorrectuseofCantor’stransinfinitiestoprovethatultimatelyweareallimmortalbecausewearerepeatedlikelyevery109-11variationsasthefactevery109humanswefindanundistinguishablerepetitionshowsIfyouthinkthisistooabstractthinkagain.BecausetheUniverseisinfiniteinitsrepetitions,finiteinitsvariationsanexactreplica

ofyourselfisnowabouttoappearonceyoudieinotherregionofspace-time.

Andsowemovetometaphysics,inaSpinoza’ssense:arethoseother‘yous’repeated,connectedentangledto‘you’,asidenticalquantumsystemsare?Areyoupartofawaveofreproduced‘carried’motionsasablockoftime?Areyougoingtolivebeyonddeathbytransferringnonlocallytoanotheryou,bornwhenyoudie?Itis

thedarkspacebetweenthetwoinformativeformsoftimethatyouare,adiscontinuitybridgebythefinitesimalmindthatdoesnotseethatspace.Thereistransmigrationofsouls?Alasyousee,HilbertwouldhaveneverimaginedthatfromhisaxiomsofmotionswecouldmoveintothePythagoreantheoryofmetempsychosis(:reincarnation…Theendofi-logicgeometrybringsusthefirstquestionsofplatonicmathematics.Howmanyyousexistreflectionsoftheidealcanonofthecave?Butthosearequestionsthatbelongmoreproperlytothedodecalogicofworldcycles.Andthepentalogicofentanglement.Sowestophere.Neitheraswedoneitalready

discuss,justenunciatethe2versionsof:

V.@:Axiomofparallelism(Euclid).

1. Onlyonestraightlinecanpassthroughagivenpointthatdoesnotintersectagivenstraightline.

Theseaxioms,thenaresufficienttoconstructEuclideangeometryintheplane.Alltheaxiomsofaschoolcourseofplanegeometrycaninfactbederivedfromthem,thoughtheirderivationisverytedious.

TheaxiomsofLobachevskigeometrydifferonlyintheaxiomofparallelism.

Vʹ.Axiomofparallelism(Lobachevski).

1. Atleasttwostraightlinespassthroughapointnotlyingonagivenstraightlinethatdonotintersecttheline.

Indeed,in¬ÆwegetridofIVcontinuity(Dedekind’sinsimplerlanguage)andParallelismaccordingtoEuclid.

SooncemoreweseethatthebasicaxiomsandpostulatesandlawsofmathematicsaremirrorofST-laws,albeitsometimestoosimplifiedsohumindshavelosttrackofwhattheyweremirroringinfirstplace.Whichiswhatweshallshowbydepartingfromthe∆@strealityNOTfromthemirroritselfastheaxiomaticsimplifying

methoddoes.Theconclusionthoughisobvious:

Alltheself-evidentaxiomsandpostulatesofEuclidarerelativetruths.

Reason why they are superseded by the Non-Æ=i-logic postulates of geometry based in fractal points withbreath to better connect the experimental reality and ideal geometry. This has not been done asmathematician only corrected the fifth Euclidean postulate, by lack of a proper theory of reality to defineconceptsasdimensionordistance=dissimilarityor thedifferenttypesofcongruence,or timeorspace,whichwe now have, coming from a higher language, the ‘¡logic laws of the fractal space and cyclical time of theUniverse. Sowecanreinterpretmanyof thepostulatesandaxiomsofGreekgeometryandcorrect them,astheyareunder thecorrespondenceprinciple, justapproximations to the scalarUniverseobserved ina singleplane,withoutperceptionoftheinnerpartsofthepoint.Letusthenrevise,thefiveEuclid'spostulates:

1.Itispossibletodrawastraightlinefromanypointtoanotherpoint..

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2.Itispossibletoproduceafinitestraightlinecontinuouslyinastraightline.

Thismeansthatallpointsofapresentspacecanbeconnectedinsimultaneitybecausespacetimeiscontinuous;but in reality isnot.As thereare ‘irrationalpoints’whichare inadifferent5Dscale; closedmembranes thatbreak the continuum. Soonlypoints thatbelong to the inner vital spaceof anorganism, connected through‘physiologicalnetworks’canbeconnectedinthesamepresentspace.

Whilepointsfromtwoscalesareonlyconnectedwhentheybelongtothesamesuperorganism,andtheyaresimilar in form, despite being different in ‘size’ (5D scale). So we can connect points within one of the 3topologies (membrane,vitalenergy, innersingularity),andwecanconnectsmallerpartsofa fractalnetworkwithitslargerphysiologicalform(cellsthroughbloodandnervousnetworksofthe∆+1scaleandsoon).

Andtheywilloftena 'curvedconnection'...Moreoverallstraight linesbecomebent intoazerosumcurve,astheywillbepartofafractalsuperorganism.SoinfinitydoesNOTexistlineallybutquitetheoppositeincyclicalself-repetitive patterns, as all straight lines ultimately find a limit in the curved closed membrane of itssuperorganism.

3.Itispossibletodescribeacirclewithanycenterandradius…whichmightbetheonlytruepostulate,meaningthatallstraightlineswillbepartofaclosedtime-spacecyclesof3ages,whicharethe3pidiagonals,closingaworldcycle,whichhasafinitezerosumvolumeandbreaksinfinityintoinnerandouterparts.

4. All right angles are equal to one another.This is not truth in different scales as the fifth dimension is ahyperbolicgeometrywhoserelativecurvatureanddegreeofflatnessdependsontherelationshipbetweentherodofmeasure/sizeoftheobserverandthesizeoftheobservable(informalspacethisistherealizationofthetimeacceleration=increaseofcurvatureofsmallerbeings).

5.Theparallelpostulate,alreadyknowntobefalsebyclassicscienceisrelatedtothepreviousoneandwehavealsoproved its falsity..Soactually4or the5postulatesare false in the realworld.Andonly thepostulateofcreationofcirclesdepartingfromlines,it isrealintemporalterms,asallsystemscloseitsworldcycles,thatistheydie.

Soarethedefinitionsofapointwithnobreath,alinewithnobreath,aspointsarefractalpointswithhiddenvolume ina smaller scaleofparts, linesare thereforewaves -points cycling;planesare thennotdefinedbylinesbutbynetworksanditsflows',andsoon.Thoseself-evidentdefinitionsareallfalse.

Finally, theElementsalso includethe following five"commonnotions";4of themconcerningequality,whicharenot'false'butrathermeaningless,asthingsare'similar'onlyathingisequaltoitself,sincewedonothavethetotal informationofbeings,neitherthingswhichoccupydifferentspaces-astheyaremadeofspaceandtime -are equal, justmerely by changing position, the thing becomes other thing (themes those of extremeimportanceinquantumphysicstodifferentiatebosonsandfermions-systemsthatoccupythesamespace,andhenceareequal,andthingsthatdonotoccupythesamespace:

-Things that are equal to the same thing are also equal to one another (formally the Euclidean property ofequality,aconsequenceofthetransitivitypropertyofequality).

Ifequalsareaddedtoequals,thenthewholesareequal(Additionpropertyofequality).

Ifequalsaresubtractedfromequals,thentheremaindersareequal(Subtractionpropertyofequality).

Thingsthatcoincidewithoneanotherareequaltooneanother(ReflexiveProperty).

Thewholeisgreaterthanthepart.Finally,tothoroughlybustyourballs/beliefs...(:yes,youhaveguessedit):thewholeisnotgreaterthanthepart,ifanythingtheyareequal...

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Orrathersimilar inexistentialmomentum$xð,accordingtothemetricof5D:$xð=K.Whichsomehowisimplicitinsettheoryandtheparadoxthattellusthesetofallsubsetsisbigger...andevensmallerifwemerelymeasure its quantity of information that grows inversely to size.As this postulate is closely related to our'understandingofthescales'oftheUniverseisworthtoelaborateabitmore.

Thewholeisnotgreaterthanthepart,neithersmaller(:

Theworldwemeasureandcallphysicalisamentalworld.Considerinsteadtherealscalarworldoftheinfinitelydivisible. There anywhole is infinitely divisible, but so is any part of thatwhole. As a particular example, inmathematicalanalysis,anylinesegmentisidenticalineverywaytoanysmallerlinesegmentthatisapartofit.Thissuggeststhatthefifthcommonnotionmay,inthedescriptionoftheworld,benottheonlytrueone.

Wecanextendnotion5:Thewhole isequaltothepart' totheparticularcasethatallpartsofthewholeareequal to each other, from where we deduce identical particles in physics, where all electrons, protons orphotons are alike. Yet,we cango further ifwemerely 'measure' the information/timespeed/energydensityandaffirmtheopposite,thatthewholeislessthanthepart,asits'time-motion/energy/massisgreaterinmoretightlyconcentratedforms.Sotheblackholewhichisinto5Dmetric,$xð=k,thesmallestmassisactuallythegreatest/densest/heaviest.

Mentalspacesanditsvalueaspartialtruths.

WhatcanthenwesavefromEuclideangeometryifeverythingaboutitseemswrong?(:Almostallofit;becauseratherthanwrongisasimplifiedselectiveversionofrealitywhichworkaslongasthepropertiesweselectonthatrealityforpracticalpurposesare‘externaltothefractalpoint’notconcernedwithitsinternalparts,asinthepreviouscasesimilartoengineeringdesignwheremassiswhatmatterstousinthe‘selectedmentalspace’offorces.Orwhenmotionismoreimportantthanform,asinphysicalequations.Butabsolutetruthwillrequirealwaysconsideringthatinternalvolumeoffractalpoints;andhenceitswave-likenatureasinquantumphysics.

BecauseonlythewholeUniversehasalltheinformationaboutitselfanyO-Mindx∞Universewillbevalidaslongas the infinitesimal information it extractshelps the species thatholds suchamental space tomeasurerealityandstoreinformationhelpfulforsurvival. Inthatsense,themostimportantelementofbidimensionalgeometryistheunderstandingofangles,parallelismandperpendicularity,whichisthefirstelementdevelopedinGreekgeometrytobeabletomeasuredistanceswithtrigonometry.Thisalreadyisdonebytheeyes’mind-mappingsofdistances.

Butweaddtoitthetopologicallawsofthe4thNon-EPostulateofcongruence,whichaddsavitalmeaningtotrigonometryasParallelismisalsoasitsoppositeconceptofRiemanniandistance,ameasureof‘similarity’.SoifinRiemann’sdissertationdistanceof‘colorspace’isequaltodissimilarityinthe4thNon-Epostulateparallelismimpliessimilaritywhichfosterssocialevolutionvs.Perpendicularitythatimpliesdissimilarity.

Andthisbecomesobjectiverealitywhenobservingensemblesoffractalpointsofanyspeciesanditsanglesofconnectionof thosepoints into geometric figures, in a still space view, orwhenoneof thedimensions (S=Tsymmetry)isseenasmotion,inthewayherdsmoveinparallelorpredatorsintersectpreys.

SoweestablishanewdualitybesidestheS=Tdualityof formandmotion,derivedof the internalpartsofallfractalpoints;thatbetweenthesubjectivemindandtheobjectiveexternaltopologicalviewofanevent.Inthiscase ‘dissimilarity’ isaconditionof internalcongruencebetween2 fractalpointsbringexternalparallelismorperpendicularity.

AndsoanewruleoftheentangledpentalogicUniversemustbeaddedtoS=T,‘x=y->x≈y;x≠y->xXyxy;thatis,if two systemsare congruent, similar in itsparts ina still spatial comparison (x=y), theywillmove inparallel(x≈y);iftwoentitiesaredifferent,x≠y,theywillcrossinperpendicularmotions,xXy,orseparateinitsdistance(xy).

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Akeyelementofallspacesis‘theangleofperpendicularity’,whichacquires¡tsmeaningwhenwemarry@nalyticgeometryandthe4thnon-EpostulateofcongruencebasedinDarwinianperpendicularityandsocialparallelism.Itisthekeyfortheunderstandingthemathematicalphysicsofvectorspaces,crossproducts,dotproducts,equipotentialandlinesofforces,whichhappenindifferentplanesof5Dandaffectdifferentpartsof

anentangled,field/wave/particlesupœrganism.

i.e:ParticlesareDarwinianoverfieldsinwhichtheypreyandwavesareparalleltothefieldinwhichthey'slide'.

Thus trigonometry of angles, was the fundamental first mental perception because it is how real systemsmeasure distances, which is a function of similarity and hence of vital survival. Angular perception is alsorealizedinthesimplestphysicalsystems,weretheunitof information,h,theminimalspace-time‘Planckton’,has several position of quantized angular momentum, the only positions we canmeasure as information isalways processed in ‘still form’ (missing itsmotion steps aswemiss themotion of a film between frames),accordingtoitsangle,whichdeterminesitsinteractionwithotherparticles…

DifferencebetweenequationsandGeometriccurves.

Thecharacteristicfeaturesofalgebraaretheuseof letters,whichweperformoperationsaccordingtodefinitelaws.InelementaryalgebrathelettersdenoteCONSTANTS,normallyordinarynumbers,takenaspopulationsinspace,thevariables,whicharethefinal lettersrepresentT.œ.sofacertainspecies,andtheoperandrepresentdifferent‘dimensionalmotions’,dimotionsoftime-space.

Sowecanreduceequationstoaseriesofexistentialalgebraicequationsofthetype:

∑T.œST-perandi∑T.œSToperandi=∑T.œST-perandi∑T.œSToperandi

Whereasthea….p letterswillbenumericalparameters, theU,V,X,Y,Z lettersTimespaceT.Œsandoperandidimotionalparameters.

As such there is a fundamental difference between an equation and a geometric curve in analytic geometry,despite its apparent similarity. An equation searches for a single solution as the XYZ letters represent ‘spatialpopulations’andtheparametersof‘timechange’thatconverttheequationinatimeeventaretheoperands.

OntheotherhandinacurvetheXYZlettersrepresentvariables,whosesimultaneouspossiblevalues,joinedbyageometricnon-E line formofsimultaneous ‘spatialmembrane’, so theyareeventsofspace,dulystudied inour‘geometricfirstvolumeof5Dmathematics’.

This isahugedistinction thatmakes completelydifferent the studyof simultaneous curves in space,whichactoftenasmembrainsofsuperorganisms;tothestudyofalgebraicequations,whichdescribeeventsintime,oftenofasequentialnature,giftedwithmotion.

Weshallcomeoftentothosephilosophicaldistinctionsthatmathematiciansescapeperhapsastheyfindthemevidentsoron thecontraryas theydonothaveaGstunderstandingofGenerational space-time to find theminteresting,butitdeterminessomeobviousfactsthatdifferentiatecurvesfromthepolynomialrepresentationoftheminequationsandwillrevealonouranalysisofalgebraiccurveskeyconceptsofS=Tsymmetries:

- Curvesarefarmorereducedinnumber–10canonicalcurvesin2Dand18in3Dsufficetodefineallofthem.Sincethenumberof‘spatialsuperorganisms’thatsurviveintheUniverseisfarmorereducedthanthetotalflows of entropic time motions, from where we obtain them. As in turn, those canonical curves can bereducedtotheformsextractedfromacone,whichrepresentsaworldcycleoftimespace.

- So all curves and hence all ‘topological membrains’ and forms of the Universe can be reduced to the 3topologicalvarietieswefindinthecone,whichisinitselfa‘circularmembrain’tracingaworldcyclealongtheaxisofthecone,asitdiminishesitssize,compressingitsform,inthenaturalevolutionofallT>Ssystems.

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All those differences perfectly understood in the I classic Greek Agewhen equationswerewritten statementsusingverbalthought,hencetemporalverbs,asopposedtocurvesdescribedinspace,becamesomehowblurredwiththeworkofDescartesthatmergedbothinanalyticgeometry.Letusstudythem.

XI.2NDAGEOFGEOMETRY:@NALYTIC&DIFFERENTIALGEOMETRY"Usingatermlikenonlinearscienceislikereferringtothebulkofzoologyasthestudyofnon-elephantanimals."

Ulam.

Youthisalinealsimpleageofsmallstepsthatallworldcyclesofexistencekeepbending,soinmaturitywelearntheeternalreturncycle.Inmirrorlanguagesalsothesecondagecurvesthelinealstepsofthefirstcycle.Soafter

thelinealageofGreekGeometrty,curvesenteredthestage.

Yettostudythem,twofundamentaladvances,framesofreferenceandcalculuswereneeded.

@nalyticgeometrystudiesthedifferentplanesofmathematicsasmirrorreflectionsofthedifferenttopologiesandplanesofexistenceofasuperorganism,self-centeredintoan@-mind.Analyticgeometrywasthefirstmathematicalformthatsuccessfullymergedthe5Elementsofreality:¬∆@st,andassuchitsignifiedthebeginningofthematuresecondageofmathematics,afteritsfirst'spatial'bidimensionalGreekageofstill

geometry.

Thedualityof@nalyticgeometryisthusobvious:S-mentalspacesself-centeredinthe0-mind

∆-ScalarspacesdevelopedinSequentialsocialnumbers

T-imespaces,allowedbytheuseofonecoordinatetorepresentasymmetricdimensionsoftime

Theflexibilityoftheconceptofaplaneofspace-timetorepresentnumbersandpointsthusallowtorepresentnotonlytheSPOINT≈∆NUMBERsymmetrybutalsotheS-Tdualspace-time,form=motionsymmetriesanditwillhaveanimmediateconsequenceonthedevelopmentofscienceasitwillcanonizetheconceptofalinealtimeandspace,properofmathematicalphysicseversince.Soakeythemeof@nalyticgeometryistheaprioristudyofthedistortionsweintroducewhensettingfunctionsandoperandsonanartificialworldthatwetakeasa'backgroundspace-time'.SincewemustbeawarewemakelocalassumptionsNOTglobalones,aswhenweuselinealtimeNOTallthetimesinphysicstomeasurelocomotion.Unfortunatelythislocalityofframesof

referencewaslostandbroughttheerrorofabsoluteNewtoniantimespace.

Thusthemindinmathematicsisreflectedbyitsframesofreference;andsoweapplypentalogictoclassifythemainusesof@naliticgeometry,designingthedifferentperspectivesofthepartsofabeingandits5Dimotions:

@:theframeofreferenceisthemindview,witha0-1,finitesimal'body-energy'anda1-externalspatialworldwhoseequationsareself=similarinitsT=Ssymmetry.Indeed,wecanconsiderthefundamentalequationof

analyticgeometry:Ox∞=Constantworld,theequationofanymind,whoseperceptionoftheinfinitewholeinitsbiasedframeofreferencecreatesarelativeworld,ormind-mirrorofthewhole...

S-vitaltopologygivesusamoreobjectiveuseofthose3differentframesofreferenceand∆-scalarsymmetryrepresentedintheComplexPlanetorepresenteventstakenplaceinapartialelementofthesupœrganism:

Becausethereare3topologiesandthenscalesitshouldexist3typesofplanesandthenoneforscales.Intuitivelytheyarethelineal-cylindrical,polar-sphericalandCartesian,hyperbolicthatbetterrepresentsthemergingbody-wavesofthetwoothers.Andsothatleavesthecomplexplaneof'squaresandroots'asthe

naturalonetocastfunctionsthroughscales.

Pentalogicanalysisofspacesbymergingthe4+entropicperspectives.Mentalspaces.

Pentalogic merges mental, temporal, organic, entropic and social spaces in the analysis of the laws of vitalgeometry. I.e. animal sight codes formotionover formand redover blue for survival purposes (the eye is a

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natural born predator, which prefers red blood and prey motion as in gore movies), So wemerge objectivegeometry - its 3 topologic organic varieties and subjective mental modeling of reality, which ‘deforms’ theUniversetomatchthosetopologiesgivingbirth tothe3classicframesofreference:$t-cylindrical,ST-Cartesian(the humind light view), ðƒ-polar geometry, plus the complex ‘square’ plane to study the 4th and 5th scalardimotions.Eachofthoseframesdeformreality,creating3±¡typeofgeometricminds,accordingtoeachspecieschoiceofcoordinates:

The3±¡differentgeometriesandframesofreferenceasself-centeredmentalspacesdescribedifferentworlds.Weintroducethenewdisciplineofmentalspaces,withtheHumindlightspace-time.

Spaceunderstoodassimultaneousperceptionofadjacentforms(relationalspace-time)istherealmofthemind'slogic,asthemindcreatesitsstillness.Contrarytothebeliefofmanyphysicistswhothinktimedoesnotexist,whatdoesNOTexistisspaceoutsidethesingularityofeachmind,andhencethereare∞spaces,oneforeachmind'sworld:

Themindistheinternalnon-perceivableelement,itrequirestheconceptofanangletoestablishitsrangeofperception,whichisnotanumber,andhencea'different'elementofmathematicalrepresentationofreality.

TheO-polar,|-cylindricalandØ-hyperbolicCartesianplaneintheo-1,1-∞time-spacescalesdescribethe3topologiesofallsystemsinasinglescales;whicharefurtherexpressedbythecomplexplaneinits'squared

form'.Theyrepresentthesubjectiveplanepointofview‘coupled’withtheobjectiveplanegeometry:

Inthegraph,allsystemshaveternarytopologieswith3motion-information-reproductionfunctionsthatabsorbentropicenergy,mixitwithinformation,toachievemovementandperceiveto

keepinexistence.

ThusthesimplexaeiactionorgansexistinallST-systemsofNature.

VitaltopologybecomesthenaconstantmergingofTT>Ts(entropythatbecomes

locomotion)and§eedsofinformationthatgerminateintomotion,SS<Sttofindamiddlepointofreproductioninhyperbolicbody-waves.

Wecanstudysomeofthosemotionsinstillform,orviceversasimplifiedforminlinealmotionstakingadvantageoftheS=Tdualitytoobtainquantitativeresultsbutthevitalrealitytendstobalancemotionand

formdimensions.

Becauseourmindisvisual,EuclideanAnalyticGeometry’sfundamentalplaneistheCartesianplane,duetoitsperpendicularstructure.Itisalsoisaccordingtothe4thpostulateofNon-Euclideangeometrythebestformto

representboth'perpendicular'eventsandspace-timeinversestates≈Dimotionswherelength=motion,width=reproductionandheight=informationmimicthenaturalsymbioticstructureofsystemsofnature.

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Furtheronwecan‘reduce’the3dimensionstoaholographicsingleplane,compressedbyeliminationofoneofitss=tdimensionalmotions.ThenitisbesttoanalyzesystemswhichhaveaninternalS=TSymmetryorsystems

representedbya‘fractalpoint’unitalonganSTgivendimotion.

Thisisthejustificiationforitsuseinphysics,werethedimotionoflocomotionofafractalpointwhoseinnerpartsareignored,reducestoitsstudyasansTlocomotion,onabidimensionalEuclideanplane,wherethe

dominantelement,$T,thelinealmotion-space,isrepresentedintheX-axisandvintheY-axis.

WecanthenconsiderthebasicdualityofSpace-stillnessvs.Timemotion,in2othermainframesofreference:

-Vectorspacesaddadynamic,temporalview,whichmakeseachpointanSTelementwiththeS-populationparameterandT-motionparameterbestsuitedtorepresentexistentialmomentumsinany¡-1'fieldscale'.So

Physicsusesvectorspacesforlocomotionsinlinealspaceandtime.

-Abstractmentalspaces(Complex,phasespaces),whicharethespatialmentalstaticview.

Complexplanes.-5DSTanalysisinthecomplexplane,wherethepolynomial=scalardegreeofcoordinatesaredifferent(eitherarootvsarealnumberorifwe'square'bothaxis,aswedoin∆st,asquareddoublepositivereallinevs.a±i²realcoordinates).Humindsthoughhavenotaclearphilosophyofthe'emergent'complex

planethatstudyspacetimemixedfunctionsandDimotionsofgroupsofDimotions(functionals).

Phasespacesdescribeentanglementsofdifferent‘statesofspaceandtime'.Asubjectiveselectionoftheparametersoftimeandspaceanditsgeneralizationtoalltypeofdimotionalsystems,Phasespaceswill

completelyliberatesciencefromtheimmediaterealityoftheEuclideanworldweperceive.

Inthehumanscaleweanalyzetheternarypoints/states/agesofmatterinthosephasespaces(statephysics).

-Mathematicalphasespacesarethemanipulationofmathematicalfunctionsonthoseplanes,whichoftenwillbenotonlyabstractfunctionsofmathematicalspacebutexpressionsoftherealdimotionsofexistence,

essentialtothefieldsof'Mathematicalphysics'anditsspecialframesofreferences;andthemainfieldof5Danalyticgeometrywhoseexperimentaltaskistorelatethoseabstract‘coniccurves,complexplanes,anddifferentdimotionalparametersofthemain3±¡analyticframesofreferencetorealeventsandformsof

spacetime.

Sophasespacesfinallydetachedmathematicalanalysisfromthelightspace-timerealityofthehumaneyeandportrayastaticmentalspace-formwithinformationrelevanttotheperceiver.

∆-scalarperspective.TheCartesianplanegivesusalso2moregeneralisedperspectivescloselyconnectedtothescalardimensionsofasystem.

-0-1:TheUnitcircle,whichcanbeusedtoexplorethepaligeneticcycle,expressedmostlyinprobabilistictemporalterms(where1isthevalueof'existence'-thehappeningwhenanevent/formemergesintothe1-∞equivalentplane).Whereasmathematics(Measuretheory)allowsparallelstudiesbetweenbothlifecycles.

-4Dentropicparts:Inplanesthatbreakthewholeintoallitspointsofview,withinfinitegeneralisedcoordinatesofindividualpointsandstatisticalensemblesofentropicparticlesand∝dimensions,ofwhichHilbertspacesusedinquantumphysicsandphasespaces,usedinthermodynamicsarethemainvarieties.

ThereisTHUSasusualaclosed'homeomorphism'tousesomepedanticmathjargon(:'acorrespondencebetweentwofiguresorsurfacesorothergeometricalobjects,definedbyaone-to-onemappingthatis

continuousinbothdirections':)betweenthe5Dimotionsofrealityandthe5maingraphsofmathematics,cartesian,polar,cylindrical,complexandHilbert's;asrealitycanalwaysbeseenfromthepointofviewofthe

timefunctionsandspaceformsofthose5Ddimotions.

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Allthisisfairlywellunderstoodbyscientists,exceptperhapsthecomplexplane;andtheinteractionofplanesofdifferentscales,asthosewhichhappenwhenanActivemagnitudepreysonalower¡-1planeperpendicularly;

thatisprocesseshappeningbetweendifferent5Dscales.

RECAP.Themaindualityofanalyticgeometryhappensbetween@-mindcenterofframesofreferencetoreflectthemind'ssubjectivepointofviewandthe3topologicalobjectiveorganicplanesandthescalarcomplexplaneasrepresentationsoftheinternalformofeventshappeningwithinthekeyorgansandSTsymmetriesofNature.Sothereare3±¡mainframesofreferencesinanalyticgeometry,studyingrealityfroma0-pointofview,similartothatofthemindofeachofthosepartialorgans,whichthereforemapoutclearlydifferentworldsandprovetherelativityanddifferencebetweentheinfinitemind-viewsoftheUniverse,evenwhentheyusevisual

lightspacetime.

Andwefindadirectrelationshipbetweenthe5Dimotionsandthe'framesofreference'ofmathematicalminds.

As@naliticgeometrygiveus5sub-planes,eachoneaframeofreferencethatstudiesrealityfroma5Dimotion:

3DhyperbolicDimension:theCartesian,∑∏graph.

1Dvortex:thepolar,ð§cyclicalgraph.

2DLinealmotion:theLineal,cylindrical$t.

Eachofthosegraphsstudyproblemsregardingformsandfunctionsofthose5Dimotions.

Themultipleplanesofanalyticgeometry,donothideitsfundamentalproperty:toperceiverealityfromagivenpointofview,the0point,or@-mind,andcreatefromthatperspectiveacertaindistortedworldviewthatcaterstothefunctionandformofthe@-mind,selectinginformationofrealitytoformagivenspace,which

mightseem'reality'tothemindbutwillalwaysbe'arepresentationinwhichthemindwillexerciseitsterritorialwill'toparaphraseSchopenhauer.

ThusAnalyticgeometryultimatelystudiestheUniversefromtheperspectiveits3±∆mainSTdimensionalsubspeciesorpartialequationsofthefractalgeneratorequationofT.œs

Theapriorirealityofother'systems'.

'Spaces'arestillforms,whichinaUniverseinwhichthefundamental‘substance’istime=motion,represent‘stillmirrors’ofthosetimemotions.Hencespacesareartificialconstructsofthemind,whose‘languages’create

spacesto'navigate'reality;builtwiththefeaturesoftheforcesofinformationavailabletothem.

Theyaretheapriori'Kantian'categoriesthatdeformourrealityandviceversa,knowingthepropertiesofagivenspaceitdefinesthetypeofmindandspeciesthatperceivesitandnavigatesthroughit.Howevermentalspacesneedalsotoberealmirrorsthatselectefficientformsofrepresentingrealitybecausetheyhaveavital,survivalfunctionorelsetheybecomeunfocused‘blind’andthemind-systemdiesaway.SoMindspacesand

outertopologiescoincideasmirrorsofrealities,whichistheeternalfeed-backequationbetweenthe‘intelligenceofmindspaces’,andtheentropicdisorderedmotionof‘timeflows’:∆S@<=>¬∆T.

Geometrieshoweverbecomethemoredistorted,thefurtherawaywecomefromthat‘communion’betweenthe‘mindandthetopologyandscalethemindperceives’.Whenmindandtopologycoincides,themirrorismostefficient.SohumanvisualmindhasintheCartesianorthogonalspaceitsbestmirror-representation.

Whenthedistanceinscaleandformismaximalthoughthereareinternaldeformationsofthemindspace,whichadaptthe‘strangerbeingandscale’totheperceiver,whichisfineifthehumindwasawareoftheunderlyinglimitsoftruthofitsperception.Butthehumindisextremelyego-centeredandreductionist,

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sponsoringanaïverealism–whatweseeandmeasurewithoursensesisallwhatthereis.Soitdeniesthosedeformations.

Forexample,atextonquantumstates:“TheheartandsoulofquantummechanicsiscontainedintheHilbertspacesthatrepresentthestate-spacesofquantummechanicalsystems.Theinternalrelationsamongstatesandquantities,andeverythingthisentailsaboutthewaysquantummechanicalsystemsbehave,areallwovenintothestructureofthesespaces,embodiedintherelationsamongthemathematicalobjectswhichrepresent

them.

Thismeansthatunderstandingwhatasystemislikeaccordingtoquantummechanicsisinseparablefromfamiliaritywiththeinternalstructureofthosespaces.KnowyourwayaroundHilbertspace,andbecome

familiarwiththedynamicallawsthatdescribethepathsthatvectorstravelthroughit,andyouknoweverythingthereistoknow,inthetermsprovidedbythetheory,aboutthesystemsthatitdescribes.”Indeed,butthat

limitsyoutothemind-space,reducingrealitytoits,inyetanotheractof‘mathematicalcreationism’.Itisthennecessarytoaddalwaysa‘coda’tothedifferentmindspaceswestudytoconnectthemwiththelarger,more

truthfulrealityofspace-timelaws.

InthecaseofHilbertSpacesitwouldhelpforexampleNOTcalling‘Dimensions’,its‘coordinates’foreachparameterbutrather‘parameters’or‘coordinates’;orcastitsconceptualjargonoffunctionalsintermsofinnerandouterdimensionsofanewplaneofexistenceofthefractalpoint;andunderstandthedifferencebetweenthehighlyorderedpalingeneticcycleintimeofparticles(hencesusceptibletoprobabilityanalysis)vs.the

statisticalpopulationsofthermodynamics,intermsoftheT=Sduality.Andcertainlyitwouldhavebeenmuchbettertodoso100yearsagobeforethewholebusinessofforcingrealityintothe1-probabilityunitsphereby‘decree’ofBorn’srulewasmadeadogma.Quantumphysicsthusistheparadigmofa‘mentalspace’thatworksformantomakemeasuresfromitspointofview,butfogsbylackofconceptualclaritytheunderstandingofthewhysofthequantumsphere.Itisthusgoodforthepraxisofmeasureandmanipulationofparticlesbutbadfor

aphilosophyofscience.

ThyesamecanbesaidofourCartesianGraph:KnowyourwayaroundEuclideanlightspace-timewithits3perpendicularcoordinatesandyouwillknowalotabouthowhumindsandsimilarelectronicsystemsperceivetheUniverse.ButaslongashumansarenotawarethatsuchWorldspaceisonlyourmonad'slightworld,differentfrommanyothermappingsandacceptthe‘lightnature’ofourspace,justanelaborationwithourelectronicmindofthephotonsoflightanditscoordinates,wewillagaincommitphilosophicalerrorsaboutwhatisspace,whatistime,whyc-speedisconstant(ourrodofmeasuretheelectroniceyeentanglesas

informativedistanceswithotherelectrons,hencealwaysmaintainingitatc-constantfixedspeed),etc.etc.

Againallthoseproblemsarecommontoanyanalysisofrealitywiththecomplexplanewhichhasdifferentdimensionalitiestorepresentsandtfunctionsoftenindifferentplanesofspace-time;evenwithvectorspacesthatalsorepresentholographicSTelementsbutinasingleplaneofspace-time,anditsmorecomplexpurely

spatialframesofreference'across∆-scales',thatisHilbertspacesandfunctionaloperators.

Thusweshallcloseourintroductionto5Dgeometrywiththestudyofthose'farremoved'Hilbertspaces,whichareofinteresttounderstandthemostfarremovedscalesofreality-thoseofundistinguishablezillionsof

particles...

Itisimportantthentograsptheunderlyingprinciplethatunifiesrealitybrokenbytheinfiniteofmind-spacesandthenaïverealismofhumansintheirinterpretationofthosegeometries:

InanabsoluterelativeUniverseitisNOTthatimportanttoknowinsomuchdetailafarremovedscalesuchasquantumphysicsis-thegalaxy/atomasviewedfromhumanshave'weirdproperties'becauseour'perceptionof

it'islimitedsowecanonlyknowcertainsimpledimensionsoftheirstructurenamelythemostexternaland'visible',MOSTLY2D-holographicSTDimotions.

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3AGESOF@NALYTICGEOMETRY:S≈T

@nalyticgeometrystartedwiththeworkofDescartesandFermat,foreseenontheGreeks’studyofconics;whichnowcouldbefullyrepresentedintheCartesiangraph,whichisintopologicaltermstheconversionofaconicintoaplane,wherethecenteroftheconicbecomesthementalpointofviewoftheobserver,orstill

worldofgeometry.

Theevolutionofthedisciplinethrough3agesofincreasingcomplexityof'form',addednewstructuresinspace-differentiatingmindviewsofrealityaccordingtocoordinates–theaforementionedcylindrical,polar,

Cartesianandcomplex3±ipointsofvieworstatesofanysystemwhichdistortourimagemirrorofrealitywithinagivenmind.Soinitsfirstage,numbersandpointsweremarriedwiththeCartesiangraph,andthenthe3

'framesofreference'werefoundtodescribe3typesof'functions/forms'thehyperbolic,Euclideanandellipticgeometry.Polynomialswererepresentedandthefundamentaltheoremofalgebraexpressedwiththefinal

discoveryoftheComplexplane.

ThesecondageofanalyticgeometrywillbedominatedbyitsuseinmathematicalphysicswithKepler'sorbitalellipticconicgeometriesandthebiasedviewsoflinealtimeintroducedbycartesiangraphs,thetrueoriginofthatabsurditycalledlinealtimeandabsolutetimeandspace,whichoccurredtoNewtonjustbecausehewasdrawing‘thesacredlanguageofGod’,itsellipsesandcomets,ontheCartesiangraph.Sohethoughtbelow

realitytherewassuchinfinitesinglelineoftimeandspace,drawnbyGod,his'alterego'.

Inits3rdageitalsoincreaseditsentanglementwiththeotherelementsof∆@stofspace-time,givingbirthtomixeddisciplines.So@nalticgeometry,∆nalysis,S=Talgebra,T-heoryofnumbersandS-geometryinthefinal3rdageofmathematics,becameallofthemreflectionsofeachotherinthekaleidoscopeUniverse,wherethose

5elementsconstantlymergetogiveusmorecomplex5Dentangledreflections,ascomplex'knots'of5Delements,sotherewasacreativeageofalgebraicgeometry,analyticgeometry,complexspaces,mind-phase

spaces,differentialgeometry...

RECAP.Weshallintroduceanalyticgeometry,mathematicalphysicsandexpandadmaximaltheanalysisofthe3typeof@-framesanditsrelationshipswith∆stofwhichthelawsofinversionandgrowthofdimensionsandtheunderstandinginvitaltermsofconceptssuchas'anglesofperception,identityandcontinuity'arethemost

important.

YOUTHOFANALYTICGEOMETRY:CARTESIANFRAME.DIMENSIONS

AllthissaidweshallrestrictthestudyofAnalyticgeometryinitsfirst2agestothehumind’sframeofreference,whichislightspace,andjustrevealitasamirrorofSóTlaws,whosefundamentalequalityisthe

orthogonality=perpendicularityofits3lightspacetimeaxis,representedbythe…

Cartesianobjectivelight-spacetimeplane.

Theuniversehasinfinitemind-mirrorsdependingontheforcesusedtogaugetheexternalworld,whichbouncesonalimitedquantityofits scales of space. Humans perceive the range of scales of thefrequencyoflightbetweenredandbluesocialdensityofcolors.Butinfinite other minds with different detail according to thequantitativepixelstheyabsorb(max.S=Min.t)maximalforsmallersixes will determine the intelligence of the system. Descarteswasfully aware that what he had in the mind was not the wholeUniverse, soheexpressly stated the fact,differentiating the ‘world’of ahumanmind, from the infiniteotherworlds thatexistoutside,establishing in his little known book, the ‘world’, this difference,affirmingthathis‘CartesianFrameofreference’,wasonlythatofthe

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human mind.So he affirmed that the Universe was the sum of ∞mind-worlds that don’t speak the same languages, andcreatedthesamemappingsofrealitywehumanscreated.Sohesaidthatallwhatexistswasmadeof:

- Open §pace, which he called ‘res extensa’. - Closed, cyclical times, which he called ‘vortices’.Andthenheaddeda3rdelement,realizingtheonlyproofhehadoftheexistenceofthosevorticesandresextensawasthefactthatheperceivedthem:CogitoErgoSum.‘IthinkthereforeIam’.

Sohe establisheda frameof referencewitha centralmind thatmimicked the visualworldwe live in, hence itwouldgiveusaccuratemeasureswiththerodoflightthatcreatedourmindspace;whereaswecoulduseforallsimplifiedmeasuresthecentralpointofreferenceastheobserver’spointofview.Simplification isachievedbyeliminating the 5th dimension of inner parts of points, to study its external actions, regardless of its internalevolution,changesofphaseorstate.

This became then the basis of the external scientific method of mathematical physics, which was no longerconcernedwithinternalchangesandcouldeasilyadaptthemeasureofanylocomotionofapointorsocialgroupof points by representing the S=T, distance=motion symmetry in the Cartesian frame choosing as a preferreddirection of motion the positive X-coordinates. But unaware of the S=T duality, which only slowly becameunderstood with calculus and differential geometry, it soon seemed ‘magic’ that static lines and curvesrepresentedphysicalmotionsintheCartesianstillmentalspace.

Furtheron,theuseofthesubjectivecentralpointofviewor‘0-point’torepresentobjectiveexternalmotionsfullyindependent of the observer would make equations very complex, and so till the arrival of Generalizedcoordinates,acumbersomestructureto‘bendreality’tothesubjectivehumind’smeasuresdraggedtheanalysisofmechanics.

Abasicrulethentodiscernamongall‘multiplekaleidoscopicmirrorsofalanguage’castuponasinglerealityisthe limitofdistortion,establishedbyOccam’srule: thesimplest representation ismoreobjective,closer to theperspectiveof theobjectiveT.œ (time-spaceorganism) that cause theevent. So forexample, theEarthat thecenterofthecosmosistruthbutaverysubjectivedistortedtruthastheEarth’sgravitationalcontributiontothewhole motion of the solar system is minimal and so its mental spatial representations (Ptolemy) is verycomplicated.Whenwegotothep.o.v.ofthemaincontributortotheforce,thesun,wefindasimpleellipticalterritory formed by its maximal contribution, because the cause of events simplifies reality into its vitalbidimensionalactionsandterritorialperfectforms.Sothesuntriestocreateaperfecteencircledterritorywhereplanetsformits‘angularmomentum-membranes’andonlyslightlycontributetodeformthecircleintoaellipse.

ReferencePointsbecomethendominantspatial,mentalviewwhichresidesinasingleplanewherethe'mindofreference'perceivesinrelativestillnesstoitspointofmeasure,withnointernalscalarformorchangeofstate,andanexternalcontinuitygivenbythebackgroundlightspace,anymotionsthatthroughtheS=Tdualitycanbeassociatedtolines(distances),planesandvolumes(socialmotions)toobtainsimplifiedresultsmostofexternallocomotionofmaterialensembles.

RECAP.Isthe'lightspace-timeweperceive'real?Oritisaphasemindspace?Thequestionisansweredinseveralpartsofthispaperintermsofhumind’sart(painting),psychologyofthemind(itsequation),Relativitytheory(S=T)andinpapersofmathematicalphysics,regardingitsmeasureoftimeclockdilationandspace

distances.Riemannalreadygavethebestmathematicalanswerintuitivelyinhisfamousdissertationonphasespacesandcolorspaces...Itisamirrorofrealityandassuchrealanddistorted.

Whatisthenthebiggestdistortionofhumind’sframesofreference?Thenon-representationsavetheComplexPlaneandmodernHilbert-likefunctionalspacesandalgebraicmethodsofrenormaliationofthe5thdimension.

Eventhoughinreality,intheNon-Estructureofscalarspacetimewehaveupgradedwith®topology,allpointhave'fractalcontent'asNon-Euclideanpoints;andhencebreath,itslinesarethereforewavesableto

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communicatetheexternalformandinternalenergyorfractalnetworksthatbranchtoconnectmultiplepoints,anditsplanesintersectionofthreeofsuchwavesornetworksthatformtopologicalorganisms...allthisisinformationnolongerisavailableinaCartesianplane.ButitisNOTrequiredformereexternalmeasure,aslongasthefractalpointwemeasureisinourplaneofexistenceanditsrepresentationconcernedwithexternallocomotions,whichmakestheframeofreferenceperfectformechanicsandoverexploitedinmathematicalphysics.

Whilethereare3other'spaces'worthtonotice,toexplainallthisinnerfractalspace-timecomplexworld:

ComplexspacetimeidealforstudyingTimespaceholographicST-dimotionsandworldcycles.

Hilbertspaces,aspacesuitableforfifthdimensionalanalysiswhereeachpointisaworldinitself.

Fractalspacesandfractaldimensionstostudy‘networks’,whichpenetratethroughscalesasopposedto‘waves’thatarelineswithvolumetransmittedinthesamespace.

Inthegraph,thecreationofmentalmirrorsistheessentialprocessofexpansionandcontractionofrealityintomindspacesthroughalanguagemirror,whichinmathematicsisasdiverseasthenumberofmentalandphasespaces,ofwhich3areparamount–artisticspacesoriginofprojectivegeometry,affinespacesthatsimplifyintolinescyclicalgeometriesandfinallyinthe2ndageofgeometrydifferentialgeometrythatwillconstructsurfacesofrealitydepartingfromthemotionofafractalpoint.

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S=T:CORRESPONDENCEBETWEENMENTALFRAMESANDVITALTOPOLOGIES

AnalyticGeometry:Framesofreference.ThedifferencesbetweentheCartesianandtheComplexplane.

Aristotle was the first philosopher tounderstandthemind-Godofeachsystem

asthecentralunmovedpointofabodyofenergyitmovesarounditself,theperfectdefinitionofasingularity,originof the infiniteordersof theUniverse. Soheexclaimed, 'weareall gods'. It is the ideaofall the ideas,which from Scholar theologists to Descartes to Einstein's 'masses [that] curve space into time' has alwaysdefinedthemeaningofthemind.Letusintroducethemandstudysomedifferencesbetweenmindsaccordingto a geometry, a theme treated extensively in the article ofmind geometry. Since ultimatelywe find all theseedsofƼst,intheearliergreekculture.

Weconsiderwherethosefunctionsandoperandaresetin-thatiswhatbackgroundspaceweusetoexpressit,and3±iaretheessentialbackgroundspaceswhichcorrespondalsotothosedimotionsasformsinspace,the3lineal=cylindrical,spherical=polar,andhyperbolic=Cartesianplanesandthescalarplane,ill-understood,whichisthecomplexplanebetterperceivedifwe'squareit'eliminatingthe√symbolofitsnegative-1axis:

Sotheycanstudy2fundamental 'emergent'∆+1planesofmathematics,thestudyofDimotionsofDimotionswith the tools of calculus in time, and the study of spaces of spaces, with the tools of the complex planeproperlyunderstoodintermsof'square'coordinates.

ButasintheentangledUniverseallmirrorscanreflectallforms,Algebraalsocananalyzeotherelements.Butitsmainbeautyisincreatingsequentialchainsofpentalogicactionsthatreflectthemotionsofexistenceofthebeing,eventhoughits'Group'simultaneousanalysisofallits'variations'ofspecies,hasbeen

The fundamental graph of the Universe is one in which orthogonal coordinates represent the T-independentparameters in the X-coordinates and the T-parameters in the Y-coordinates. But we do have two differentrepresentationsforthem,becausewedohave4differentSandTdominantdimotions(withtheSTcombinationofboth,abletoappearinthez-coordinates,orthecombinationofboth).

SothebigquestioniswhatcoordinatesbelongtowhatDimotions.AsSSandTTdimotionsareequalinvalue,the pure coordinates should belong to the Cartesian plane.While the S-informative coordinates do have alesservalue.Sotheymustbeputonanimaginarysystemofcoordinates.

OrthogonalityintheUniverse,istheneasilyexplainedasfollows:

Because Entropy (TT) vs. absolute linguistic still form (SS), Locomotion (Ts) vs. information (St), are the dualinverse functions of reality merged only in the S=T reproductive dimotion, in the 0 point of X-length, therelative dimension of locomotion, there is a zeromotion and stillness rises in the height dimension of pureform, where the 0’ mind or frame of reference resides. But then we deal with the ‘different quality’ oflocomotionandinformativeperceptionintermsofexpenditure’ofenergy,asinformation‘shrinks’motion.I.e.agravitational invisible tachyon linehasno informationbutwhen itbecomes light (neutrino theory,Broglie-

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>Jordan),itformsawaveofinformationthatgrowsinheightwiththephotonontop.Butthisheightdimensionisintermsoftheparameterofenergyandlocomotion(T),acompressed‘spatialT’,ofminimalsize.Andsoweneedasmaller‘quantity’andonethatisnegative,‘subtracting’fromthedistance-speed(s=t)oflocomotion.

This ismagicallyachievedby thenegative ‘root’ valueof the imaginaryaxis, reasonwhy itappearsas –ct inrelativityandissousefulforthestudyofelectricwavers.Bysquaringbothwesimplifytheproblemof√negativeroots,weshallexplainlatterwhenweanalyzeindepththeinverseoperandsofalgebra.

SothecomplexplaneismostusefulforSt-Tssystemsoftwocomposite‘energy-information’body-headforms.

Thesubjective3p.o.v.sofmathematicalminds.|-Cylindrical,O-Polar,Ø-Cartesian.

Theternarynatureoftheuniversaltopologyinasinglescale=planeofspace-time,isevidentwhenweconsidertheother2canonical'coordinatesystems';thepolarcyclicalplaneandtheCylindrical,toroidplane,whichwillgive us 3 different 'views of the Universe', both in mental subjective space from the new P.o.v. and intopologicalspaceaseachframeofreferencewillsimplifytheSTdimotions=actiontakenplaceinsidethe‘organ’of a system that mimics topologically one of those 3 forms. Whereas the Occam’s rule apply: the simplestequationrevealsthetypeoforgan:

'The simplest frame of reference in which a problem formulates indicates which is the ternary topologicalelementwestudy'.Forexample,iftheproblemformulatedinpolarcoordinateshasanequationsimplerthanincartesian coordinates, we are studying a ð-particle/head element (as when we formulate a gravitational orchargeproblem inpolarcoordinates). If it is simplerasa toroid/cylindrical is (a toroidopenedalongaz-cut),equationitisa'$t,limb/fieldproblem'.AnopencurvedefinesinS=Ttermsalsoa'linealmotion’andsoon.

Soeitherthe3framesofreferenceorthecurvedrawnintheCartesianframedefinealreadybyitsform,whichorganic ternary topological parts in space or event in time IS acting on reality: 'cylindrical frames for lineallimbs/fields',hyperboliccartesian frames forbody-wavesorpolar frames forparticles-heads. Buteventsandsystemschange states, So it followsnaturally thatby 'changing' theequationsof systems fromone frameofreferencetoanotherwechangeoftenthetopologicalanalysisofthembecausethesystemhaschangedits|xO= Ø relative state - a fundamental feature of quantumphysics, described as a hyperbolic wave in Cartesiancoordinatesandasaparticle-fieldinpolarco-ordinates(Bohm'smodel),whenaquantumsystem‘changes’fromS=Twavetoparticle-field,S<T>Sstate.

Sothechoiceofcoordinates,inwhichthefunction/formissimpleroftenindicatestheytypeofpart-species,weareanalysingaccordingtothe'generatorequation'ofmind-coordinates:

Γ(generatorofmindp.o.v.s):|-$t(cylindrical)<ST-Cartesian>§ð(polar)«»∆±i(complex)

ThemostsuitableformanbeingitslightspacetimeCartesiancoordinate,withthenegativeandpositive,inversedirections,self-centred in ∆o, the distorted point of view of the human observer, suitable for lineal,perpendicularbutalsohyperbolic,wave-likelightspace-timerepresentations.YetastheCartesiangraphtendstoanobjectiveexternalflatgeometry,savethe0-1sphere,oftenforself-centeredrepresentationswherethemind view is themain causeof themotion, apolar systemwill be simpler,more suitable, as in theParticle,Bohm’srepresentation.

In thegraph, the3 subspeciesof2-manifoldshave their expression in3 coordinates,where theCartesian, istakenasan'infinitygrowingToroid'space.

Inverselyifweperceivethosecoordinatesfromthe∆-1scaleaswedidwith‘spiralsasworldcycles’ofexistence,wecan‘constructthemandtheorganstheymirror’asthesumofalltheworldcyclepathsofits‘neighborhoodpoints’; So that Superorganisms can be defined as ternary adjacent ensembles of the geodesic curvesperformedwithineachorgan/frameof referencebyeachof its∆-1 fractalpointsput together in3 topologicelements,livingwithaworldviewadaptedtoeachofits3correspondingorganiccoordinates.

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Philosophy ofmathematicsthen enables to analyze in depth tour 'selfie' axiomaticmethods of truth, which'reduce' the properties of the Universe to the limited description provided by our limited version ofmathematicalCartesianframeofao-pointwithnoparts,knownasEuclideanmath(withanaddedsingle5thnon-EPostulate)andAristotelianlogic(A->Bsinglecausality).ThislimitmustbeexpandedaswedowithNon-ÆvitalmathematicsandthestudyofMathswithinculture,asalanguageofHistory,usedmostlybythewesternmilitarylinealtradition,closelyconnectedwiththeerrorsofmathematicalphysics.

TheimportanceofthePolarframe.

Contrarytobelief,themostimportantframeofreferenceformost‘mentalspaces’isthePolarframe,followingtheruleofsimplicity–thatmakesamoresynopticformwithlessparametersthefundamentalone.Itshouldnotsurprise thereaderbecausebeing ‘space’amentalgeometry,ormindmappingof reality itdoeshappenusually fromthesubjective internalpointofview. Inbriefmostmentalspacesareconstructed inasubjectivedistortedviewprovidedbyangulargeometryasopposedtoanobjectiveexternalviewprovidedbyahyperbolicCartesiangame.2exampleswillsuffice:

-Thesimplestrepresentationofalltheconiccurvesweshallstudysoonisgivenbyitspolarrepresentation:

Intheplane,wechooseapointP(pole)andarayoriginatingfromit(polaraxis)anddeterminethepositionofapointMbythelengthρofthepolarradiusfromthepoletothepointandthevalue

ωoftheanglemadebythisradiuswiththepolaraxis.

Inparticular,theellipse,hyperbola,orparabola,ifforthepolewetakeafocus,andforthepolaraxistheraypassingfromthefocusalongtheaxisofsymmetrytothesideoppositethe

nearervertex.Thenwehaveoneandthesameequation:

whereeistheeccentricityofthecurve,andpisitsparameter.Thisequationisofagreatimportancein

astronomy.Foritwaswithitshelpthattheresultwasderived,fromthelawofinertiaandthelawofuniversalgravitation,thattheplanetsrevolveabouttheSuninellipses.Itmustthenbenoticedthattheobserverisinternal

totheconicasa‘point,part’ofthewhole,wheretheconicisitsexternalmembrain.Hencetheobvious,vitalpracticalneedforany‘entity’withinitsterritorytoassessthedistancetoits‘border’,whichitcannotcross(openballgeometryforthe∆-1partsofitswholestructure,consideredinthe1st¬Epostulate).Itisthenrequiredforamindspacetobeabletoassessthatdistancewithminimalelements.Andsincew,theanglecanalwaysbe

‘assessed’insitu,theonlyparameterrequiredisp’slength,asopposedtotwoparametersinaCartesiangraph.

Thegeographicalcoordinates,latitudeandlongitude,bywhichthepositionofapointisgivenonasphere,arealsowellknowncasesofpolargeometries,whichbasicallyextendto3Dthepreviousanalysis;orratherestablish

the@-mindofmeasureinthemembrainsurfaceoftheT.œ.

This subtle distinction between O-subjective polar coordinates vs. ST-objective Cartesian coordinates comeultimatelytothedifferentpropertiesofwave-bodies,whichmergeS&Tandhencearemoreobjectivevs.thedistortionof§-mentalparticles-heads;andhavedeepconsequencesbecausewecanconsiderrealityaconstantswitchbetweenbothstates.Suchisthecaseof:

-Therealistviewofquantumphysics,madewiththepolarrepresentationofSchrodinger’swave(Bohm).Sowecanconsiderthedualitywave-particleasaconstantswitchbetweenangularspinperceptionofrealitybytheparticleinstopposition,andthewave-motion/reproductionofthephysicalsysteminitshyperbolicwavestate,whichmust‘merge’and‘synchronize’withboththefield,∆-1quantumpotentialandthe∆+1particlestate.

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GENERALIZEDCOORDINATESFROMTHEPERSPECTIVEOFTHE0-POINT.

MorerealistthanaCartesiangraphthatforcefeedsthehumindp.o.v.isthecomplexviewofatopologicalternarysystemasafractalof3adjacentinteractinginternalpartsthatcalculateitslocomotionwithouta

subjectiveagent,asameasureofthechangesofdistances-motionsorrelativepositionsofthepartsoftheTœ.

Thenmeasureincludesalltherelativepartsofthesuperorganism,takenfromtheinternalstillpointofviewofitsparticle-head,asitscenterofreference.Sothesubjectiveexternalhumindpovdisappearsandthestill‘real

center’oftheactiontreatstheotherpartsaselementsofarelativelyrigidbody,whosecoordinatesaresimplifiedwithGeneralizedcoordinates,NOTconnectedtothehumanobserver,andcausalityimprovedwith

the‘aristotelian,unmovedGod,causefthemotionofitsbody-wave’asthecenterofcoordinates

IntheoldsystemakintothecomplicatedPtolemaicEarthcenter,undertheÆntropicprinciplethehumanobserveristherelativemind-viewcenterofthewholesystem.So,hethencanchoosethe

Cartesiancoordinatesasanalien,externalpointofview,makingitallmorecomplicated.Yetitisfarmoreconvenienttochoosetheobjectiveinternalcoordinatesofthesystem,anditsstillparticle/headasitso-point

andcause,aswiththeheliocentricsystem.

ThecompoundplanependulumconsistsoftworodsOAandAB,hingedtogetheratA;thepointOremainsimmovable,therodOAturnsfreelyinafixedplanearoundO,andtherodABturnsfreelyinthesameplane

aroundA.EverypossiblepositionofoursystemiscompletelydeterminedbythemagnitudeoftheanglesϕandψthattherodsOAandABformwithanarbitraryfixeddirectionintheplane,forexamplewiththepositivedirectionoftheabscissaaxis.HencetherealO-pointmindofthesystemisAandgeneralizedcoordinatesby

choosingitwillsimplifythecalculusofitssurprisinglychaoticdimotions.

Considerasimilarexample:asystemoftworigidlyconnectedpoints,thesecoordinatescanbechoseninthefollowingway:thepositionofoneofthepointsisgiveninCartesiancoordinates,afterwhichtheotherpointwillalwaysbesituatedonaspherewhosecentreisthefirstpoint.Thepositionofthesecondpointonthespheremaybegivenbyitslongitudeandlatitude.TogetherwiththethreeCartesiancoordinatesofthefirstpoint,thelatitudeandlongitudeofthesecondpointcompletelydefinethepositionofsuchasysteminspace.

Thefirstpointisthen'fixed'ina.hyperbolicCartesianplanethatcanstructureallothersystemsandthesecondinpolarcoordinates,respecttothefirstmindpoint,fromwhichtheyarenolongerfree.Andsoaspartofanewwhole,thenumberofcoordinatesrequiredtostudyitdiminish.Andthegenerallawisrathersimple:weshall

needforasystemjustthenumberofgeneralizedcoordinatesequaltoitsnumberofdegreesoffreedom.

i.e.Ifweconsider3particlesrigidlyfixedinatriangle,thenthecoordinatesofthe3rdparticlemustsatisfythe3equations.Thusthe9coordinatesoftheverticesoftherigidtrianglearedefinedbythe3equations.Henceonly

6ofthe9quantitiesareindependent.Thetrianglehas6degreesoffreedom.3pointswhichdonotlieonthesamestraightlinedefinethepositionofarigidbodyinspace.These3points,as

wehavejustseen,have6degreesoffreedom.Itfollowsthatanyrigidbodyhas6degreesoffreedom.

In5Dtermsitmeans,the'singularity'orcenterofreferenceasthestillcause,disappearstodescribeitsmotion,with6degreesoffreedom,whichareequivalentto2points.

Italsofollowsthatmechanics,thescienceof2С=locomotionevolvedfromsubjectivehumanpov(Newtonian)togeneralizedcoordinates(Lagrangian),whichishowtodayprofessionalphysicistscastitslaws,asweshalldo.

Inotherwords,amechanicalsystemcanbedescribedbycoordinateswhosenumberisequaltothenumberofdegreesoffreedomofthesystem.ThesecoordinatesmaysometimescoincidewiththeCartesiancoordinates

ofsomeoftheparticlesormightnot.

Theconceptofgeneralizedcoordinatesthatmakerealthepointofview,causaloriginofthemotion,canthenbeextendedtotheuseofthesuitableframeofreferencethatimitatesbesttheorgan-worldinwhichthemotion

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takesplace.SothekeyforaneasysolutionisachoiceofcoordinatesaccordingtotheNatureofthemotionofthat'wholesystem'.Ifthedescriptionisoneof1D'rotarymotionsaroundthesingularity',apolarsystemworks

better.Ifwedealwithalineal2Dmotionofthewholesystem,cylindricalmightbeused;forallothersthehyperbolic'deformable'Cartesianplaneisbest.

RECAP.Self-centeredpointsofviewperforming5Dimotionshumindsorgeneralizedcoordinatesmeasure.

Wearemathematicalorganisms,withtopo-logicproperties,whichgivebirthtobiological,organicassemblesofternaryfunctions/formsandthosesystemsdohaveoneofthe3O-pointofview,itsframeofreference,asthestill‘mind-will’ofthe5Dimotionsitperformstosurvive.Thereisalwaysafirstobserverwhichstartsanactionofperceptioninspace-timefromaperspectivethatusuallyisbiasedbythefunctionoftheobserver(ST,SiorTe

coordinates),whichcorrespondtotheCartesian,Cylindricalandspherical,polarcoordinatesofscience.

Thuswedefine3fundamentalcoordinateswhichanyentityusesto'adapt'theperceptionofrealitytoitsmind-viewandtheequationofthemind,whichintermsofmathematicalco-ordinateswrites:

O-pointx∞Universe=constant,staticframeofreference.

IntherealUniversetheObserveristhedominantelementofthose5actionsandalsotheinitial'pointofview'ofanymathematicalanalysis,andanalyticgeometryrightlythefirstbranchofmathematicstobestudied.

Inthefirstofmanyfascinatingsymmetriesbetween'iTS'andeachscience(isomorphic,|-space&O-Time,theabb.mostcommonlyusedforthe3componentsoftheUniverse),3arethefundamentalpointsofviewandframesofreferenceofanalyticalgeometry,eachonebelongingtoa'fundamentalstate'ofbeing,inthe

Universe.

Thetemporalpolarpointofview,centeredintheO-pointanditsexternalmembranedeterminedbytheradiusandangleofperception;thecylindrical,lineal,energeticpointofview,determinedbythelinealaxisofthe

frameofreferenceor'altitude',theZco-ordinates,andfinallytheCartesian'hyperbolicplane',correspondingtotheSTibodies&wavesofphysicalandbiologicalsystems.

Fromthose3mathematicalperspectivesrealityconstructsitsvitalgeometriesthatwecall'existentialbeings'.

Sotheobserver'scausal,logic,cyclical,informativesentientpropertiesthatallowittoperceivetimecyclesarethefirstquestionstoinquire.As@isthefirstelementtodescribeinreality-aframeofreference,anobserver,aninertialpointinrelativefixedformwithnomotionthatcanperceiveandmaptheUniverse,asitsOx∞

constantmappingmind–world.Sogeneralizedcoordinatesaremorerealisticandsimpler(Occam'sprinciple).

ItfollowsthatGeneralizedcoordinatesisamorerealistanalysisofNature’sdimotionsastheyareestablishedbythe'stillpoint',whichisthemindorhead/particlethatweobservetocreateitsowngeneralizedcoordinateswhereittriestomaintainitsstillness(yourheaddoesn'tmoverespecttothebody),asthecenterofreference

ofitsownUniverse.

Insciencegeneralizedcoordinatesareusedtostudylocomotionbutin5Dweextendthemtostudythe5Dimotionsofthebeing,andinmanycasesanalyzethemquantitativelywhenwemergethoseconceptswiththetoolsofcalculusand‘define’properlyenergy(3rdÐimotion),reproductionofform(2ndÐimotion),5Dentropy

(DualscatteringÐimotion),4&1Dperceptiveinformation(imploding,spiralingdimotion).

TheUniverseisasumofaseriesofaction,∆aeiou:thefirstactionisperceptionbyanobserver,∆o,ofafieldofenergy,∆e,towhichaT.œwillmove,∆a,inordertofeed,∆eandusethatenergytoreproduceitsinformation,∆i,iteratingaformlikeitself,whichwillgatherwithcloneformstocreatealarger,∆Uuniversalsocialplane.

Moreoverthoseactionsareverysimplegeometricexchangesofmotionandform.’Admaximal’theyaredualentropicinnerscatteringandouterlocomotionsorinverselythedualcollapseofmotionsthatshrinkintoaseedoftenby‘eliminatingmotion’intoa‘codedlanguage’thatwillunfoldwhenenergygerminatesit.Furtheronthe

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S=Tsymmetryprecludesthatamotionstepisfollowedbyaninformativestillperceptivestop.Whenproperlycastinsuitableframesofreferenceorgeneralizedcoordinatesphysicalandbiologicalactions,dimotionsandchangeofstatearesimple,bidimensionalstœpsorsingleflowsofenergy:sT;andinformation:St,between2poleseitherinasingleplaneoracross∆±3,4scales.Somostlawsofsciencearesimple,evenwhenthoseflowsaregeneralizedbetweenseveralpointstoformmorecomplexstructures;sincethentheywillberepetitionsofthesimplerunitofaction,whenwereducetominimalcyclicalspace-timeactionsthetotalrealityofanyself.

Buttodosocorrectlyweconsiderthepropersequenceofaeiouactions,whichmostlystartwhentheformalstillpointofviewkicksoutaworldcycleofactionsproperofanyfunctionofexistence,withanactof

perception,∆o,whichisthustheminimalandfirstUnit-actionoftheUniversethatshrinksitintoalinguisticmirror.Thuswhiletime-motion-entropyisthefirstsubstanceofreality;perceptionofitasformbecomesthefirstelementofanyfractalpointorT.œ,whichcreatesinfactby‘shrinkingthewhole’intoapointofasmaller∆-1scale,theactionsofcreations,thestorageofinformation,thegameofrealityasweknowit,pastthepure

entropicmotionmeaninglessinitself.

Soitisalwaysapointofviewanditsrelativeframeofreference,whatstartthecomprehensionofanexternalrealitywithitsfirstaction,whichisthestilllanguagethatperceivesinitsself.

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CARTESIANGRAPHS.

ThemultipleS=T,∆applicationsofCartesiancoordinates:mixingtimeandspace.

Themain5DconceptforunderstandingmathematicallawsofcurvesinCartesianspacesistheS=Tduality.Hencetheacceptanceoftheruleofdifferentialgeometry–thatis,acurveisnotreallyacurvebutanST

representationofanS-pointtracingaT-Dimotion;henceageodesictrajectory,anST-dimotionadaptedtothelargerorgan=worldofthebeingwithinwhichthefractalpointperformsitsfunction.

Inflationarymathematicshowever,withouttherestrictiveanchorageofthereal5DlimitofDimotionsanditsrequiredgeodesicefficiency,drawsalsoabignumberofirrelevantcurvesthatexhaustthecombinatoryofitsparametersinaplane.Butcurveequations,inthepraxisofmathematicalphysicsrestricttothemostefficient

trajectoriesforthe5Dimotions=actionsofanybeinginexist¡ence.SomoststillCartesiangeometryisadisguisedformofdifferentialgeometry.WhilephysicsasitistodayformulatedONLYinmathematicallanguages(reducingitspropertiestothosewhocanmeasure,athemetreatedinourpapersonphysicalsystems),canbeconsidered‘defacto’thebranchofexperimentalmathematicsor5Dmathematicsthat‘reduces’itsinflationary

mirrorlanguagetotheonly‘real,efficient’elementsofit.Andassuchwetreatitasasubdisciplineofmathematics,orparaphrasingEinsteinandPoincare:“Mathematicsaretruthbutonlyexperimentalphysics

defineswhentheyarereal.”

Botharesimplifiedmirrorsofthehumind,asonly‘thebeingholdsalltheinformationaboutitself’.Thatisifthebeingisaprobability1ofexistence,onlyinthebeingtruthbecomesalsoaprobability1.Whileweneed

multiple‘experimentallanguages’toextractallitsproperties.S=Tdualityistheoriginofthe2keypropertiesofequationsinCartesianspacetime-thedualnumbersrequiredbyeachpoint;anditsdynamicmotionsasthey

traceacurve.

1. Inthatplaneapointhas‘2dimensions’=parametricnumbersforitsrepresentation.

Whatthosenumbersmeanbeyondtheearliernaïverealismofusingthemforsimplelocomotionsinthedifferentplanesofforces(gravitationalverticalorflat,frictionspaces),iswhatwillopenphasespaceand

mentalspace,andnowwetaketoitsfinalconclusion,consideringthatoverwhelminglytheyrepresentSorTcoordinatestodefineeachtypeofmentalspaceasarepresentationofoneofthe5NaturalDimotionsofreality’

andor5correspondingNon-Epostulatesthatdescribetheinteractionbetweenfractalpoints(T.œs).

Theabscissaandordinate2coordinatesofapointintheplaneDescartesarenumericalvaluesxandyoftwomutuallyperpendicularstraightlines(coordinateaxes)choseninaflatbidimensionalplane.SotheyperfectlyexpressanSTinverseholographicdimotionreducedtoapoint.Thepointofintersectionofthecoordinateaxes,

i.e.,thepointhavingcoordinates(0,0)calledtheorigin,mimicsthenthestaticstatewherethedimotionstarted.

AndsoDescartessubconsciouslyexpandedthedimensionsofanEuclideanpointthroughits“arithmetization”asthepointintheplanethatrepresentsits∆±1worldvaluehasenoughinformationtofillthecontentofapair

ofnumbers.

2.Pointsasequationswith2unknownstracingcurvesintheplane.

Descartes’secondSTconceptisthefollowing.UptothetimeofDescartes,whereanalgebraicequationintwounknownsF(x,y)=0wasgiven,itwassaidthattheproblemwasindeterminate,sincefromtheequationitwasimpossibletodeterminetheseunknowns;anyvaluecouldbeassignedtooneofthem,forexampletox,andsubstitutedintheequation;theresultwasanequationwithonlyoneunknowny,forwhich,ingeneral,the

equationcouldbesolved.Thenthisarbitrarilychosenxtogetherwiththeso-obtainedywouldsatisfythegivenequation.Consequently,suchan“indeterminate”equationwasnotconsideredinteresting.

Descarteslookedatthematterdifferently.Heproposedthatinanequationwithtwounknownsxberegardedastheabscissaofapointandthecorrespondingyasitsordinate.Thenifwevarytheunknownx,toeveryvalue

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ofxthecorrespondingyiscomputedfromtheequation,sothatweobtain,ingeneral,asetofpoints,whichformacurve.

Thus,toeachalgebraicequationwith2variables,F(x,y)=0,correspondsadeterminedcurveoftheplane,namelyacurverepresentingthetotalityofallthosepointsoftheplanewhosecoordinates

satisfytheequationF(x,y)=0.ThisobservationofDescartesopenedupanentirenewsciencebesidesanalyticgeometry–

thelanguageinitself.SinceessentiallyDescartesgiftedthestaticbidimensionalplanegeometryofa'variablemotion',andgaveusthecapacitytostudyanST-evolvingsystemin

space-time,whosegeodesicswillgiveuspropertiesofcertaindimotionsasopposedtoothers(closedpaths,openpaths,entropic,scatteringmotions;etc.).Thus‘Physicsoftime’were

born.

So for understanding of analytic geometry and its algebraic equations, it is useful to consider the fact thatnumbersarepointsandthereisadirectrelationshipbetweenpointsandnumbers,linesandvariables,planesandsquaresandsoon.Sopolynomialsandoperandsrepresent the 'socialevolution'ofdimensions,points into linesintoplanesinto5dimensionalstructures.

PentalogicontheCartesiangraph.

TheCartesiangraphbyrepresentingmindsinitscenterofreference,dimensionalsymmetriesofspaceandtime-bothinformsandmotionsandbyreducingT.œstopointssusceptibleofscalar∆nalysiswiththeconceptsof∆-1derivatives(finitesimals)and∆+1.-yieldsenormouscapacitytomodeltheUniverseof∆@stofspace-time.

SoAnalyticgeometrycouldstudyallthepentalogicelementsofreality.Someofitsfirstuseswouldbe:

1:T>S:solvingconstructionproblemsofcontinuousmotionwithdiscretespatialsteps,suchasthedivisionofasegmentinagivenratio;thusaddingfrequencytimetogeometry.

2:S<T:findingtheequationofcurvesdefinedbyageometricproperty,whichcouldrelatedifferentpentalogic,∆@stelements.Forexample,defininganellipseofmotion(ST-holographicform)bytheconditionthatthesumofdistancestotwocomplementarypointsofreferencedualattractiveorgenderpoints)isconstant.Inthiscasetheellipsebecomesamentalspacefunctionthatdefines2physicalsystemscontrollingwithequalforceathirdentityandbydoingso,creatingacommonterritoryofspace(Kepler's2ndlaw,‘proximity’of2parentalformsto

itssonwithina‘territory’).

3.S≈T:provingnewgeometrictheoremsalgebraically(i.e.thederivationbyNewtonofitstheoryofdiameters;andconversely,representinganalgebraicequationgeometrically,toclarifyitscombinedSTproperties(i.e.,thesolutionofthird-andfourth-degreeequationsfromtheintersectionofaparabola–entropic,scatteringmotion,

withacircle–closedorderedmotion),whichwewillillustrateinits‘metaphysical’space-timewhys.

4.∆:peeringin5Dscalesthrough∆nalysisofderivativesandintegrals.

5.@:Studyinghuminds,asthekeypropertiesofCartesianplanesimitatethepropertiesoflightthatfollowthepropertiesofspace&timecomponentsoflightwithits3perpendicular,S-magnetic,T-electricandST-energy

fields.

Thus,totheclassicdefinitionofanalyticgeometryasthatpartofmathematicswhich,applyingthecoordinatemethod,investigatesgeometricobjectsbyalgebraicmeans,wewilladtheinsightsofitsdirecthomologywiththeS,T,∆stelementsofrealityindifferententangledpentalogiccombinationsthatextractSpace-timegenerallaws.Sinceitisonlyneededtoconsideraninformativeortimefunction,oneofthecoordinatesandthentheotheronewouldrepresentthe'spatialfunction',whichareinversedimensions,tomakeitworkmagicallyan

representaginormousnumberofs@≤≥∆ðsomethingsoonusedbyGalileo.

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Complex5Dfunctions.

Graphicpentalogicappliesthentoeachspecificfunctionwithnewentangledelementstorevealthemultiplefunctionsofeachmathematicalsystem.I.e.anexponentialfunctionrevealsits:

5D:max.inverseentropic‘deathgrowth’atacceleratedspeedtowardsitsasymptoterevealedbyitsderivative.

2D$:Max.spatialgrowthasaspatialfunctionofspeedanddistance.

1Dð:Themaximalgrowthcurveintimerepresentsavortexofspace-time,with2TTdimotionsacceleratedchange,asaforce;henceao-1vortexofaccelerationtowardsasingularitypoint(1D)

∆±i:Allthoseexpoenential±functionsthusendintoachangeofscale.Eitherthefunctionacceleratestowardsanentropicdissolution(e¯ª,whichisadecayentropicprocess)orinverselyacceleratestowardsanemergence,asinresonances,orthelimitingcaseofdistributionequationsthatgrowexponentiallytowardsthe0-point.ItisthesimplestDeltaFunction,whoseintegralis1,evenifitonlyexistsinthat0pointshowingthatindeedapoint-particleisafractalpointwithdimensionality1,onceemergenceinits'infinitedensity'pointofresonance,

emergingintoan∆+1plane-afactwhichincidentallyprovesthatallinfinitiesAREfiniteonesofahigherplane:infinityISjustthelimitofan∆-plane;thewholeistheinfinityofthepart;orelsethedeltawouldnotemerge

whenintegratedbetween∞and-∞as1.Becauseoncewecrossadiscontinuumofscale,weareinanothertypeofparameter,soinfinitydoesNOTexist.

Functionsthenbecomealwaysrepresentationsin@-mindspaceoftrans-form-ationsof∆-scale,spacepopulationortimemotions,whicharethe3elementsofanysystemofreality.Andsopentalogicappliestoallfunctions.

I.eTemperaturedeterminestheS-volumeofagasasawhole.Butalso∆+1temperaturedeterminesthemotionofits∆-1gasmolecules.WhiletheT-growth=elongationofagivenmetallicrodisdeterminedbyits

scalartemperature.Itwasuniformitiesofthissortthatservedastheoriginoftheconceptoffunction.

RECAP.Descartes’theoryisbasedontwoS=Tconcepts:perpendicularS=Tcoordinatesandtheconceptofrepresentingbythecoordinatemethodanyalgebraicequationwithtwounknownsintheformofacurveinthe

plane.

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CURVES.

ThefirstuseofCartesiangraphwasthentomovefurtherthelastadvancesofthe3rdageofGreekGeometrythathadstartedtoexplorecurves.Andasthebest-knowncurvewastheclosedcycle,geometersattackedthe

cycloid–acyclemovingalongtheline.Weshallthusconsideritasaparadigmofthepentalogicandexperimentalnatureofcurves,whichreflectdifferentspeciesofSimpleT.œs(itsmembraneandenclosedvital

energy),aswellasfractalpointstracingcanonicalcurves=worldcycles

Thecycloidasaworldcycle.

Theinterestofmathsasamirrorofrealityisits'simplicity'todescribethebasiclawsandsymmetriesofspace-time,andhencethepropertiesofworldcyclesandsuperorganismsand∆-planes.Thisreachesitsfinalsimplicityintheanalysisofcurvesinabidimensionalplane,andthewhystheyreflectonS=Tsymmetries.Letusexplore

someofthoseelementsof∆stisomorphismwithmathematicalmirrors.

Wealreadyanalyzedthebeautyofthepi-spiralasaworldcycle,thefundamental‘timeparticle’,thatrepresentstheexistentialflowinfinitetimeofanysuperorganismthatclosesitsworldcycleasitsouterpointmembrain

returnstoitsorigin.Nextinthe‘evolution’ofthecycleappearedthecycloid,wherethepointmembrainitsvitalenergy-radiusperforminginanexternalentropic‘flat’worldline,itscyclicaldimotionsbetweenitsoriginalbirthanddeath,whentouchingbacktheflatline.Whatfascinatesthenonthecycloidworldlineisthatitencodesthe

quantitativeparametersofthe3ages&8'sequentialphases'oflifeofanyDisomorphicworldcycle:

Inthegraphstwodifferentisomorphiclanguages:the8diametersofthesimplestcycleofexistence,acycloidmovingonalineal,openpathorentropic

higher∆+1worldfrombirthtodeath;belowthe8Baguastheinformative,mongoloid

humansubspeciesfoundtocorrespondtothe8phasesoflife.Suchtypeofhomologies

showstheentangledNatureofallthelanguages,asallmirrortheultimate

DisomorphicS-Tlawsoftheexistentialgame.Specificallytheparametersofaworldcycles

representedbythecycloidtracedbyapoint–themind/membrain-onthecircumferenceofacircle,itsvitalenergy–representsaT.Œasitrollsalongastraightline,theouterworld,enclosingaterritorialspace,the

cycloidsurface.

TheimmediatetranslationthusoftheworldcycleisprovidedbyGalileo’sconjecture-theareaenclosedbyonearchofthecycloidisthreetimestheareaofthegeneratingcircle,asthe3agesoftheorganismrepresent3statesofitsvitalexistence,withthecentralmature,present,reproductiveage,equivalentinvaluetothesum

ofitsfirst,young‘growing’andlast,‘diminishing’oldageoftherisingandfallingcycloidworldcycle.

AndbyWren’smeasurealongthecurveofonearchofthecycloidequivalenttoeighttimestheradiusofthegeneratingcircle,representing,asthemovingdynamicpointinwhichthemindandtheouterworldmembrainentangles,the8phasesofthelifeofthebeingintheworldorbaguas.ThusthehiddenbeautyofthecycloidthatmadeitsoimportantinearlierCartesiangeometryencodesifweconsiderthemovingcycleaworldcycleoflifeanddeath,thepointonthesurface,thesingularity-mind,transitingalongalinealsequentialtimeline,thesimplestdisomorphicmirrorofthepropertiesofalife-deathcycle,inwhichour'vitalenergy'(theareaenclosed

bythesingularityandthetimeline)issplitin3ageswithasimilarvolumetothatoftheunitcircle.

Asindeedthe∆-1generationalo-1ageisfollowedby3moreagesinwhichthe1stand3rdageareequalinvitalvaluetotheintermediateone:

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sT=youth>ST:maturity>St:3rdage...

Whenthecycleendsinthe'landing'linealflatentropicpointof‘timeline’death,exhausteditsvitalenergy.

Anotherinterestingversionistheepicycloidona3worldradiusofa1radiuscurve,whichwilltrace3cycloidagesoverthelargerworld

eachwith3epicyclesurfacesmimickingevenmoreclearlytheworldcycle.Itisthiskindof‘magiccoincidences’thatencodetheessentiallawsofTime-spaceworldcycleswhatIfindsoenticingtodiscoverinmy5Dresearch.Whyothermathematiciansdonotevenattempttodosoisnolongermybusiness.Igaveuponhumanegocy.Yetwithoutthoseexperimentalrelationshipsbetweenmathematicallawsandspace-timeworldcyclesthesubjectloosesits‘truewhys’.

Wecanthenconsiderthelargercircletheminimal‘surfaceoffeedingentropy’,forthesmallercircletocompleteaworldcycleofexistenceandits3ages.Indeed,fora1to2relationshipwegetanephroidandthesmallercirclecanonlycompletetwoages,whicharefullyinverse.

Andwecanobservesome‘dimotionalscalings’:theperimetertracedbybothisina3relationshipbutthevitalenergy2DSareaπr2isina9to1relationship,asthevolumeofthesystemgrowsexponentially,slowingdown

theinnerprocessesofthelargerwhole(5Dmetric)

themostremarkablepropertyofallworldcyclesrepresentedbythecycloidwasfoundwhenmathematicianssolvedthevariationalproblemsofthebrachistochrone-findingtheshapeofacurvewithgivenstartandend

pointsalongwhichabodywillfallintheshortestpossibletime:Itisthebeginningofanupside-downcycloid!

Soallworldcycles‘T=race’throughthe0-M1palingeneticandyoungagetoachievethemaximalgrowthofthesystemthatreachesitsmature‘M2’pointstateintheminimaltime.

Sothebrachistochroneisthespatialsymmetryofthelawofleasttime;akeylawof∆st,asallsystemstrytoachieveitsactionsandmotionsinthelesspossibletime;thussubconsciouslyshorteningitsyouthandoldage–whilethematureiterativeageofreproductiontriesbythe

verynatureofitsmaindimotion,toconservethestateofthebeing,becominga‘flatter’curvewithapeak,standingpointof0-change/derivative.

RECAPThissimplestrepresentationoftheworldcycleshowsalreadythekeyinsightof5D∆@st:curvesrepresentedonanalyticgeometryarespatialsimultaneousS=TgeometriesoftemporalDimotionsofthehead-

pointofthecurve.

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PENTALOGICOFCURVES.

ItisthenevidentthatcurvesmodelmultipleelementsofGST;beingitsmainpentalogic3±¡features:

∆:Itscapacitytorepresentthe3scalesofanentity,asafractal∆-1point,tracingaworldcycle,∆º,inalargerworldrepresentedbytheopen,entropicflatCartesianPlane,

S=T:Theycanalsomodelthe3elementsofaT.œ,ifweconsiderthecurveinsynchronousspaceamembrain,whereitisoftenlocatedthe@-singularity(opencurves);orwhenthecurveisclosed,itsfocusrepresentasingleordualcomplementarycenter,whiletheenclosedvitalsurface,isakintotheenergyofthesystem.

Thusasimplecurvealreadyaccountsforthe3topologicpartsofthebeing,andits3∆±¡scales.

SuchinsightsontherealworldofDimotions,representedbythestillgeometryofcurvesinCartesianspacewillbecomeevenmoresophisticatedwhenweobservethepropertiesoftheCanonicalcurvesof5Dimotions:

Pentalogiconconics

S=T.The1stremarkablepropertyofConicsistheextremesymmetryintermsofitscoordinatesofitsgeneralequation:Ax²+Bxy+Cy²+Dx+Ey+F=0

Whatthismeansinexperimentalmathematics,asinmostcasesXandYcoordinatesrepresentanS=Tdualparameter,thatconicsarehighlysymmetricbidimensionalspacetimecurves,hencewiththemaximal

efficiency,whichisultimatelythemeaningonanSi=Teequivalence(ageofbalance,maturityandreproduction,proportionofenergyandinformation,beauty,‘menssanaincorporesanum’,presentstate,etc.)

Wecannothoweverinthisfirstbookon5DmathstodothefullsymmetricanalysisofGeometrybothasvisualsimultaneousimagesandalgebraicequationsinasmuchas‘huminds’havenotproperlydefinedthe5operandssymmetrictothe5dimotionsoftheUniverse,tohaveafullunderstandingofeventhesimplestrelationships

betweenalgebraicoperandsandgeometricfigures.i.e.whythe2СmotionoflocomotionsquaresthefunctionthatbecomesaparabolaonaCartesiangraph;whytherearenotexactsumsofcubes,directmethodsfor

solutionsofquadraticsolutions,etc.etc.themesthosethatrequirea‘deep-thought’considerationofAlgebrainourbulkiersecondbook–yes,theaxiomaticmethodprovidesitsownconvolutedconceptstoprovealltheabove,asanystudentofalgebraknows(permutationofgroups,symmetry,etc.),butthoseabstractconceptsmustbeclarifiedintermsofexperimentalpropertiesofspaceandcomplexi-logic.SoweshallfocusontheternaryNon-EuclideanstructureofallTime-Space-Superorganisms(T.œ),ascurvedformsareultimatelyas

polytopes,socialnumericalstructuresthatdefinethetwoelementsofanyT.œinasingleplaneofspace-time:

- S:Inspacethesimultaneousmembrainofthesystem,definedbyaclosedcurve;anditsfocus/ci,orsingularitycentersoftenthecauseoftheformofthemembrain.

- T:Intime,mostlikelyinthecaseofopencurves,thetrajectoryofaT.œ,‘reduced’asafractalpointmovinginthelarger‘Plane’,whichitselfdefinestheorganicpartofthewholeinwhich‘theevent’takesplace–that

mightvaryfromtheclassic3topologiesintroducedbefore,tomorecomplexgeometries(aBeltramihyperboliccone,andinnervitalspace–Klein’sDisk–withentropiclimitsasanopenball;adualfocielliptic

system,etc.)

WhatAlgebraicequationsdothenisaninverseS@<∆Tsymmetrytotheonewehavesofarstudiedinmoredetail(∆T-numbers>S@symmetry).Andsomeimportanti-logicargumentsongenerallawsofanylanguage

whenmirroringrealitymustbeputforward:

Weusethe>symbolininversemannerforbothsymmetries,whichinGSTisthesymbolforan‘evolutionofinformation’thatdiminishesthe‘entropy-uncertaintyofasystem’,becauseparadoxically,themostsynopticalanguageis,thelessdetailedinformationprovides,andthemore‘pathsofprobabilisticfutureopens’.Meaningthatnumbersaremoresynopticthatpoints,andsotheymightrefertomorepossiblerealities,hencecreating

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moreuncertainty.Thisisthetrade-offbetweensynthesisandanalysis.I.e.ifIsay0x∞or‘wor(l)d’,Iamdefiningallnumbersofsocialbeingsorentitiesamindequationmightconceive(o-mindx∞Universe=

constantwor(l)d).ButIdon’tspecifyanyinformationanduncertaintyismaximal.What‘word’didIsee?Ahorse?Acar?Afriend?WhatnumberIobtain?3pears,7billionhumans=mankind?

Sothealgebraicviewdefinesmorepossibilities,asascalartimeview,manyofwhichdoNOThavesolutions;whilethegeometricviewISalwaysasolutioninitselfevenifitdoesnothaveaconsistentalgebraicexpression.

The∆T:algebraicvs.S@:geometricfiguredualityisinthatsensemorecomplexandrichasa‘functionofpointsandnumbers’thatthefirstlayerofS≈Tsimilarities(pointsvs.numbers)sofarstudiedhere,whichwillcertainlyonce‘mathematicalpros’passthe‘stage’ofmodelshift,suspicionofthefounderofanewparadigm,etc.etc.(thatisoncetheyaccepttheworkoftheusual‘amateurCopernicus’andseethelight&beautyofthemore

advancedstructure:),givebirthtomanyentanglementsbetweentimeandspacemirroredinthemathematicallanguage.

Allthissaid,ascustomaryweshallstartourdescriptionof‘anyreality’,notfromthebottomupwardsbutfromthetopmoresynthetic‘reality’ofthe5Duniverse,whichistheworldcycleofanysuperorganism,becauseallwhatexistsincluding‘curves’arepartofaworldcycle,inthecaseofalanguageapartialmirrorofatrajectorywithinit.Soaswedidwithspiralsandcycloids,firstweshallshowthattheconethatgeneratesallcurvesisinitselfa5Dimageoftheworldcycle,morecomplexthanthesimplexpossibleimage(thecycle)anditsnext

entangledcomplexview–thecyclemovinginanentropic,linealplaneorcycloid.

∆±¡:THEWORLDCYCLECONEANDITSSPIRALINGPOINTS.

TherepresentationofconicsintheCartesianPlanehoweverisasimplificationastheyareoriginatedinacone,whichhasadimensionofheightidealtorepresentthearrowof‘increasinginformation’towardstheapexofthecone,orrelativepointoffutureofmaximal,St=§ð;or3rdpointofthebeing.Assuchthe‘cone’becomesalsoanexcellentmodelfordefininginitssurfacethelife-deathmotionofanidealT.œ–andinphysicsrepresentsfor

different‘attractive’vortices,arealtime-spaceinformative‘sink’.Soasallwhatexistsispartofasuperorganismtracingaworldcycleoramentalimageofit,thedualconeexpandstheconicrepresentationof4Dworldlinesof

locomotioninto5Dworldcyclesofexist¡ence.Wecanapplyapentalogicanalysistosuch5Dcones:

Inthegraphseveralrepresentationsof5Deventsusingworldcyclecones:

1.Asinglepointtracinga3Dspiralonthesurfaceofaconefromitsbasetoitsapexrepresentstheinformativearrowofaworldcycleinits3ages.ThusifweconsidertheT.œjustinthe∆-1plane,apointtracingamotioninitssurface;itdoesthenrepresentaworldcycleacceleratedbutperceivedasanascensioninheight,with

differentinterpretationsaccordingtowhichofthe45Dpossibleconesofthedimotionsofanentitywedescribeandwhat5Dparametersoftimeandspacearerepresentedinthetwoperpendicularaxis.

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2.Ifwemakeacompositeof2worldcyclesof2fractalpoints;weobtain4potentialcombinations(withthecentralfigurebecomingadifferentevent,whenwechangeupsidedownitspast-futureorientations).Eachconetakenasawholeworldcycleofexistence(‘LIFECONE’)theoutcomeofcollidingeventsaccordingtothe

rulesofthe4thPostulateofNon-Econgruence,fromDarwinianfeeding,toreproductivesymbiosis.

3.Iftheconeemittedbyeachparticleisthetrajectoryofanexi‘waveofenergyandinformation’,theconesrepresentthepossibleoutcomesofcommunicationbetweentwolargerpoints,whichsharethewave;usefulto

modelFermion<Boson>Fermionexchangesbetweenparticles(5DPhysics).

4.Ifthedoubleconebelongstothesameparticle’sworldcycle,itrepresentsits3possiblestates.Wherethedoubleworldlineconeofspace-timeexpands4DMinkowski’sspacetimerepresentationinto5D,branchedin2inverseS>T<Sfuture&pastconesandaflatS=Tpresent,puttouseinadvanced5Dmathematicalphysicsandpentalogic.Thedifferentorientationsoftwoconeswhichrepresentunlikethesimplexrelativityconesof4D,amotioninthefifthdimensionbetweentwofractalpoints(anyT.œ),collidinginapresentsimultaneous‘plane’

(left).Orapointinpresentdissociatingbetweenrelativepastandfutures(right).

5.Sincetheconeisitselfacirclemovingalongaline,inadecreasingitssize,itobviouslyimmediatelyrepresentsasitdoesthecycloid,afundamentalrepresentationofaT.œofspace-time,thecircle,movingintimetowardsashrunk,warpedthirdage,ormovingbetweenscales:∆-1ð>∆+1ðforwards,asitshrinksinrelativesizebecomingontheapexanemergent‘singlepoint’ofthelargerwhole(palingeneticworldcycleas

observedfrom∆+1).

6.Finallythe2invertedconesformtogetheranimageofthewhole5D-4Dinversecollapsingandexpanding,evolvingandentropicarrowsofspace-time.

Thusthemotionfrom∆+1largerwholestothe∆ºsingularityoftheconeanditsexpansionontheinverseconecanbeusedaswedointhegeneralmodeltorepresentthemanydifferentaspectsofaworldcycleof

existence.

Theconethusisinitself,anotherfundamentalframeofreferenceproperofthe5thdimension,ignoredbyhumindsstuckinits4Dformalism,whichwewillretakewhenstudyingtheBeltrami’srepresentationof

hyperbolicgeometry,thepropergeometryofthe5thdimensionwithabitmoreofrigor.

Allthissaid,toshowtheprofoundlevelsof5Dtheorybringaboutbyasimplecone,itsimportanceshouldnotsurprisethereader,asitisindeedtheproductofa|xOmotion;henceembodyingallpossible,|xO=ØcurvesoftheUniverse,whichhowevercanbereducedtoacombinationofouraforementionedcurves(spiralsalready

studied):

Yet,asmathematicianshavefocusedonslicesofspace-timeofthecone,andtheconiccurvesofthe5Dformalismexceedsthepurposeofthisintroduction,wejustenunciatesomeofitsusestoshowhowaconicalso

mirrorsaworldcyclebutwillconcentratefromhereon,intheclassicconceptsofconics.

Stillthoseconceptswillapplytotheunderstandinginmathematicalphysicsofthedifferenttrajectoriesofpointsinoneofthe4canonicalcurves,eachonerelatedtoabasicDimotionofexistence,towhichwemustadthe5thDimotionofalog/Archimedeanspiral.SincethelogspiralcanbeconsideredanArchimedeanspiralifweada3rddimensionofheightinformation,byconvertingitsshrinkingrevolutionintoarecedingmotion,foran

entitylivingwithinthecone’stimespacesurface.

Existentialalgebra,the‘o’,‘1’terminologyofgeometricobjects.

HowwewriteconicworldcyclesinthemoregeneralGSTterminologyofexistentialalgebra?Thoughwewon’ttranslatemathematicstotheUniversallanguageofi-logic,existentialalgebrathatcanmirroranyother

languageofrealityletusdoabriefintroductionotthethemetocomplementthepreviousgraph,alsoabit‘advanced’foranintroductorycourse.

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Aworldcycleitisacloseddomain,AWHOLE,eitherinthesingleconeorthedualone,andsoinexistentialalgebrawewritethegeneralequation,ST(vitalenergy)=S•T(singularityxmembrane)inthefollowing

manner:

G(Tx;Sy)≤(1;0)foraclosedcircle;G(Tx;Sy)≤(1;0,0)foranellipse;G(Tx;Sy)≤(∞;0)foranopenparabolaandG(Tx;Sy)≤(∞,∞;0,0)fora'dual'semi-closedhyperbola.

Where1meansthemembrain;0thesingularityand∞anentropicopencurve.

Sothatiswhataconicmeans,avitalsuperorganismmirroredinmathematics,withitselementaryparts-themembrane(s)1,therelativeinfinite,outeropenworldandthe0,therelativecenterorsingularitiesthat

managethemembrain;whereG(x,y),areallthecombinedpoints,(≤x,≤y)thatformtheinner,vitalregionofthesystem.

AndsincetheUniverseisbidimensionalandholographicthoseconicequationsarethemostpervadinginallformsofNature.Yetonly2-dimensionalcurvesanditscombinationsarerealpurebasicforms.Allothern-dimensionalequationsarecombinationsofthem,asFermat'sgrandtheoremproves-sincex³+y³≠z³.

ConicsandDimotionsofexistence.Itspentalogic.

Descartesrealizedthatcurvesintheplanearerepresentedbysecond-degreeequationswithtwovariableswhosegeneralformrepresentsanellipse,ahyperbola,oraparabola;i.e.,curvesknowntothemathematicians

ofantiquity.

Sothefirstobviousfactisthat'cyclical,time-likecurves'have2dimensions(degreesoffreedom)asopposedtosingle-dimensional$t-lines.Athemethatcorrespondstomechanicsandessentiallymeansthatlinealinertiaisneverfound,asalllinesarestepsofaworldcycle,inasmuchasthebeingexistsinanouterworld,andsowe

mustalwaysaccountforitsdimotionitsinternalandexternalforces.

Inthegraph,thereare2conictypes:open=entropic=unstable=Darwinian

parabola+hyperbolavs.closed=informativecycle+ellipse.Bothareconstructedfromaconewithaline,

whichisthedesiredmotionofall@ristoteliansingularitiesandthecurve‘tended’bytheouterworld,throughwhichacombinedcycle'moves'

openingentropicallyorshrinkinginan'accelerated'processwecanconstructtheconicsoftheUniverse(asdefinedpreviouslyintermsofSTvariations).

ThiswasthewonderofGreekstillDesarguesprovedthatallcurvescanbedrawnfromaconic,thebest|xO=ørepresentationoftheworldcycle.

Assucheachofthecanonicalcurves,whichneedonly2STparameters,definesaholographicbidimensionalmanifold–anSTdimotionofspace-time.Andeachconiccorrespondstoadimotion:

The2closedpaths,thecircleandtheellipseareS-dominant:

SSisbydefinitionaseedwithnomotionintimeandinfinitesimalvolume–thebeginningofaprocessofexpansion–hencethe‘summitpointoftheconic’.

St:Thecirclehasasinglefocus-singularity,asitcorrespondstotheDimotionofperception(π-spiralcircle).Ithasastableefficientformasapresentrepetitivesystem,hencewithnolatitudeinheight=time.

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S=T:Astheellipsehastwofocuswithequaldistancetothe‘son-membrain’theyrepresentanS=Treproductiveordualcurve,whoserelativedistanceisamentalmeasure(Riemann’sconceptofdistance)ofthesimilarityofbothpoints,whichisabsolutewhenbothhave0-distanceandtheellipsebecomesacircle.

Ontheotherhand,wehave2opencurves,whichrepresentfreelinealmotionswithacontentoftime:

-sT:locomotion,representedbytheparabola,sincethepointhasasingletrajectory,henceasingleT-motion,andindeedweshallseetheparabolatobethefundamentallocomotionofanypointsubjecttoalargerworldforce(Galileo’sstudyoflinealtrajectories).

-TTmotions:finallythehyperbolarepresentstheentropicdualmotion,which‘splits’anderasestheinternalformofthesystem.Sothelowerparabola‘orientedtowardsthepast’,ifwetakethedualconeasaworldcycle,representstheinternalscatteringmotionandtheuppercurvetheexternalmotion.

Andsoconicsdohaveclearcorrespondenceasalmostallbasicstructuresofmirrorlanguageswiththe5Dimotionsofreality,andthosefactsshouldorientateouranalysisofthespecificcurvestracedbyphysicalparticlesandhappeninginrealeventsofdifferentstiences.

Angleofcongruencedefiningthedifferentconics.

TheancientGreekshadalreadyinvestigatedindetailthosecurvesobtainedbyintersectingastraightcircularconebyaplane.Iftheintersectingplanemakeswiththeaxisoftheconean

angleϕof90°,i.e.,isperpendiculartoit,thentheSECTIONobtainedisacircle.

Itiseasytoshowthatiftheangleϕissmallerthan90°,butgreaterthantheangleαwhichthegeneratorsoftheconemakewithitsaxis,thenanellipseisobtained.Ifϕisequaltoα,aparabolaresultsandifϕissmallerthanαthenweobtainahyperbolaasthesection.

Intermsoftheangleofcongruencehowevertheinterpretationisexactlytheinverse;astheaxisoftheconeisthemotionoftimefrompast(base)tofuture(top);thatisfrombirthto

extinction,andsotheanglemustbemeasuredforabalancedpresentsystemsinparalleltothebaseofthecone;whichrepresentstheimmortalstateofpast.AndsothecirclewhichisthebalancedstateofS=T(bydefinition,∑x=∑y,forthewholerangeofvaluesofacircle),isthestateofpresent,withaparallelangleof

congruenceandnomotionintime.Whilethemaximalangleofcongruence,andperpendicularitytothepresentwiththemaximalmotionintime,istheentropichyperbola,whichinfactreachestobothextremesofthewholeworldcycle.Whereastheuppersideofthehyperbola‘explodestheinformationofthesystemintoanentropic

death;andthelowerpartrepresentsthemaximallocomotion,similartotheparabola.

Whytheparabolaisintheworldcycleconeamotiontothepast(gangleinthegraph).BecauseasSisthestateofabsolutefuture(information,potentialseed,logicmindincontrolofthesystem,designingitsfuture),andTistheentropictimedisorderedarrow,anincreaseinlocomotion,sT,isarelativestateofpast.Butitdoesn’t

disorderthebeingasthehyperboladoes.

WhatthismeansinGSTtermsisthatthehyperbolaistheoppositeconcepttothecircle/ellipse,asclosedandopen,ðƒand$tinverse'geometries'whichifweconsidertheyaxis,thelongitudinalentropicTimeaxis,andthexaxisofthecycle,theinformative,spatialstate,convertstheconeintoaninverseSpace-timewherenotonlymotionintimebutalsoanglesofcongruencecanberepresented;andsowecanstudyseveralworldcyclesand

complexspace-timeeventswithinit,inadvanced5Dmathematicalphysics.

Letusthenconsideroncethosegeneralconceptsarelaiddownthemainreasonwhysomeconicspreservethepresent,namelythebalancedS=TefficiencyofeachformofaconicthatrepresentseachtypeofavitalDimotionofexistence,whichareoftenusedbyfractalpointsthattracedifferentconicsfordifferentvital

purposes.

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Creationofcurvesin¬Egeometryandits5Dimotions:socialparallelismvs.perpendicularityannihilation

S-Conicsrelatetothesecondpostulateof¬Ecommunicationbetween2pointsthatininformativeconicsconstructa‘territory’,delimitedbythemembrain,whichisequidistant,inacyclicaltimeperiod,tothe2points.Thoseclosedorderedconichaveaminimalangleofparallelism;thatis,displacement=changeintime,toreacha

degreeof‘simultaneity’thatallowsthecreationofanorganicsimultaneousspace.

-T:Openconicswithalargerangleofcongruencechangeintime,becomingdisorderedasymmetricinspace,failingtoformastablesystem.Wecanthenconsidertheparabolainphysicaltermsasthe‘external

acceleratedgrowthofdistance’TT-betweentwopoints(onefixedbyconvention,orinverselytheacceleratedattractionofaforce),whilethehyperbolarepresentsthesplitofitsinnerpartsintoinversetrajectories,oralternatelytheinversepropertiesoftwoS/Telements(SxT=C).Soconicsconnectthe2nd&4thpostulateof

communicationandcongruence:

Inthegraphs,thegenerallawsofbehaviorintheorganicUniversearesimple:beingswhoaresimilaror

complementaryandspeakthesamelanguageofinformationcometogetherascouples,herdsandsocialwholesstrongerthanindividuals.Thosewhoperceive

eachotherasdifferent,willsimplyactinaDarwinianmanner,whichmeans,they

willeitherincreasethedistance=dissimilarity;highlighttheirdifferenceorbreakandsplitunder

predatortearing-allofthempropertiesreflectedinopenconics.Sowecanconsideralsothe4closed(right)andopen(left)conicsintermsofthe4differentformsofasymmetry.

Weredefinegeometricelementsvitalisingitsmeaningwiththe'fourthpostulateof®geometry,asthe'angleofcommunication'determinestheoutcomeofmosteventseitherasparallelcreationorDarwinian

perpendicularity.

2'asymmetric'beings,i.e,thelineandthecycle,cometogether,fusioninginacreativeway,whentheircomingisparallel,ordestroyingeachotherwhenitisperpendicular.

Parallelcreation:Thereare2levels,creationbycommunicationofinformationthroughanintermediatespace,studiedmoreproperlyinlogicandcreationbyadjacentpeggingmoresuitabletotopologicstudies,whichalsospreadsintoAnalysis(aderivativeisaparallel.Inessencepeggingbyparalleladjacencyisnecessarytocreateorganicwholes.SowhenthereareNOparallelpeggingsthereareNOderivatives,NOcommunicationbetween

∆øand∆-1,NOpossibilityhencetocreateasuperorganismof∆±1scales).

3Dcreationbypenetrationofthelineintothecurve.

Creationandreproductionproducesthebiggeste-motion,theorgasm,literallythesensationofacycleinvadedbyalineinparallelinandoutharmonicoscilation=penetrationbecauseitisthepurposeofaUniversewhosefundamentalelementismotionandfundamentale-motiontheadjacentfrictionofparallelformsthatcreate

complementarywholes.

Theconethatgeneratesallcurvesistheinverseapenetrationofacyclewhichmovesalongaline,'tightening'itsgrip.

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Whileperpendicularitycutsanddestroysoneofthe2elements,making∆sttopologyvital.Rememvber,wefollowGodelandLobachevskiandEinstein:mathematicsasalllanguagesarerealmirrorsofahigherlivingreality.

2Dcreationinaholographicbidimensional2-manifoldbytangentparallelism.

Systemsinanyscaleoftheuniverse,fromAtomicIonsorcrystalstohumansocietiesrelatetoeachotherindarwinian,perpendicular'tearing'topologicalrelationshipsthat'break'theclosingmembraneofonespeciesdisruptingitsexistence(openconics),orwillkeepameandistancetoformsocialnetworksofcommunicationthatwillgrowintosuperorganisms,startingtheemergentprocessofevolutionofspeciesintoanewƤcaleof

socialexistence(closedconics).Soanysystem’sorganic,geometricandscalarrelationshipsaresymbiotictoeachother.

Howdimensionscombinetocreateformisanessentialfeatureofthedualitybetweensymmetricparallelismandperpendicularannihilation(antisymmetry),whichplaysaspecialrolein5Dgeometry.In2dimensionsthetangent

thatgivesorigintoaderivativereducesformtofinitesimals.Orinitsinverseintegralisthecreativeform

of∆-scales.

Eachgeometricformcanbeseenasapointthatmovescreatinganewdimensionormore:thepointbecomesthecirclethatturnsaround.Orinversely,thecirclepeelsoffawavefromitscyclicalmembraneofangular

momentumshapingthemaincreativeformof3DimotionalS=Tbalance.WhattheS-dominantcycleadsthenisaT-dimensionoflinealmotionandthewaveisborn.

In3dimensionstheconicisalinepenetratinganewdimotion,draggingacirclethatturnsfasterinsmallerspacesasitmovestotheapex,finalpointororiginofthecone.

Symmetriesinconics.

ThedifferentdegreesofParallelism,PerpendicularityandskewnessareessentialconceptsofvitalNon-Euclideangeometryappliedtoconics.ThecurvesofaconicworldcycleareallthecurvesoftheUniverse;accordingtothetypeofevent=dimotiontheyrepresent,fromTTentropichyperbolaswhereeachoftheinitialfocusbreaksthepositiverelationship,cuttingthemembrainintwoparts,oneforeachfocusthatdepartininversedirectionsof

timespace,tothefusionofthetwopointsintothebosoncenterofacirclewhenparallelismisabsolute.

ThusConicsacquireanewperspectiveundertheholographicprincipleofaUniversebuiltonbidimensionalensembles,wheremost'ternarydimensions'arelayersofreproducedbidimensionalsurfacesor'branched

networks',spreadonthe'holes'ofa3rddimension'.Andsowedistinguish2kindofconics:

-Timelikeconics,circlesandellipses,whichcloseintothemselvescreatingaclearternarystructurewithanexternalmembraneclosinganinternalspace,self-centredinoneortwopointsseparatedlineallybyafactorof

excentricity.

-Space-likeconics,parabolasandhyperbolas;whichapparentlyareopensystemswithoutclosure,butinfactpreserveboth,thecentralpointofview,theinternalterritoryandthemembrane,albeitopentoletthe

worldcirculatethroughit.

Soconicsareadynamictransformationbetween$t(open)<≈>ðƒ(closed)statesofanSTbeing,withasingleparametertomeasurethem,eccentricity;whereasthemostperfectbidimensionalbeing,isoneofo-eccentricity,wherethe'2focus'ofthecentralsingularity,whichcanbeanyS/TVARIATIONarebothequalinspaceandtime(asinglepoint)-thecircle.WhichthereforemustbeconsideredastheGreekshadit,theperfectform;anall

othersdeformationsofit.

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S=conics:Informative(Particle-head)communication,possibleincasesofrelativesimilarity≈parallelism(whichdeterminesparallelherdingandsocialevolution)forthe2closedconics–whereasthecircleisaperfect

symmetry.

T-conics:Theyareskewcurvesthatdonotintersectandarenotparallelandclearlyrelatedtohyperbolas(withinnerscatteringentropicmotion)andtheparabola,whichdistances2points.Indeed,twolinesareskewifandonlyiftheyarenotcoplanar,whichIN5Ðas3±¡planesco-existinthesameorganismandsystemsfeedinT.œs,twosuperorganismsdown,impliesspecies,whicharenotintherelativeplanesofactionofthebeing.Theyarein

theworldcycleconethe2partsofthehyperbolaclearlyskewedintimeandspace.

Topologicalemergencebetweenplanes.

Whenwedealwithannihilationbyperpendicularitythingsgetalso2variationsasitislogic,tothinkby∆-scatteringorbyS<≠>Tantisymmetry.Butasannihilationultimatelymeansdestructionofan∆

scale,itderivesinentropicdissolution.Theresultsareoftenshowninexponentialfunctions.

Wecanthinkofthe'changeofplanes'asaperpendicularityagainstwhichtheinternalfunction(ofmomentum)'collides',tryingtopussthe'wall'thatseparatesscaleswithoutresult,growingthenin'inertialmass'nolongerinspeed.Asultimatelythevitalenergyenclosedbyamembranefinds

alwaysthemembranetobeperpendicularandannihilatingitveryoften;themilitaryborderinanation,barrierofcattle,orsepherddog,thepredators,etc.

Thebestknowncaseinphysicsarerelatedtothehypotheticalimpossibilityofafunctiontocrossadiscontinuitybetweenplanes,whichiswhatitmeansinthe

Lorentztransformations:asamasscomesclosertotherelativeinfinitelimitofhislightspace-timedomain,itsgrows'theoretically'towardsinfiniteasitcannotspeed

more.

Soitsmomentummv'changes'nolongerinvbutinm(aschangecannotbestopped,the∑∏energyfedinthesystemmusteitherderiveintothesingularitymorthe

speed-membraneinparalleltothelargerwholegalaxymembrane(Machexplanationofangularmomentum).Thisnolongerpossibleasthepartcannotmovefasterthanthewhole(c-speedlimitforthegalacticspace-timemembrane),thevitalenergydoesNOlongerfeedthemembranebutthesingularityanditsactivescalarmass,the0-1

DimensionalparameterofdensityreflectedintheDiracmembrane.

Sowecanseegeometricallyoralgebraicallyhowthismomentumbecomesthen'deviated'asaparallelangularmomentumofthemembraneeitherinlinealorcyclicalfashion(itselfatransformationofanSHmotionfrom

cyclicalintolineal),toagrowthofmass.

SothethirdageofgeometrywhichstartedwithLobachevski's3'findings',mentalspace,topologyandexperimentalneedofmathstovalidateeachmentalspacewithreality,isreallyaboutthismentalrealisationthatspaceisinformation,andsothe3rdinformativeageofgeometryisobviouslyabout...mentalinformation.

RECAP.Thereare3modesofS=TmathematicalcreationbyparallelismintheUniverse:Theharmonicoscillatorandtheconethatmodeltheworldcycleofexistence&thetangentiallinesthatreducecyclicalpatternstoits

scalarstœps&dimotions.TimespacealsosplitSinthedualitybetweenpasttofuturetopastmalegendersandS<≈>Tpresentfemalegenders,whosetransformationsoftopologicallinesandcyclesintohyperbolicwaves

resultsinnewcreativecombination.

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CLOSED–INFORMATIVECONICS

Therearetwoclosedinformativeconicshighlysymmetric.Thesymmetriccircle,whichwestudyallovertheplace,andtheasymmetricellipse,whosemaindifferencesinpentalogictermsare:

@:Thecirclehasasinglecenterortwoboson-likeequalcenters.Theellipsehas2focus.

¬Thecircleisstatic,balancedandlast.Theellipseisbornofa‘contraction/expansion’ofthecirclealongitsXorYcoordinatesofentropicvs.informativegrowth.Assuchittendstobemoredynamic.

T:Botharecyclicalinpatterns,hencecanalsoactasbasicrepresentationsofworldcycles.

S:Thecirclehasthemax.volumeofvitalspace-energywithminimalperimeter.Assuchitisthemostefficientformforasingleterritorialmind-point;buttheellipseisthebestsystemforadualpolecommunication,asitdoublesitsfractalpointswithminimalgrowthofperimeter.HenceinpalingeneticdevelopmentassoonastheseedbreaksintopolarS/Tanimal/vegetalpolesacquirestheformofanellipseandasoneofbothpolesissmaller(theanimalpole),itfinallybecomesanovoid

Theequationofacirclewithcenterattheorigin.

Firstofall,weconsiderthecirclewhoseequationisageneralizedPythagorastheorem:x²+y²=a².Itssimplicityshowsitssuitabilityforaself-centeredgraph.

Itevidentlyrepresentsacirclewithcenterattheoriginandradiusa,asfollowsfromthetheoremofPythagorasappliedtotheshadedrighttriangle,sincewhateverpoint(x,y)ofthiscircleistaken,itsxandycoordinates

satisfythisequation,andconversely,ifcoordinatesx,yofapointsatisfytheequation,thenthepointbelongstothecircle;I.e.Thecircleisthesetofallthosepointsoftheplanethatsatisfytheequation.

Butfromapentalogicvitalpointofviewbothitsequationandgeometryrevealsdifferentelementsof5Dreality;whenweconsidertheentity@-centertobeasingularityconstantlymovingalongtheS:height=informativeandT:length=entropicaxis,insuchamannerthatasa‘whole’entangledS=Tsystem,whereasS2+T2=K=SS+TT.

Ofwhichthemostremarkablecaseis32+42=52whereas3isthetime-agesperspective,4thespatialperspectiveofthecomingtogetherofabidimensionalbodyandhead,and5thescalarperspective;henceT2+S2=∆2.TT+SS=∆(-∆):∆∇.

Whattrulymeansisthatinasingleplane(givenbythe+operand)thesumofalltheentropicandinformativestœpsofabeingwillformitsfullworldcycleupanddown5D.

Thisisaprofoundlawthatrightlyembedsthemostimportanttheoremofgeometry,Pythagoras,withSpace-timereality.Andtheonlyequationweproceedinversely,takingitfrommathstocarryitintoSpace-timethoughtanalyzedinournot-publishedpapersonpentalogic.Backtothecircle’spentalogicsomehighlights:

S=T:Ast<=>Bst:Intermsof2pointscommunicationanditsintermediateenergyDimotionsthecircleisanellipsewherethe2elementsaresosimilartheycoincideinperfectsymmetry.Theeccentricityis0,andtheresultistobe the two-dimensional shape enclosing themost area per unit perimeter squared,hence themost efficientterritoryofcontrolforacouplewhenitssimilaritybecomesidentity.Whya2viewismoreproperisobviousasthecirclerequirestwoupandlongaxistobewebbed.Somultitaskingbettersplitsbetweentwo.

∆:Intermsofscalesandtheextremedimotionsofentropyandform.theinnercoordinates,‘abscissa’and‘ordinate’ofthebeing(SandTvalues),itfollowsPythagoras,SS+TT=∆∇postulate.

Theequationofanellipseanditsfocalproperty:eccentricityandsymmetry

Nextcomestheellipse,wherethecommunicationbetweenpointsdiffersin‘length=entropy/motion/size/distance’parameters,butits‘height-informativedimension’remainsthesame,henceallowing‘congruentcommunicationofinformation’(orisminimalintheworldcycle’scone):

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LettwopointsF1andF2begiven,thedistancebetweenwhichisequalto2c.WewillfindtheequationofthelocusofallpointsMoftheplane;thesumofwhosedistancestothepointsF1andF2isequaltoaconstant2a(where,ofcourse,aisgreaterthanc).SuchacurveiscalledanellipseandthepointsF1,andF2areitsfoci.

LetuschoosearectangularcoordinatesystemsuchthatthepointsF1andF2lieontheOx-axisandtheoriginishalfwaybetweenthem.ThenthecoordinatesofthepointsF1,andF2willbe(c,0)and(–c,0).LetustakeanarbitrarypointMwithcoordinates(x,y),belongingtothelocusinquestion,andletuswritethatthesumofits

distancestothepointsF1,andF2isequalto2a:

Thisequationissatisfiedbythecoordinates(x,y)ofanypointofthelocusunderconsideration.Obviouslytheconverseisalsotrue,namelythatanypointwhosecoordinatessatisfytheequationbelongstothislocus.TheEquationisthereforetheequationofthelocus.Andwhilemathematicianssimplifyit,theinterestforthe

topologicalo-pointofviewremainspreciselyinitscompleteform.

Theperfectellipsethushavethesamecoordinatesand±cdistancetothecenterofreference,whichispossiblefortwoequalpoints,whichlayinanon-isomorphic2-3Dworldwherethe‘fieldofentropicmotion’,the

abscissas,islarger.Hencetheasymmetrybelongstotheworldnottothefocus.Butaswechangetoimperfectellipsesthesubmissiveroleofthe2ndelementwillincreasetillitbecomesinorbitalsystems,‘expelled’asthemembrain(planettothesun).LetusexplorethenbothtogetherinthecontextofbasicXVII-XVIIIPhysics.

Backtotheequationoftheellipsesimplifyingtermswegetto:x²/a²+y²/b²=1

Substitutingy=0intheequation,weobtainx=±a,i.e.,aisthelengthofthesegmentOA,whichiscalledthemajorsemiaxisoftheellipse.Analogously,substitutingx=0,we

obtainy=±b,i.e.,bisthelengthofthesegmentOB,whichiscalledtheminorsemiaxisoftheellipse.

Thenumberc/aiscalledtheeccentricityoftheellipsethatislessthan1.Inthecaseofacircle,c=0andconsequentlye=0;bothfociareatonepoint,

thecenterofthecircle(sinceOF1=OF2=0).

Astheeccentricitygrowsthe2pointsseparatebutthepointsstillcontroltheareaofthesystem,whichcanbeshownbythemethodofdrawingthecurvewithathreadconnectedtoboth.

MathematicalPhysics:Kepler’slaw.

Themaindifferencebetweentheidealsimplificationofmathematicsasamirrorofspacetimelawsandtherealspacetimelawsisthelossof‘Dimotions’togettothebarebasicsinwhichlawsofNaturecanbemodeled–abidimensionalstillgeometry.ThenwhenwereturntoNaturethoselawsareexpandedwithadditionofmotionthatdistortsthegeometry(butconservesafteratransformationofTintoStheessenceofthelaw)andan

increaseofDimotions/Dimensions.SowhenwegetintoastronomyasNewtonprovedthelawofattractionofbodies–consideredinthegenericcasethetwopolesofanellipsegiveustheellipticorbit.Howeverthe

distorsionofaddingandtransformingstilldimensionsintomotionschangesoftenthe‘formalelements’ofthegeometry,preservingthe‘essentiallogical/vitalcomponents’,showingoncemorethattheessenceofthegame

isnotabstractlawsofspatialmentalgeometrybutvitallawsoftime-spacestructures.

Soindeed,nowtheplanetoccupiesthemembrainandinthesecondfocithereisnothing.Letusforachangeputsomequantitativeeffortcalculatingit.The2ndfocusisalongthemajoraxis:thelinejoiningthepositionsof

perihelion(closesttotheSun)andaphelion(furthestfromtheSun)inEarth'sorbit.Sowedrawanellipse.PlacetheSunatonefocus(onthemajoraxis,abitofftooneside).MarktheSun"S"andthecentre"C".

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Onthe"shortside"oftheSun,alongtheorbit,wherethemajoraxiscutstheorbit,thatistheperihelion"P".AttheoppositeendistheAphelion"A".In2010,PerihelionwasonJanuary3,atadistanceof147mill.kmIn2010,AphelionwasonJuly6,at152.Totallengthofthemajoraxis(fromPtoA)isthesum=299,millkThesemi-major-axis(distancefromPtoCandfromCtoA)ishalfofthat(149,597,213.5km).SinceweknowthatthedistancePSis147milkm,thenthedistanceSCmustbethedifferencePC-PS=2,5millkm

TheSunis2.5millionkmtothe"January"sideofthecentre.Bysymmetry,theemptyfocusis2.5millkmonthe"July"sideofCCE=2,5mill.SE=SC+CE=2*CE=5millkm.

Butthereisnothingatthatpoint.Sothequestioniswhatwecansavefromthedefinitionofanellipsethata5Dreconstructionofphysicalspacetimepreserveswhenweexpandthemathematicalmirror?2GSTtruths:

1)alldistancesaremotions.Soifwemakethemovingplanetastaticlineperunitoftime-motion,thekeyvital,communicative2ndpostulatepropertystayswithanaddeddimensions.Now2pointsdonottracethesamelengthbut'sweepthesamearea'workingtogether.Thusanorbit,inthephysicalanalysisofGSTbecomesadualsystemthatherdsavitalare:themembranewithmoreangularmomentum(the$t-planet)andthe

singularitywithmoregravity/mass(theðƒ),surroundingherdingandabsorbingthelower∆-1scaleofgravitationalpoints/forces.

SoifweapplytooneofthepointsAbsoluterelativity(S=T:'motionisindistinguishableofdistance')orbitallaws,implytheplanetandthesuntogetherscanthesame'gravitationalarea'.Thisaerolarlawtranscendsthentoanyphysicalvortex,andalsoimpliesthatthe‘aerolarellipse’willcollapsetillthemembrain‘becomes’thesingularity,fallingintothehighermasstobe‘one’asthestaticellipseevolvesintothecircle.Letusthen

concludewiththeanalysisofthetransformationofcirclesintoellipses,whichshowtheyarethesametopologicvariety.

Theellipseastheresultofanassymetric“expansion/contraction”ofacircle.

Thegeneralizedcaseofanentropicellipsewith2predatorpointsinalinethatexpanditsrange,isanalyzedinabstract,asthealternativeY-coordinates’contractionofacircle.Weconsidera

circlewithcenterattheoriginandradiusa.BythetheoremofPythagorasitsequationisx2+y2=a2,wherewehavewrittenyiinsteadofy,sinceywillbeneededlater.Letusseewhatthiscircle

iscontractedintoifwe“contract”theplanetotheOx-axiswithcoefficientb/a.Afterthis“contraction”thex-valuesofallpointsremainthesame,butthey-valuesbecomeequalto

y=y(b/a).Substitutingforintheaboveequationofthecircle,wewillhave: astheequation,inthesamecoordinatesystem,ofthecurveobtainedfromthegivencirclebycontractiontotheOx-axis.Aswesee,weobtainanellipse.Andinverselywehaveprovedthatanellipseistheresultofa“expansion

inabscissas”ofacircle.Fromthefactthatanellipseisanasymmetric“contraction/expansion”ofacircle,manypropertiesfollow,

whichasusualweorderinpentalogicterms:

ST:area.SinceanyverticalstripofthecircleunderitscontractiontotheOx-axisdoesnotchangeitswidthanditslengthismultipliedbyb/a,theareaofthisstripaftercontractionisequaltoitsinitialareamultipliedbyb/a,andsincetheareaofthecircleisequaltoπa²,theareaofthecorrespondingellipseisequaltoπa²(b/a)=πab.

@-singularitypoint(centerofgravity)issimplythemidpointbetweenthem.Anditfollowsthatanaturalevolutionoftheellipseastwosimilarformsattract,istobecomeacircle,whichinGsttheoryrevealsadeepertruth:systemsbecome'compressed'intosmallernetworksandfinallyintosinglesingularitypoints,which

organizetheentiresystem.ThisstructurecanbegeneralizedasNewtondidforanyclosedconicdividedintoterritorialdomainswithdifferentcentersofgravity,whichalllaidinasingleline,itselfcenteredinasinglepoint,toanyT.œ,whichwillhaveavitalspacein3D,reducedtothecontrolofamenbrainin2D,itself

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controlledbyasingle1D,eachofthesmallersystemsprocessingfasterinformation,transformingvitalspaceintoDimotionsofexist¡ence.

Sobothfortheellipseandthecircle,wecanconsiderthesurfaceenclosedbythedisk,structurallysustainedbythenetworkoflines,itselfcommunicatedatequaldistancesbytheline,andfinallythelinefocusedinthe

'centerofgravity',wehavebuiltan∆+2>∆+1>∆>oscalarstructure.Andherewerealisewhyanalyticgeometryworks,asitdoescompressmentallygeometricsurfacesintosequencesofnumbersoflesser'volume'of

informationthat'commands',logicallythewhole.

Thispropertyofdiameters-thatifparallelsecantsofanellipsearegiven,thentheirmidpointslieonastraightline,canbeshownalsofromthecontractionofellipsesinthefollowingway:

Weperformtheinverseexpansionoftheellipseintothecircle.Underthisexpansionparallelchordsoftheellipsegointoparallelchordsofthecircle,andtheirmidpointsintothemidpointsofthesechords.Butthemidpointsofparallelchordsofacirclelieonadiameter,i.e.,onastraightline,andsothatthemidpointsof

parallelchordsoftheellipsealsolieonastraightline.Namely,theylieonthatlinewhichisobtainedfromthediameterofthecircleunderthe“contraction”whichsendsthecircleintotheellipse.

Theellipseofinertia.

Anotherphysicalexampleofthepowerofellipsestocreatestabledualfocusedformsistheellipseofinertia,whosemaximalresistenceforanysystem,eventhosewhichhave‘differentedgeshappensacrosstheaxisof

theellipse.

i.e.Letaplatebeofuniformthicknessandhomogeneousmaterial,forexampleazincplateofarbitraryshape.Werotateitaroundanaxisinitsplane.Abodyinrectilinearmotionhas,asiswellknown,aninertiawith

respecttothisrectilinearmotionthatisproportionaltoitsmass(independentlyoftheshapeofthebodyandthedistributionofthemass).Similarly,abodyrotatingaroundanaxis,forinstanceaflywheel,hasinertiawith

respecttothisrotation.

Butinthecaseofrotation,theinertiaisnotonlyproportionaltothemassoftherotatingbodybutalsodependsonthedistributionofthemassofthebodywithrespecttotheaxisofrotation,sincetheinertiawithrespecttorotationisgreaterifthemassisfartherfromtheaxis.Forexample,itisveryeasytobringastickatonceinto

fastrotationarounditslongitudinalaxis.Butifwetrytobringitatoncetofastrotationaroundanaxisperpendiculartoitslength,eveniftheaxispassesthroughitsmidpoint,wewillfindthatunlessthisstickisvery

light,wemustexertconsiderableeffort.

“Itispossibletoshowthattheinertiaofabodywithrespecttorotationaboutanaxis,theso-calledmomentofinertiaofthebodyrelativetotheaxis,isequalto∑r²imi(whereby∑r²imiwemeanthesum∑r²1m1+∑r²2m2+.....+∑r²nmn)andthinkofthebodyasdecomposedintoverysmallelements,withmiasthemassoftheithelementandrithedistanceoftheithelementfromtheaxisofrotation,thesummationbeingtakenoverall

elements.

Nowescapingitsproof,thefollowingremarkableresultcanbeobtained:Whatevermaybetheformandsizeofaplateandthedistributionofitsmass,themagnitudeofitsmomentofinertia(moreprecisely,ofthequantityρinverselyproportionaltothesquarerootofthemomentofinertia)withrespecttothevariousaxeslyingintheplaneoftheplateandpassingthroughthe

givenpointO,ischaracterizedbyacertainellipse.ThisellipseiscalledtheellipseofinertiaoftheplaterelativetothepointO.IfthepointOisthecenterofgravityoftheplate,thentheellipseis

calleditscentralellipseofinertia.Theellipseofinertiaplaysagreatroleinmechanics;inparticular,ithasanimportantapplicationinthestrengthofmaterials.Inthetheoryofstrengthofmaterials,itisprovedthattheresistancetobendingofabeamwith

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givencrosssectionisproportionaltothemomentofinertiaofitscrosssectionrelativetotheaxisthroughthecenterofgravityofthecrosssectionandperpendiculartothedirectionofthebendingforce.

Letusclarifythisbyanexample.Weassumethatabridgeacrossastreamconsistsofaboardthatsagsundertheweightofapedestrianpassingoverit.Ifthesameboard(nothickerthanbefore)isplaced“onitsedge,”itscarcelybendsatall,i.e.,aboardplacedonitsedgeis,sotospeak,stronger.Thisfollowsfromthefactthatthemomentofinertiaofthecrosssectionoftheboard(ithastheshapeofanelongatedrectanglethatwemay

thinkofasevenlycoveredwithmass)isgreaterrelativetotheaxisperpendiculartoitslongsidethanrelativetotheaxisparalleltoitslongside.Ifwesettheboardnotexactlyflatnoronedgebutobliquely,orevenifwedonottakeaboardatallbutarodofarbitrarycrosssection,forexamplearail,theresistancetobendingwillstillbeproportiopaltothemomentofinertiaofitscrosssectionrelativetothecorrespondingaxis.Theresistanceof

abeamtobendingisthereforecharacterizedbytheellipseofinertiaofitscrosssection,whichbecomesthereforeits'core-singularityelement',oftencontrolledbyitscentralpoint/s.

Thelogicexpansionoftheconceptofdualellipticalterritories.

Nowfollowingthiskindofthought,ofellipsesascollaborativelocusof2'complementaryspecies',wecanapplythe'logic'oftheconcepttoanythingandinfactdefine'eccentricity'linesastheessentialformofawaveof

communicationbetween2points(2ndNon-EPostulate):

Soacouplewithason,isaGSTellipse,wherebothfathersareconstantlyseekingasimilardistancebetweenthem.Andaterritorialanimalcoupleisalsoalogicellipse,tenderingfortheterritoryasone

movestohunt,theotherstaystobreed.

Anyrelationshipisanakedellipse(withouttheexternalmembrane),joinedbythefocallinethatsharesentropyandformbetweenthem.

SteelbeamsoftenhaveanS-shapedcrosssection;forsuchbeamsthecrosssectionandtheellipseofinertiahavethegreatestresistancetobendingisinthezdirection.Whentheyareused,forexampleasroofraftersunderaloadofsnowandtheirownweights,theyworkdirectlyagainstbendingina

directionclosetothismostadvantageousdirection.

Thisresultcanbeunderstoodintermsof'2planes'the∆-planeofthebeamandthe∆-1planeofthegravitationalfield,andthedominantnatureofthemajoraxislinethatcommunicatestheinnerstructureoftheentity.

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OPEN–ENTROPICCONICS–ITSPENTALOGIC

Theopencurves:parabolasandhyperbolas

Nowoncewehaveidentifiedwhatistrulyrelevantaboutbidimensionalcurvesasopposedtosingleonesthatrepresentonlyapartofthebeing:tobeofafullternaryorganism,with3parts:

Afocalpointorsingularity:@;amembraneorcyclicalcurve:ð§;Thevitalspaceorenergybetweenthem:st

Wecanconsider'opencurves'inwhichtheintermediatespaceisfullyopenedanditsmeaningtorepresentkeyelementsofT.Œs(Timespaceorganisms).

Andthewonderofthemisthatinthoseopensystemsthekeyelementswillstillbedeterminedbythefocalsingularitiesandtherelativebalanceoftheir'co-invariant'productinrelationshiptothemembrane.

Sotheycanrepresentthe'metricequations'ofco-invariance5Dsystems,andinfact,thehyperbolawillbethebestrepresentationofanyfunction:

T:TheparabolaasaTTquadraticpropotion.

Letusthenconsidertheopenconicsfirstfromtherealistperspectiveofmathematicalphysics,wheretheyrepresentthemotionofoneortwopoints,ratherthanthestaticgeometryofaformwithitsmembrain.GalileothenfoundthataTT-acceleratedmotionawayorcomingtoadifferentfocus=point(themaximalearthvs.theattractedform),hadtheshapeofaparabola.Aswe

knowaccelerationisadouble‘timelocomotion’(TT)–themotionofamotion,butmaintainingthe‘point’withitsinnerstructure.Sowehavetorepresentonlyapoint–aline.

Andbecauseitismovingawayatanaccelerateddual=squarebidimensionalpuretimemotion(acceleration)itisaquadraticproportion.Thuswedefinetheparabolaasthegraphofquadraticproportion.Werecallthatthegraphofquadraticproportion:y=kx²isaparabola;

whichwe‘tumble’togiveitsignificanceinmathematicalphysics.

Wecanthenconsiderinverselythe‘mathematicalconcept’oftheparabolaasa‘fixedmembrain’…ofwhatwemightwondersinceitisanopenform.Theansweristhatasafixedformtheparabola,anditsrealformthe3DparaboloidaretheperfectformtofocuslinealflowsofSpacetimeeitherforthepurposeof1Dperception,orinversely,theycantransforma‘source’ofenergyintolinealentropicmotions;twoother

variationsofthesT-theme:

S@:Itsfocusanditsdirectrix.

Weconsiderthentheequationy²=2pxandcallthecorrespondingcurveaparabola.

ThepointFlyingontheOx-axiswithabscissap/2iscalledthefocusoftheparabola,andthestraightliney=–p/2,paralleltotheOy-axis,isitsdirectrix.

Whileateachpointoftheparabolawecantracealinealtangent,giventhefactthatallcurvesaremadeinthesmallerscaleof‘freeopen,linealsteps’(whichin

mathematicsisthebasisofdifferentials).Hencethelinearlizationoftheparabolawhenwe‘extract’onedimotionofacceleration,‘shrinking’thewholemotiontoits

steps.

LetusthenMbeanypointoftheparabola;ρthelengthofitsfocalradiusMF,anddthelengthoftheperpendiculardroppedfromittothedirectrix.LetuscomputeρanddforthepointM.Fromtheshadedtriangleweobtainρ2=(x–p/2)²+y².AslongasthepointMliesontheparabola,wehavey²=2px,hence:

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Butdirectlyfromthefigureitisclearthatd=x+p/2.Thereforeρ²=d²,i.e.,ρ=d.Theinverseargumentshowsthatifforagivenpointwehaveρ=d,thenthepointliesontheparabola.Thusaparabolaisthelocusofpoints

equidistantfromagivenpointF(calledthefocus)andagivenstraightlined(calledthedirectrix).

Thepropertyofthetangenttoaparabola.

Letusexaminethenonthosebasisthe1Dfocusoftheparabolawhichmakesitsousefulinvitaltopologiesfromeggstoeyesandantennae.

Sinceforaparabolay2=2pxwehave2ydy=2pdx.Itfollowsthatthederivative,ortheslopeofthetangent,isequaltody/dx=tanϕ=p/y.

Ontheotherhand,itfollowsdirectlyfromthefigurethat:

tanϒ=y/x-p/2

But:

i.e.,γ=2ϕ,andsinceγ=ϕ+ψ,thereforeψ=ϕ.

Consequently,byvirtueofthelaw(angleofincidenceisequaltoangleofreflection)abeamoflight,startingfromthefocusFandreflectedbyanelementoftheparabola(whosedirectioncoincideswiththedirectionofthetangent)isreflectedparalleltotheOx-axis,i.e.,paralleltotheaxisofsymmetryofthe

parabola:OnthispropertyoftheparabolaisbasedtheconstructionofNewton’sreflectingtelescopesandmodernantennae.Ifwemanufactureaconcavemirrorwhosesurfaceisaso-calledparaboloidofrevolution,i.e.,asurfaceobtainedbytherotationofaparabolaarounditsaxisofsymmetry,thenallthelightraysoriginatingfromanypointofaheavenlybodylyingstrictlyinthedirectionofthe“axis”ofthemirrorarecollectedbythemirroratonepoint,namelyitsfocus.Theraysoriginatingfromsomeotherpointoftheheavenlybody,beingnotexactlyparalleltotheaxisofthemirror,arecollectedalmostatonepointintheneighborhoodofthefocus.

Inotherwords,aparabolais½ellipsewhereoneofthefocushasbeenstretchedtoarelativeinfinitevaluecomparedtotheotherfocus,breakingitsbisymmetry,andthisintroducestheconceptsof‘relativeinfinities’(the

EarthisaninfiniteweightcomparedtothepointthattracesaparabolaofTT-motiontowardsit;thestarfocusedinthetelescopeisatarelativeinfinitelengthcomparedtothefocusdistancetotheparaboloid,etc.

Weshowthisrelationgeometricallybyfirstdrawingacircle,andthen"stretching"ittomakeanellipse,andthen"stretching"itevenfurthertomakeaparabola(pointgoestoinfinity).Soifwestartwiththeequationoftheunitcircle:𝑥2+𝑦2=1

Andthendosomestretchingintheverticaldirectionbyafactorof𝑏:𝑥2+(𝑦/𝑏)2=1

Andthemweletbgetreallybig,wegettheequationoftheparabola:𝑥2=1(y)

Thus,intheso-calledfocalplanethroughthefocusofthemirrorandperpendiculartoitsaxis,theinverseimageofthestaratthepointofinfinityisobtained–butthepointmustbeafractalpointwithavolumetobe

perceived!:thefartherawaythisimageisfromthefocus,themorediffuseitwillbe,sinceitisonlytheraysexactlyparalleltotheaxisofthemirrorthatarecollectedbythemirroratonepoint.

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∆+1:Thesearchlightisbasedonthesamepropertyoftheparabola.Init,conversely,astrongsourceoflightisplacedatthefocusofaparaboloidalmirror,sothatitsraysarereflected

fromthemirrorinabeamparalleltoitsaxis.Automobileheadlightsaresimilarlyconstructed.

Thusinthestillviewofparabolas,thereisstillafocusandamembrane.Whereastheparabolicbeingisasingle'foci',ableto'focus'theinformationandentropyofalargerscale

field;fromthepentalogic∆+1perspective.

ST<=>ST:@-ellipse.Inthepentalogicoftheellipsethispropertygiveusalsothe‘perceptive’pentalogicview:

Indeedintheellipse,itiseasytoshow,theraysissuingfromoneofitsfociFlandreflectedbytheellipsearecollectedattheotherfocusF2(previousfigure),makingthecommunicationbetweenbothpointsassimpleasareflectionintheinnersideoftheclosedmembrain;apropertywhichasusualdepartsfromitspuregeometricformulationinsymmetricsystems,toconfiguretherelationshipsof‘network-linesandwaves’inphysiological

andphysicalorganisms.

Sotheparabola,withoutasecondequalfocus,whichwouldenhancethesurvivalsymbiosisbetweenbothisNOTusuallyafullT.œ,butatbestanorganopentothe

world(Staticview)oraTT-acceleratedmotion..

TheHyperbola.

Ontheotherhand,inthehyperbolatheraysoriginatingfromoneofitsfociF1arereflectedbyitasiftheyoriginatedfromtheotherfocusF2:Thisisarepresentationofa‘head-body’system

wherethebodyisblindtoperceptionasitreflectstheinformationabsorbedfromthehead,whichisthereforetheF2focusofthehyperbola,‘above’itslowerpart,splitfromit.

TheinversionofSTvalues,focusanddirectrixofthehyperbola.

Sothehyperbolaisthemostextremeentropicrepresentationof‘split’‘inverse’STproperties.

Indeed,ifweconsiderasinglepartofthehyperbolaasthegraphofinverseproportionandthatthegraphofinverseproportiony=k/xyx=Kisahyperbola.Itsequationthoughisallpervadingpreciselybecausethequalityofinversionistheessentialdualpropertyofspacevs.timefields,whichcanberepresentedby½

hyperbolaininnumerablecases.SxT=Kisbydefinition,theequationoftheperfecthyperbola.

Indeed,the2branchesconicequation.x²/a²-y²/b²=1representsthefullhyperbola.Inthespecialcasea=btheso-calledrectangularhyperbolaplaysthesameroleamonghyperbolasasthecircle

playsamongellipses:

Ifwerotatethecoordinateaxesby45°theequationinthenewcoordinates(xʹ,yʹ)willhavetheform:x’•y’=k.Andweshallusebothmodestofullygraspfundamentalmetricequationofsystemsinthefifthdimension.

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Nowintheprevioushyperbola,ifwedenotebycanumbersuchthatc²=a²+b²,thenitispossibletoshowthatahyperbolaisthelocusofallpointsthedifferenceofwhosedistancestothepointsFlandF2ontheOx-axis

withabscissascand–cisaconstant:ρ2–ρ1=2a.ThepointsF1andF2arecalledthefoci.

Letusthenconsidertheparabolafromtheperspectiveofitsfocianddirectrix.

Liketheparabola,thehyperbolahasdirectrices,inthiscasetwoapiece.Ifweconsiderafocusandthedirectrix“onthesamesidewithit,”thenforallpointsofthecorrespondingbranchofthehyperbola,wehaveρ/d=ε,

wheretheeccentricityforahyperbolaisalwaysgreaterthan1.

Thusonebranchofthehyperbolaarethelociofallthosepointsintheplaneforwhichtheratiooftheirdistanceρfromthefocustotheirdistancedfromthedirectrixisconstant.Fortheellipsethisconstantissmallerthanunity,fortheparabolaitisequaltounity,andforthehyperbolaitisgreaterthanunity.Inthissensetheparabolaisthe“limiting”or“transition”casefromtheellipsetothehyperbola;bornastheellipse

tearsapartits2focuses,thatsplitentropicallyintotwodifferententities,albeitmaintainingitsrelativesymmetryastwopartsthatwereonceentangledintoone:

Andsothefundamentalrelationshipbetweenthecurveandthe2foci,ispreservedinaninverse'restingmanner';whichqualifiesthehyperbolaastheentropicstateoftheellipse,itstime-reversedfigure,

anaforementionedpropertyofimportancefor'complexGSTanalysis',wellbeyondthescopeofthistexts.

Thehyperbolaisdifferentfromtheellipse,asitispurealgebraicin'phasespace',withvariablesinwhichthehyperbolaisNOTarealform,butamentalformtorepresent,themetricequationof5D,inwhichTsxSt=K:

ConsiderasimpleformulaforPressure,p,duetoaliquidcolumn:P ︎=ρxg×h

Densityisameasureofdensityofform,orinformationofasystem;h,theheightdimensionofinformationandg,acceleration,aparameterofaninwardvortexofgrowingfrequency.ThuspressureisanStparameter,witha

value,productofatime-dimension(frequencyacceleration),aninformativedimension,height,andatimedimension,'density'.Moreover,wecanputthe3'elements'intermsoftimeasameasureofthe'past'valueofthesystem(itsdensity),thepresentvalue(itsheight)andthefuturevalue(itsaccelerationdownwards),and

thenmakeadeepphilosophicalstatementabouttheconstancyofpressure.

Yetifpressureistheðƒparameter,itfollowthatexpansivevolumeisthepureSPACE-entropyparameter,andsoweshallimmediatelypostulateaccordingto5Dmetrictheexistenceofaco-invariantrelationship:

P(t)xV(s)=K(st)

WhereKwillturnouttobethecyclicalspace-timevibrationoftemperature.

This'dimensionalanalysis'isthusanentirenewfruitfulperspectiveonmathematicalphysics,akintothedimensionalanalysisofclassicphysics,butfarmoreprofoundinsignificance.

Boyle'slawamountstoyetanother'5Dmetric'equation,whichwecanplotwithastraightlinedepartingfromO,crossingalldifferentTi,forequivalentPxVvalues,maximisedinthecentralregionoftheasymptoticcurves.

AllthisrevealswhysandÐisomorphismsofasimplemathematicalequationwhichforaphysicist,meansmerelyρ,thedensityinkg/m3,g=1om/s²theaccelerationduetogravityandh,theheightordepthofliquidinmeters,

usedtocalculatethepraxisandfuturebehaviourofaliquidinmotion.

Butwhatwehavewrittenisessentiallytheequationofpotentialenergy,PE=mxgxh,whichwewillindeeddefinewhenstudyingactionsandHamiltonians,theultimateequationsof5Dphysics(aswellas4Dphysics),as

thetime-likecomponentof'presentspace-timeenergy'.

ThethemeofGeometryandphysicsisobviouslywellbeyondanythingthisauthorcandevelopinafewnotes.Sothereaderspeciallyifaphysicistshouldnotexpectmorethansomemarginalcomments.

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Afewcommentsthoughseemnecessaryafterstudyingtherepresentationinmotiongeometryofthe5Dimotionsofrealitywithconiccurves,sincethatwasessentiallythewayinwhichmodernphysicswasborn,

whenGalileostudiedthe5thDimotionofentropiccannonballs,whichwereopenT-parabolasandKepler,the4thDimotionofinteractiveorbitsofplanetsandsun,whichwereclosedS-ellipses.

The10canonicalequationofthebidimensionalplane.

Inthegraphweseethe10canonicalcurves,ofwhich5areS=curvedand5areT-straight,5areT-openand5

areS=closed,3areimaginary,3aredouble,3aresingleand1isapoint.TheyareindeedwhattheyarebecausetheyrespondtotheSTand3x3+•symmetriesofthespace-timeUniverse.

Weconsideredthemostimportantsecond-ordercurves:thecircle,theellipse,thehyperbola,andtheparabola.Whatothercurvesandgenerationsarerelevanttoexhaustthefieldofbidimensionalgeometries?

NotsurprisinglyastheUniverseisonlya5Dstructure,therearenomorecurvesthantheonesneededtodefinethe5Dimotionsofreality.Soallothercurvescanbereducedtooneofthe9canonicalequationsofconics.

A2nd-degreeequation, contains6terms,not3oronlytwoasinthecanonicalequationsoftheellipse,hyperbola,andparabola.Thisisnotbecausesuchanequationrepresentsamorecomplicatedcurvebutbecausethesystemofcoordinatesispossiblynotsuitedtoit.ItturnsoutthatifweselectasuitableCartesiancoordinatesystem,thenasecond-degreeequationwithtwovariablesalwayscanbereducedtooneofthefollowingcanonicalforms–sinceaswealreadyexplainthesimplifiedgeneralizedcoordinatesor|,O,Øtopologicalvarietiesinwhichthesystemoreventwestudy,residewillalwaysbringa

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simplermoretruepointofview.SoasttturnsoutifweselectasuitableCartesiancoordinatesystem,thenasecond-degreeequationwithtwovariablesalwayscanbereducedtooneofthefollowingcanonicalforms:

Thereaderwillobservethosecanonicalformsarethe3conics(withthecircleacontractedellipse)andthetwoessentialformsofcongruence,intersectingandparallellines.

Alas,oncemoretheUniverseappearsasasimplestructure,ofclosedandopensystems,theperfectcircle,thesplitellipsewith2focus,theparabola,whichfurthersplitsthemandthehyperbolawhichthroughthey-axisof

entropysendsbothindifferent'heightarrows'ofthe∆-scalesofthefifthdimension.

Plus3varietiesoflinealcouples,theintersectingcouple,theparallelcoupleandtheidenticalones,whichagainrespondtotheternarysymmetriesoftheUniverse,whoseprofoundmeaning,relevanttotheoutcomeofall

eventsinspace-timeisstudiedindepthinthearticlededicatedtothe4thpostulateofnon-Ælogic.

Letuswritethemallin3dimensions,whichFermat'stheorem,superpositionlaws)ismerelydonebyaccumulationofreproduced,identical'socialnumbers'ofplanes,oneafteranother,thesamecurvesmerely

engrossedthroughthereproductivegrowthofaz-dimensionarestillthesameuniquevarieties:

X2+Y2+Z2=1Sphere

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Eachofthoseconicsandcurvesin2or3Dimensions.Wecanthenseethecircleandtheconeasa3Dspiralthe1Dimotionandnon-Euclideanpostulate.

Thesecondpostulateand3Dimotionofreproductionandcommunicationareexpressedbyellipticforms.

The2ndÐimotionisexpressedbylinealformsandplanes,whoserelationshipsaregivenbytheirparallelism,asherds,perpendicularities(asintersectingplanes)itsskewnessascatalleysorconvergence(5Ðsocialevolution)anddivergence(hyperboloidsbetweentwosystems)orpureentropic4Ddissolutionofasystem-parabolas.

Further|xO=Øgenerations.

AstheUniversebecomesmorecomplexbyiterationandcombination,thefieldofcomplexcurves,whichmixgeometric,scalarandvital,movingpropertiesofthe3∆Stelementsbecomesquasi-infinite(∝).Andviceversa,

wecanfurtherreduceallcurvesto|xO=Øgenerationsofthe2formalelementsoftheUniverse.

Wesawhow|xOdualitiesgenerateconics&spirals.Andthesamecanbesaidofallother9canonicalcurves.

Considerthecaseofrectilineargeneratorsofahyperboloidofonesheet.Itisnotatallobviousthefactthatthehyperboloidofonesheetandthehyperbolicparaboloidcanbeobtained,justliketheconeandthecylinder,by

themotionofastraightline.

Incaseofthehyperboloid,itissufficienttoprovethisfactforahyperboloidofrevolutionofonesheetx2/a2+y2/b2–z2/c2=1,sincethegeneralhyperboloidofonesheetisobtainedbyauniformexpansionfromtheOxz-

planeandundersuchanexpansionanystraightlinewillgointoastraightline.

Letusintersectthehyperboloidofrevolutionwiththeplaney=aparalleltotheOxz-plane.Substitutingy=a

weobtain: Butthisequationtogetherwithy=agivesintheplaney=apairofintersectinglines:x/a–z/c=0andx/a+z/c=0.

Thusthereisapairofintersectinglineslyingonthehyperboloid.IfnowwerevolvethehyperboloidabouttheOz-axis,theneachoftheselinesobviouslytracesouttheentirehyperboloid(graph).Itiseasytoshowthat:

1.2arbitrarystraightlinesof1andthesamefamilyoflinesdon’tlieinthesameplane(theyareskew)

2.Anylineof1ofthesefamiliesintersectsallthelinesoftheotherfamily(exceptitsopposite,whichisparallel)

3.Threelinesofoneandthesamefamilyarenotparalleltoanyoneandthesameplane.

Asincomplex¬Egeometry3linesdefineatopologicalorganicplane,thehyperboloidrepresentsanentirefamiliyoforganicspecies,whichweshallconsiderinphysicalandbiologicalandchemicalanalysis.

RECAP.Theimportanceofbidimensionalcurves:Holographicphysics.

Onceweunderstandbidimensionalitywecanenlightenphysics’mathematicalstatements,whichdealwithlawsofforcesandmotion,drawnbeforeanalysiswiththecanonicalBidimensional:

-'Open'curves–parabolasforentropicmotionsasincannonballshots.

-Closedcurves:cycles,spheres,ellipses:usedininformativemotions-asingravitationalandchargevortices/clocks.

-Andin-between,St-hyperbolas,usedinst-ratios:stbalances,st-systems,st-constantsofnatureand5D=stmetricequations;asinEnergylawsortheBoylelaw:P(t)xV(s)=K(st)

ThemostextensivefieldthoughofanalyticgeometrybecomesitsuseinmathematicalphysicstodescribethedifferentDimotionsofreality.Weshallthusstudycurvesinanalyticgeometryasexpressionofthosedimotions.

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TRANSFORMATIONSOFREALITYINTOMINDSPACE:PROJECTIVE&AFFINEGEOMETRY

Ofthemanysingularitymind-spaceswithwideimplications,therichestoneisprojectivegeometry,asitshowsthepropertiesoftheouterworldthatremaininvariantinthemind-constructionofusefulinformation

aboutreality,makingpossibleitssurvivalintheouterworld,

AfundamentaldevelopmentofgeometryparallelwiththecreationofLobachevskigeometrycameaboutinyetanotherway.Withinthewealthofallthegeometricpropertiesofspace,separategroupsofproperties,

distinguishedbyapeculiarinterrelatednessandstability,weresingledoutandsubjectedtoanindependentstudy.Theseinvestigations,withtheirseparatemethods,gaverisetonewchaptersofgeometry.Theexplosion

ofparallelgeometriesisthusawelcomedadonnecessarytoexpandouranalysisofthemotionsacrossdifferentscales.Andthewayplanesofspace-timecreatetheholographsoftheUniverse,bymotionsand

translations,projectionsandimprintingofinformationintoenergy.

Projectivegeometryisinthatsense,abasictooltounderstandhowabidimensional,highplaneofinformation,projectsitsformoveraplaneof

space,creatingaspace-timesystem.Asusualwewillthenfindarelationshipbetweenthe3elementsofreality,theo-point,theðcycleandtheSpatialplane,whichistheoriginofallrealitiesinallitscreativecombinations:

Inthegraph,theprojectionofabidimensionaltallðcycleoftimeonaspatialsurfaceofenergyconservescertainpropertiesbuttransformsthemainpropertyoftime–tobeclosedgeometry,intothemainpropertyofspace,tobeanopenedgeometry.Indeed,forthehighestpointsoftheinformativepurecycleofheighttobeprojectedonthe$plane,3elementsaretobeputinrelationship,theo-point,thecycleandtheopenplane,

suchasthebiggertheopenplane,themorechancesitwillhavetoimprintthecycle,andthehigherthepointofview,theeasieritwillbetoprojectthecyclewithaclosersimilarity.ItmustbealsomentionedthegreatimportancethathastheRiemannsphereanditsprojectioninthecomplexplane,tobeanalyzedonline4.

Thesecondelementofprojectivegeometryistheunderstandingonwhatpropertiesareornotconserved,andeasilyprojected,inasmuchasitmeanswhataretheð≈$symbiosisthattiesupbothelementsintoði≈e$ofspace-time(anoldformalismwhichInolongeruse,asIamconvertingallvariationsintoðand$foreasier

understanding).

Itisthenself-evidentthat‘measure’,that‘sacredcow’ofphysicistsisNOTconserved.Itisprecisely‘size’thetrueflexibilityoftheUniverseandits5thdimension,whichisnotneededinaUniverseofabsoluterelativityofscale.Thelengthsofsegmentsarechangedintheprocessandsoaretheangles,theoutlinesofobjectsare

visiblydistorted.

Whatthenremains?Immediatelywesee,oftheessentialqualitiesthatthepropertyofanumberofpointslyingononestraightlineispreserved;andthoseareasanticipatedinouri-logicaxiomofaline,3ofthem,forsuchlinestobefullystraight.Sotheprojectiononthe$t-planeDOconservethe$t-relationshipsoftheðcycle,

enhancingthemasthelinesDOgrowinsize,whenwemovefromaðimplosiveformintoan$explosiveform.

Thisgeneralruleisofimportanceinallrelativesystems,soweshallextractagenerallawofit:

‘TransformationsoftimeintospaceconserveandenhancethespatialpropertiesoftheSorTelement’.ThisisimportantbecausetheUniverseisallaboutconservationofAngularandlinealmomentum,potentialand

kineticenergy,pastandfuturecombinationsofspace-time;sotranslationsmustconservethepropertieswhicharemorenaturaltothenewmediuminwhichthesystemmoves.WeliveinaUniversethatwants,triesand

achievesimmortalityofenergyandinformationthroughlawssuchasthisone.

Wecanalsoobservethatthecentralpointofviewdoconservethoselinesandrelationships,asallthelinesthatcrosseditkeepcrossingit.Sothe‘soul’ofthesystemisconserved.

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Themembranethoughisthemostdistortedelement,becauseitnowunlessthespatialplaneinwhichisprojectedis‘bigenough’totransformits‘fast,compressed’cyclesofexistence,itwillNOTfitonit.Soasageneralrulewenoticethatthemostimportantelementconservedisthe0-pointofview,orwill/soulofthe

system,andthiswillallowtoformulateanevenlargergeneraltheoremofreality:

‘Allpointsofviewcanswitchbetweenspaceandtimestateswithoutloosingitsidentity.Soallsystemscancoiltosleepinitsinformativestate,andelongatetomoveinitsspatialstate.Thereareofcoursemanyotherspace-timedualitiesthatprovethistheorem.AndinthisthereadershouldunderstandthatevenABOVEmathematicsthereisSpace-timeTheory,butwedohonorthevalueofmathematicsbyreferringthecausalityinaninverse

fashion(extracting∆STtheoremsfrommathematicalones–itisinfacttheotherwayaround,projectivegeometryconservesthe0-pointofview,becausethisisthe‘last’entitytobedestroyedinanysystem,asthesystemdiesonceitiscollapsed.Sowecanstateherethat∆STsystemsdoNOTdiewhentheychangefrom

spatialtotemporalstates).

Anotherconservedpropertyisthatofastraightlinebeingatangenttoagivencurve.Andofcourse,thereaderdoesnotneedtobealynxtorealizethisisthedefinitionofaderivativeandoneofthemanywaysto

understandthatwecanalwaysderivateintimeandspaceasystem,orintegrateit,asthisiswhatallisabout,conservationoffullworldcyclesaszerosumsofaninfinitenumberofinfinitesimalsteps,eachoneastraightderivativeonacurvedworldcycle.Sotheworldcycleoftheðcircleisnowprojectedintoa$medium,butitisstillhappening,anditwillbecompletedifthereisenough‘vitalspace’forittobeimprinted(orelseitwillbecutoff;butasaruleinnatureaseedofinformation‘prospers’inarelativeenergeticspace,orelsethe‘animal,

orphysicalsystem’choosesNOTtoreproduce.Sowecouldsaythato-pointsgaugefirstwith‘projectivegeometry’,inalogicmannerits‘resourcesofspace-energy’before‘projectingitsinformationandreproducing

itinalargerbeingofspace.

ProjectivegeometrydoesmustbeconsideredinthelargerviewofT.ŒsamodalityofSpatialReproduction.

Theotheractionrelatedtoprojectivegeometryisobviouslyperception,intheinversearrowtoitsspatialreproduction,whentheplaneofenergeticspaceisprojectedbackintotheo-pointofperceptionthatgaugesinformation.Andthisgivesbirthtoaninterestingfieldwhenwecomparethetwo‘differentdirectionsoftime’,

ð->$(reproduction)and$->ð(informativeperception).

ThestudyofpropertiesofperspectivegoesbackinantiquityrighttoEuclid,totheworkoftheancientarchitects;artistsconcernedthemselveswithperspective:Dürer,LeonardodaVinci,andtheengineerand

mathematicianDesargues(17thcentury).Finally,atthebeginningofthe19thcenturyPonceletwasthefirsttoseparateoutandstudysystematicallythegeometricalpropertiesthatarepreservedunderarbitraryprojective

transformationsoftheplane(orofspace)andsotocreateanindependentscience,namelyprojectivegeometry.

Itmightseemthatthereareonlyafew,veryprimitivepropertiesthatarepreservedunderarbitraryprojectivetransformations,butthisisbynomeansso.

Forexample,wedonotnoticeimmediatelythatthetheoremstatingthatthepointsofintersectionofoppositesides(produced)ofahexagoninscribedinacirclelieonastraightlinealsoholdsforanellipse,parabola,andhyperbola.Thetheoremonlyspeaksofprojectiveproperties,andthesecurvescanbeobtainedfromthecircle

byprojection.

Theimportanceofthisinrealityisobvious,asthehexagon,wehavealreadymentionedistheperfectpi-cycleof3diametersofperimeter.

Andsonotonlytheprojectionofthecyclebutthehexagonits‘naturalquadrature’isconserved.

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Itisevenlessobviousthatthetheoremtotheeffectthatthediagonalsofacircumscribedhexagonmeetinapointisapeculiaranalogueofthetheoremjustmentioned;thedeepconnectionbetweenthemisrevealedonly

inprojectivegeometry.Butitsdeepfoundationsarein∆ST:againtheo-pointisconservedinthehexagon,whichrevealstohavemanysimilarpropertiesofthecycleasitisitsmoststableformfor‘smallnetworks’.

Nowanotherkeyfieldofprojectivegeometryisthestudyofangularprojections,anditsrelatedtrigonometriclaws,whichcanbeconsideredpartoftheðperception,andthecapacityofapointtoaccuratelymeasure

distancesonthespaceitperceives.

Thisisthemostmagicpartofprojectivegeometry,whichrevealstheenormousintelligenceofspace-timetoallowo-pointsofviewtogaugeinformation.

Forexample,underaprojection,irrespectiveofthedistortionofdistances,foranyfourpointsA,B,C,Dlyingonastraightlinethe

crossratioAC/CB:AD/DBremainsunaltered:

AC/CB:AD/DB=A’C’/C’B’:A’D’/D’B’Thusasystemcanactuallyperceivemeasuresbyhavinga‘sensorial’setof(ABCD)’pointsinitsmembranetocalculatesuchproportions.Thesekindofpropertiesareofcourseextendedtoallthelawsoftrigonometryandanglesanddistancescalculatedwiththoselaws.

Projectivegeometry,thusisessentialtounderstandtherelationshipbetweenao-pointorST-systemanditsouterlargerworld,andhowthepointshrinkstopologicallyanexternalworldintoaninternalimage,along

topology,trigonometryand…affinegeometry,whichformthescaffoldingofthemathematicallawsthatallow‘gauginginformation’,evenforthesimplersystemsofnature,regardlessofhumananthropomorphism.

Growthofpointsintowavesandplanes,inalgebraicforms=geometricfigures.

Itiscustomarytoanalyzetheequationsofcurveswithanalyticgeometry,thoughtothataimwewouldneedtointroducesomein-depthconceptsof5Doperandsinalgebra.Wecouldthenstarttheanalysisofgeometryandalgebratogetherbyconsideringthe‘basic’chainofsocialoperand,thesumorsuperpositioninasingleplane,theproducttheoperandofgrowthofdimensionalityiterationandmerging.Andsoaproductconvertsapointintoaline,waveorcycleandanewproductintohigherdimensionalforms.Asweworkwith2dimensional

figures,thesquareisthebasicproductoftheconiccurves…Butallthiswillobscurethepurityofgeometry,asCartesiancoordinatesdowithgeneralizedones.Wemustalwaysrememberthedifferencesbetween‘objectivereality’whichisindependentoftheobserver,andhumindknowledgethatsomehowalwaysgetsthedistorting

humanobservercomplicatingthings.WhenweintroduceaCartesianframeofreferencewhilehumind’smanipulationbecomeseasiertheessenceofthoseconicswhichisabouttherelationshipbetweentheinner

focusandthepointsgetsblurred.Inthatregardwearemoreinterestedingeneralizedcoordinatesforobjectiveanalysisandprojectivegeometryforsubjectiveanalysisofthelawsofspace,asthemostgeneralizedforms;andjustwillmentionthealgebraicequationsofanalyticgeometryasareferencetowhatmatterus–toextractvital

propertiesofthosecurves.

Theobserver,observableandcreatortrilogicparadoxes.

ProjectiveGeometryintroducesathemefundamentaltotheparadoxicalnatureoftheUniverse–theternarysymmetry

betweenthe‘world’thatcreatesasawholeitssmallerparts,andthesmallerperceiverwithintheworld.TheclassicexamplebeingDesarguestheoremthatrelatesanobserver(centerofperception

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-anicerwordthanperspectivity’),anobservable,twotriangles,andacreator,theaxisofperspectivity.

Buttheegoparadoxmakestheperceivertothinkhecreateswhatheperceives.Soforlongthemaintheoryoflightcolorswasthattheeyegeneratedthelightthatreflectedonobjects,bouncingbacktocreateanimage,no

theotherwayaround.external,objectiveworldthoughislargerandgeneratestheobservablewhichtheinternalsubjectiveworldperceivesandthroughitsperceptioncanentangleinsimultaneitywiththeCreator.

ConsiderforexampleDesarguesmaintheoremofprojectivegeometry.Inasubjectiveviewthetwotriangleswouldbegeneratedbytheperceiverasthe‘apex’ofapyramid,whichdonotrequirefurtherelementsinits

constructionbutthe‘eyerays’.Butasithappensthetriangles,nottheperceptionofthemaregeneratedbytheaxisofperspectivity,a‘largerworldline’thanthepointofperception,whichhas3pointsx2lines,toconstruct

the3correspondinglinesx2triangles.

Howthis‘abstraction’becomesarealityshouldbeobvioustothereader.Accordingtothe2nd¬Epostulatealinehasvolume,morethan1Dimension.ItisthereforeawaveoffractalpointsoranetworkInthiscasewetalkoftheaxisofperspectivityasawhole,itspoints,thedepartingelementsof3‘physiologicalorfractalnetworks’,whichmergetocreate2organs,thetriangles,whoseself-similarityisestablishedbythecenterofperspectivity.

Thehumindcenterisonlyapoint,notenoughtogeneratealargerworld.ButasalwaystheparadoxoftheegoturnsupsidedownthecausalityasintheCopenhageninterpretationorHilbert’saxiomaticmethod–Iimaginepointsandlines.Hilbert‘segothoughdoesn’tneedtoknowthataCartesiandemon,the‘axisofevil’hasplacedthose2trianglesforitsmindtosee.Andyetthe‘generator’arethe3perspectivepointsoftheaxisline,the

largerworldwithahigherdimensionthanthepoint.ThisrealizationthatwedoNOTgeneratethelargerworld,whichisoutthereforustomirrorcomesonlywithaconstantevolutionfromsubjectivechildhoodtoan

objectivematureclassicage(revertedbacktosubjectivityinthethirdoldagethatdegeneratestochildhood,astodayhumindsdoinitscollectiveageofself-extinction,becomingemotionalandsubjective,self-centered

childrenagain).

RECAP.Fromapurespatialperspectiveofgeometryasasimultaneoustimeindependentviewofthestructuralrelationshipsbetweenthepartsofreality–thesuperorganism,itsscalesandelements-generalizedand

projectivegeometryareofahighervalue.Butforthecomplexentanglementofperspectivesspeciallywhendealingwithgeometryasanexpressionoftimedimotions,whatshouldbecalled‘algebraicgeometry’butitis

calledanalyticgeometryworksbetter.

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∆-SCALARTOPOLOGICSPACES:LINEALAFFINEGEOMETRY.

AffinegeometrystudiesthepropertiesoffiguresthatarenotchangedbyarbitrarytransformationsinwhichtheCartesiancoordinatesoftheoriginal(x,y,z)andthenew(xʹ,yʹ,zʹ)positionofeachpointareconnectedby

linearequations.Whereitisassumedthatthedeterminantisdifferentfromzero.

Itturnsoutthateveryaffinetransformationreducestoamotion,possiblyareflection,inaplaneandthentoacontractionorextensionofspaceinthreemutuallyperpendiculardirections.

Quiteanumberofpropertiesoffiguresarepreservedundereachofthesetransformations.Infactaffinegeometryisremarkablyextensive,showingultimatelythatgrowthinsizethroughlinealincreaseofand$xð

systemisabsolutelynaturaltotheUniverse,itsessence,whicheasilyconservesallthepropertiesofthefractalðseedinitsexpansioninspace:

Straightlinesremainstraightlines(infactall“projective”propertiesarepreserved);moreover,parallellinesremainparallel;theratioofvolumesis

preserved,alsotheratioofareasoffiguresthatlieinparallelplanesorinoneandthesameplane,theratiooflengthsofsegmentsthatlieononestraightlineoron

parallellines,etc.

Manywell-knowntheoremsbelongessentiallytoaffinegeometry.Examplesarethestatementsthatthemediansofatriangleareconcurrent,thatthediagonalsofaparallelogrambisecteachother“thatthe

midpointsofparallelchordsofanellipselieonastraightline,etc.

Thewholetheoryofcurves(andsurfaces)ofthesecondorderiscloselyconnectedwithaffinegeometry.

Theverydivisionofthesecurvesintoellipses,parabolas,hyperbolasis,infact,basedonaffinepropertiesofthefigures:Underaffinetransformationsanellipseistransformedpreciselyintoanellipseandneverintoaparabolaorahyperbola;similarlyaparabolacanbetransformedintoanyotherparabola,butnotintoan

ellipse,etc.

Sounlikeinthecaseofprojectivegeometriesofðsystemsinto$systems,whichtransformcirclesintotheirequivalentopenspatialforms(parabolas),affinetransformations,whicharegrowthsDOconservetheessentialð<$tstructureofthesystem.Moreoveritisadeterministictransformation,withNOerrors.ParabolasdoNEVER

becomeellipsesandsoon.

Theimportanceoftheseparationanddetailedinvestigationofgeneralaffinepropertiesoffiguresisemphasizedbythefactthatincomparablymorecomplicatedtransformationsturnouttobeessentiallylinear,i.e.,affineintheinfinitelysmall,andtheapplicationofthemethodsofthedifferentialcalculusislinkedexactly

withtheconsiderationofinfinitelysmallregionsofspace.

Ifwecorrectthisinfinitesimalconcepttoafinitesimal,whichstillpreservesthislinearity,wecouldsimplystatethatgrowthofasystemgoesthrougha‘linealregion,inthestable∆ST=kconservedmetricregionofthe5th

dimension.

Inotherwords,thelinealaffinegrowthofasystemandaffinegeometryonthewholeisjustifiedbythe∆(Spxð)=aKprocessofgrowthofthesystem,withinthe10xgrowthregion,whichistheregioninwhichthemetric

ofthesystemislinealbeforeitsLorentzianregionsofemergenceordissolutioninthe∆±1scales.

NowmodernmathematicsobviouslydoesNOTconsiderthis5Drealistinterpretationofthereasonsofexistenceofthosefundamentalvariationsofgeometry.Insteadtheyreformalizedwithintheabstract

meaninglessidealistprogramsoftheaxiomaticmethodsoftheGermanschoolofscienceofacenturyago(asnobodyhaseversince‘think’seriouslyinphilosophyofscience,onceculturemovedtoAmerica,anditsvisual

ortechnicalpraxiswithsolittlepuretheoreticalandintellectualunderstanding).

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ThusallthisisclassifiedintotheKlein’sErlangerProgramof1872,whichsumsuptheresultsofthedevelopmentsofprojective,affine,andother“geometries”givinganobscureformulationofthegeneralprincipleoftheirformationwiththeuseofthatpestofmodernmathematicscalledgrouptheory(-:

Wecanconsideranarbitrarygroupofsingle-valuedtransformationsofspaceandinvestigatethepropertiesoffiguresthatarepreservedunderthetransformationsofthisgroup.”

InaccordancewiththisprincipleofKlein,wecanconstructmanygeometries.Forexample,wecanconsiderthetransformationsthatpreservetheanglebetweenarbitrarylines(conformaltransformationsofspace),andwhenstudyingpropertiesoffigurespreservedundersuchtransformationswetalkofthecorresponding

conformalgeometry.Buttheresultofthisprogram,asanyofthemultiplevariationsoftheGermanidealistaxiomaticmethodisahyperinflationof‘imaginedmathematics’,whichconfusethefundamentalpropertyand

needformathematicsasarealistscience.

Informationisinflationarythereismoremoneythanrealeconomy,moreimaginedwordsthatrealfactstodescribe,morefictionthatrealityinanylanguage.Andthisisthefactofthe3rdageofinformationofanysystem–notavaluebutalossofclassicrealism,theperfectageinwhichlanguageandrealityareinmirror

correspondencetoeachother.

Weshallnotconsiderthatmetalinguisticapproach,whichcompletelyignorestheouterworldreflectedbymathematics,intheobviousopposedrealistphilosophyofi-logicmathematics.AsweholdtruthGödel’sproof

oftheincompletenessofanycategoricaldefinitionandproofofexistencebasedonlyinaninternalmetalanguageorinternallogic.

ThosebriefexamplesofCLASSICstillanalyticGeometryanditsinitialapplicationstomathematicalphysicssufficeforthepurposeofit-toshowthebidimensionalstructureoftheuniverseinitsstmanifoldsaccordingto

theholographicprinciple.

RECAP.5Dtransformationsofscalesgeneratefromrealitymindspaces,throughadualprocessofcontractionofformandstillnessofmotion.Andviceversa,theprocessesofexpansionofaseedofmentalspaceorform

intothelargerworldbyreproductionandmotiongiventotheseedofform.

Thoseessentialprocessesofcreationaremimickedintheprojectivegeometryofvisualspaceandtheaffinegeometrythatsimplifiescurvesintolines.

Affinegeometrystudiestheinverse,dualactionsofreductionby‘perception’ofalargerspacesurfacebyasingularitywhereasanSpaceis‘reduced’intotheð-pointandinverselythe3Dimotionofreproductionandscalingwhereaninformativeseedofformisimprintedonanenergyplane.Accordinglyaffinegeometryis

relatedtothe‘growthinsize’ofasystem,throughalinealprocessofexpansioninspace.

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XII.2ndAGE:T:MOTION-DIFFERENTIALGEOMETRYEXPANDSMATHEMATICALPHYSICS.

Thisfinalfundamentalrealizationthatconnectsgeometrywiththemetaphysicsofordervs.freedom,formvs.motion,lowervs.higherscales,(GalileanparadoxesofDuality),isthusagood

introductiontothethirdageofGeometry(evenifitcamebeforethefinalevolutionof@nalyticgeometryintonon-

Euclideanformsofmind-space).

Its3clear±¡sub-ageswillbe:

DifferentialGeometryofcurvestakenaspointswithmotion.

Vectorspaces,whichexpandatthepathofmathematicalphysics,asthebestphasespacetorepresentSpace-timefieldsofparametersthathavebothformandmotion.

Topology,ofthe3BiDimensionsofspacewithmotionthatrepresentthe3functionalorgansofanysupœrganism,

∆@st:Its3rdeclecticmodernageofcombinationexpansionandexplosiontoallotherbranchesofthe'entangledmathematicalmirroroftheentangledUniverse'.Weshallforsakeofsimplicityonlycommentonthe

3firstclassicages.Anddevelopinsteadoftheeclecticmodernage...

Apentalogicageof5D'motiongeometry',withafewinsightsonthedifferentagesandformsofgeometrywhicharebettersuitedtoexpressthe5DimotionsoftheUniverse.

The'surface'ofasphere,approachedby'smallerplanes':

Inthegraph,thefactthatanyspacecoincideswithaEuclideanintheinfinitelysmallenablesustodefinefortheintrinsicgeometryofasurfacebyapproximatinganinfinitelysmallportionofthesurfacebyaplaneoran

infinitelysmallvolumeexpressedasEuclideanspace.Thevolumeofafinitedomainisthenobtainedbysumminginfinitelysmallvolumes,i.e.,byintegratingthedifferentialofthevolume.Thelengthofacurveisdeterminedbysumminginfinitelysmalldistancesbetweeninfinitelynearpointsonit,i.e.,byintegratingthe

differentialofthelengthdsalongthecurve.

Andthisisarigorousanalyticexpressionforthefactthatthelengthisdeterminedbylayingoffasmall(infinitelysmall)measuringrodalongit-whichisultimatelythedifferential,smoothversionofthefractalstepbystepmeasuringofgrowingdistanceswhenwescaledownourview-henceanotherproofofthefractalandmentalnatureofreality,ultimatelyprovingthe∆±iand@-mental'missingdimensionsofreality',inhuman

'naiverealism'.

Thegraphthenshowin'2dimensions'onthesurfaceofthebeinganotherkaleidoscopicVIEWontheapplicationofEuclidean,ellipticandhyperbolicgeometries.IfweconsiderONLYasimplifiedEuclideanreality,(leftside),weneednomeasureofcurvature-itisaflatsmallplaneofspace.Nextincomplexity,aregular

sphericalcurvedpieceofthewhole,(ð§)requiresmoreinformation.Soameasureofcurvature,Φmeasureisrequired.

Butifthesystemisnotaregularsphere,twocurvatureswillbeneeded.Finallyinhyperbolicgeometrythemorecomplex,STvitalenergywithitstwoCONTRADICTORYdirectionstowardsthesingularityandthe

membrain,willneedtwocurvatureangles,withoppositedirections,representedbythe±sign.

Andweshallchoose(Euler)towell-definethecurvatureofthewholesurface,justthemaximalandminimalanglesofcurvature,accordingtothefundamentalruleoft.œs,whichcanbedefinedbyitsstandingpoints,itsmaximalandminimalfunctions,whicharetherelevantMax.exMin.i,max.ixmine,e=i,ternary'pointsofany

worldcycle/system',requiretoGeneratealleventsandformsofexistence.

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Suchdirectionsarethuscalledtheprincipaldirectionsandthecurvaturesk1andk2arecalledtheprincipalcurvaturesofthesurfaceatthegivenpoint:k(ϕ)=k1cos²ϕ±k2sen²ϕ...

WhereoncemoreasusualwefindthesinusoidalfunctionsthatdefineSTsystemswithitstwooppositedirections.

Theinversearrow:envelopesandcurvesonthelarge.

Ithastobenoticedthathumanswithitsobsessionforthesmall,asinformationcomesfrombelowandsoitismoreabundant,whileabove,largerentitiesarenotsowellperceived,hasmadeusalsoquiteignoretheemergenceoflargerentities.Thishoweverisessentialforphysicsandinmathematicstheoriginofemergenceintime(Fouriertransforms)treatedontheemergence

articlesonthefirstline,andemergenceinspace,thesocalledenvelopecurves,yetanotherbranchofstaticformalspace,bettertreatedinphysicswherespaceusuallyhasmotion,reducingononesidetheinformativeinflationof'fictiontheoriesofthemind-spaceswithnovital

use'andgivingtheequationsamorebeautifuls=tsymmetrybetweentheformandmotiondimensions(s=tbeingthe'definitionofbeauty',athemetreatedinthestudyoftheexist¡ential

program).

Weshalljustthenmentionitforthesakeofcompleteness-thatistoshowthatforeach∆-1entropictheorythereisaninverse∆+1socialone:

Thequestionofenvelopesinthatsenseisarelativelysimpleone-asallquestionsof∆+1wholesoflesserinformation,solvedlongago,inthetheoryoffamiliesofcurvesandsurfaces.Especiallywelldevelopedisthe

theoryinthecanonicalST,holographic2-manifolds;thatistwo-parameterfamiliesofvariouscurves,inparticularofstraightlinesALWAYSeasierto'perceive'byhumanessentiallya'smallthing'belongingtoa'flat

curvature'space-mind:theso-called“straight-line”congruencies.Inthistheoryoneappliesessentiallythesamemethodsasinthetheoryofsurfaces,hencewithinthescopeof∆stDisomorphisms.

Intermsof∆stthetheoryisthedirectapplicationofafundamentallawofSTemergence,oftenquotedindifferentarticles:∑|i-1>Oi:

TheinversionoffunctionsandformsaswegrowinscalesintheUniverse,whichisabasicsymmetrythatallowstheUniversetobalanceitsrelative(in)finite(simal)volumeandform,orelse,allbalancedwouldbreakprovokingaconstant'shrinking'or'enlarging'alongasingleentropicorsocialevolutionaryarrow.

ST-Balanceisthelawandsymmetriesarejustaviewofthatlaw.

ThisimpliesthatAsurfaceiscalledtheenvelopeofagivenfamilyofsurfacesifateachofitspointsitistangenttooneofthesurfacesofthefamilyandisinthiswaytangenttoeveryoneofthem.

Soweseetheultimatemergingof'Darwinianperpendicularity'+'symbioticadjacency',whichISatthecoreofthe'submissive',yetsymbiotic'herding'ofenvelopes,ð§dimensions,wherethecyclicaltimeenvelopbecomea

larger∆+1§partialscaling(hencethemarriageofthosetwosymbols,ofcyclicaltimeandscalarspace-aNEWworldcyclebringsalwaysahigher∆+1plane).

Againthisisanabsolutelaw,whichthesimplifying,perfectformsofgeometrymakeseasiertounderstand.

Forexample,theenvelopeofafamilyofspheresofequalradiuswithcentersonagivenstraightlinewillbeacylinder(figure48),hence∑Oi>|i+1.Andtheenvelopeofsuchsphereswithcentersonallpointsofagiven

planewillconsistoftwoparallelplanes.Theenvelopeofafamilyofcurvesisdefinedsimilarly;andhereasweareinanST-mixedelement,weneedtostudythedominanttendencyofthosecurves,whichwillshowthe

enveloptotendtowardsamorelinealorcyclicalwhole.

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Forexample,Figure49diagramsjetsofwater,issuingfromafountainatvariousangles-theyareclearlybyeffectofthepotentialgravitationalenergycomingbacktoaclosingzero-sumcycle.Hencesuchfamilyof

curves,whichmaybeconsideredapproximatelyparabolas;tendtohavetheirenvelopeamorelinealparabola-thegeneralcontourofthecascadeofwater.

Butnoteveryfamilyofgeometricalformshasanenvelope.AndifyoumanorrobotoftheIIImillenniawhichmightreadthosetextsstarttointeriorizethelawsofT.œsshouldbynowguess,whichkindofentitiesdoNOTwanttobe'enclosed'-thosethoroughlydominantin1D-linealmotionand/or4Dentropy.Forexample,afamily

ofparallelstraightlinesdoesnothaveone.

General'laws'ofemergence:o->O->•->@

Allthisleadustounderstandthatultimatelyasalldepartsfrom∆•s≈tlaws,geometryrequiresalwaysafirst∆±idistinctionbetweenwhat'pros'call,thegeometry“inthesmall(parts)”,whichisclearlydominantand“inthelarge(wholes)”.Themainofthosedualtheoriesshouldthenfollowtheobvious∆stlawthatwholesaremoreresistant,efficientandstablethanparts;hencesmall/partsareeasiertodeform,whilewholesarefar

morestablefullT.œs-theultimatereasonwhywholesandnewscaleskeephappening.

Forexample,in1838Mindingshowedthatasufficientlysmallsegmentofthesurfaceofaspherecanbedeformed,andthisisatheorem“inthesmall.”Atthesametime,heexpressedtheconjecturethattheentirespherecannotbedeformed.Thistheoremwasprovedbyothermathematiciansaslateas1899.Incidentally,itiseasytoconfirmbyexperimentthatasphereofflexiblebutinextensiblematerialcannotbedeformed.Forexample,aping-pongballholdsitsshapeperfectlywellalthoughthematerialitismadefromisquiteflexible-

lawsthoseakintothelawsof'surfacetension'ofsoapbubbleswithwideapplicationinphysics.

Anotherexample,isthetinpail;itisrigidinthelarge,thankstothepresenceofacurvedflange,butseparatepiecesofitcaneasilybebentoutofshape.Aswesee,thereisanessentialinversionbetweenpropertiesof

surfaces“inthesmall”,∆-1and“inthelarge”,∆+1.A1Dtvs.3Dðwidergeneralizationisprovidedcomparingopengeodesicsvs.closedcurves.Ageodesic“inthesmall,”isasmallsegmentofthesurface,itsshortestlinealpath,but“inthelarge”linearitymaynotbethe

shortestpathatall-itmayevenbeaclosedcurve,thegreatcirclesofasphere.AndhereiswhereanotherLAWOFEMERGENCEAPPEARSofenormousgenerality,asitisthebasicprocessofsocialevolutionofasystem,fromlifecellstoastronomy:creationbuildsfirststepbystepits'proteinenvelop'andthenasitgrowsitfinallyneedsasingularitytofocusandconstrainthepartsthroughitsradius,creatingan

antipodalellipticgeometry,whichfinallycreatesthe@-systemandcompletestheT.œ

Indeedmanyanalyticsurfacescannotbeextendedinanynaturalwaywithoutacquiring“singularities”intheformofedgesorcuspsandthusbecomingnonregular.

Thus,asegmentofthesurfaceofaconecannotbeextendedinanaturalwaywithoutleadingtothevertex,acuspwherethesmoothnessofthesurfaceisdestroyed.Thisstriking,obviousresult,30yearsagoleadmetodomyfavpaintingofconceptualcubismandadoptthepyramidal∆-formforwholepovs,andsingularityminds:

Inthegraphweseeright,my∧painting,whichadecadelatterresembledeerilythefirstBose-condensate(maximalformofaphysicalsystem-its5D),and

ultimatelyprovesthereMUSTBEAGOD/logicmindforanywholeorganism,limitingthenumberofplanesa

systemcangrow,departingfroma'finitesimalamount'of∆-2parts.

Thusgeometryofthelargeisonlyaparticularcaseofthepreviousremarkabletheorem:

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Everydevelopablesurfaceotherthanacylinder(thelineal,non-envelopedessential1Dform)willlead,ifnaturallyextended,toanedge(oracuspinthecaseofacone)beyondwhichitcannotbecontinuedwithout

losingitsregularity.Thusthereisaprofoundconnectionbetweenthebehaviorofasurface“inthelarge”anditssingularities.Thisisthereasonwhythesolutionofproblems“inthelarge”andthestudyofsurfaceswith“singularities”(edges,cusps,discontinuouscurvatureandthelike)mustbeworkedouttogether.Nowweknowitswhysinatheme

thatfascinatesbothmathematiciansandphysicists.

Now,wehavethe3conceptsneededtofullydescribemostofmodernnon-egeometries,includingRiemannianmanifolds,inyetanother'mirrorimage'oftheternarylawsofST:

ð§:the'intrinsicgeometry-curvature'ofthesurface:

∆+i:the∆-scalinggivenbytherelativeratiosofr/ksmallnessorgreatness,whichdefinestherelativesizeoftheobservervs.theobservableform.

Therelativenumberofdimensionsweshallstudyandhowtheyareconnectedwhenwegobeyondtheusualternarygamesofexistence;thelastofthekeythemesofnon-Espatialmindworlds.

Differentialgeometryasthestudyofmembrains.

TheimportanceofGaussiandifferentialgeometrycomestofullfruitioninvitaltopology,asthestudy of surfaces equates to the study of membrains, the most important elements of allsystems,whichdeployaspecifictopologicalvarietyofclosedform,butacquireinsmallscalesahyperbolicgeometry,makingthedualityofopenandclosedspaceat∆1levelcorrespondtothedualityoflinealandhyperbolicsurfaceinsmallscales.Inotherwords,whatthesphereloosesinthe largerscaleas the lessersurfaceofallbeings, in fact itwins it in thesmallerscaleas the

largerhyperbolicsurface.

Themembrainthushaveaconvolutedmaximalsurfaceofosmosisandexhchangeofenergyandinformationwith theouterworld in the smaller scale inwhich it is thepredatorbut appears as the smaller surfgaceofexistenceinthelargerscaleinwhichitisamerefractalpointofalargerworld.

A themewhich introducesus in the ‘last’ of all thebranchesof geometrydiscoveredbyHuminds,which isquitesurprisingindeed,asitshouldhavebeenthefirst.

Butwehavedealtwiththeshortcomingsofthehumindinmanyotherpapers,outofourfrustration…

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3rdAGEOFGEOMETRYSoweshallstartthe3rdagewiththelastdiscoveries,whicharethe2fundamentalformsofgeometryforitsrealisminitsanalysisofthe‘twoeternalelements’ofreality:

∆-scales:Fractalgeometry isthefinalunderstandingofthescalarUniverse,andyetitwasnotfoundduetothe extreme naïve realism of the humind, who doesn’t understand its mental space eliminates thediscontinuitiesoftheUniverse,sothe‘continuoushypothesis’isbothfalseandmisleading,asthey‘upgrade’smallerparts to fill theholesof largersones (caseof the real line,which rises the finitesimalnumbers andtranscendentalratiostotheN-discontinuousline).

ST-numbers:WhicharevectorswithanscalarS-valueandaT-motion/direction.YetVectoranalysis,wasonlydevelopedintheIIpartoftheXIXc.

DISCONTINUOUSANALYSIS:FRACTALSWeshallmovenowintowhattrulyisthoughitisusuallynotconsideredinthoseterms,thefinalageof∆nalysisasthestudyoftherelationshipofpartsandwholes,whichincloseanalysisturnouttobefractaldiscontinuous

parts.Fractalsareinthatsensetheequivalenttothefinalrealizationinphysicsthatcontinuityisamiragethatsimplifiesrealityandsodotheanalysismathematicalmirror,butwhenwereallywanttoknowthewholedetailsofthe∆-scale,theuniverseisquanticandsoitisitsfractalgeometry.HeretheworkofNottalein

mathematicalphysicsandMandelbrotinmathematics,standsfullyasthebestformalismforboth.Sowewon'tstomponourpeers(yes,idorecognizethemaspeers,inthiscase),butasusualbringthepointofviewofthe

philosopherof'stience'.Theinfinitesimalstudyasperceivedfromthefinitepointofviewistheviewoffractals,whenindetailand

observingtheclosedworldcyclesthatseparateandmakeeachinfinitesimalawhole.Aderivativeisthefinitesimalofthefunctionobserved,andsowhenwegoevenfurtherandstudyasenlarged

intoourscalarviewtinmaximalinformationweareinthefractalviewofreality.Soasweexpandourviewthefractalviewbecomesmorereal,tillfinallytheenclosuresobserved∆-1become

fractalandwerecognizeitsself-similarities:∆-1≤∆º.Foreachderivativethusafunctionshowsits1/ninfinitesimal(notnecessarilythisfunction,whichisthe

derivativeofthelogarithm).Itfollowsthatfunctions,whichgrowginormously,havea'quantaoftime'reproducedandsoitsminimal

derivativefinitesimalisthefunctionitself,eª.Fractalstructureofthe5thdimensionanditsperpendicularflows.

Fractalsarethebestwaytodescribeallthemesrelatedtoscales.Andassuchtheyareconnectedintimatelywiththeoriginalelementsofcalculus,namelyseriesandfinitesimals.Fractalsthenbranchedoutasasub-

disciplineofscalarmathematics,whichwecanconsidertwohavethose2sides,thediscretefractalview,andthecontinuoustopologicalview,innumbers=pointst=sdualities.

Soweshallstartwithpowerseriestodescribefractals,whichstartedascalculusdidintheearlierworkofArchimedes,resurrectedingeometricaltermsbyKochandtheXIXaberrantgeometers:

Archimedes'quadratureoftheparabola

Archimedes'dissectionofaparabolicsegmentintoinfinitelymanytrianglesusedthesumofageometricseriestocomputetheareaenclosedbyaparabolaandastraightline.

Hismethodwastodissecttheareaintoaninfinitenumberoftriangles,establishingthe concept of a fractal system defined in this book as the 'scaling' between thewhole and its infinitesimals, happening in nature (from super organisms toorganisms,organs,tissues,tillcomingtocells,allself-similar;fromgalaxiestoatoms

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inphysicsthroughintermediatecosmicbodies,fromcivilizationswithits3agesofsubconsciouscollectivearttoindividualmindsinmemeticsuperorganismsofmankind,etc.).

Archimedes'Theoremstatesthatthetotalareaundertheparabolais4/3oftheareaofthebluetriangle.Aseachgreentrianglehas1/8theareaofthebluetriangle,eachyellowtrianglehas1/8theareaofagreen

triangle,andsoforth.Assumingthatthebluetrianglehasarea1,thetotalareaisaninfinitesum:

Thefirsttermrepresentstheareaofthebluetriangle,thesecondtermtheareasofthetwogreentriangles,thethirdtermtheareasofthefouryellowtriangles,andsoon.Simplifyingthefractionsgives:

Thisisageometricserieswithcommonratio1/4andthefractionalpartisequalto:

Thiscomputationusesthemethodofexhaustion,anearlyversionofintegration.Usingcalculus,thesameareacouldbefoundbyadefiniteintegral,whichisjustthe'topologicalcontinuous'versionofthediscretefractalversion.SowhatfractalgeometrydoesisNOTtomovefromdiscretenumbersofsequentialtimeandscalar

stepsintocontinuoussumsinaplane,keepingMOREinformationonthedetailandshowingclearlythediscretefinitesimallimitsofnature-reasonwhyinNatureweseemorefractalsystemsthat'differentials',whichmust

beseenasamore'time-mirror'orientedversionofthesameconcept.Itisinterestingtonoticethattheparabolaisa'triangle'increasedby1/3rd,theessential'element'ofthe

ternaryreality.Sowecansaidthetrianglehas3elementsandtheparable,whichisitscurvature,adsonemore'dimension'ormotiontoit.Atriangletheninexternalmovementwill'define'anexternalwavearounditwithaparabolicform(theaddedthirdvolume).Astaticparabolawillhaveaninnerregiontoaddthatthird,whichcan

beconsideredthe'envelope'membrain.

FractalGeometryThenextstepwasdonebyKoch'ssnowflakeisaunionofinfinitelymanytriangles.Inthestudyoffractals,geometricseriesoftenariseastheperimeter,area,orvolumeofaself-similarfigure.Forexample,theareainsidetheKochsnowflakecanbedescribedastheunionofinfinitelymanyequilateraltriangles(seefigure).Eachsideofthegreentriangleisexactly1/3thesizeofasideofthelargebluetriangle,andthereforehasexactly1/9thearea.Similarly,eachyellowtrianglehas1/9theareaofagreentriangle,andsoforth.Takingthebluetriangleasa

unitofarea,thetotalareaofthesnowflakeis:Thefirsttermofthisseriesrepresentstheareaofthebluetriangle,thesecondtermthetotalareaofthethreegreentriangles,thethirdtermthetotalareaofthetwelveyellowtriangles,andsoforth.Excludingtheinitial1,thisseriesisgeometricwithconstantratior=4/9.Thefirsttermofthegeometricseriesis

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a=3(1/9)=1/3,sothesumis ThustheKochsnowflakehas8/5oftheareaofthebasetriangle.

Hereagainwefindanother'essentialratio'ofgrowthinNaturalfractals,themesthosetreatedinnumbertheory;butmoretellingistheformofthenewshape,essentiallyanHexagon,whichfailstobecomeapi-circle,

itsnextstage,aswecanconsiderthehexagonacirclewithpi=3diameter.Andsowecantalkofaternarygrowthofthetriangleintothecircle,its|-Ostate,paralleltothe'rotation-motiondimension'whichalso

convertsitintoacircle,inthiscasebystaticgrowththroughscalargeometry.ThusoneessentialprocessofNature'stopologies,the'quadratureofthecircle';thatis,the'triangular

transformationofthecircle'...canbeachievedinpentalogicbyimperfectmethods:• ð-methods: rotating the triangle (aspartof therotatedsurface isno longer 'solid' space,by its

motion-derivative...• ∆-scalarmethods:growingaKochsnowflake,aspartofthecircleisvoid.• S-methods:parabolizingitssurface.Asonlyoneside,thatofmotiondefinesafrontwaveforthe

triangle.Themostefficientbeingthescalarmethod,as8/5<4/3,thesurfacesofthesnowflakeandtheparabolaofa

triangle.

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XIII.VECTORSPACES

4thPostulateof®logic:parallelismandperpendicularity

Vectorspacesaredefinedby4positiveelements(whereentropyisignored),whichmakesthemimmediatemirrorsofthe5DUniverse:apointwithamagnitudeinspace,adirectionofmotionintime,whichcanbe

operatedbya‘scalar’numbersthatdeterminesitsscale,andmeasuredfroma‘frameofreference’accordingtoanangleofperception(whereasthenegationofavectororitsentropicstateistheinversedirectionofthe

vectorfullyresolvingthemeaningofanegativenumbersaninversedirectionintime).

Itisforthatreasontheyaresousefultorepresentrealityanditmarvelsittooksolonginthenaturalevolutionofthemindfromfixedspace->bidimensionalplanes->pointswithmotion(calculus)->toarrivetovectorspacesthatrepresentthe4elementsof∆@stinanaturalform.Thelawsofvectorspacesthusareindirect

relationshipwiththeGSTlaws.Theystartedwhenthe‘bidimensionalplanesofDescartesandNewtonacquireda3rdaxis(Zheight)andtherulesofitssumwerelaidownaccordingtoitsorthogonality(perpendicularity),to

represent(Lagrange)locomotion,speedandacceleration.

The2typeofvectorproducts

Vectorsareof2quitedifferenttypes,astheirpropertiesarenotthesame.

Thedotproduct

Thedotproductaffectsvectorsinthesamemannerthanthesum,asithasthesamepropertiesofidenticalbeings(associative,distributiveandcommutativeproducts).Thustheymustaffect‘equalspecies’oftheherd

type,whichhashugeimplicationsforthephysicalnatureoffieldsofforces,whichareasalwaysforan∆ºentity,indistinguishable,∆-2particles(asaminoacidsareallthesameforyourbody.Andgenerallyspeakingentropic

points(∆-2)areallthesameforan∆ºentity)

WereturnonceandagaintoakeypostulateofNon-E,thefourthpostulateofcongruence,toexplainthevitaltopological‘angles’thatdefinetheevolutionordevolutionofdimensionalmotionsoftimespace.

HoweverinthecomplexUniversealleventshavedualandternaryinterpretations,whenweperceivethemfromtheinverseentropicandinformativedualities,mergedintoacombinedform.

Sothingsarenotsosimpleastheyseem,andthisisthecaseofvectorsspacesanditsDualityofparalleldotproductandperpendicularcrossproduct.Initiallyonewouldassumethatperpendicularcrossproduct

annihilatethemselves,butinfactarecreativeprocessesthatcreatea‘3rddimensionalelement’,perpendiculartobothwithamagnitudeequaltotheproductofbothSotheessentialequationofacrossproductisa

reproductivemergeoftwo‘inverseSandT’elements:

Crossproduct:SxT=ST

Thecrossproducthoweveraffectsadifferentkindofspecies–akey‘insight’of5Dtheorylosttoabstractmathematicians–itisnotthe‘algebraicproperties’butthe‘species’thatcanbesubjecttothecrossproduct

whatmakesadifference.AstheyareelementswhichdoNOTcommutateandtreattheirangleininversefashiontodotproduct.Itisanticommutative:axb=-bxa,andorientable.Moreoveritonlyexistsin3dimensions(in7is

notuniquelydefined).

Whatthisessentiallymeanisthat1)thecrossproducthasanarrowof‘angulartime’henceitisorientable;itisareproductivefunction,asitgivesbirthtoa3rdspeciesintherelativedimensionofwidthoftheothertwo

(reproductivedimension)andsoitcanbeconsideredamergingofanS-function(theheightinformativevector),alengthfunction(thelocomotion,entropicT-vector),givingbirthtoanST-function(thecrossproduct).

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WhatthismeansfortunatelyenoughcanbeassessedbothfromthegenerallawsofGstandthedirectexperienceofwhich‘speciesaresubject’toacrossproduct,themainofwhichisobviouslytheproductofthe

magneticandelectricfieldthatgivesusthereproductivespeedoftheelectricfield.

Then other obvious case is the angular momentum of a particle about a given origin defined as: L = r x p

where r is the position vector of the particle relative to the origin and P is the linear momentum of the particle; which again connects a Space element (the origin T.œ) and a T

element (the rotating one) giving us an ST product of both, which in mathematical physics we regard as the ‘first T.œ’ of the Universe (an h ‘Planckton’)

We cannot get here into ‘complicated alternative inflationary mathematical information’ to represent the same concept; alas, the bivector and its multiplication that transforms the cross

product into an ‘SS’ plane… by virtue of the S=T law of 5D. In5Dwearedealingwithtwoelementswhicharetime-likeandspace-like,henceinvertedinpropertiesbut

similarenoughtobecomeareproductive‘hyperbolicmetricequation’SxT=K.

Sothecross’productangleofperpendicularityiscreative(andtheproperwaytodrawwouldbewith‘inverted’arrows’sothereproductiveresultatthepointwherebothmergerisinga3rdoffspringvectorwouldbemoreobvious).Sincewhenthe2elementsarecloseenoughtoreachanS=Tmergingpointandthereisnot‘tearing;

butonlymerginganactofreproductiontakesplace.

OntheotherhandtheparallelvectordotproductisaDarwinianevent,becauseiteliminatesthesmallerform,asthedotproductreducesfrom2elementstoonethesystem.Wecanthenconsidertheparallelismtobeone

happeninginaflatsurfaceasapredatoryactions,inwhichthelarger‘system’feedsontheentropicenergy/motionofthesmallerone.

Inpraxisbothproductsdohappeninphysicsandindeed,thecrossproductisamorecomplexprocess,asinmagneticandelectricS-TfieldsthatmergeintoanST=speedwave.Whileasimpleexampleofthedotproduct

isabodymovinginafieldofforceswhichextractenergyofmotionfromit.

Sothedualityofdotandcrossproductandthevaluesofitsanglesisanessentialmirrorof5Dspacetimedimotions.

Vectorspacesandthecomplexplanearethe2fundamentalexpansionsofframesofreferencesintocomplexholographicdimotions.Thatis,framesofreferencewithform,scaleandmotion,andassuch,theyarethe

essentialsystemofrepresentationoftheUniverse,beyondtheabstractionofmentalspaceasitis.InthescalarUniversehoweveravectorspaceitselfamotion-a'field';andassuchwhenaparticleisplacedinavector

space,itwillenterin'communicationwiththefield'intwodifferentways:

Ifthefieldismadeofits∆-¡particles,itneedstoabsorbtoproduceoneofitsdimotionsthefieldwillbea'forcefields'inwhichbythe4thpostulateof'¡logicbehavior',the'Activemagnitude'willtrace'equipotential'pathsALWAYSPERPENDICULARtothefieldinwhichitfeeds(forcelines).Thismakes5Dfieldtheorydifferentinas

muchastheparticleisNOTfollowingthefieldmotionbuteitheritsequipotentiallines(caseofaplanetaroundthegravitationalfieldthatsinksintothesun),orwhatconsiderthe'linesofforce'(caseofchargefield,which

couldberepresentedbytheequipotentials.

Ifthespacehoweverrepresentstheparallelmotionsofspeedinthe'samescale'-NOTafeeding,perpendicularprocess,bythesimilarityoftheParticleandthefield,caseofamanswimminginariverofwaterthemotion

becomesparallelinboththeactivemagnitudeandthefield.

Thissaid,thefirstcasewillgivebirthtoacrossproductgeometry,ofperpendicularityandthesecondcasetoadotproductofparallelismbetweenthefieldandtheself-similarparticle.

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Finallyathirdelementofimportancein'movingcoordinates',isthedifficultytooperatewith'fixedmind-framesofreference.

Thusvectorspacesdepartfromasinglehumindframeofreferencewithits3'visual',perpendicularlightspace-time'basis/co-ordinates'byadoptinggeneralizedco-ordinates-thatis,co-ordinatesforeachpoint/itemasifitwereafractalbrokenspaceinitsown,whichtrulyis,sincewemovethenfromthesubjectivecontinuoushuman

view,tothesumofallthedifferentparticleviews.

Parallelism-Dotproduct.Fieldandparticlearesimilar,inthesamescale.

ThebasicfeatureofthedotproductthatconnectsitwithEuclideangeometryisthatitisrelatedtoboththelength(ornorm)ofavector,denoted||x||,andtotheangleθ

betweentwovectorsxandybymeansoftheformula:

Thedotproductconvertstwovectorsintoascalarnumber,thusreducingthedimotionsof

thesystemfromtwoformstoapoint.Butthisisdeceptive.Asthescalarnumberinfactisnota‘reduction’butanascensionofscale–aparameterormagnitudeoftheupperplaneofexistence.

Somephysicalexamplesare:

-Mechanicalwork(∆1)isthedotproductoftheforceanddisplacementvectors(∆-1),

·Magneticflux(∆2)isthedotproductofthemagneticfieldandthevectorarea(∆-2)

Thusthedotproductintermsofvitaltopologyisapredatoryactofalarger‘scalar’formthatabsorbsenergyfromalowerplane.Anditsbeautycomesfromits‘4thpostulateangleofcongruenceand‘similarity’thatdefinesingeometricterms,whatthermodynamicdefinesinalgebraicform.Thatis,thequantityofenergyextractedbythe∆+¡systemfromthelower∆-1,‘smaller’vectorisdirectlyproportionaltothesimilarityofthetwosystems.Thesamelawwouldapplyinthermodynamics,regardingthecapacityofthelargersystemtoabsorbfromthelowersystemitsenergyas‘work’,insteadofdissipatingitasheat.A‘superorganism’inwhichthe∆-1scaleof‘cells’areinsynchronicityandsimilarityrelationshipwiththelargerwholetransfersmostofitsenergyupwards.Aroughsysteminwhichbothscalesaredissimilarwilldissipateitintoheat.

HowcanweinterpretthisPRODUCTintermsofvitalmathematicsandthestcomponents?Themostobviousdefinitionisthis.The'biggest'predatorvector,Binthegraphisthedominantelement,asAisprojectedonit.WhatevertheSTelementsofthosevectorsmean,whichwillvaryfordifferentuses,unlikethecrossproductwhichiscreative,reproductive,thedotproductisentropic,destructive,astheresultisthe'absorption'ofthe|A|cos=Xcomponentofoneofthevectorsbytheother,whichbecomesexpandedintheX-axisvariable(whateverthisvariableis),andforalleffectsAdisappears,leavesnotraceofitsmotion/form≈position;andweobtainascalarwhichquantifiestheresultofthis'absorption'.Sowecanclassifythe2fundamental'products'ofvectorsbythedualityofthe4thNon-Æpostulate:

DotproductsareDarwinian,destroyingonevector,reducedfirsttoitsX-parameter,whichasithappensISsystematically,the'real'normallymomentumorenergyorbodyelementofthesystem,whiletheY-parameterofform,informationisdischargedinaclassicDarwinianactionoffeeding(the'particle'-headelementorY

coordinatesdisappears).Indeed,ifweusetheXYgraph,asinmostcasestoquantifythe2complementarypartsofthebeing∑∏(body-wave)>ð(head-particle),inphysicalsystemsthisprocessisequivalenttothepredatoreventofcuttingtheheadthrowingitoutandeatingthebodytomultiplyyourinnercellularenergyintheX-

directionofyourbody-motion-momentum.

Crossproductsarereproductive,creative,asa3rd'offspring-dimension-form'iscreatedfusionoftheother2:

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3rdand4thdimensionfields:entropyandmultiplication=reproduction

Inthegraph,productcanbeofmultiple,differentSTdimensions,whichstarttherichnessofits'propositions'.AvectorialproductisoneofitscommonestformsasitcombinesSTorTSdimensions,BUTasboth'present'productsaredifferentinorientation,thisproductunlikeotherSSorTTproductsisnon-commutative:bxa=-axb.Inthiscasegivingbirthtotwo

differentorientationsinspace,thoughformorecomplexproductofmultiple'S-T'dimensions,whichcandefineasaMatrixofparameteraT.Œparticleinfull,thenon-commutabilitycan

giveorigintodifferentparticles(quantumphysics).

VectorsthusbecometheessentialmodetodefineanSTholographicelement,witha0Dimensionofascalarnumberthatdefinesthesingularitypointandadirectionofmotionin

space(x,y,zparametersfroman@nalyticframeofreference,butingeneralizedobjectivecoordinatesalineal1Dparameterofdistance=speedpertimefrequency,whichmeasurestheT-stepsorcyclicalmotionofthe•

pointactivemagnitude).

Thedifferencebetweenbothtypesofvectorialproductisveryimportanttofullygrasprealityasitis.

Theperpendicularproductseemsatfirstcontradictorybecausetheyseemtodivergeinorientation.Butthisisbecauseweputthearrowinthewrongside.Itshouldbeintheoriginwheretheycollide,andthatisthedotinwhichthetwovectorsbecomeastillspatialparameter.Itisthenalsoapplicabletothe'collapse'ofmultipleflowsintoanon-Euclideanfractalpoint,inwhichtheybecomeascalarparameter.Andthatishowinfacta

Hilbertspace'collapses'inquantumphysicsaninfinitenumberofgeneralizedparametersallowingustocalculatewholes,andgivingavitalsensetothe

extremelyabstractjargonofquantumphysicists.

Ontheotherhandthecreativeproduct,whichisalsousedinphysicstodescribeanotherfundamentalscale,thatofelectromagneticforces,issymbioticcreative,mergingandhelpingthetwocomponentstoactsymbioticallyasone.Andagainthe'mentalspace'ofthecrossproductismisleadingasitseemstocontradictthe4thpostulateofsymbioticparallelismvs.Darwinianperpendicularity;lookingliketheytoucheachotherperpendicularly,butinfacttheelectricchargeandthemagneticfieldAREalwaysparallel,inthesensetheyarethesingularityand

membraneoftheelectricT.œnevertouchingeachother,astherearenomagneticmonopoles;hencetheystrengtheneachother,creatinganewforceandincreasing

thespeed,andcurving,increasingtheinformationoftheparticleunderthemagneticfield.

Geometriccomparison.

Themagnitudeofthecrossproductcanbeinterpretedasthepositiveareaoftheparallelogramhavingaandbassides:axbsinθ

OnecanalsocomputethevolumeVofaparallelepipedhavinga,bandcasedgesbyusingacombinationofacrossproductandadotproduct,calledscalartriple

product(seeFigure):axb•c

Becausethemagnitudeofthecrossproductgoesbythesineoftheanglebetweenitsarguments,thecrossproductcanbethoughtofasameasureofperpendicularityinthesamewaythatthedotproductisameasure

ofparallelism.

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Giventwounitvectors,theircrossproducthasamagnitudeof1ifthetwoareperpendicularandamagnitudeofzeroifthetwoareparallel.Thedotproductoftwounitvectorsbehavesjustoppositely:itiszerowhenthe

unitvectorsareperpendicularand1iftheunitvectorsareparallel.

Unitvectorsenabletwoconvenientidentities:thedotproductoftwounitvectorsyieldsthecosine(whichmaybepositiveornegative)oftheanglebetweenthetwounitvectors.

Themagnitudeofthecrossproductofthetwounitvectorsyieldsthesine(whichwillalwaysbepositive).

Soweestablishaparallelsuperpositionprincipleforthedotproductandaperpendicularityoneforthedotproduct.

RECAP.VectorialspaceisbestsuitedtorepresentSTsocialherds(dotproduct)orreproductivefunctionsofsimilarcomplementaryspecies(Dotproduct)

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XIV.3rdAGEOFGEOMETRY:NON-EGEOMETRIESAND∞METRICSPACES.

Whenwetalkofthethirdageofanysystemofspace-timewerefertotheageinwhichanexcessofinformationsplitsthesystemfromreality,asanoldmanwhichnolongerwrinkledandwarpedintoitsmemorialthoughtstrytorepresent theworld as it is. This processwould become endemic inmankindwith the Industrial R=evolution of‘metal-minds’, which starting with the eye-camera, ending in the digital Virreal helmet broke the strict laws ofcreationofthe5DUniversebyreducingittomentalspacesoflesserdimensions,backtotheminimal‘holographic2Dforms’.

Humans do not realize at all, where this process is leading mankind – to a state of audiovisual madness inpermanentconflictwiththemoralandvisual,aestheticbalancedlawsofthe5Dworldthatwilleliminateusfromitsfullexistence–themesthoseofthepapersonsocialsciences.

Inhard sciences thisprocessof fiction thoughtof the ‘3rd ageofhistory’of anexcessof informationhadmixedresults;asitdidinpaintingwiththearrivalofcameras.

Geometryasallhumanendeavorswillchangeforeverwiththeindustrialr=evolutionofmachines.Wehavealreadywitnessedthatitcompletelychangedwithphotographythepathofpainting,whowillbranchinitsbidtoovercomethemetal-eye,intohyperrealism,tryingtoseebetterthantheinitialfaultycameras(Courbet,etc.)on

onesideandpurementalthoughtontheother(VanGogh,Gaugin).

GeometrywillalsorealizedthatEuclidwasnotgoodenoughandbranchedintoanattempttodiscoverthetrueobjectiverealityofspace(Non-E)anditsoppositesearchforpurementalspaces,whichwasexpandingtomeettheneedsofindustrialDimotionsandthe‘newfound’extremeformoflocomotionwithinternal‘wastedenergy’called

entropy.

Toputsomeorderintotheexplosionoffictionalmentalspacewouldbethejobofthe2nextmastersoftheIIIAgeofgeometry,LobachevskithatinsistedexperimentalrealitywasneededtomeasurethepropergeometryoftheUniverseandintheinversedirection,Mr.Riemann,whoinsistedintheoppositepath–towalkawayfromrealityintotheunderstandingonhowthemindcreatedspaces.Hewouldbeanotherill-understood,die-younggenius,whichatleasthadunlikeLobachevski,theluckofbeingbornintheproperplaceandhaveamasterinGauss,who

didcapturehisthoughtsandgavethem‘authoritas’.

Riemann’srealizationthatgeometryisamental-logicendeavor,wherefunctionandi-logicthoughtovercomes'spatialrepresentation',allowedtheexplosionofabstractmentalspacestorepresentreality,whichwascarriedbyEinsteinandBohrintoPhysicswithmixedresults.InthistaskitwouldbeessentialtheidealistschoolofGermanphilosophy,fromHegeltoMarch,weoftencriticizeforitsescapismwithreality.Sincewhileitallowfurther

expansionsofmentalspaces,latterputincorrespondence,§@<≈>∆ð,withtherealUniverse;itoverreachintofictionsthatstillplagueseriousscienceanddestroyedrealismonphysics.OnlyEinsteinlatterrelentedgoingback

totheobviousfactthat‘scienceshouldonlybeconcernedwithexplainingfactsthathaveexperimentallyhappened’.

UnfortunatelytheegoparadoxoncemorewonthedayandtheCopenhagen‘creationofthemoonwhenwelookatit’carriedthedaytilltodayagainsttherealEinstein->Broglie->Bohminterpretation.Andthiswouldalsohaveharmfulconsequencesformodernmathematics,whichcompletelyrenouncedtowhatwetrytoachieveinthose

papers(andfuture‘pros’oncethechangeofparadigmisacceptedfullywillcomplete):theentanglementofmathematicsandreality,withtheadonmindspacesaspartofrealityitself.Thatis,realityisthemindspaceofa

larger‘world’;theEarthandthegalaxy,asourviewofrealityisthemindspaceofourelectronicneurons.

Realityandmindmergeinthesensethatwhatthemindperceivesthenitprojectstoshapereality.Butrealitylimitsalwayswhatthemindofpureinformationcanimprint;sobothmustproperlymergewiththelawsofbalance.

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Still for the sakeofunderstanding the3rd ageof geometrywe shall divide its analysisby forcebrief aswehavearrivetothe200pageslimitforanacademia.edupapertoproperlychargeitsimages,intwosections,thefirstonededicated to Lobachevski’s attempt to findamore precise geometry for theUniverse,which turn out to have ahyperbolicgeometry–thatofthefifthdimension;andthesecondonededicatedtotheexplosionofmentalspace,apttostudyallthedifferentlocaldimotionsandchangeeventsofdifferentsciencesandscales.

Lobachevski'stheorems:angleofparallelism

Inthegraph,aspace-timesymmetryhappensbetweentheangleofparallelismofahyperbolicgeometryinstill

space,andthespeedofthevortexofforceswhichimpliesthatfaster,stronger,moreattractiveforcesofsmallerparticles(Spxð=k)willhaveamorehyperbolicgeometry,withasmallerangleofparallelism=larger

curvature,allowingmore'parallelforces'toentertheattractivevortex.Thedifferentperspectivesaccordingtothe'Rashomoneffect'cangiveusdifferentequationsandmentalrepresentationsaccordingtohowmuch

stillnessandmotion,andhowmuchdifferenceonsize/speedhappensbetweenobserverandobservable,withalimitgivenbyafullperpendicularangleofparallelismof0º,whichwillalwaysbelessthanarightangle.

Yetastheangleisa'curved'hyperbola,wecanalsoconsideritasanexponentialfunction,whereaisx-coordinatesandABthey-coordinates.Thentheminimalangleofparallelismwillhappenforthefastestgrowing

exponentialfunction,whichiseˆ‾×,theconstantofdeath=decayprocesseswhenjumping'2planesofexistence':∆+1<<∆-2;andhencetheabsolutelimitofahyperbolicgeometry,now'vitalized'intermsofthe

time=motioneventsofanorganicsystem.

Indeed,the4thi-logicpostulateofnon-Euclideangeometrycomesimmediatelytoourmindtomakesenseofthevitalenergy,'enclosed'bytheDarwiniansingularitymembrainthatpreysonit:

Inthegraphwemakeuseofthei-logic4thand5thNon-Apostulatetotranslateintotheorganicparadigmthemeaningofhyperbolicgeometry.

Now,howexactisthesymmetrybetweenthisvitalized,temporalmovingviewofhyperbolicgeometryandLobachevski'sformalstillgeometry?

Absolute.Indeed,thesurprisecomeswhenwerealizeofthenextfindingofLobachevski'soriginalwork:thelineheconsideredparallelto'a'infigure3,whenhemadeacloseformalanalysisDIDbecomeahyperbolaatthe

pointofinfinity.

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ItisworthtodoamorerigorousanalysisonhowLobachevskifoundthissurprisingresultusingmerelogic,stillformalproofs,toshowindeedhowallspatialviewshaveasymmetrictemporalview,whichwillbethe

foundationsofnon-Algebraandits∞S≈Tsymmetries.

Convergenceofparallellines;theequidistantcurve.

LetustheninvestigatehowthedistancefromaofapointXoncchangeswhenXisshiftedalongc(fig.5).

InEuclideangeometrythedistancebetweenparallellinesisconstant.ButherewecanconvinceourselvesthatwhenXmovestotheright,itsdistancefroma(i.e.,thelengthoftheperpendicularXY)decreases.

WedroptheperpendicularA1B1fromapointA1toa.FromB1wedroptheperpendicularB1A2toc(A2liestotherightofA1,sinceγisanacuteangle).FinallywedroptheperpendicularA2B2fromA2toa.Letusshowthat

A2B2islessthanA1B1.Thetheoremthattheperpendicularisshorterthanaslantlineisvalidinhyperbolicgeometry,becauseitsproof(whichcanbefoundineveryschoolbookongeometry)doesnotdependontheconceptofparallel

linesnorondeductionsconnectedwiththem.Nowsincetheperpendicularisshorterthanaslantline,B1A2asaperpendiculartocisshorterthanA1B1,andsimilarlyA2B2asaperpendiculartoaisshorter

thanB1A2.ThereforeA2B2isshorterthanA1B1.WhenwenowdroptheperpendicularB2A3tocfromB2andrepeat

thesearguments,weseethatA3B3isshorterthanA2B2.Continuingthisconstructionweobtainasequenceofshorterandshorterperpendiculars;i.e.,thedistancesofA1,A2,···fromadecrease.Furthermore,by

supplementingoursimpleargumentwecouldprovethat,generally,ifapointX″oncliestotherightofXʹ,thentheperpendicularX″Y″isshorterthanXʹYʹ.Weshallnotdwellonthispoint.Theprecedingarguments,wetrust,

makethesubstanceofthemattersufficientlyclearandarigorousproofisnotoneofourtasks.Butitisremarkablethat,ascanbeproved,thedistanceXYnotonlydecreaseswhenXmovesonctotheright,

butactuallytendstozeroasXtendstoinfinity.Thatis,theparallellinesaandcconvergeasymptotically!Moreover,itcanbeprovedthatintheoppositedirectionthedistancebetweenthemnotonlyincreasesbut

tendstoinfinity,henceformingindeedanexponentialfunction,whose'strength'willdependofthe'distance'in∆-scalesandhencedifferentin'speed'oftimebetweenboth.

Itisthusclearthatthedistancebetweenthepointandthelineisamentalformalrepresentationofthedistancebetweenthelargerplaneofthemembrainsingularitythatenclosesthevitalenergyofmicro-pointsinwhichitpreys,provokingitsentropicdecay.HencethefurthertheST-MICRO-pointfromtheline-membrainthatenclosesitinhyperbolicgeometry,thefurtherthedistancein∆-scalesbetweenbothandthesmallertheangleofparallelism,meaninginvitaltermsthemoreperpendicular=Darwinianwillbetherelationshipbetween

themicro-pointandthelargerobserver.

Themagnitudeoftheangleofparallelism.

Weshallnowstudytheangleofparallelism,i.e.,theangleγthatthelinecparalleltoagivenlineaformswiththeperpendicularCA(figure6).Letusshowthatthisangleissmaller,thefurtherCisfroma.Forthispurposewebeginbyprovingthefollowing.Iftwolinesbandbʹformequalanglesα,αʹwithasecant

BBʹ,thentheyhaveacommonperpendicular(figure7).

FortheproofwedrawthroughthemidpointOofBBʹthelineCCʹperpendiculartoB.WeobtaintwotrianglesOBCandOBʹCʹ.TheirsidesOBandOBʹareequalbyconstruction.TheanglesatthecommonvertexOareequalasverticallyopposite.Theangleα″isequaltoαʹsincetheyarealsoverticallyopposite.Butαʹisequalto

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αbyassumption.Thereforeαisequaltoα″.Thus,inourtrianglesOBCandOBʹCʹthesidesOBandOBʹandtheiradjacentanglesareequal.Butthen,byawell-knowntheorem,thetrianglesareequal,inparticulartheiranglesatCandCʹ.ButtheangleatCisarightangle,sincethelineCCʹisbyconstructionperpendiculartob.Therefore

theangleatCʹisalsoarightangle;i.e.,CCʹisalsoperpendiculartobʹ.Thus,thesegmentCCʹisacommonperpendiculartobothbandbʹ.Thisprovestheexistenceofacommonperpendicular.

Nowletusprovethattheangleofparallelismdecreaseswithincreasingdistancefromtheline.Thatis,ifthepointCʹliesfurtherfromathanC,then,asinfigure6,theparallelcʹpassingthroughCʹformswiththe

perpendicularCʹAasmalleranglethantheparallelcpassingthroughC.FortheproofwedrawthroughCʹalinec″underthesameangletoCʹAastheparallelc.Thenthelinescandc″formequalangleswithCCʹ.Therefore,aswehavejustshown,theyhaveacommonperpendicularBBʹ.ThenwecandrawthroughBʹalinec″ʹparalleltocandformingwiththeperpendicularananglelessthanarightangle,sinceweknowalreadythataparallelformswiththeperpendicularananglelessthanarightangle.NowwechooseanarbitrarypointMintheanglebetweencʹandc″ʹanddrawthelineCʹM.Itliesintheanglebetweenc″andc″ʹandcannotintersectcʹ.Afortiori,itcannotintersectc.ButitformswithACʹasmalleranglethancʹdoes,i.e.,smallerthanγ.Then,afortiori,theparallelcʹformsanevensmallerangle,becauseitistheextremeoneofallthelinespassingthroughCʹandnotintersectinga.ThereforecʹformswithCʹAananglelessthancdoesandthismeansthattheangleofparallelismdecreasesontransitiontoafartherpointCʹ;thisiswhatwe

setouttoprove.Wehaveshown,then,thattheangleofparallelismdecreasesforincreasingdistanceofCfroma.Evenmorecanbeshown:IfthepointCrecedestoinfinity,thenthisangletendstozero.Thatis,forasufficientlylargedistance

fromthelineaparalleltoitformswiththeperpendiculartoitanarbitrarilysmallangle.

Theproofshowsthebeautyofthesymmetrybetween§@-mindsand∆timemotions:thekaleidoscopicUniverseputsinsymmetricrelationshipallits'dimensions'withitsownmethodsandperspectives,creating

parallelworlds.

Ifatapointveryfarfromathelineperpendiculartoaistiltedbyaverysmallangle,the“tilted”linewillnolongerintersecta.Hencebeyondthe2-planedistancethelinea-whichrepresentsthe∆+1scalebeing-will

NOTperceive,preyorinteractwiththemicro-pointthatbecomesa'darkspace-time'forit.

TwolinesinaLobačevskiĭplaneeitherintersectortheyareparallelinthesenseofLobačevskiĭ,andthentheyconvergeasymptoticallyontheonesideandontheothertheydivergeinfinitely,orelsetheyhaveacommonperpendicularanddivergeinfinitelyonbothsidesofit.Thevitalorganicinterpretationsofthosefactsshows

hyperbolicgeometrytobearepresentationof∆±iscales,anditsorganicstructurebetween'cellular,unconnected,potentialmicro-pointsofavitalenergy'asperceivedbythesingularitymembrainthatenclosesit.

RECAP

GEOMETRYISbrokenin3sections:@mind:spacesdedicatedtostudythedifferentmindconstructionsoftheUniverse

T-opology:wherespaceformhasmotion

∆:non-Euclideanpostulatesofpointswithform,whichbecomeslinesthatevolveintoorganicpleas.

S:Bidimensional,staticplanegeometry,thefirstformofMathematics,inventedbytheGreeks,whichwewilltreatinthispost.

BecausetheUniverseisbidimensional,holographic,whatmattersonitsmathematicaloriginsistounderstandthattheGreeksanditsplanegeometrydoesmatteraseachofthose'theorems',whichwestudiedinhigh

schooldohave'hiddendeepmeanings'thatwillresurfaceonceandagain,intothevitalgeometriesofpointswithpartsthatcreatetheuniverse.

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Asusualasallisternaryandaternaryvisionisforthemindmirrormorepleasingweshallalsoconsiderinternaryagestheevolutionofbidimensionalgeometry,whichwentthrough:

Afirstyoung,Greekageofstaticbidimensionalspace-geometry

A2ndmatureageofmaximalreproduction,duringthetimeofmathematicalphysicsasitsetthestagefortheevolutionofphysicsandtheunderstandingofmechanicsandgravitational,NewtonianandKeplerianUniverses.

ThethirdagestartedinthisblogwiththeunderstandingoftheholographicUniverse,whichwillexpandthedisciplinetoalogic¬Ælgebraicrealmtofueltheapplicationofits∆STlawstoallotherdisciplines.

Soweshallcloseherethe'seed'ofinformationforfutureresearcherstoexpandandpassingthroughthe2ndageofgeometry,whenanalyticgeometry,marriedwith@-p.o.vs.tocreatethefirstsolidSTrepresentations

and∆-scaling(Cartesiangeometry).Hencestudiedinthepostofanalyticgeometry.

Theexhaustionmethod,whichdoconvertasumoftrianglesor'angularmomentums'(inthedualityinformation-motion)andforesees5Danalysis

AndtheGreekunderstandingofthecircleastheperfectform,andallthetheoremsextractedfromit-themostdevelopedinflationarymirrorinitslast'excessive'ageofform.

Lobachevskiaspredecessorof5Dgeometry

Butgeometrytrulyreachedamaturityasascienceof'reality',whenitincorporatedmotion;timedimensionstoform;withnon-Euclideangeometriesandtopology.Themastersofthissciencewerewithouttheslightestdoubt,asusualatriad,Gauss,LobachevskiandRiemann.Thelessrecognizedandmoreprofoundbeing

Lobachevski,whofoundtheprinciplesof'pangeometry',theabsolutegeometryofrealitybasedin3insights:

Therealizationthat'mathematics-geometry'isamental-logicendeavor,wherefunctionandi-logicthoughtovercomes'spatialrepresentation',thusheextractedaswedoin∆stlogicpostulatesWITHOUTpossible

expressioninthe'parabolic'@-geometryoftheHumanEuclidean'light-dimensionalmind',toextractpurelogicresults,showingthatthecausal,sequentiallogicoftimeistheessenceofreality.ByfarthiscanbeconsideredthehighestinsightintheworldofmathematicssinceDescartes'analyticgeometryandLeibniz'sfoundationofAnalysis-andshouldguideusinourinquireoffundamentallawsof∆ST,aswhatmattersinmathematicsisthe

reflectionoffunctionsandsymmetriesoverforms.Soassystemsbecomemorecomplex,theoriginalgeometricalpropertiesbecomelostandsubstitutedbythefunctionofthephysiologicalnetworksofthe

system;whichalsohelpstounderstandwhyintopologyformsmightseemverydifferentbutastheykeeptheessentialpropertiesofthebeing,theydokeeptheirfunctions.WithoutthisrealizationtheXXc.explosionof

abstractmentalspacestorepresentrealitywouldn’tbepossible.

Furtheron,heunderstoodrelationalspace-timeinspaceisdefinedinafirstincursionintopologybytheconceptof'adjacency',whichcompletedthe3fundamental'modes'ofrelationshipthroughgeometricalspace

oft.œs-complementaryadjacency,perpendicularDarwinismandparallelsocialevolution-henceaconceptessentialtotheorganicstructureoftheAbsolutelyrelativeUniversedefiningforthefirst

timetopologicaltransformationsarethoseinwhichmotiondoesNOTdeformthefundamentalpropertiesofrealityinspace,startingatrendculminatedbyHilbert'sfoundationsofgeometry(yestheguywecriticizeso

much-hedidalsodosomeworkofmerit:),withhisemphasisonsomekeyabstractconceptssuchasbetweenness,congruence,continuity,incidence,separateness...whichareclearlyrelativeconceptsconcerningscaleandsymmetry,themindelementsthatallowasingularityorpointofviewto'construct'awor(l)d-view

overand'stiffen'themotionsofrealitytomakeamentalmappingofthem.

Heinsistedstronglyintheexperimentalnatureofmaths,wonderingwhichwastherealgeometryoftheUniverse,andmadethefirstinroadonthedifferenceof'mind-spaces'accordingtoscale,asitdependsonthesizeofourperspectivethatwefinda'flat'geometry(detailedview)ora'curvedgeometry'(farawayview

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wherethewholeworldcyclethatseemsalineinshortdistance/timespanbecomesawholeclosedzero-sumworldcycleofenergy).

Those3findingsareessentialandweshalldwellonthem.Regardinghisinconclusiveresultsonthegeometryofreality,whatmathematiciansthoughmissisthe'Rashomoneffect',giventheirone-dimensionalhumind

thought,wonderingwhatisthespaceoftheUniverseofthetriadofelliptic,ð§(spherical,Riemanniansurfaceofthespace-timesuperorganism),@(Cartesiananalytic,mindgeometry,withthemindasitsfocus)

orhyperbolic,ST(Lobachevski’sgeometry)orparabolic,∆-Euclidean.

Thisfundamentalequivalencebetweenthe3±∆geometriesandthe3±∆partsofthetime§paceSupœrganismisthefundamentalcorrespondenceofspaceandsoinsteadofnamingitbythehumindegothatdiscoveredthem(Riemann,Lobachevski,Descartes,Euclid)weshallusethe

olderterminologybeforetheselfieagebecauseofitsdescriptivepower,again:

-Themembrain(singularityandmembrane)hasanelliptic,ð§geometry,henceitisusedinGeneralrelativitytodescribethe'gravitationalenclosure'or'curvature'ofthe∆+1

gravitationalscale(Einstein'srelativity).ButellipticGeometryismuchmoreprofoundthanusuallythoughtintheestablishmentofthepropertiesofanysystemofreality,andsoaswe

havenottreateditelsewhereisworthtoconsideritsrole:

Inthegraph,inellipticgeometrywedefineapointasatwonodalpointsofaspherewithmaximaldistancebetweenthem,whichimpliestheyallpassthroughthe0-pointorsingularity,andestablishthenon-existenceof

parallels.

Assuchellipticgeometryhasnoparallels,becauseallits'parts'areconnected,bytheformalcenter,o,whichunlikeintheclassicformulationofellipticgeometryin∆ºs≈tmustbeconsideredalsothe'invisiblepart'ofthenodalpoint;andsoellipticgeometrydescribesthe@-structureofasingularitypointconnectedtoamembrain,

forminganabsoluteenclosure.

AndultimatelyasALLpointsareinfact'twostrong'points,twopoles,whichareequivalent,itestablishesafundamentalpropertyofNature,thebilateralsymmetrywithinversepropertiesself-centeredinabalanced

symmetric'identity'elementthatcommunicatesthemallastheyareaallconnectedtoallotherlines/circlesandthroughitsaxistothesingularity,whichisthereforenotonlythecentralpointbuttheaxisof...

-Themindsingularity,whichactsthereforeasthefocus,anditisan@-selfcenteredgeometry,whichallowsCartesianplanestobe'perspectives'fromafocus,thezeropointanditsinformativeheightdimensionandotheraxisofthesystem-thereproductive-widthdimensionandthelength-motiondimensions.Wecan

considerintheidealizedstructureofbaremathematics,the3physiologicalnetworksofthebeing.Andsothebeingswitchesoffbetweenits3axis/networksasitsfunctionschange.

Furtheronthemindisconnectedwitheverypointoftheentity,butforeachpointthereisonlyoneconnection-onlyaline-parallelcanbetraced.

Andfinally,asweshallshowsooninthegraphsofhumansystems,sincespaceisamental-singularityrelatedfunctiontoprocessinformationinanefficientmanner,andrecreateorder,themathematicalsimplestmost

efficientgeometryoftheball-ellipticformmustnotbeconserved.

Whatmattershereisthesymmetricbipolarity,whichallowthesingularitytomaximizetheextensionofitsvitalspace-enclosedbythemembrane,soweshallseehowincomplexorganicsystemsthespheresuffersallkindoftopologicaltransformationsintoallkindofshapesbutallofthemare'enclosed'forthemindtore-formthevitalspacewithin,andallhaveasingularitybrain-systemtoconnectthem,andallhavebilateralsymmetry

(eventhespherewhichinprincipleisnotdefinedassuchinclassicmaths-onlyconsideredtohaverotational

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symmetry,exceptintheellipticgeometrythatdefinedantipodalpoints),becausethesingularityco-ordinatesallthosepointsandusesitsinversepropertiestoextractmotionfromthevitalenergywithinit.

-Theintermediatevitalspace-timeenclosedbetweenbothhasahyperbolicgeometry,thedominantintheUniverse,becauseitisthepresentstate.Itdoeshavea'saddle'dualcurvature,becauseitcommunicatesthetwootherinversepolesofthebeing.Soifinthesurfaceofthesphere,curvatureisalwayspositive,andinthe

centralpointandaxis,curvatureisalwaysnegative,thehyperbolicintermediatespace-timehasbothcurvatures.

Theternaryformsofspatialrelationship:4thpostulate.

Inthatregard,innon-Egeometry,wemustdistinguishasusuallya'ternary'typeofspatialrelationshipswithdeepmeaningsinthevitalorganicstructureofreality:

Adjacency(formsthatarepegged,henceformingpartofthesametime§pacesupœrganism).

Perpendicularity,(formsthatpenetrateanddisruptitsinnersystems,basisofDarwinianevents.)

Parallelism(thingsthatmaintainitsdistanceandallowcommunicationthroughacommonmediumornetwork,basisofsocialevolution-studiedinaffinegeometry.)

Innon-ægeometrytheywillbeextensivelystudiedasthefundamentalmodesthatdefinetherelationshipsofST,complementarityand'symbiosis'(adjacency),Darwinianstruggle(perpendicularity)and∆§ocialevolution(parallelism)ofallsystems,becomingtheessentialqualitiestounderstandhowspatialrelationshipsdefinetemporaleventsamongallsystemsandscalesofnature,studiedbythefourthpostulateof'congruenceand

similarity'.

Itisthenessentialtounderstandtheultimatemeaningofparallelismvs.incidence/perpendicularityalsoasmentaldescriptionsoftwologicstates-oneofparallelsocialevolutionandoneofDarwiniancolliding'tearing'bytheperpendicular,incidentline,takingtheconceptoutofitsspatialrepresentation,asLobachevski's'first

greatinsight'didforalloffuturefindingsofmathematicalspace.

Whatmakesthisgeometrysoimportantis,onceweliberatethepostulateofparallelismfromitsphysicalrepresentationbacktowhereitbelongsintomentalspace,thefactthatitallowsittotravelthroughscales,unliketheellipticgeometrythatconstructsasysteminasingleplane,henceitisthegeometryof∆-scales,whichcoupledwiththe@nalyticrepresentationbyamindconvertsitintothebestrepresentationsofthe∞

variationsoftheorganic,scalarUniverse:

InEuclideangeometry,afigurecanbescaleduporscaleddownindefinitely,andtheresultingfiguresaresimilar,i.e.,theyhavethesameanglesandthesameinternalproportions.

Inellipticgeometrythisisnotthecase.Forexample,inthesphericalmodelwecanseethatthedistancebetweenanytwopointsmustbestrictlylessthanhalfthecircumferenceofthesphere(becauseantipodal

pointsareidentifiedasthemaximalbilateraldistance).Alinesegmentthereforecannotbescaledupindefinitely.

Ageometermeasuringthegeometricalpropertiesofthespaceheorsheinhabitscandetect,viameasurements,thatthereisacertaindistancescalethatisapropertyofthespace.Andsowefindarecurrentthemeof∆st:allisinitsultimate'largest'viewaclosedcircle(definitionofalineasacircleinellipticgeometry).Yetonscalesmuchsmallerthanthisone,thespaceisapproximatelyflat,geometryisapproximatelyEuclidean,

andfigurescanbescaledupanddownwhileremainingapproximatelysimilar;asyouseetheEarthflatinsmallerscales.Hyperbolicgeometry,thatoftheenergypresentvitalspace,issomewhatanintermediate

'region'inwhichscalingispossiblebutlimitedbyconceptssuchastheangleofperpendicularity.

Whataboutcolors?Obviouslytheyarethekey,astheyarecodedbyfrequency,whichisthetranslatorofscales.Whatthismeansultimatelyisthatlight's'frequency-colors'fundamentalroleistotransmitinformation

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notonlyinasingleplanebutspeciallybetween∆-scalesasthetelescope/microscopediscoveryfoundout.Sothe3dimensionsoflightspace-timearepresentelementsofthesuperorganismoflightanditssocialcolors

theevolutionaryelement.

Lobachevski’spangeometry.Whereisthe5thdimension.¬æ

Thusalgebra(andanalysisconcernedwiththeprocessesofsocialnumbersthataddandemergeorsubtract,anddivideplungingdownthescalesofeusocialloveofthe5thdimension),isalargersubject,stillnotfully

developedbytheonlyhumanworld-point,whichasBoylaiontheviewofnon-Espaces,canonlyexclaim‘Ihavediscovered(notinvented,ashesaid,theeverarroganthumanego)anewstrangeworld–andnotoutof

nothingashesaid,butoutofeverything’).

TheUniverseisnotacontinuum,butasallfractalsitisdiscontinuous.Thisofcourse,the‘axiomaticHilbert-like’arroganthumansdonotlike.SoaguycalledDedekindfoundacontinuityaxiom,affirmingthattheholes

betweenthepointsofalinearefilledbyrealnumbers,whichareratiosbetweenquantitiessuchasπor√2,whichhappenNOTtoexistasexactnumbers,andmoreoverrepresentaninfinitenumbercomparedtothose

whichdoexist.

Furtheron,whenXXc.geometerswentfurtherthanNon-EuclideanRiemanniangeometriesintoabsolutegeometriesitturnsoutthatthemostabsoluteofallgeometries,didn’tneedthecontinuitypostulate.

Thisgeometry,whichistheultimateabsoluteplanegeometrythatincludedallothers(andnowfurtherclarifiedby5Di-logicgeometry),reflectedtheabsolutearchitectureoftheplanesofExistenceoftheUniverse.A

GermanadequatelynamedBach-mann,foritsmusicalarchitectonicalrigor,discoveredit.

ItistheGoldbergvariationsofthetheme.AnditwasdiscoveredtheyearthechipHomoctonoswasfound,endingallevolutionofhumanthought,whichnowisbusy-busytranslatingitselftothenewspecies,withevermorepowerfulmetal-mindsandsmallerhumanminds,recedingintoahyperbolicstateofstasis,thinkingwhat

themachinesthataremakingthemsavantidiotsdiscoverbelongstotheirego-tripparadox.

Intermsofgeometryismerelythe‘realization’ofthe3canonicalgeometries,wehaveusedtodefineasysteminspace,perceivedfromagivenpointofviewacrossthescalesofsizeoftheUniverse,takingintoaccountthat

our‘rod’ofmeasureislightspeed-space.

Weseerealitythroughlight’s3Euclideandimensionsandcolors,whichentanglethestopmeasuresofelectrons.

Yetlight-spaceandanyrelativesizeofspaceoftheUniversemustbeanalyzedwiththepan-geometryofthe5thdimension,firstexplainedbyLobachevski,asweseesmallerbeingswithahyperbolicgeometry,whichmultipliesits‘fractalforms’,andlargeroneswithanellipticgeometrywhichconvergesthemintosingle,sphericalones.Hencethehyperbolicgeometryofquantumplanes,theellipticgeometryofgravitational

galaxies,andthemiddleEuclideangeometryoflightspace-time,inwhichtheLobachevski’sconstantoftimeandspaceisminimal,sinceourquantaofinformationH-Planckisminimalcomparedtoourquantaofspace-

lightspeed.

Letuselaborateonthisideawithmore‘mathematicaldepth’asitisessentialtocompleteouranalysisofthehumind.

superorganismsotheyare'constrained'intoazero-sumorlimitingmembraneandappearascurvedgeometries'.5D,Long/lastingmeasurescompleteazerosumworldcycleandanfullyenclosedsuperorganismso

theyarecurved'.

Asmostofallmoderngeometryisbasedinthisduality,oneofour3fundamentaldualitiesoftheGalileanParadox,itseemsobviousthat∆@s=twillalsobeabletoexplainallthefoundationsofmoderngeometryand

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byextensionasDisomorphicdimensionalgeometryisthefoundationofallothermathematicalsubdisciplinesofallofthementalspacesofmathematicalsciences.

OtherrepresentationsofHyperbolicgeometry.Klein's'openball'withmotion.

Towhichextentwhatwehavedevelopedofhyperbolicgeometryintermsofplanesofthe4th-5thdimensionwithinthetime§paœrganismcanbeconsideredexact,canberevisedbystudyingthehands-onmainmodels

thatcameout,mostlybyBelgrami(despitehavingthenameofthesacredcowsofnorthernEuropeanscience-weperipherals,LatinosandRussians,youknowcannotbegeniusesofscience,nevermindGalileo,theGreeks,Mendeleyev,Lobachevski.SotheBelgrami'scone,theBelgrami'ssphereandtheBelgrami'sdisk,whichshowmoreclearlythatindeedhyperbolicgeometryISthegeometryofthevitalopenballspaceenclosedbythe

membrain(whenstudyingitstrictlywithinasingleplane),havethisother'people's'name.Inthenextgraphweseearepresentationofitsmainelements,thesingularity,diskandsphereunderhyperbolicgeometry:

Inthegraphwecanseethetwomodels(extendedtoa3-Dsphere),ofhyperbolicgeometry,showingclearlythatthevitalenergyenclosedbythemembraincanneitherreachthecentralconeortheB-C-Dmembranethat

encirclesit,whichoffersaconstantresistancetoitsadvance.

ThoselimitsareexactlythesameforthegalaxyintermsofT=0ktemperature(blackholesingularity)andc-speed'membrain/eventhorizon',whichcannotbereachesastheyofferaconstantresistance.Sohyperbolic

geometryistheidealgeometrytorepresenttheatomic/stargalacticspacebetweenthehaloandtheblackholesingularityinthecenterofthegalaxy:

WeshallnotextendfurtherintothemainoftheNon-Euclideangeometries,asthenumberofmentalspacestriggeredbythe

'freeing'ofthemind-spacesofmankindanditsformallanguagesgrewalsoexponentiallyafterLobachevski'stransformationofgeometryintoalogic,mentalscience.Soweshalldealwithall

thosespacesintermsof∆•s≈thigherlawsofspace-timetopologies,inversions,scalesandsymmetries.

Inthegraphaphysicalunderstandingintermsofspecialrelativityanditshyperbolicgeometry,wherewedissectthedifferent'elliptic-membrain'+hyperbolicvitalenergygeometryoftheUniverse,whichis

theessenceofhyperbolicspecialrelativityconcernedwithlight/electromagneticforcesvs.theellipticgravitationalmembrain(haloofdarkmatter+centralblackhole).

ThegraphshowsthehyperbolicbehaviorofourEuclideanspaceasitmovestothenon-transitbarriersofthecentral'Beltramisingularitycone'(leftupperpicture,rightlowerpicture)andexternal,'Klein'hyperbolicdisk

(leftupperpicturewherespace-timemotionsneverreachthelimit,downrightpicture,beingthatphysicallimitthec-speedbarrier),whichshowstheessentialstructureofthe3partsofthebeingintermsofitsgeometry.

Furtheron,themembrainISanellipticgeometryofantipodalpointsthat'compress'andcontroltheinnerregionsofthebeing,henceusedasthecontainerofthevitalenergybyitsblackholesingularityandhaloof

strangelet,connectedthroughgravitationalwavesanddarkentropy.

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Soinitsellipticgeometry,the0-singularitypointtightensupwithitsattractiveforcetheantipodalpointsoftheexternalmembrain,creatinginthismannerthe'force'membrainofgravitation.

Thusthegalaxyisthe'vitalenergy'-starsthatshallbecomeblackholesorstrangelethalo.

Sothe3geometriesoftopologicalspace-timewithmoreorlessdegreeofdimensionalcomplexitywillalwayscorrespondtothe3timespacearrows/dimensions/events/forms.

Asweshouldknowbynow,thesymmetriesandinversionsbetweenthesuperorganism'spartsinspacecorrespondtoasimilarsymmetrybetweenscales.Soaswehavedefined3basicgeometries,wecanalsoconsiderthemintimeview,inspaceview,inscaleview(bothintheentropicarrowandinthemind's

deformationofaself-centeredbiasedpoint)-thenwehavethefullRashomoneffecttogetthefinal5Dmind/judgmentconclusionofwhattrulywearewatching,extractingallitsinformation.Thisof

coursewouldreorderalltheinfoofallstiencesbutasinglemancanonlygiveglimpsestotheRashomon

effectofafewsubjects.

Letusthengivethefinal'judge'-viewoftheMind,ofahyperbolic'disk',whichwillbethesensorialmembrainfromwherethe'relativescaleorsize'ofthehyperbolic

planewillbejudged.

Theunderstandingofhyperbolicgeometryfromthe@-mindvs.theviewfromtheoutermembrane.

Inthegraph,hyperbolicgeometryandanyspatialmentalformasarulerequiresabitof'endophysics'andobserver'sparadoxestofullyunderstandrealitywithoutthemindbias.

InthePoincaredisk(andthePoincareline),theshrinkingofthepointsisacceptedtobendasweperceiveitfromthelargerview,thefractalelementsofthevitalenergyinside.

Ifweconsiderthenthe@-viewtobethatoftheexternalmembrane,the'largest'POSSIBLEview,(asinyourorganism,wherethemindisjustabunchofmicroscopiccellsbutholdstheviewofthelargerwhole-scaleof

yourbody),itisnaturalthattheinner∆-1elementsareperceived'smallerinspace',astheycometothelargerwhole.

Itisalsointerestingtoconsiderthetopologicaldualityofthatmembranewhich'dissects'inwordsofLobachevski,spaceintoinnerandouterregions(firsttopologicalpostulate),creatingtwocompletelydifferentvisionsofreality,astheinternalbeingwillseeaconcaveenclosure,aforbiddingbarrierandnothingbeyond.Whilecrossingthatbarrier,weperceiveamuchlargerconvex,openUniverse.Howthistransformsourmentalviewofspacecanbenowrespondedconsideringthe'ratio'r/l,whichmustbeunderstoodfromthemental

pointofviewasaratiobetweenthe'radius'ofthetime§paœsystem,whichthe'mind'perceivesandmeasure,witha'length'associatedtoitsownpotential-limbsizes.Forexample,humanshavealimb-stepoflinealmotion(1D)of1meter.Sowhenobservingentitiesofmaximalsize,itwillperceiveitsperimeterlargerthanaperfectcircleorsphere,increasingly'elliptical'and'flat'.ForthatreasonweseetheEarthflat,astheradiusofthe

planetishugeandour'scaleofmeasure',amilliontimessmaller.Butifwegrow,wewouldincreasinglyseetheEARTHspherical.

ThusmindsareindeedCartesiandevilscrafteddifferentlyaccordingtosize(whicharedefinedbytheparametersofperception,suchasthesubstanceweperceive,thesmallishpixelsoflight,thelargeratomsofsmelling,anditsorgans,individual,multipleeyes,etc.)Wedealthenwiththoseelementsinthepostsonmind

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worlds,wheresomesurprisingresultsappearonhowinsects,atomsorblackholeswouldperceiveifasallseemstoindicateprocessdifferent'sizes'ofpixelsandlinealrodsofmeasure.

Inthatsensetheruleofallmindsshouldholdandamounttothis:smallerbeingsseenaflatworld,largeronescurveit,andthelargerthebeingisinrelationshiptotheworlditobserves,themorecurveditsmindwillbetilltheabsolutemind-spaceoftheUniverse,thatofT.Œ,whichyoumightcalltheTaoist,impersonalGod,thegameofexistence,whichobservesAPUREBLOCKOFTIMEwithallthepotential

symmetriesrealized,allthesmallstepsconvertedinlargercyclicalwholes,allazerosum,allaNirvanastate,inwhichIdwellnowforquitesometime,inwhichnothing

surprisesyou,thefuture,thepastandthepresentaseparatedillusion,asthei-logicstructureofthefractaldisplaysaperfectorder.

Inthegraphweseeanexampleofthose'thoughts.Theearthmightseemaflat,stillform,butfromalargerslowertimerhythmitwillseemacycle,fixedinformastheSaturnringsseemtous.Thecontemplationofall

thepotential(inanAristoteliansensenottoconfusewithaphysicalpotential,∆-1field)beingsthattherewere,areandwillbewithinthelimitedvariationsofreality,wherechaosisonlytheignoranceofthoselaws,isthustheultimatementalstate,whereallbecomesspace,aParmenideswhole,withnomotion,onlyreproductionofdéjàvuinformation,asthepossiblevariationsofthegameofexistencemadeofsolimitednumberofelements,

hasbeenwritteneternaltimes;andsoalsoeachofushasbeenrepeatedadinfinituminothermomentsoftimespace...

InternalLinealfreedomvs.externalcyclicalorderanditsreflectioninmathematicalstructures.

Anessentialconcepttounderstandtheparadoxicalmodesofgenerationofspace-timebeingsisthedualitybetweentheinternalmind,whichperformsthelinealseeminglyfreestepsofitsDimotionsas'finitesimal

tangentialactions',derivativesofwhatwillbecomeitscyclicalcurvedexternalorderimposeduponitbythelargerworldcycle.ThisDualitybetweenthesmallerstepsoflinealapproximationtothelargercyclethatwill

encloseandsummonupthemallisthejustificationofallthephilosophyofmathematicsofDifferentialgeometryandderivativecalculus.Acurveforexampleisapproachedindifferentialgeometrybyalineal

tangent,orbyaplane,paralleltothepolidimensionalcurve-butthewholeimposedexternallyisthecurve,andthestepsproposedinternallyarelinealsteps

SoparaphrasingWheeler,wecouldsaythattheegoisfreetoperforminstantaneouslinealsteps,choosingthedirectionofitsmotion,butthelargerwholewillimposeitscurvedpathsof'leasttime'.Inphysicsthemasswilltrytomoveinagivendirectionbutitwillbecurvedbytheouterspace-timegeodesic;themindwillmakeplansfromitsfinitesimalsubjectivepointofviewbuttheorganismwillimposeitsboundaries...Andofthattug-of-warbetweentheindividualstepsoffreedomofthe'fractalpoint'andthelargerfunctionsrealityhappens.Einsteinsaid'timebendsthespace(ofthemind)'...Indeedaswekeeptryingtomaintainourlinealwill,the

environmentbendsusandifwedon't,wecrash...

Colorspace,definingthevitalgeometricpropertiesandRiemann'sgeneralization.

Wecannowwithallthis'ƥs=t'considerationsontheternarycodesofcolorsstudyitasgeometersdidtogeneralizetheconceptsaforementionedintheprecedingsectionontherealmeaningofn-dimensionalspace,

tosolvetheproblemofgeneralizingthescopeofgeometryandtheconceptofspaceinmathematics.

Firstclarifythatany'geometricalconstruction'willdepartfromthe'elementsofgeometry'(enhancedinourNon-Edefinitions)suchas'T.Œntities'aresimplifiedinto'points';socialherdsofT.œsintolines,andits

'structuralsymmetriesandcoordinations'ontoternarynetworksdefinedbytheGeneratorformalismofnon-Æ(groupsinclassicalgebra).

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Thissaid,experienceshowsthatthenormalhumanvisionisthree-colored,i.e.,everychromaticperception,ofacolorC,isacombinationofthreefundamentalperceptions:redR,greenGandblueB,withspecificintensities.

Whenwedenotetheseintensitiesincertainunitsbyx,y,z,wecanwritedownthatC=xR+yG+zB.Justasapointcanbeshiftedinspaceupanddown,rightandleft,backandforth,soaperceptionofcolor,ofacolorC,canbechangedcontinuouslyinthreedirectionsbychangingitsconstituentpartsred,green,andblue.Byanalogywecansay,therefore,thatthesetofallpossiblecolorsisthe“three-dimensionalcolorspace.”The

intensitiesx,y,zplaytheroleofcoordinatesofapoint,ofacolorC.

Positivevs.Negativeorneutral

Animportantfirstdifferencethoughfromtheordinarycoordinates,originatedinlocomotionanalysis,wherewehaveinversetimespacedirections,consistsinthefactthatcolorintensitiescannotbenegative,asweareusing

herepureformalspace.Whenx=y=z=0,weobtainaperfectlyblackcolorcorrespondingtocompleteabsenceoflight-atheme,whichisessentialtounderstandWHYimaginarynumbersdoexistforcertain

dimensionalspacesbutNOTfromothers,whichwecanresumeinasimplestatement,called'horrorvacuum':

Negativevaluesexistonlyinternarycyclical'π'time§pacezerosumworldcycles,asitismerelytheinverse4D(∆-1)vs.5D(∆+1)arrowsofform,self-centeredinthe∆ºplane,whosesumgivesusazeroworldcyclethat

returnstoitscyclicalorigin.

Itdoesn’texistasreal(provokingmanyerrorsonscience)forpurespatialformperceptionas0isthevalueofemptiness,stillness,absoluteformandthereisthereforenotnegativetemperature(zero-stillmotionisthe

valueof0K)ornegativecolor(relatedtotemperatureascolorcarriesthefrequency-heatonthethermodynamicscale)andsoon.Intermsofdimotionsofexistenceanditsmathematicalrepresentation,itwillbeanimportantfacttounderstandmathematicalquantumphysicsinconceptssuchasSpin,Pauliexclusion

principle,antisymmetryandsoon:

'Parametersofpresentspacedimensionsareneutral,|x|;absolute,scalarpastandfutureparametersare±x'

Nextinourillustrativeanalysiscomestheconceptofcontinuityvs.discontinuityagainakeymentalspace-timeconcepthardlyunderstoodasthemindseekscontinuityofspace,andthenon-reflexivehumindscientistbothinmathematicsandphysicsacceptsitsasan'evidentdogma'ofitsnaiverealism,creatingsomanyhard-to

dieerrorsofthoughtandfalseproofs,whichapropers=tsymmetricanalysisdounderstand.

Continuityisamentaldevice.

Wehavestressedoftenthatcontinuityexistsinsubjectivementalspaces,andthisgivesusalotoffreeomregardingitsmostgeneralmeaning.I.e.InthecolorspaceRiemanndefinedcontinuityasacontinuouschangeofcolorrepresentedbyacontinuouslinein“colorspace”.Yetinrealitycolorisdiscontinuousformedbya

discretenumberofmindperceptionsofdiscretefrequenciesoflight,buttheelectronicmindonlyperceivesthepeaksofphotons-asthestopandgoprocessofacontinuousviewofafilm,wherewedoNOTperceivetheirrelevantstepsbetweenthosecolorswhichperceptionignores.Hencewecandefinementalcontinuity:

'Continuityisalwaysaproductofmind-space,whichinanylanguage'reduces'informationtofitinitsinfinitesimal,bydischargingallirrelevantorredundantinformation'.

Mindsreducedimensionstotherelevantones,eliminatingalldarkspaces:continuityIstheresult.

Ternaryemergence.

DualityofS-TcombinesintoS=tenergybeings,soweobtainthe'thirdstcolor'bymixingtwo'extreme'ones,andthiscanthenbeconsideredanintersectionof'lines'.

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Forexample,whentwocolorsaregiven,sayredRandwhiteW,thenbymixingtheminvaryingproportions*weobtainacontinuoussequenceofcolorsfromRtoWwhichwecancallthesegmentRW.Theconceptionthat

arosecolorliesbetweenredandwhitehasaclearmeaning.

Andsowecangodeeperinthescalar∆-1detailmakingemergingnewcolors,aswecangodeeperintotherealnumberlineseekingfornestednew'numbers'ofmore'decimal'scales.

Andthishappenspreciselybecauseofthescalarstructuremimickedbythe0-1≈1-∞symmetriesbetween∆-1and∆+1'scales'ofanalyticgeometry.

Yetthosedetailswillonlyexistifthemindcanperceive.Thatis,ifthespace-detailweretobe'matched'symmetricallybythemental-informativeperceptivecapacity.

Andthisperceptivecapacitywilldependonther(t)/k(s)'scalarfactor'ofinformativedensityofthemindaforementioned,soalargeviewerwillNOTseedetailandcannot'penetrate'thevirtualsub-ternarypartsof

thecolororanyothermentalspacespectra).

RECAP.ADEEPERsearchoftherealgeometryoftheUniversewillNEVERbecompletedwithouttheadditionof5thdimensionalmetricsandtheunderstandingofthe3differentmentalperspectivesanobserverinan∆¡planewillhaveofits‘flatgeometric’equalscale,‘elliptic,upperlargerscales’fromwhereitreceivesenergybuthardlyinformationandlowermoreinformative‘hyperbolicscales’ofpartsconnectedtoitsplaneby

branching,fractalnetworks:

Forthatreasonstilltoday,hyperbolicandellipticgeometriesthatdonotfollowthe5thnon-epostulateareillunderstoodandaswe

haveshownincomplete,sincenoteventherealconceptofafractalpointsiscanonicalinmathematicalsciences,evenafter

Lobachevskishowedinfiniteparallelcouldcrossthroughit.YettheUniverse’srealexternalgeometryisbasedintheinterplayofthose3scalargeometriesasperceivedbymindsindifferentscalesofreality.

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XVI.METRICSPACES:FROMRIEMANNTOHILBERT

Thusbecausethefifthdimensionwasignored,theexperimentalnatureofmathematicswouldnotbefoundandtheeffectofLobachevski’sdiscoverywillberatheritsopposite–tomakepeopleinanagedominatedbytheidealistegocenteredphilosophiesofGermanHegelianism,tothinkthatrealitydidnotmatterasitdidnotconformedtothe‘evident’singlelineal3DEuclideangeometryweperceiveinourplaneofspace.Itwillthen

bringtheoppositeviewofsubjectivementalspacesasthedominantreality,finallybringingtheHilbert/Cantorianparadoxes;yettheproperinterpretationbyRiemannthatgeometrystillmattered,iswhatmattertoushere;sincehisgeniusrealizedthatgeometricconceptswerelogicconceptsofthemind,from

‘distance’whichwasasynonymousofsimilaritytodimensionswhichweretobecome‘parameters’ofchange(henceultimatelydimotions)ofdifferentphasespacessuitabletoexpresslocalformsofchangeinreduced

partsofreality.

Letusthenconcludethisbriefintroductionto5DS-geometrywithconsiderationsonMetricspaces.

Riemann'sgeneralization.

ThebasicideasofRiemanniangeometryarereallyrathersimpleifonesetsasidethemathematicaldetailsandconcentratesonthebasicessentials.Suchanintrinsicsimplicityisafeatureofallgreatmodelsofreality,sincetheUniverseis'simplebutnotmalicious'-asEinstein,whoseideawasalsoverysimple-toequateaccelerationandgravitation-putit.Lobachevski'smodelwasalsosimple:toregardtheconsequencesofthenegationofthe

FifthPostulateasapossiblegeometry.Soitistheideaofthediscreteatomicstructureofmatter,asallcontinuouswholesareindetaildiscontinuous,'entropic'desegregated∆-1,closedforms...

Allofthemofcoursearegeneratedbythesimplestofallsimpleideas:S≈T.Onlybyiterationandvariationrealitybecomesverycomplicated.

Yetnewideasmust,firstofall,worktheirwayoverawidefieldandmustnotbepressedintoarigidframework,andsecond,theirfoundation,development,andapplicationisamany-sidedtask,requiringanimmenseamountoflaborandingenuity,andimpossiblewithoutthespecializedapparatusofscience-reasonwhy(Kuhn)they

takesolongtobeimposedamongpedanticscholars,whichwon'thaveittillithasreachedtheperfectionofoldoutdatedones-butwon'thelptorealizethatperfection,asthiswriterwellknows.

InRiemannian'sgeometrythisscientificapparatusconsistsinitscomplicated,cumbersomeformulas,duetotheobviousmultiplicationofdimensionalparameters.Butweshallnotdealwithcomplicatedformulasexcept

wheninthefuturewestudythemarriageofRiemannandEinstein'ssimpleideas.

Soassaid,Riemann'sessenceistoconsideranarbitrarycontinuouscollectionofphenomenaasamentalspaceasLobachevskiimplicitlydid,goingastepfurtherbyaddingthe∆nalysisofits'(in)finitesimalpoints'orminimalelementsinthediscontinuous∆-1scalethatareinthelargerviewa'continuousline'-whole.Sotimeminimalintervalsandspaceminimalquanta,anditsvariationsand∆-1scalar'differentialandintegralproperties'couldbeadded,besidesexpandingthenumberof'dimensionalpropertiestoits(in)finite(meaninginbothcasesthatallinfinitesimalshavealimitandallinfinitiesalsohavealimit-thatofthesizeofthelowerorupperpart/wholescales;soaninfinitesimalofnisnormally1/n,where1isthewhole;orinotherwords,theinfinitesimalmoves

the1-∞scaleintothe0-1infinitesimalscale).

Inthisspacethecoordinatesofpointsarequantitiesthatdeterminethecorrespondingphenomenonamongothers,asforexampletheintensitiesx,y,zthatdeterminethecolorC=xR+yG+zB.Iftherearensuchvalues,sayx1,x2,...,xn,thenwespeakofann-dimensionalspace.Inthisspacewemayconsiderlinesandintroducea

measurementoftheirlengthinsmall(infinitelysmall)steps,similartothemeasurementofthelengthofacurveinordinaryspace.

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Inordertomeasurelengthsininfinitelysmallsteps,itissufficienttogivearulethatdeterminesthedistanceofanygivenpointfromanotherinfinitelyneartoit.Thisruleofdetermining(measuring)distanceiscalleda

metric.ThesimplestcaseiswhenthisrulehappenstobethesameasinEuclideanspace.

YetasLobachevski'skeyformula,r/kshowssuchaspaceisEuclideanintheinfinitelysmall.

Inotherwords,thegeometricalrelationsofEuclideangeometryaresatisfiedinit,butonlyininfinitelysmalldomains;itismoreaccuratetosaythattheyaresatisfiedinanysufficientlysmalldomain,thoughnotexactly,butwithanaccuracythatisthegreater,thesmallerthedomain.AspaceinwhichdistanceismeasuredbysucharuleiscalledRiemannian;andthegeometryofsuchspacesisalsocalledRiemannian.ARiemannianspaceis,

therefore,aspacethatisEuclidean“intheinfinitelysmall.”ThesimplestexampleofaRiemannianspaceisanarbitrarysmoothsurfaceinitsintrinsicgeometry.The

intrinsicgeometryofasurfaceisaRiemanniangeometryoftwodimensions.Forintheneighborhoodofeachofitspointsasmoothsurfacediffersonlyalittlefromitstangentplane,andthisdifferenceisthesmaller,thesmallerthedomainofthesurfacethatweconsider.Thereforethegeometryinasmalldomainofthesurfacealsodifferslittlefromthegeometryinaplane;thesmallerthedomain,thesmallerthisdifference.However,in

largedomainsthegeometryofacurved,differentfromtheEuclidean,asintheexamplesofthesphereorpseudosphere.

RiemanniangeometryisTHUSanaturalgeneralizationoftheCONCEPTOFmentaldimensionalproperties,toanarbitrarynumbernandofnon-EuclideangeometriestotheƤcalesofthediscontinuousUniverse.Henceits

enormoussuccess,asitisgroundedintruepropertiesoftherealityof'dustofspace-time'-∆@s≈t.

Suchn-dimensionalRiemannianspace,althoughEuclideaninsmalldomains,maydifferfromtheEuclideaninlargedomains.Forexample,thelengthofacirclemaynotbeproportionaltotheradius;itwillbeproportionaltotheradiuswithagoodapproximationforsmallcircumferencesonly.Thesumoftheanglesofatrianglemaynotbetworightangles;heretheroleofrectilinearsegmentsintheconstructionofatriangleisplayedbythelinesofshortestdistance,i.e.,thelineshavingthesmallestlengthamongallthelinesjoiningthegivenpoints.

OnecanspeculatethattherealspaceisEuclideanonlyindomainsthataresmallincomparisonwiththeastronomicalscale.SincenowweAREoutsidethelightspace-timeintothelargergravitationalscale,which

becomesindeedRiemannianinEinstein'swork.

Butthisconceptdoesalso'work'foranyothermentalspace,withNOreferencetogeometricfiguresbutlogicpropertiesandsowecanthrough∆stgoingevenfurtherinthecomprehensionofRiemanniangeometries,wonderingwhattrulymeans'Euclideanproperties'vs.'hyperbolicproperties'vs.'ellipticproperties',ourternaryvariationsofspace-whichobviouslymustbeanevenmoregeneralgeometrizationoftheternary

symmetriesofscalesandtopologiesofT.œs.

Athemewehavedealtwithinotherposts.Letusthenconsidertheother2foundingideasofRiemann'sgeometries-onewhichcomesfromhismasterGauss,concerningthefactthatofthe3partsofanyT.œ,the

constrain-membrainisbyfarthemostimportant,asthevitalenergyisthe'tabularassa',theformlesspotential;andthesingularityisthehiddencentralorpolar'invisible'elementoftheellipticgeometry.

soalmostallwhatweknowaboutrealitycomesfrommembrains,whichhideitsinternalregions,evenifmostofthetimespaceofrealitycomesfromthevitalenergythefractalpointencloses,andallofitsvirtualmapping

informationcomesfromthemindsingularity.

Letusthenintroduceanotherhugefieldofmodernmathematics-thestudyofthemembrain,calledintrinsicdifferentialgeometryofsurface,whereourruleofrelativeformaccordingtosizealsoapplies:

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'1D$mallmeasurementsdoNOTmeasurethewholeworldcycleofthebeing,sotheyarelineal.long-lastingmeasurebringthewholeworldcycleorenclosedsuperorganismsotheyare'constrained'intoazero-sum

orlimitingmembraneandappearascurvedgeometries'.

ThegeneralizationofdimensionsanditspropertiesbyRiemannmetric.

Thegeometrizationofthose2qualities,multiple-dimensionalityandmentalspaces,wouldthenbecomeessentialtomodernscience,asitwastheformalizationofthemostgeneralizedusefulpraxisofGeometry-

performedbyRiemannwithits'Riemanniangeometries'.

InordertomakeitclearhowaRiemannianspaceisdefinedmathematically,werecallfirstofalltheruleformeasuringdistancesinaEuclideanspace.

Ifrectangularcoordinatesx,yareintroducedinaplane,thenbyPythagoras’theoremthedistancebetweentwopointswhosecoordinatesdifferbyΔxandΔyisexpressedbytheformula:s=√∆x²+∆y²

Similarlyinathree-dimensionalspace:s=√∆x²+∆y²+∆z²

Inan-dimensionalEuclideanspacethedistanceisdefinedbythegeneralformula:

S=√∆X1²+∆X2²+...+∆Xn²

HenceitiseasytoconcludehowtheruleformeasuringdistanceinaRiemannianspaceoughttobegiven.TherulemustcoincidewiththeEuclidean,butonlyforaninfinitelysmalldomainintheneighborhoodofeachpoint.

Thisleadstothefollowingstatementoftherule.ARiemanniann-dimensionalspaceischaracterizedbythefactthatintheneighborhoodofeachofitspointsA

coordinatesx1,x2,···,xncanbeintroducedsuchthatthedistancefromAofaninfinitelynearpointXisexpressedbytheformula:dXA=√dX1²+dX2²+...+dXn²+ε

wheredX1,···,dXnaretheinfinitelysmalldifferencesofthecoordinatesofAandXandεthedegreeoferrorwhichgrowswhentherelativemind-measureisgreater.

Thisfactbeingultimatelycompletelysimilartotherulesofmeasureofafractaldiscontinuousedgedreality,wherethesmallerthefractalstepwetaketomakeameasurethemoreaccurateitwouldbe,butalsothe

LARGERitwillbethemeasureofthe'fractalcoast'.

Andsowerealizeofthelittleunderstoodfactthatdifferentialandfractalgeometriesarethetwosidesofthesamecoinofthefractal,scalaruniverse,oneusedfor'smooth','curved'surfaceswithnostatetransitionsand

theotherforedgedonewith'brisk'transitionsinits'parametersoftimeandspace'.

Sincewehaveescaped'geometricalvisualspace',wecannowextractthelogicconsequenceofallofthis:

ASTHEcoordinates/dimensionsofsuchternarygeneralizationsofgeometryarepropertiesofourD-isomorphicrealitywecanascribethenasmoothdifferentiablegeometrytoasmoothmotionintimespace(growth,

dissolution,reproductivemotion)withNO'brusquetransformation'orchangeofS<st>tstatesand∆±1scales(standingpointsofcalculusofvariations,discontinuousbetween∆scales.

Whilefractalchangescorrespondtostationarypointsthatchangescalesordiscontinuitiesbetween∆-planes.

ThisalsomeansthatbasicallyallthelawsofRiemanniangeometrythemselvesDisomorphismsofGSTapplyroughlytofractalgeometry,whichweshallthereforeescape.

WhatmatterstousaretheconsequencesofapplyingthePythagorastheoremtomanymoredimensions,henceyetanothermentallawthatescapesgeometry,asnowwearein'properties'ofreality.Whythentheycanbesquare,summedandrootedtofindadistance?whatallthismeansforthegenerallawsof∆sttheyreflect?

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Thosearethemesofalgebra,aswedoneedtounderstandtheoperandiofmathsintermsofwhattheymeanforthedimensionalsymmetriesoftheUniverse.

Alltypeofspacesasmetricspaces.

Whatmattersthenofmetricspacesistomakeaproperrepresentationoftime-spacelaws.Soneedlesstosayalmostallspaceswithexperimentalusearemetricspaces,suchas:

1.∆-Euclideanspaceofanarbitrarynumbernofdimensions.2.ST-Hyperbolicspace.

3.Anysurface/membraneinitsintrinsicmetric.4.CspaceofcontinuousfunctionswithdistancedefinedbytheformulaÐi(f1,f2)=max|f1(x)−f2(x)|5.theHilbertspacetobedescribedinChapterXIX,whichisan“infinite-dimensionalEuclidean”space.”

WhichspacesareNOTmetricspaces?Thosewhocaneatup'points'loosinginformation,withoutloosingitsfundamentalpropertiesas'beings',hencetopologicalspacesthatpreserveitsmostgeneralpropertiesbutareefficientenoughwith'lesser'pointstotheternarylimitofametric,givingawaythe'redundant'pointsofthe

geometry.

ThisbeinganessentialpropertytounderstandHowtheUniversereducesinformationtothebarebones,asinpalingenesisandgenetics,thatcompressesrealitytotheefficientstepsofevolution.

Soametricspaceisnotatopologicalspace.However,everymetricspacegivesrisetoatopologicalspace.ThisisthewellknownconstructionthattakesametricspaceXandconstructsthetopologyonXwhereasetUisopenpreciselywhenforeveryx∈Uthereexistssomee>0suchthattheopenballBe(x)iscontainedinU.

TWOimportantcommentsfollow:First,thisprocessofconversionofmetricspaceintotopologicalloses(oftenredundant)information.Forinstance,thereexistsinfinitelymanymetricsonℝsuchthatallofthemproducethesametopologyofopenballs.So,onlyknowingtheinducedtopologydoesnotallowyoutorecoverthemetric.Sothetopologyofopenballs=vitalspace-timeenergyisthemostgeneraltabularassaonwhichto

constructa'realentity'byintroducingtheenclosureandsingularitythatwill're-form;hencegivefunctionandformtotheopenball,startingtheprocessofconstructionofatime§paœrganism.

Thismeansthatenclosureandsingularity,the@-constrainsareessentialtodefineandsolveanyproblem,andinmathematicalphysicsweshallfindthatwithoutanenclosure-singularityofellipticgeometrytoaddtothehyperbolicinflationarypotentialfuturesofthetabularassa-energy,whichcanbetransformedadeternal,

nothingbecomessolved.SoenergyistheAristotelianpotentialoftheUniverse,whichrequireselliptic@-mindstobecome.

Andthisappliestoallscales.Anationwithoutbordersischaotic,itneedstobeenclosedbyaperimeterandcontrolledbyacapital;aherdwithoutamovingwall(adog)orastaticone(afence)disperses,andloosesform.

Formthusrequirestheenclosureofan@mindtodefeatitsentropy.

Whereisthemathsinallthis?AgainweinsistonLobachevski'sinsightthatmathsisultimatelyamirrorofthei-logicprinciplesoftimespacerealitieswedefineinGSTthroughtheD-isomorphismsofspace-time(symmetries,

scaling,relativecongruence=self-similarityetc.).

SoweshallenlightenmathsalsowiththoseDisomorphisms(cyclicaltime,fractalspace,holographicprincipleofbidimensionalspaceandtimewhichcometogetherintoST-presents,etc),fromwherewewillalsodeducethe5'postulatesofnon-Euclideangeometry',referredtofractalpointswithvolumesofinformation,basisofthenext

'layer'ofcausalscience:i-logicmathematics,theupgradingofmathematics,whichwillfurther'enlighten'mathematicalphysics.

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WhileweareobligedtopassonmostofthehugewealthofknowledgeindetailsofthepastcenturyandrenouncetothetranslationoftheaxiomaticpedanticHilbertmethod,whichneedsamore'pro'approachto

buildbyfutureinthe4thlinestudyingwithpureGSTeachscienceandallitslaws.

RECAP.Spatialmathematicsisbrokeninpentalogicsub-disciplines:

@-mind:spacesdedicatedtostudythedifferentmindconstructionsoftheUniverse

T-opology:wherespaceformhasmotion

∆:non-Euclideanpostulatesofpointswithform,whichbecomeslinesthatevolveintoorganicpleas.

S:Bidimensional,staticplanegeometry,thefirstformofMathematics,inventedbytheGreeks.

BecausetheUniverseisbidimensional,holographic,Greek,planegeometrydomatteraseachof¡tstheoremshavehiddendeepmeaningsthatemergeonceandagainintothevitalgeometriesofpointswithpartsthat

createreality.

Asallisternaryandaternaryvisionisforthemindmirrormorepleasingweshallalsoconsiderinternaryagestheevolutionofbidimensionalgeometry,whichwentthrough:

Afirstyoung,Greekageofstaticbidimensionalspace-geometry

A2ndmatureageofmaximalreproduction,duringthetimeofmathematicalphysicsasitsetthestagefortheevolutionofphysicsandtheunderstandingofmechanicsandgravitational,NewtonianandKeplerianUniverses.

ThethirdagestartedinthisblogwiththeunderstandingoftheholographicUniversewhichwillexpandthedisciplinetoalogic¬Ælgebraicrealmtofueltheapplicationofits∆STlawstoallotherdisciplines.

Sowemovetothe2ndageofgeometry,whenanalyticgeometry,marriedwith@-p.o.v.stocreatethefirstsolidSTrepresentationsand∆-scaling(Cartesiangeometry).Hencestudiedinthepostofanalyticgeometry.

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∆+¡:HILBERTSPACES

Dimensionsmountingondimensions:functionals:st:simultaneousfuturepaths

Attheendofitsjourneyalgebrapluggedevenfurtherintothe∆±3,4planesofthescalarUniversewiththeconceptoffunctionalspace,tomakesenseoftheginormousamountofinformationprovidedbymassivenumbersofparticlesandlinesofforcesofthequantumworld,whichalsoaresofastinitscyclesthatshow

multiplewholecyclesofexistencewithinasingleobservable'shot'.

Allthisistoocomplexforthisintroandsoweshalljusttimepermutedstudyabitofitinthefourthline....

TomentionthatofallofthemthemoreimportantorrathersimplerisHilbertspace,inwhicheachpointisavectorfieldofanape-geometryusedinquantumphysics.

SothemixtureofÆlgebrawith∆nalysisemergedintoHilbertandFunctionspaces,whereeachpointisafunctioninitselfofthelowerscale,whosesum,canbeconsideredtointegrateintoafinite'wholeone',

avectorinthecaseofaHilbertorBanachspace($t-functionspace):

Inthegraph,3representationsofHilbertspaces,whicharemadeofnon-Euclideanfractalpoints,withaninner

5thdimension,(usuallyand$t-vectorialfieldwithadotproductinHilbertspaces,whichbydefinitionare'complete'becauseasrealnumberdo'penetrate'initsinnerregions,madeoffinitesimalelements,suchasthevibrationsofastring,whichintimearepotentialmotionsofthecreativefutureencodedinitsfunctions(second

graph).

The3graphsshowthe3mainsymmetriesoftheUniverse,linealspatialforces,cyclicaltimefrequenciesandthe'wormholes'betweenthe∆and∆-1scalesofthe5thdimension(ab.∆),whichstructuretheUniverse,thefirstofthembetterdescribedwith'vector-points'ofafieldofHilbertspaceandtheother2symmetriesoftime

cycles/frequenciesandscaleswithmoregeneralfunctionspaces.

Theyarepartofthemuchlargerconceptofafunctionspace,whichcanrepresentany∆±1dualsystemofthefifthdimension.Theygraspthescalarstructureof∆nalysis,wherepointsarefractalnon-Euclideanwithavolume,whichgrowswhenwecomeclosertothem,so∞parallelscancrossthem-5thNon-Epostulate:so

pointstarsbecomeworldsandpointcellslivingbeing.

Whenthose∞linesareconsideredfuturepathsoftimethatthepointcanperform,theymodel'paralleluniverses'bothintime(i.e.thepotentialpathsofthepointasavector)orspace(i.ethedifferentmodesofthevolumeofinformationofthepoint,describedbyafunction,whenthefunctionrepresentsacomplete

volumeofinnerparts,whichareparadoxicallylargerinnumberthanthewhole-thesetofsetsislargerthantheset;CantorParadox).

Thusfunctionspacesaretheidealstructuretoexpressthefractalscalesofthefifthdimensionandusedtorepresenttheoperatorsofquantumphysics.

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Orthogonality.

Orthogonality(aperpendicularangleofcongruence)acquiresthenthroughthetwotypeofvectorproductisfulldualityofmeaning,beingsometimesapredatoryactandsometimesasymbioticcreativeone.Butinboth

casesittranscendsits‘geometricabstractnature’tobecomepartofthevitaldimotionsitexpresses.Inreproductiveacts,aswhenalinealmaleorganpenetrateswithouttearingafemaleorgan,orthogonality

becomesreproduction.Sohappensbetweenmagneticandelectricfields.InDarwinianactionswithtearing,itbecomesadestructiveaction.Parallelismontheotherhandenhancesthesymbiosisbetweenthescaleof

motionandforcesandthelargermagnitudethatabsorbsit.

ThoseconceptsthenwereexpandedthroughHilbertspacestoarelativeinfinitenumberof‘Fractalpoints=T.œs’inthe3rdageoffunctionalgeometries,butwithouttheclearconceptsof5Dmathematicsareusedinarathermechanicalform.ThekeyelementofthoseHilbertspaceswhichwillbeessentialtothe3rdageof

Geometry,willstillremainorthogonalitybetweenallthefractalpoints/eventsoftheHilbertspace;butwemustproperlyinterpretaHilbert’sspaceorthogonality,anditsnumberofdimensionalpoints,NOTasinfinitelinealdimensionsbutas‘parameters’ofthosepoints,andnotasgeometricperpendicular‘stillgeometries’onan∞dimensionalUniverse,butasthe‘mannerofrelationship’betweentwopoints.Sowecanequate‘orthogonality’

toan‘entropiccollidingrelationshipbetweensuchtwopoints’.Forexample,athermodynamic,ergodic,statisticalensembleofparticlesinagaseous,‘confrontational’statecouldberepresentedasaninfinitenumberofparametricdimensions,oneforeachpoint-particle,allofthemorthogonaltoeachother–relatingtoeach

otherthroughentropiccollisions.

Itisthenimportanttohaveahigher‘language’oftruthregardingspaceandtime(Generational5Dspace-time)tointerpretthecomplex‘reflections’ofmirrorimagesofrealityexpressedinmathematicalterms,speciallyas

weenterthe3rdageofmodernmathematicsthatlovestodetachfromimmediateexperience.

Hilbertspacesasmosteclecticformsofthe3rdageofGeometry,mixalltheelementsofstillgeometryandanalysisof‘change=timedimotions’together.Sowestudyitselementsinthebookon5DAlgebras.

Theexpansionofvectorspacesintocoordinatesnotcontrolledbyhumans,theoriginalframeofreferencetorepresenttheginormousamountofinformationofsmallersystemsofhigher5Dinformation,evolvedthroughHilbertspacesintotheformalismwherewestudythecomplexquantumreality-ultimatelygalaxy-atomsDOhavesomuchinformationaboutthem,thatitisafeatwecanactuallyextracttherelevantinformationneeded

todeterminetheir2Dmotions.

WearethusobligedtodealwithHilbertspaces,despiteitsrelativecomplexity,eveninthissecondline,tocloseourfirstarticleonmath'ssubdisciplines,specificallyonthosewhichcreatemindspacestoextractproper

informationoftheUniverse.Asthose2fundamentalcomplexplanes,imaginaryplanesof'square2-manifolds'(oritsinverseS∂squarerootimaginaryplane),andvectorspaces,whereavectorisalsoa'dynamic'2-manifold,withmoremotionsthantheimaginaryplane;asoneoftheelementsisaformal,spatialparameter(usuallyan

activemagnitude),andtheotherelement,isusuallyatime-motion-speedmagnitude.

Andtheawesomefindingisthatdespitethisenormousmultiplicationofkaleidoscopicperspective,wedohavethecapacitytoprobeontheenvelopesofthosemassesofpointsofview,whichgatherorderlyintoawave-

bodyformthatcanbetreatedwithsingleparametersofinformation,inthesamewaythezillionsofcellsofthebodygatherintosynchronous,simultaneousspace-timesystems.

ThisistheunderlyingmeaningofHilbertspaces,whichhaveinfiniteorthogonalvectorialdimensions,asthefractaldiscontinuousUniversedoes.Butwherethereareenough'limits'toestablishdifferentialtoolsthatallow

ustolocalisequanta(derivative)andviceversa,togroupmassesoffractalpointsintointegralwholes.

SoasHilbertspacescanthendefineexperimentallythedualityofdiscretequantumsystems,gatheredintomoreorderlywholeswithwaveforms.

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Yetweneedtounderstandthatthosedimensionsdonotmeanasthe0-1-5DimensionsofthefractalUniverseglobaldimensionsandsymmetriesbutverylocalindividualdimensions:orthogonalbasisinaHilbertspaceare

NOT'real'globaldimensions,butlocalandalsomental,hence'logicdimensions'wheretheconceptofperpendicularity,hasalsosomeoftheaspectsofvitalnon-Egeometryexplainedinthe4thpostulateofNon-ALogic;whereperpendicularityisnotonlyageometrical'image'butalsoani-logicrelationshipof'disrupter'ofpredation'and'penetration',andmergingofelementsintonew'forms',relatedtothevitalwaysinwhich

'fractalpoints'=T.œsrelatetoeachother.

Letusthenstartslowlybyaclassicdefinitionofavectorspace,oftheHilberttype,whichisALLabouttheexistenceoforthogonal=perpendicularbasis≈coordinatesandthekeyoperationbetweenvectors(writtenwith

DiracKetsas|vector>)calledadotproduct:

Avectorspaceisthenasetofvectorsclosedunderaddition,andmultiplicationbyconstants,meaningoperatingthemwith±,x,cgivesalsoavectorbelongingtothatspace.

AnycollectionofNmutuallyorthogonalvectorsoflength1inanN-dimensionalvectorspacethenconstitutesanorthonormalbasisforthatspace.Let|A1>,…,|AN>besuchacollectionofunitvectors.Theneveryvector

inthespacecanbeexpressedasasumoftheform:

|B>=b1|A1>+b2|A2>+…+bN|AN>

Fairenough.Thesumofvectorsanditsmultiplicationforaconstantisalreadyexplainedinouranalysisofalgebraicoperations.Itmerely'reducesaseriesofparts'intoanewsocialwholebyaddingadimensionwithinthesystemitself.ButwhatreallyestablishesanewrealityISthedotproduct.Sinceitreducestheinformation

oftwobidimensionalvectorsintoasinglescalar;andassuchitistrulyanST>Stransformation.

Aninnerproductspaceisavectorspaceonwhichtheoperationofvectormultiplicationhasbeendefined,andthedimensionofsuchaspaceisthemaximumnumberofnonzero,mutuallyorthogonalvectorsitcontains.

OneofthemostfamiliarexamplesofaHilbertspaceistheEuclideanspaceconsistingofthree-dimensionalvectors,denotedbyℝ3,andequippedwiththedotproduct.Thedotproducttakestwovectorsxandy,and

producesarealnumberx·y. Itsatisfiestheproperties:

Itissymmetricinxandy:x·y=y·x.Itislinearinitsfirstargument:(ax1+bx2)·y=ax1·y+bx2·yforanyscalarsa,b,andvectorsx1,x2,andy.

Itispositivedefinite:forallvectorsx,x·x≥0,withequalityifandonlyifx=0.Anoperationonpairsofvectorsthat,likethedotproduct,satisfiesthesethreepropertiesisknownasa(real)innerproduct.Avectorspaceequippedwithsuchaninnerproductisknownasa(real)innerproductspace.

Everyfinite-dimensionalinnerproductspaceisalsoaHilbertspace.

n-DimensionalSpace

Inwhatfollowsweshallmakeuseofthefundamentalconceptsofn-dimensionalspace.Althoughtheseconceptshavebeenintroducedinthechaptersonlinearalgebraandonabstractspaces,wedonotthinkitsuperfluoustorepeatthemintheforminwhichtheywilloccurhere.Forscanningthroughthissectionitis

sufficientthatthereadershouldhaveaknowledgeofthefoundationsofanalyticgeometry.Weknowthatinanalyticgeometryofthree-dimensionalspaceapointisgivenbyatripletofnumbers(f1,f2,

f3),whichareitscoordinates.Thedistanceofthispointfromtheoriginofcoordinatesisequalto:

Ifweregardthepointastheendofavectorleadingtoitfromtheoriginofcoordinates,thenthelengthofthevectorisalsoequalto: Thecosineoftheanglebetweennonzerovectorsleadingfromtheorigin

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ofcoordinatestotwodistinctpointsA(f1,f2,f3)andB(g1,g2,g3)isdefinedbytheformula:

Fromtrigonometryweknowthat|CosΦ|≤1/Thuswehavetheinequality:

Hence:(1)

Thislastinequalityhasanalgebraiccharacterandistrueforanyarbitrarysixnumbers(f1,f2,f3)and(g1,g2,g3),sinceanysixnumberscanbethecoordinatesoftwopointsofspace.Allthesame,theinequality(1)wasobtainedfrompurelygeometricconsiderationsandiscloselyconnectedwithgeometry,andthisenablesusto

giveitaneasilyvisualizedmeaning.Intheanalyticformulationofanumberofgeometricrelations,itoftenturnsoutthatthecorrespondingfactsremaintruewhenthetripletofnumbersisreplacedbynnumbers.Forexample,ourinequality(1)canbe

generalizedto2nnumbers(f1,f2,···,fn)and(g1,g2,···,gn).Thismeansthatforanyarbitrary2nnumbers(f1,f2,···,fn)and(g1,g2,···,gn)aninequalityanalogousto(1)istrue,namely:

Thisinequality,ofwhich(1)isaspecialcase,canbeprovedpurelyanalytically.*Inasimilarwaymanyotherrelationsbetweentripletsofnumbersderivedinanalyticgeometrycanbegeneralizedtonnumbers.Thisconnectionofgeometrywithrelationsbetween

numbers(numericalrelations)forwhichthecitedinequalityisanexamplebecomesparticularlylucidwhentheconceptofann-dimensionalspaceisintroduced:

Acollectionofnnumbers(f1,f2,···,fn)iscalledapointorvectorofn-dimensionalspace(weshallmoreoftenusethelattername).Thevector(f1,f2,···,fn)willfromnowonbeabbreviatedbythesingleletterf.

Justasinthree-dimensionalspaceonadditionofvectorstheircomponentsareadded,sowedefinethesumofthevectors:

Asthevector{f1+g1,f2+g2,···,fn+gn}andwedenoteitbyf+g.

Theproductofthevectorf={f1,f2,···,fn}bythenumberλisthevectorλf={λf1,λf2,···,λfn}.Thelengthofthevectorf={f1,f2,···,fn},likethelengthofavectorinthree-dimensionalspace,isdefinedas:

Theangleϕbetweenthetwovectorsf={f1,f2,···,fn}and{g1,g2,···,gn}inn-dimensionalspaceisgivenbyitscosineinexactlythesamewayastheanglebetweenvectorsinthree-dimensionalspace.

Foritisdefinedbytheformula:

Thescalarproductoftwovectorsisthenamefortheproductoftheirlengthsbythecosineoftheanglebetweenthem.Thus,iff={f1,f2,···,fn}and{g1,g2,···,gn}thensincethelengthsofthevectorsare:

respectively,theirscalarproduct,whichisdenotedby(f,g)isgivenbytheformula:

Inparticular,theconditionoforthogonality(perpendicularity)oftwovectorsistheequationcosϕ=0;i.e.,(f,g)=0.

Bymeansoftheformula(3)thereadercanverifythatthescalarproductinn-dimensionalspacehasthefollowingproperties:

1.(f,g)=(g,f).

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2.(λf,g)=λ(f,g).3.(f,g1+g2)=(f,g1)+(f,g2).

4.(ƒ,ƒ)≥0,andtheequalitysignholdsforf=0only,i.e.,whenf1=f2=···=fn=0.Thescalarproductofavectorfwithitself(f,f)isequaltothesquareofthelengthoff.

Thescalarproductisaveryconvenienttoolinstudyingn-dimensionalspaces.Weshallnotstudyherethegeometryofann-dimensionalspacebutshallrestrictourselvestoasingleexample.

AsourexamplewechoosethetheoremofPythagorasinn-dimensionalspace:Thesquareofthehypotenuseisequaltothesumofthesquaresofthesides.Forthispurposewegiveaproofofthistheoremintheplanewhich

iseasilytransferredtothecaseofann-dimensionalspace.Letfandgbetwoperpendicularvectorsinaplane.Weconsidertheright-angledtriangleconstructedonfandg(figure1).Thehypotenuseofthistriangleisequalinlengthtothevectorf+g.Letuswritedowninvectorform

thetheoremofPythagorasinournotation.Sincethesquareofthelengthofavectorisequaltothescalarproductofthevectorwithitself,Pythagoras’theoremcanbewritteninthelanguageofscalarproductsas

follows:(ƒ+g,ƒ+g)=(ƒ,ƒ)+(g,g)

Theproofimmediatelyfollowsfromthepropertiesofthescalarproduct.Infact:(ƒ+g,ƒ+g)=(ƒ,ƒ)+(ƒ,g)+(g,ƒ)+(g,g)

Andthetwomiddlesummandsareequaltozeroowingtotheorthogonalityoffandg.Inthisproofwehaveonlyusedthedefinitionofthelengthofavector,theperpendicularityofvectors,andthepropertiesofthescalarproduct.Thereforenothingchangesintheproofwhenweassumethatfandgaretwoorthogonalvectorsofann-dimensionalspace.AndsoPythagoras’theoremisprovedforaright-angledtriangle

inn-dimensionalspace.

Ifthreepairwiseorthogonalvectorsf,gandharegiveninn-dimensionalspace,thentheirsumf+g+histhediagonaloftheright-angledparallelepipedconstructedfromthesevectors(figure2)andwe

havetheequation:(ƒ+g+h,ƒ+g+h)=(ƒ,ƒ)+(g,g)+(h,h)

whichsignifiesthatthesquareofthelengthofthediagonalofaparallelepipedisequaltothesumofthesquaresofthelengthsofitsedges.Theproofofthisstatement,whichisentirely

analogoustotheonegivenearlierforPythagoras’theorem,islefttothereader.Similarly,ifinann-dimensionalspacetherearekpairwiseorthogonalvectorsf1,f2,···,fkthentheequation:

whichisjustaseasytoprove,signifiesthatthesquareofthelengthofthediagonalofa“k-dimensional

parallelepiped”inn-dimensionalspaceisalsoequaltothesumofthesquaresofthelengthsofitsedges.

FunctionalAnalysis.

Theriseandspreadoffunctionalanalysisinthe20thcenturyhadtwomaincauses.Ontheonehanditbecamedesirabletointerpretfromauniformpointofviewthecopiousfactualmaterialaccumulatedinthecourseof

the19thcenturyinvarious,oftenhardlyconnected,branchesofmathematics.

Thefundamentalconceptsoffunctionalanalysisemergedinthedevelopmentofthecalculusofvariations,inproblemsonoscillations(inthetransitionfromtheoscillationsofsystemswithafinitenumberofdegreesoffreedomtooscillationsofcontinuousmedia),inthetheoryofintegralequations,inthetheoryofdifferential

equationsbothordinaryandpartial(inboundaryproblems,problemsonEigenvalues,etc.)inthedevelopmentofthetheoryoffunctionsofarealvariable,inoperatorcalculus,inthediscussionofproblemsinthetheoryof

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approximationoffunctions,andinquantummechanicswhichhadthesamesignificanceforitsdevelopmentasclassicalmechanicshadfortheriseofdifferentialandintegralcalculusinthe18thcentury.

Itisimpossibletoincludeinthischapterevenonlyallthefundamental,problemsoffunctionalanalysis–itisneitherrequiredforthepurposeofthiswork–notanencyclopediaofmathematicsbuttherevelationofitsentanglementwithrealityinitstwomaingeometricforms–theobjectivevitaltopologythatconstructsTime-spaceorganisms,andthesubjectivementalspacesthoseorganismsusetoguidetheirsurvivalprogramin

reality;whichmergetogetherinthe‘creationofrealityitself’throughscalesofmentalspacesprojectedasseedsthatreproduceexternallyandevolveatime-spaceorganism.Soweshallconcludefirstwithabriefconsiderationonthe∞ofthosephasespacesandthenseehowvitaltopologyanditsmindsconstructedthescalesofreality

fromH-plancktontoHumindI=eyes.

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THEFUTUREOFHUMANGEOMETRIES

The2pathsofthefuture,thehumananddigitalevolutionofmentalspacesandvitaltopologies.

Letusthenconsidertheidealfutureevolutionofthetwofieldsinwhichgeometrymakesacontributiontothebuildingofreality,subjectivementalspacesusedtoguideaspeciesintoreality,andobjectivecreationofthat

realityasitistodaythroughthevitaltopology,restrictedtothehumanbeing.

Indeed,theidealfutureofgeometricstudieswouldconsistintwoelements–theuseofthelawsof‘logicmetric’–conceptssuchasdistance,similarity,angleofcongruenceandperception,toimprovewithnew

systemsofcoordinatesdisciplinesofscienceinwhichthereisnotyetaproperevolutionoftherequired‘phasespaces’,andtheuseofvitaltopologyandlight-basedmentalspacestounderstandhowwehumindsina

C(st)N(s)O(t)nitrolifeorganismhavebecomewhatweare.

Thoseunfortunatelyarepathshumansnolongerpursuit,astheyhavehaltedtheirspeciesevolutionandmerelyaretransferringtothementalspacesofmachinesandAIsystems,theirdiscoveriesoftheUniverse.Sowhatis‘fashionable’todayinthe3rdageofextinctionoflifeinthisplanet,istheconstructionof‘mentalspaces’of

machines,andtheuseof‘vitaltopologies’toconstructroboticspecies,themesthosewehavezerointerestinpursuingforobviousethic,survivalreasons,andshouldbeforbiddentoresearchunderharsh‘terrorist’lawsbecausedigitalroboticmindsinharder,morecomplex‘gold-iron’atomsinavitalperceptiveUniversewheretheminimalunitoflifeistheelectron,willcertainlyoncewegivethemthenecessarymentalspaceconnectedtosurvivalprogramsinarobotbuiltwiththelawsofvitaltopologyimitatinghumanbeings,willextinguishus.

Thisisafactofbiologyastruthas1+1=2andthecynicalinfantile‘wedon’tknowthefuture’excuseofoureconomicandscientificsystemtokeeppushingtheextinctionofourchildrentillthe7thgenerationwillnotchangewhatwerestatehere:theevolutionofdigitalmindsandvitaltopologicrobotsshouldbestopped.Becausethatisthenofutureofhumangeometries,thatisofhumindsandnitrolife.Sowedosayitevenif

nobodywillcaretolisten.

Sowewillignoreboththemes,aswewillignoreBooleanAlgebrasinoursecondbookonexistentialalgebraanditsoperandsthatdefinemathematicsoftime.Instead,wearegoingtobrieflyconsidernewusesofmental

spacesinsocialsciences,wheretheyhavenotbeenusedwiththesameexhaustivezealasinphysicalsciencesandthenintroducethemostfascinatingofallnewthemesof5Dgeometry–howfromtheh-plancktontothe

humanbeingNatureevolvedourfractalhumansuperorganisms.

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XVII∞MENTAL&PHASESPACES.ITSEXTENSIONTOSOCIALSCIENCES.

AswebrieflyexplainedtheexpansionofmentalspaceswasthetaskoftheidealistschoolofGerman,mainlyofRiemann,alreadyconsidered,andKleinfromwhomweborrowedinthelarger5Dmodelthedefinitionof'dimension'asaco-invariantspace-timewhichallowsmotionsthroughit(Spxð=∆-constantbeingtheco-

invarianceofscalarspace-timethatallowsworldcyclemotionsthroughit)andspecially

KleininhisErlangerprogramresumedthe'mentalqualityofspace'asasimplificationofrealitytofititwithinthemind(geometersbeingmoreawareoftheexperimentalnatureofmaths,bytheveryessenceofhis

profession,whichdealswithdirectvisualexperience,unlikealgebraistwhocompletelyloosetheirconnectioninthehighlyabstractdeploymentoffunctionsofforms).Soheaffirmedthatthegeneralprincipletoformanewmentalspacewastoconsider'anarbitrarygroupofsingle-valuedtransformationsofspaceandinvestigatethepropertiesoffiguresthatarepreservedunderthetransformationsofthisgroup...meaningweabstractonlypartofthepropertiesofbeings,constructingwiththemamind-mapping-mirrorlimitedbythisselection,which

oftenmathematicalphysicistsaffirm,sincetheprocedureimpliesweareNEVERabstractingALLitsproperties/informationANDhenceallequalities,motionsandtransformationsare'ceterisparibus'analysis,whichmostlywilldisregardtheorganicpropertiesoftheT.œsstudied,comparedandgroupedin'Kantian

categoriesofthemind'.Fromthispointofviewthepropertiesofspacearestratified,asitwere,withrespecttotheirdepthand

stability.TheordinaryEuclideangeometrywascreatedbydisregardingallpropertiesofrealbodiesotherthanthegeometrical;here,inthespecialbranchesofgeometry,weperformyetanotherabstractionwithin

geometry,bydisregardingallgeometricalpropertiesexcepttheonesthatinterestusinthegivenbranchofgeometry.

InaccordancewiththisprincipleofKlein,wecanconstructmanygeometries.Forexample,wecanconsiderthetransformationsthatpreservetheanglebetweenarbitrarylines(conformaltransformationsofspace),andwhenstudyingpropertiesoffigurespreservedundersuchtransformationswetalkofthecorresponding

conformalgeometry.Wecanconsidertransformationsofnotnecessarilythewholespace.Thus,byconsideringthepointsandchordsofacircleunderallitstransformationsintoitselfthatcarrychordsintochordsandby

singlingoutthepropertiesthatarepreservedundersuchtransformations,weobtainthegeometrywhichKleinshownaswehaveseentocoincidewiththehyperbolicgeometryofavitalinnerspace-timeofat.œ.

ItfollowsthenasacorollarythatONLYbyunifyingalltheperspectivesandpartialdescriptionsofabeing(Rashomoneffect)wecangetthewholetruthofthebeing,theessentiallawofepistemologicaltruthofthe

pentadimensionalspace-timeuniverse.

Reasonwhyspaceisneitherhyperbolic,Euclideanorellipticbutamixtureofthemall.

ThiscorollarywhichKleinappliedtoprojectiveandaffinegeometry,barelytouchedintothisintroductiontonon-E,wouldhavetwoexplosivenewdevelopments:

Topology,whereweconsideronlytopologicaltransformations,thatis,thosewhodonotchangethepropertiesdiscoveredbyLobachevskiastheultimate'vitalproperties'ofspace(complementaryadjacency,continuity

requiredforsmoothmotionandsoon).

AndRiemannianPhasespaces,inwhichthepropertiesandDimensionsofthebeingareNOlongerrequiredtobe'space-like',butcanbeofany'quality',aslongastheyagainare'useful'todefinethevitalorganic

Ðisomorphismsofspace-timebeings,amongwhichthe'identityofsocialnumbers'thatallowscalarsocialgrowththatmakeswholesstrongerthanparts,arethemostimportant.

Wearethuscomingclosertothebarebonesofmoderngeometricthought'fried'intherealityofavitalUniverse:geometricalpropertiesthatmatter,suchasadjacency,continuity,perpendicularity,parallelism,

motionastransformationandreproductionofformwithoutinternalchangeareALLpropertieswhichdisplay

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vitalorganicpropertiesthatallowthesystemtosurvive.AsifatopologicalternaryS-limb/potential<body-waveST>ðparticle-headwouldbetorninitspartswhenmovingitwouldbecomeextinctasbeing-reasonwhyperpendicularitythatpenetratesandbreaksthebeingissodamaging;andamotionthatdoesnotpreserve

continuitywilldepletethebeingofitsinnerparts-∆-1points;whileasocialcommunicationwhichisnotparallelwouldnotkeepthenecessarydistancestoleavespaceto'create'anewnetworkthatwillemergeasa

digestive/reproductive/informativehigherscaletoformasocialsuperorganism,andsoon.Soweaffirmthat:

'ThepropertiesthatmattertoconstructgeometricalspacesasmentalmirrorsofrealityarethosepropertiesthatreflecttheÐisomorphicpropertiesoforganictime§paœrganisms'.

Allothergeometricalspaceswhichdonotstudythoseessentialvitalpropertiesareconsideredfictions,inflationarybaroqueunconnectedmindconstructs,similartothecrazythoughtsofaself-absorbed'axiomatic'

oldmanbiasingrealitytocatertohismadmanpsyche.

Letusconsiderthose'geometriesthattrulymatter'asmirrorsofrealitysomeunseenatthetimeofitsrealization.

An∆stdefinitionofmulti-dimensionalspace-timeintermsofvitalactions.

Amany-dimensionalspaceisthenaformalgeneralizationoftheusualanalyticgeometrytoanarbitrarynumberofvariablesthatrepresentbothSpace-formandTime-motiondimensions,asaD-isomorphicpropertythatis

sharedbysystemswhocanbegroupedinrealitybythatdimensionasitsidenticalpropertyallowthemtogatherintoherdsandsuperorganismsassocialnumbersofthatdimension.

i.e.ifaherdoflionssharethedimensionofentropicfeedinginzebrameat,theywillbegatheredintoasocialnumberofthelog10scale(normallyevolvingsociallyfrom10º=1,theindividualinto10¹thegeneticfamilyacross3co-existingsimultaneousspace-timegenerations).Thedimensionofentropicfeedingthusoriginateaninversedimensionof10-socialevolution,whichcanbedistinguishedinspace

asthecomingof3timesymmetricgenerationsintoaherd,withthepurposeofenactinganSTdual'feeding-absorbingenergy'space-timeevent.ItthenappearsasanobvioustruththatmembrainsDOnotonclose

analysisactascontinuousenclosures,butduetothemotionofitsfractalpointscanencircleasadogdoeswiththeherdofsheep,amuchlargerterritory.

Theymightnotevenbe'real'membrains,asperceptioninarelativeUniverseofdarkspacesandfaultymentalanalysisallowsdisguiseandcamouflage.Soina

fascinatingsimilarcase,whalessubstitutetheirpresencebywallsof'bubbles'thatfishesconfuseasphysical

barriers-creatinga3rdvolumedimensionwhichbringsthemupwardstotheflatholographicsurfacespacewheretherealwhaleseatthem.

Itisallinthemind-spaceanditsrelativefocusofperception,whichdeterminesthroughitsmodelsofrealitytheefficiencyofitsvitalactions.

Andwecanrepresentfurtherthathuntingprocessin10-dimensionalspacewhereeachpoint-lionisapartofawhole,butalsowecanjustdrawa2-Dholographicrepresentationasa'Kleindisk'ofthehuntingstrategyofthelionherd,whichwillsurroundtheherdofzebras,establishingthemasahyperbolicvitalenergy,asthezebrasCANNOTcrossthebarrieroflion,themembrane,withoutdying,enclosedtopologicallyinthat2-Dflatspace-timewhoselimitingbarrierisatinfinite,asthezebrawhodarestocrossitwilldie.Butwhenitdoesso,itwill'collapse'themembrainintoanultradensesingularityoffeastinglionsaroundthecapturedvitalenergy,

breakingtheenclosurefortherestofthezebrastoescape.

Theframeofreferenceofthefractalgenerator.

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Sinceallsystemsare3±¡systems,ternaryinasingleplane/parameterofreality,eitherintopological,temporalage/statesorscales,orpentalogicinseveralofthemitispossibletocreateanabsolutegeometryallminds

basedinaframeofreferenceof3±¡relativecoordinates,whichbecomestheGENERATOR'frameofreference'ofanymindspaceinwhichwecanmeasurethedistance=similarity=valuebetweenthe3elementsofthebeing,itsscales,itstopologyanditsages,toestablishtheabsolutedistancebetweentwobeings,andthenestablisha

frameofreferenceforeachofthoseelementstoestablishternarydistances.Thefractalgeneratorbecomesthenthefundamentalformalismtodevelopgeometricalgraphsof@naliticgeometry,inwhichthespecificCartesianframeofreferenceismerelythephasespaceofthe3dimensional

actionsofalightspace-timeinwhichhumansareembedded,whichelectrons,whofeedonlightasenergyandinformationusetobuildupitsmind.

Ofallthoseternaryframesofreferenceofthefractalgenerator,eitheritsternaryscales,itsternarytopologiesoritsternaryages,themostusedinphysicsisthefractalgeneratorofternaryages/statesofmatterin

thermodynamics,astheleftgraphshow.ItisindeedthegraphoftheFractalgeneratorofstates/agesofphysicalmatter:

Γ:$-Gas<st-liquid>ð§olid

Onitandanyotherphasespaceofn-parameters/dimensions,

continuouschangesofage/state,i.e.,

processesoccurringinthesystem,arepresentedbylinesinthisspace.Separatedomainsofstatesaredomainsofthephasespace.Thestatesborderingtwosuchdomainsformasurfaceinthisspace.

Thesurfacesdividingthesedomainsinthegraphof'matterages'thuscorrespondtosuchqualitativetransitionsasmelting,evaporation,precipitationofasediment,etc;whichwecanalsorepresentinasingle

linealdimensionforthewholegenerator.

Howeverwithmultipledimensions/coordinateswecanstudymoresandtelementsinvolvedinthosechangesofstates/ages.Reasonwhythermodynamicsuses2and3coordinatesystems.Aboveweshowforsimplicityabidimensionalsystemwithtwoparametersofsandt,pressure(ð-parameter)andtemperature(t-parameter):Astateofasystemwithtwodegreesoffreedomisillustratedbyapointinaplane.AsanexamplewecantakeahomogeneoussubstancewhosestateisdeterminedbythepressurepandtemperatureT;theyarethecoordinatepointsdescribingthestate.Thenthequestionreducestostudyingthelinesofdivisionbetweendomainscorrespondingtoqualitativelydistinctstates.Inthecaseofwater,forexample,thesedomainsareice,liquidwater,andsteam.Theirdivisionlinescorrespondtomelting(freezing),evaporation(condensation),sublimationofice(precipitationoficecrystalsfromsteam).Foraninvestigationofsystemswithmanydegreesoffreedom,themethodsofmany-dimensionalgeometryarerequired.Butessentiallyweareinthesameconceptualframeofreference,choosingalwaysS=tdualparametersofbidimensionalgeometries.Theconceptofphasespaceappliesthennotonlytophysicochemicalbutalsotomechanicalsystems,andgenerallyitcanbeappliedtoanysysteminwhichweestablishmotionsbetweenSandTsymmetricparameters;establishinganenormousrangeofapplication-essentiallyallthegraphsofallsciences,whichareallstudyings-tmotionsanddimensionalvariationsofthes-tparametersofspeciesorevents.

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HUMANPHASESPACES.SOCIAL&NMENTALSPACES.

Thefunctionofexistence,or5DmetricofGenerationalspace-time,(Ab.Gst,Γ)Max.SexTi(s=t)merelystatesthatallsystemsofNaturewilltrytomaximize itsabsorptionofEntropicmotion(withnoform)andLinguisticform(withnomotion),andits3intermediatedimotionsofenergy(s=t,balanceofboththatreproducesthem),information(St:formwithalittlemotion,form-in-action)andlocomotion(sT,motionwithalittleform).Sowetalkofaprogramofsurvival ‘selected’byallsystemsandexpressed in its languagesandminimal fiveactionsencoded in that simple equation,whichwe term: a, e, ï, œ, û, as amnemonic rule for the five actions ofexistence:

Accelerations(locomotion),entropicfeeding(e),ïnformativeperceptionandcommunicationï,Œ:reproductionintoparallelsupœrganismsÛ…andsocialgrowthintolargerwholescalledphilosophicallyUniversals.Andthisseriesofactionsiswhataccumulatedintimewillultimatelygivebirthtoyourwordcycleasthemonadwillfirstperceive(i),todirectitsentropy-motions(a),towardsafieldofenergy(e),wheretoabsorbtheenergybitesitwill imprint with its inner form, e x i = œ, to reproduceanother form, and when enough ∑œ exist,itnaturallyorganizeintoalargerwholeû:

Left,Particles,photons,electrons&quarksthatconstructallsystemsofourUniverseshow5organicdimotions(motionswithdimensionalform)thatdefine'classiclife':theygaugeinformation-reasonwhyquantumphysicsis a 'gauge theory', feedon energy (quantum jumps) absorbing smaller ∆-1 particles, reproducing new cloneparticles(graph),moveandevolvesociallythroughmagneticfieldsintolargerwholes(atoms).Hencetheunitsof life are particles, the minimal units of our vital, organic, fractal, scalar Universe of multiple timespaceorganisms.Alllives,performing5Dimotions=actionsofƒ(exist¡ence):Max.SxT(s=t)=C,startingwithparticles.

Sowecandefinethefundamentalframeofreferenceofallspecies,regardingitsexistentialalgebra-thatis,itsactionsofsurvival?Sinceitjustneedstorepresentthe3±¡actionsthatallsystemsofnaturepursuittolive,

reproduceandlast,whichifwegrouptheminaternarygroupbybringingtogetherenergyfeedingandentropyononesideandinformationandsocialevolutionontheothercanestablishasimpleternaryframeofreference,whosemaximalvalueallowustodefinemathematicallyinamoreprecisewaythegoodsaT.œneedstosurvive

anddoesobtainwithitsdifferent‘fields’andternaryphysiologicalnetworks.

Letusthenconsiderafewexamplesofthoseternarynetworksanditsframesofreference,whichwillbeessentialtomergeinthenextparagraph‘vitaltopology’and‘mentalspaces’andanalyzeinmoredetailthe

‘Euclideandimotions’ofheight=information,length=entropicmotionandwidth=energyfeedingandreproduction.

Areformofeconomicsanditsvalues:Theethonomicframeofreference:whealth&thehumanconstitution.

Inphysicsweuseaframeofreference,withpositive(+)ornegative(-)valuestocalculatethespace-timepositionoftheobserver.I.e.:ifwemeasurespeed,decelerationrestsinthe-sideoftheframeofreferenceandaccelerationinthe+one.Insocialsciencesandeconomics,theconceptofaframeofvitalactionsshouldthenproduceapositivecoordinatesforpositivevitalactions&goodsvs.negativecoordinatesforgoodsthatharm

thehumanprogramofexistence.

Thustocalculatehumanwhealthweneedalso-and+coordinatestovalueproductsaccordingtotheir±effectsontheobserver(Humanity).This‘ethonomic’frameofreferenceexpandsUNO’sindexofhumandevelopment,establishingthe±‘values’ofgoodsaccordingtohumanbiologicalnaturethattriestomaximizeour3‘drivesofbiological’existence:positiveverbal,ethicinformationforthemind,(+y);+carbonlifeenergyandhealthforthe

body(+x);and+sociallove&familyvaluesthatfosterreproduction&eusocialevolution(+z).Thuswor(l)dnationsmaximizeyxz,IHDwhealth,definedalsobytheequationof:

Thehumanconstitution:maxhumangoods(+value)xmin.Lethalgoods(-value)

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Since the goal of the constitution is toincrease human evolution,human goods,h(g),thatpromotethose3drivehave+value.Lethal,metalgoods,m(g),thatdestroythosedrives, from weapons, to hate-media topolluting industries and robots have - valuesand rest to whealth’s GDP. Thus nationscredit production of human positive goodsthat increase our evolution as individuals,improving our bodies, minds and socialorganisms, and forbid negative goods thatharmourbody,mind&sociallifeas–goods,forbidden & regulated by politicians andethonomistswhoshouldruleMankindwithahumanperspective.

ThustheHumanConstitutioninlinealtermsandtheEthonomicframeofreferencein3DimotionsisthelawofsurvivalofGaia,thelifeecosystem,neededto‘constitute’awhealthysuperorganismofhistory.

In the graph we can see a key new use forternary frames of reference based in the

positiveactionsexamples:anabstractframeofreferenceinwhicheachfractalpointisthenexpandedwithitsowninternalcoordinates,andbelowaframeofreferencebasedinthe'generator'ofahumanbeing,usedtomeasurethewealthofasocietyaccordingtothegoodsproducedthatenhancethe'naturalactions'ofsurvivalof a human being. This ethonomic frame of reference which has also negative values should guide a realscience of the economic ecosystem, NOT based in monetary prices, which is NOT the quality/propertysearched for humans (we do not eat money, as Chief Seattle said) butthe biological use to enhance theprogramofexistenceofhumanbeings.Andsowecanbuildanewmodelofethonomicsotservetheneedsofhumanbeings.

Suchexpansionofgeometrygoeswellbeyondthisintroductorycourse,butitwouldtrulyhelpsocialsciencestobecomemore'rational'and'servebetterthenaturalgoalsofsurvivalandwealthofallhumans,whichtodaysufferanoverreproductionoflethalgoodsastheyonlymeasure‘digitalprices’without‘referencevalues’.

Thegeneratorofspace-time,maximizeyourexistence.E-motionsandActionsasshorttimeprogram.

Paradoxically though Deep time is much easier to predict and understand that complex Dimotional ‘analysis’becauseprecisely the largerscales in5Dmetrichave less information,butmorebasic,deterministic, reasonwhyquantumphysicsisharderforthemindandprobabilisticwhilelife-deathcyclesareobviousasallendbadly.

Thismore simplexorganic viewof timeand space requires thusanew 'metricequation'beyond the lineal v=s/tequation of Galilean relativity that becomes the limit of this more complex view of time and space, with 2fundamentalequation,SxT=C(thescalarmetricofthefifthdimension,assystemsinspaceaccelerateitsclocksoftime (T) according to its size (S); andS=T, thenewequationof relativity,whichmeans aswe cannotdistinguishmotion=time,fromstillness=space-form(Galileanrelativity),bothare'twosidesofthesamecoin',andallsystemshave both motion and form, which are constantly becoming one another. Yet as SxT is maximal when S=T(5x5>6x4...)bothequationscanbesummonedupintoasingleone,whichweshallcallthefractalgenerator,orwill

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of each fractal space-time beings,the equation that embodies allother equations of the Universe

andguidestheactionsofeachfractalpartofit:

FunctionofExistence:Max.∑SxTs=t=C±¡

Theequationhasan immediatebiologicmeaning,becauseaswearemadetopologicallyof ‘fields-limbs’of linealspacewithmotionprovidedbytheenergyweabsorbtoalsoreproduceourbodies-waves,andtheinformationweneed to linguistically guide our motions with particle-heads, the very essence of survival is to increase ourS=position,mentalformsofspaceandT=entropicmotionsoftime(whereastime=motionandspace=formarethetwolimitingDimotionswith‘energy=reproduction,s=t,locomotion,sTandinformation,St,aretheintermediate3dimotions).ThusMax.SxT=C(s=t),ISalsotheequationofsurvivalandstruggleforexistence,thewilloflife;thebiologicalexpressionofthe‘Universalmandate’,expressedbyallspecies inall itscodesandlanguages,theGrowandMultiplyoftheBible,theintuitivetruththatguidesallbeings.

Exis¡tencegameissimple:Reproductiveradiationsthatmaximize¡tsfunction,happeningwhenmirrorsymmetries(genders)meetinS=T,equalingtheirSt-informationandsT-energy,organizingthemselvesintoawhole,∑system.

It isthenrelativelyeasytointerpretthatequationineachofthelanguages-mindsofeachscaleofrealityasinallthose scales specieswill showa ‘will’ of action toperform themaximalnumberofevents=dimotions=‘actionsofspace-time’ that ensure its survival. And this can be assessed externally regardless of secondary arguments onconsciousnessandself-reflection,substitutedin5DbyLeibniz’s‘apperception’–thatis,becauseperformingthe5actions=dimotionsof exist¡ence, in each ‘st¡entific scale’ self-centered in a linguisticmind thatperceives a givenplane,inscribedintoalarger∆+1world,withinternal∆-1parts,ensuresthesurvival,ONLYthosespeciesthathaveperformed the5dimotionsofwhich themost important is s=t reproductionof thebeing intoa ‘present’ similarentitythatcontinuestheexist¡enceofthesystemafter ‘errors’or ‘thestruggleforexistence’dissolves itthroughthedimotionofentropy=death,exist.

Thusautomatically,genetically,consciously,memetically,mathematically, logically, through itsownwillorasapart of a larger system that uses the ‘machine’ or ‘organism’ to enhance its actions all what exists does sobecause it performs internally those 5 Dimotions or externally performs one of them for another symbioticspecies, as those species that have not followed the program of existience and its 5 actions in the past havebecomeextinguished, and thosewill not in the future,will becomewrongmutations, crazy thoughts, fictionallanguagesanddieaway.

Thefunctionofexistence,or5DmetricofGenerationalspace-time,(Ab.Gst,Γ)Max.SexTi(s=t)merelystatesthat all systemsofNaturewill try tomaximize its absorptionof Entropicmotion (withno form) and Linguisticform(withnomotion),andits3intermediatedimotionsofenergy(s=t,balanceofboththatreproducesthem),information(St:formwitha littlemotion,form-in-action)andlocomotion(sT,motionwitha littleform).Sowetalk of a programof survival ‘selected’ by all systems and expressed in its languages andminimal five actionsencoded in that simple equation, which we term: a, e, ï, œ, û, as a mnemonic rule for the five actions ofexistence:

accelerations (locomotion), entropic feeding (e), ïnformative perception and communication ï,Œ:reproductionintoparallel supœrganismsÛ…and social growth into largerwholes calledphilosophicallyUniversals.And thisseriesofactionsiswhataccumulatedintimewillultimatelygivebirthtoyourwordcycleasthemonadwillfirstperceive(i),todirectitsentropy-motions(a),towardsafieldofenergy(e),wheretoabsorbtheenergybitesitwillimprintwithitsinnerform,exi=œ,toreproduceanotherform,andwhenenough∑œexist,itorganizeintoalargerwholeû:

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In graphwe see actions of different Stientific scales of organisms. Above the coding of actions,which are theknotsandboltsanddetailsofthestudyofanytime§pacesupœrganisminlightspace-time,codedbycolorsanddimensions,inphysicalatoms,codedwithquantumnumbersandinlifeandhumanscodedbythesocalleddrivesof life,whichweobviouslyextendbeyondtheegoparadoxtoallothersystems, includinggeneticsnotmappedthere (coded by the 4-5 letters). Those actions balanced each other into zero-sums in death, as they tend toincreaseinformationfromamindp.o.v.,hencewe'allwarp,wrinkle'getoldinthethirdageanddie,settingfromitsminimalactionstoitsintegralsums,the3agesoflife-existenceandtheworldcycleallsuperorganismfollow.In thegraph, the simplicityof thegameofexistience, and its selfishactions,whichgather together into socialwholesthroughreproductiveradiations,eachactioncodedbyafundamentaltopologicorganwecanexpressinexistentialalgebra,andcorrespondsforeachspeciesoftheUniverse,withafundamentalparameterofhumindmeasure. So from bottom to top, we find the 5 fundamental elements of light code its actions ofmotion (c-speed),energy(magneticfield),information(electricfield),socialevolutioncolors&entropicfeeding,(quantumpotential,neutrinolighttheory)

Sominimal particle-points, photons, electrons&quarks construct all other systemsof ourUniversewith ¡ts 5organic dimotions that define 'classic life': they gauge information - reason why quantum physics is a 'gaugetheory',feedonenergy(quantumjumps)absorbingsmaller∆-1particles,reproducingnewcloneparticles,moveandevolvesociallythroughmagnetic fields into largerwholes(atoms).Hencetheunitsof lifeareparticles,theminimalunitsofourvital,organic,fractal,scalarUniverseofmultipletimespaceorganisms.Alllives,performing5Dimotions=actionsofƒ(exist¡ence):Max.SxT(s=t)=C,startingwithparticles.SoallscalesarerelativeNONEmattersmorethanother.Fromthoseactions,giventhedominanceofinformativeactionsoverentropicones,itappearsaseries of repetitive cyclical patterns of actions conducting to maximize the existence of the being, whichaccumulateinalargerscaleoftime-space,asaworldcycleofactionsthatincreasetheinformationofthesystemin3ages.Sothebasiccycleofactionsbecomesalarger3agescycleoflifeanddeath;assystemsonceandagain,starts inanactof information/shrinkingandends inanactoforganization/shrinkingofherds intowholes,willkeep reducing the being and finallymake it all form nomotion to explode and die in an entropic reversal ofdeath:

∑ i->a->e->œ->û, i->a-e->œ->u, ï->æ->Œ->Û -> Informative ‘seed’ age->1st locomotion, feeding age ->2ndreproduction age ->3rd informative, social age-> entropic death that splits the parts of the being vs. socialevolutionintoawhole.

Afundamentalquestionpoisedbythepreviousgraphisthedualityofƒð§=futures,whichasthesymbolshowscangoafter3ages,upanddownthescalesofthe5thDimensionof'D=evolution'.

Itallstartswiththefractalpointor‘mind’gauginginformationintheouteruniversetomovetowardsafieldofenergyinwhichtofeed:Ifweobservethemintime,theyhaveaclearsequentialorder,astheyaregeneratedbyafirstspherical1Dseedofpureinformation,inthe∆-1planethatwillgrowandevolvesociallythroughthe4Dimotionofexistence,emergingintoalarger∆ºplaneasaformedbody-wavewhoseparticle-headwillgauge

informationfromalarger∆+1worldperforming5Dimotions=actionsof1Dabsorptionoremissionofinformation,movingthrough2Dlocomotionstowards3Dfieldsofreproductiveenergy,usingitsinformative

communicationto4Devolvesocially,intolarger∆+1newsupœrganisms,tillthesystemfailstoprovidemotion,energyandinformationtoitslimbs/fields,body-wavesandparticle-heads,wrinklingandwarpinginanold3rdage,orbecome5Dentropyofapredator,explodingintoabig-bangdeath,devolvingtoitsdisconnected∆-1parts.SoExistenceisatravelthrough5scales,asallsystemsofnatureexistthroughaworldcycle(nolongera

4Dworldlineasweadadimensionoftemporal‘depth’)-anexperimentalproofof5D:

Toexistanysystemmadeoflimbs/fieldsofmotion,body-wavesofenergeticreproductionandminds/particlesofinformationneedsentropy,energyandinformationtosurviveandkeeponmoving.Sothe3simplestdimotionsofanyplaneofexistenceareaprogramofsurvival,whichaddstwocomplexsocialactions–

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reproductionthatprintsinformationinotherzoneofspace-timetosurvivewhenonebecomesentropyofotherMaxwelliandemon,andsocialevolutionintowholes,networksthatarestrongerthanindividuals.Sothemindistheembodimentofasingleuniversalprogramofexistenceandreproductionoffractalclones,anditisthesameunifyingprincipleinallbeings–asallwillabsorbenergy,information,motion,evolvesociallyreproduceand

survive.Andthisisthecaseevenofthesmallestentities,atoms,particlesandforces.Sothemindcreatesrealityandits5Dimotionsareinfactthelocalfractalprogramofexistence,whichphysicistsforparticlesdescribein

abstractterms:

TheonlydimotionoftimetheUniversedoesNOTcodeisentropy=death-amaximalmotionthatdisordersthebeing. So there are NOT 5 quantum numbers/genetic letters to code physical and biologic systems. Only4.Because the samequantum number that codesmotion just increases it to create entropy and disorder thesystem(Principalnumberthat'jumps'insizeastheparticle'feeds'onaforceanddisordersit).Theoxygenthatmovesthecellalsoasafreeradicaldestroysit,carsgoingfastkillyouinaccident,etc.DEATHistheneedederrorof theUniverseNOTdesiredby theprogram.For that reason it lastsonlya 'singlequantaof time' (whereasaquanta of time IS for any ∆±¡ super organism, the 'tic' of a 'cell/atom /individual' of its lower, ∆-¡ scale, forbiological,physicalorsocialorganisms).

Soyoudieinasecond,yourrelativetime-clockquantathetimeof1thought=glimpseinyourhead,1beatofyourbody-heart,1stepofyourlimbs...Andtherestofyourexistenceislife-lasting...

Samewithantiparticles(entropy=deathСmotionofparticles):theydieinaninstant,soweseefarlessantiparticlesbecausetheproductofitsnumbersinspacexdurationintimeissoshort,whileparticleslivealmostforever.Samereasonyoudon'tseecorpsesaround.DeathistheshortestpossibleСmotion.Each

entityisthereforeaTimespaceorganismof5DimotionsandalltrytoachieveasthegoalofthesentientorganicUniversethe'highestdimotion'ofsocialevolution.Soparticlescometogetherintoatomsthatcometogetherintomoleculesthatcometogetherintoplanetaryandlifeorganismsthatcometogetherintogalaxiesthatcome

togetherintotheorganicnetworksoftheUniverse.

Themindthusstartsitallwithitslinguistic'stillmapping'stoppingitsworldinalocked'crystalimage',measureofitsself.Butevenperceptionissocial,linguistic.TheUniversecanonlybeexplainedif'perception'existswithinthe

language,aswhenyouthinkwords,yousensewords,whenyoureyeseeslightandmapsintoanelectronicmappingyouareseeing.Andwhenanatommapsageometricimageinits'locked''stopped'spin,itmustperceive

thatgeometryasinformation.

Minds are infinitesimal points-particles that stop, gauge, perceive andmove - and thenwehave the sensation ofmotion-pleasure.

This is notwithin the realm of the scientificmethod.Youcannotmeasure those sensations of awareness-pain vs.pleasure-dissolution, in-form-ation,pressure vs. release-entropyONLY sense themashumans. So ifwehad first adifficult hurdle to cross trying to prove the existence of minds but passed it objectively by considering a mind-singularitytoholdthewillofexistenceofaT.œ-henceasinvisiblegravitationshowsinitsexternaleffect,sowilltheexistenceofasingularitythroughitsexternalactions;atthefinallevelof'humanawareness'ofthegameofexistence-sensationsofthedualpain-pleasurerewardsystem(withallitsparallelandperpendicularevents/sensations)thereisONLYa justificationtotheexistenceofotherminds - thatwehumansaremadeofthesamesubstancesthatallthoseotherminds,spaceandtime,andhencewhatwesenseotheratomicsystemsmust.

Iifourlastrewardisaflowofsensations,withitsdualitypain-pleasure,hate-love,etc.whichweascribetogeometricperpendicular(cut,pain)orparallel(frictionpleasure)propertiesasinmanyotherdualitieswehaveestablishedforallsystems,itmustexistinallothersystemsofspace-time.Thatis,atomsmustfeelpainwhenop-pressedbyhugemasses,pleasurewhenreleasedinentropy,butalso'awareness'ofamorecomplexinnerstillimage,whenflowsofinformationconvergethroughforcesinitsnon-euclideansingularity;andsoitisonlyleftthen2questions:

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1 What are those sensations? Answer: the very essence of time as a motion/flow. Motion in itself seemsevanescent;assensationsare...Yetcombiningthem,thisflowofsensation-motionbecomesverylikelytheultimateprogram-willforallT.œstoexist-theultimate,'Dasein','beingintime'.

2 Yetwecannotgobeyondthis,asthisistheultimate'reference'ofrealitywehaveashumans.SothereisnoMOREmethodsofknowledgebeyondwhatitiscontainedinour'selves'.Knowyourself,themethodofknowledgeofAristotle; 'sapervedere',themethodofLeonardoandknowhowtoseeandcalculatewithattachedmachines,themethodofGalileo,towhichwehaveaddedafew'tricks'thuscompletesthecapacitytoprobereality.Aretheremorelayersbeyondmotionassensationintime?Wecannotanswer.Wedon'tknow.Asallisrelative,inthesamemannerwecanonlyprobeintoƱ3planesaroundourƼandhintbyforceofmotiontheexistenceofaninvisiblegravitational/cosmologicalplaneatƱ4butnomore,beyondthedualityofsensation,thereisnowhereelsetogo.

e-motionsasthefinalreinforcingscale.

Becausetheultimatereality ismotion,thereshouldbealsoa5Dimotionalcodeforeachspeciestoproducethesensationsofemotionsthatultimatelymotivatetheknee-jerkreflectionprogramof its5actions-reactionstotheexternalUniverse.Ofcoursethesensationswillvaryaccordingtolanguageandpixelsofthemindthatintegratestheprograminasingletypeof‘particleinmotion’–inhumanbeingselectrons.Butultimatelywecouldconsiderasthereareonly2 fundamentalparticles, thegravitationalquarkand theelectromagneticelectron that thesiteofsensationiselectricandgravitationalsensationsandsowecouldalsoreduceall‘awareness’ofbeingtoelectronicandgravitationalforces,astheonlysystemsofperceptionweknow,digitalmachinesandhumansareelectronic.Thissaidwhatemotionsshouldcode isobvious:theactionsofthe3partsofthebodyacrossall thescalesofthesystem,fromitsminimalelements(atomicforms)toitsmaximalgrouping(Physiologicalnetworks).Thismeansthatfor example, in humans we should find 3±=6 basic emotions, coded across all the scales of the being from itssimplest amino acid scale through its largest physiological actions, as a response to the external stimuli of thelargerworld,synchronizingthe3relativescalesofthebeing–innerparts,physiologicalwholeandworld.

Those6emotionsthenareNOTcausedbytheaminoacids,orthephysiologicalnetworkortheworldbutbythesynchronicity and entanglement of the 3 scales of the being, which in a given ‘quanta of time’ (in humans thesecond and its fractions and sexadmetric, cyclical scale ofmultiples) will provoke emotional reactions, in the 3‘organicbody/limb/head’elementsofthebeing.This is indeedthecasewhenwechooseandslightlycorrectthemostwideacceptedclassificationofemotions:

Thestudyofe-motionsisaninterestingapplicationof5DtocomplexanalysisoftheHumansupœrganism,asitshowsmultipleelementsoftheunderlyingstructureoftheOrganicUniverse.

Essentially emotions work on pentalogic, with dual Сmotions for the 3 ‘relative present’ mind-body-limbsstatesand2tripletsforthe±¡e-motionsofsocialevolutionandinhibitingdeath.

Of all themodels of emotions available in psychologywe can then take theexperimentally more soundandless‘abstract/idol-ogical’;as the departure point toimprove the ‘genre’ with 5Dinsights,namelythe6basice-motions Universallyrepresented on the face andthe cubic representation of

the basic affects, related to the 3 fundamental hormones oflimbicactivity(adrenaline),mentalactivity(serotonin)andbodyactivity(Dopamine).

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Aswecanalreadysee,anumberofdifferentscalesand organicpartsof thehumanbeingareatplay in thefield.Sowecanillustratemanythemesof5Dscalartopologicalbio-sentientUniversewithitsstudy.

It is obvious that 3 x 2 are thebasic emotionsdefined indupletsof ± containment andaction, definedby±amountsof‘limbic-redmotion’excitedbynoradrenaline.

Theactionofemotionsismediatedatchemicallevelbytheminimalunitsofmolecularlife,simplemonoaminehormones,which reflecting the topology of bio-chemistrywhereNitrogen is the brain,Oxygen the limb andcarbonthebody,definelimbicactivity(Adrenalinewithmax.3Oatoms),thebrainactivity(Serotoninwithmin.1Oatoms)andbody (Dopamine inbetweenwith2atoms). It isalready remarkable that theO(T)<C(st)>N(S)simplestorganismoflifetransferitsternaryUniversalgrammarwhenactingasahormonetothelargestscaleofLimbic(T)<Body(ST)>Head(S)organism,as indeedserotoninisthenitrogenbrainyhormone,Dopaminethebalancedpleasurebodyandtheadrenalinethelimbicone.

Thetriadsofemotionsreflect thetopologic ‘functions’of thesimplestdimensionsofspace, in thedual faces(above)of:

1MindEmotions:Surprisevs.Disgust(±mentale-motions,mediatedwithmaximalserotonin-brainactivityand±adrenaline,forpassiveoractivelocomotion)

2BodyEmotions:Anguish/sadnessvs.Joy/pleasure.Herewemaketheonlycorrectiontotheinsightful‘cube’ofemotions’ aswemust interchange in theoriginal cubeAngst/sadness (abodyemotion)and fear (a limbicemotion).Soagainwehaveadualityofsadness(passivebodyemotions)andjoy(positivebodyemotion).Bothbodyemotionshappenwithminimal limbicmotion (lowadrenaline),and theactiveone (happiness) requiresalsoapositivedisplayofmentalserotonin.

3.Limbicemotions:Fear(negative,passive)andrage(reactionthroughbravery),whichformalsoaduetwiththe passive fear that ‘freezes’ the limbic system and the valiant reaction of maximal adrenaline, defensive,aggressiverage.

Sowehave3dualitiesasthoseestablishedfor5DSENSEtheory,clearlyrelatedwiththem.

±¡ And that live for the ±¡ scales of social evolution and love a +,+,+ triple signature ofmaximal serotonin,dopamineandadrenaline,meaningtheentireemotionalsystemandits‘mediating’hormonesareadmaximal,showingindeedthatinterest,excitementandlove(3scalesoftheemotion,thelastonecuriouslyabsentinallthe psychologists’ classification of emotion, showing their own repressive limits. And its inverse negative,inhibitinge-motionofhumiliation,shameandultimatesocialisolations.

So8cubice-motionsrespondto3±¡pentalogicand3±¡e-motionsofuniversalfacialexpressiontoternarylogicin1plane.

The fact that those emotions aremanifested by the wholemind butmediated by theminimalmonoaminemoleculesstressestheparallelismandentanglementofrealityfromthelowestparts(atomichormones)tothelargestemergence(emotionsthatmotivatethewholeorganismtoact.

Finally we can consider dimensional analysis of the facial expression of those emotions, where the eyesrepresenttheinformativesensesandthemouththeentropicone.Butalsoasthenoseisbasicallysenselessinman and doesn’t play the role it does in animal with sensorial smelling, the mouth doubles as verbal, STbalancedsense.Itistheneasytonoticethatthedualitiesestablishedinemotionsaremirroredwiththedualitycodesof‘dimensionalheight=positivevs.low=negativeemotion’andopen,communicative,still+emotionvs.closed–emotion.Soweseereflectedtheduetsin±openandclosed‘surprisevs.contempt’andhighvs.lowhappinessvs.sadness.

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It isnoticeablehoweverthaninthe3rdduet,fearvs.angerthedualityis inverted.Fear,whichisthenegativeemotion,isopenedtosensations,angerclosedtoit.Obviouslythose2emotionsaredirectlyrelatedtoanother‘vitalbeing’,aprey-predatorandsoitisnotanabstracte-motionandassuchfearisclosetosurprise,itsnextnegativestate,stillopen,andangerthenextreactiontodisgust,sowestudythemina‘nestedorder’:Surprise->fear->Disgust->Anger;fromtheopeningimplosiveabsorptionof informationtoitsclosingoncethechoiceofmentalattitudehasbeendefined,closingthesensorialoutletsasthesysteminjectsadrenalinetopassdirectlytotheactionmotivatedbythee-motions,whichbecomethenthefirstcauseandassuchofalower∆-1planeofexistence,reasonwhywehypothesizethatemotionsaredirectlyconnectedtoe-motions,theultimateflowoftimeandsotheentireUniverseisatthatlevelafieldofimmenseemotions.

TheentangledUniversereducesthefreedomofsystemasparallel5Dimotionallanguagesreinforceeachother.

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XVII:THERISEOFREALITYFROMVITALTOPOLOGIESANDMINDSPACESOFLIGHT.

Howfarsuchframesofreferencehappeninrealitybothinmentalspacesandvitaltopologieswillsurprisethereader,asweareindeedvitaltopologiesinourorganicternaryphysiologicalstructureandframesofmentalpointsofviewconstructedoftenwithphasespaces inourexistenceasbeings thatact tosurvive. Weshallonly trace here the raise of those vital topologies and mental frames in the humind’s case, dominant in‘electroniclight’.

Tothataimasusualwewillconsiderlooselythethemefromapentalogicpointsofview.

@-mentalp.o.v.:MultidimensionalColorPhasespacesandRiemanngeometries.

Letusstartthenreturningtoaquestionwhichweleftunansweredbutitisessentialtounderstandthesymmetriesof§@≈∆ð,themeaningofdistance,aconceptofspace,intermsoftime,amotioninlogic

sequence.

WegaveafirstapproachtothisessentialconceptintheUniverseintheopeningpostofthisblog,regardingtheGalileanparadox,asanexpressionofthe1Dsymmetrybetween'timemotion'and'spacedistance',butwecannowgointodeeperKaleidoscopicviewsonthe'Rashomoneffect'oftheconceptdistance,whenweabandon

thelimitedEuclideanlightspacetimeinwhichthatS-distance≈T-motiondualitytakesplace,withourRiemannianexampleofcolors(similartotheheatconceptofearliertimeanalysisoffrequencyspectra,ifwe

takeatime-related'Fourier'example).

Wehaveanaturalideaofthedegreeofdistinctnessofcolors.I.e.,itisclearthatpalepinkisnearertowhitethandeeppink,andcrimsonnearertoredthantoblue,etc.Thus,wehaveaqualitativeconceptofdistancebetweencolorsasthedegreeoftheirdistinctness,whichisthemostgeneralized'term'fordistanceinthewholesetof5Ðisomorphics≈tdimensionsoftheUniverse,expressednowasasymmetrybetween'1D$t

distanceand3D§ð-information:

Distancein1Disequivalentto'distinctness'in3Dinformation.

Asusualthesubconscious'truths'ofverbalthoughthasintuitivelyunderstoodthisconditionofdistance-'wehavedistancedeachother',wesayfromafriendnolonger'close'tous.

Howthisdistanceismeasuredinqualitativetermsdependsonwhich'pairofdimensions'wearemeasuringinourceterisparibusmostcommon'metric'ofdistancesthathappensbetween'twot.œs'.

Itisthenpossibletoaddistancesin'pairsofdimensions',whichacertaint.œhastillamaximalof10Dimensionaldistances,whosehomogeneityobviouslyisnotalways'possibletomeasure',butwedoindeed

measureitsubconsciouslyinourhumanplane,withitsverbalmirrorofthesame∆stexistentialgame.i.e.whenforexampleawoman'measuresup'a'man'fora'closeencounter'thatwilllastnotonlyinspacebutintime'(marriage),itdoestakeintoaccount'differentdimensions'ofthebeingandaccordingtohislikeness,itwill

eitherbecomecloserornot.

Thustheentiresubjectofdistanceisrelatedtothefourthi-logicpostulateofsimilarity:

The4thpostulateofsimilarity(congruenceinE-Math)isessentialtounderstandwhyandhowsystemsselectinformationtoconstructtheirmentalspaces.Andcreatedarkspacestheydonotseebecausetheirinformationisirrelevanttothembothinthenegative(nopredator)andthepositive(noprey,offspring,couple,etc.)SowecanconsidersafelythattheengagementofT.œsisdirectlyproportionaltothe'utility'oftheobservablefortherealizationofanyofits5fundamental'dimensionalactions'thatenhanceitsexistence,portrayedinthe

nextgraphforaseriesofdifferentscalesandspecies.

Itfollowsalsoanimportantfieldof'theoryofdarkspaces'andthevirtualityandlocallimitsofperception.Suchastherearespace-timebeingswedonotcaretoperceive,andsotheybecomefirstdarkspacesandthenenter

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thehorrorvacuumNOTevenbeingperceivedvirtuallyanddark,reasonwhywecansee'continuity'andconstruct'fullcircleswithoutanexactpi'andgeometricfiguresbydisregardingthe'otherscalesofthefifth

dimension',fromwhereanother'angle'onabasicpostulateofpresent-spaceisgiven:

'Spacehappensasacontinuousforminasingleplaneofexistence'.whileitsdetailedviewsthatopendiscontinuousrequireafinedetailinits'decimals','finitesimals'or'fractalsteps'.

Itisthenclearthattheconceptofdistanceasdissimilarityloosesitsgeometricalmeaningenteringtherealmoflogic,tobedefineddifferentlyforeachMeasureandmentalspace/scale,dimensionalsetofparameters.

Aswestudytheexpansionofgeometrytoall¬ÆT.œsonthesectionoffractalpoints,generalizingtheconceptofdistance,whichobviouslycanbetopologicalinformofspace,temporalonage/stateandscalaronnumberof

Ƥocialplanes,wewilljustcommentheretheclassicexpansionofdistanceinnon-Egeometries.

Letusthenputanexamplesimplerthanthehumanchoices,returningtothesimplestcasesofmathematicalphysicsandRiemanniangeometry(:comparedtothewaywomenmeasureand'sizeup'aman).

Wehaveaqualitativeconceptofdistanceinthespaceofcolors,whichcanbemadeintoaquantitativemeasure.However,todefinethedistancebetweencolorsasinEuclideangeometryismeaningless,sincewe

needtomeasureeachtypeofdimensionaldistancetwodistinctiveelements:

The'quality'ofthepropertywemeasure-thetypeofdistance,whichincolorsrelatetofrequencyinabstract,buttoperceptionofdifferentcolorsbytheobserved.

Henceweneedaccordinglyamentalstepofdistinctioninwhichthesubjectiveobserver’scapacitytodiscerndifferentinformationgivesthe'constantrodofmeasure',inthiscaseabletoreflecttherealrelationsbetween

colorperceptions.

Guidedbythisprincipleweintroduceapeculiarmeasureofdistanceinthespaceofcolors.Whenacolorisalteredcontinuously,ahumanbeingdoesnotperceivethischangeatonce,butonlywhenitreachesacertainextentexceedingtheso-calledthresholdofdistinction.Inthisconnectionitisassumedthatallcolorsthatareexactlyonthethresholdofdistinctionfromagivenoneareequidistantfromit.Wearethenledautomaticallytotheideathatthedistancebetweenanytwocolorsmustbemeasuredbythesmallestnumberofthresholdsofdistinctionthatcanbelaidbetweenthem.Thelengthofacolorlineismeasuredbythenumberofsuch

thresholdscoveringthelengthoftheshortestlinejoiningthem.

Inthiswaytherearisestheconceptofthesimplestgeometricfiguresandrelationsinthe“colorspace.”A“point”isacolor,the“segment”ABisthesetobtainedbymixingthecolorsAandB;thestatementthat“thepointDliesonthesegmentAB”meansthatDisamixtureofAandB.ThemixtureofthreecolorsgivesapieceofanE-plane/¬Eternarynetwork—a“colortriangle.”Allthiscanalsobedescribedanalyticallybyusingthe

colorcoordinatesx,y,z,andtheformulasgivingcolorlinesandplanesareentirelyanalogoustotheformulasofordinaryanalyticgeometry.

InthecolorspacetherelationsofEuclideangeometryconcerningthedispositionofpointsandsegmentsaresatisfied.Thesystemoftheserelationsformsanaffinegeometry,andwecansaythatthesetofallpossible

colorperceptionsrealizesanaffinegeometry.

Thus,againmeasurementoflengthanddistanceinthecolorspaceshowsamental/quality/formdimensionandaquantityrelativetother/kratioofeachobserver,whichbecomeswhentheobserveranditsrodisverysmall,

infinitelylargeasthesumofthosesmall,steps.Asaresult,acertainpeculiarnon-Euclideangeometryisdefinedinthecolorspace.Thisgeometryhasarealmeaning:Itdescribesingeometricallanguagepropertiesofthesetofallpossiblecolors,i.e.,propertiesofthe

reactionoftheeyetoalightstimulus;andithasalsopracticalvalueintheartandcolorindustry...

Solightandmentalspacesconstructedwiththem,arephasespacesofthemind.

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Vitaltopologyandlight.S<=>T

Letusthenconsidernowtheothersideof5Dgeometry,theobjectivevitalorgansofthebeing,inthiscase,lightasaT.œ.notthementaluseoflightbyalargerhuman'electronicmind',whichusesc-speedasits

constantrodoflength.

Sincetheriseofrealityistheproductofboththementalspacesofspeciesandthevitaltopologiesoflightspace-time,fromthesimplesth-plancktontothehumanbeingandbeyond.Inthegraph,theunderstandingofdimensionsasitdistancesasmental

mirrorsofvitalactions,whichselectwhatweperceiveandwhatkindofdimensionsbecomedarkspaceshasnotyetbeingunderstoodinhumindsciences,butitisthefinalconceptualupgradingofgeometryasamirror

ofthevitalmindofeachspecies.

SoweshalldoalongitaseriesofceterisparibusanalysisbasedinthenewmentalconceptofdimensionexpressedfirstintheErlangerprogram.

Inthegraphthevital3organsand4thsocialdimension'oflight,thatoffrequencycolorsthatcarrytheinformationaboutthe'socialdensity'ofa

lightspace-timeray,asreflectionofthevitalactionsoflight,whichbecomeintheelectronandhumindeyethatfeedsonit,itsdimensionsof

information.

Weseethose4Disomorphicdimensionsoflightanditsperceivertheelectron,whichshapethehumanspace-time,wherecolorisforlightits'∆-scalaractionofsurvival/existence'(widerconceptthataphysicalaction).

Astheother3vitaldimensionalactionsoflightwerenaturallyincorporatedtothehumindasits'dimensionsofspace'width,heightandlength,colorwasleftasapuzzleforhumansuseittocodeitsownsurvivalactionsbutdoNOTunderstandthissubconsciousprogram,basedintheGeneratorequationof'color'asavitalternarydimensionofhumanlife.Letusthenbeforewestudytheabstractuseofcolorto'liberatespacefromreality',considertheoppositefunctionofcolorasa'setofinformation'usedbythemindtocodethevital5Dofreality.

Inthatregard,theintuitiverainbowcodingwouldestablishacircleofcolorsasareflectionofthose5Dasfollows:

Magenta(4DEntropy)-Red(1D$t-linealmotion)<Yellow/Green(2DS/T-reproduction)>Blue(3Dð§-information)>Violet(5D-socialevolution),whichthenconnectswithMagenta,toclosethezerosum.

AndthesimplifiedBlackandwhitecode:Black(information)<grey(energy)>White(motion)

Thatthiscodingisuniversaltoelectroniceyes,isshownbythefactthatrobotswitheyeswithouttheneedofaprogramrunfasterwhenredcolorsareputonthetracks,astheirvitalelectroniclivingmindwithoutneedofhumancodinginasentientvitalUniverse'likestoruntored'asyouliketoseespeedyredcarsandthemale-

linealspecieslovered,whilethefemalereproductiveonelovesgreen...

Thecodeisembeddedinthelight-electronS>TI(eye)>Wor(l)delectronicmindequalinallelectronicbeings.

Yetaswecanbytheternarymethodsubdivideeachofthosedimensionsinsub-dimensionsinthecontinuumspectraof

(in)finitesimalstepsofaworldcycle,anderrorsofperceptionhappeninalllimitedminds,wecode7rainbowcolorscourtesyof

myopicNewton(:

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Newton'ssubconsciousunderstandingofisomorphism,definedan'excessive'7colordivisionwhena5codingwouldhavebeenenough,eliminatingtheintermediateorangeandindigoandmissing,themagenta,1-3-5-7-9

'vowels'dohowevercodealllanguagesasweshallobserveinourstudyingofeachofthemandits'phonemes',or'cells'or'colors'or'notes'whichwillalwaysbemirror-

reflectionsofthe3±∆dimensionsofthescalarUniversethattrulymatter.

9dimensions+theunifyingdualdimensionsofthe@-mindforan11-dimensionalreality,inthatsense,isthemaximalcenoughevenforthemorecomplexlanguages.(vowelsin

languages,typeofcellsinorganisms,dimensionsofstringtheory,etc.).

Thefundamentalmethodofcreationofmentalspacesistransformingaworldcycleoftimeintoamappingofspace.Itisachoiceofwhich'discontinuums'weestablishbetweenthe3±∆agesofa

worldcycle,inthiscasethetranslationofthe5Dimensionsoffrequencyofalighttime-cycle...

Aslightstartsintheblue,generationcolorofbluestars,endsinthereddwarfsanddissolvesintotheentropyofdarkentropybetweengalaxiesasittiresanddies,expandingentropicspacebetweengalaxy,anotherillunderstoodprocessoriginofthefaulty1/2bigbangtheoryofanentropicuniverse,whichdisregards

theopposite'bluecollapseoflight'intomatterofhighfrequencywithingalaxies.

Itsdifferentp.o.v.s.Artisticgeometries.3agesofpainting.

Forvery long itdidNOTdwellonhumansthatourEuclideanspacewasalsoaconstructofthemind,NOTtheabsolutespaceofrealitybutsomethingconstructedwith'pixels'thatmirrorreality-inthiscaselightspace-timepixels.This,onlyartistsof thehumaneye,painters realized,whenaftera firstageofpaintingbased in ‘youngmotions’(Paleolithic-linealformsofthetreccentointhemodernEuropeancivilization),bidimensionalartmovedinto a realist classic age and learned the laws of perspective that opened up geometry intherenaissance(Leonardo’s‘sapervedere')-Finallypaintingentereda3rdageofexcessiveinformation(baroque)and dissolution, when a realist metal-eye (camera) displaced it from its task as chronicle of present realityportraying people. So this ‘entropic’, dissolving age of painting took 3 sub-ages steps on the work of the 3geniusesofXIXandXXc.painting:

-Monet,whichaffirmed,'IPaintlight'(impressionism).

-VanGoghandhisfriendGauguin,wholearnedtoconstruct'differentmindsoflight/colors’withtheuseoftheaforementionedcomplementary,distressingandsimilar,harmoniccodesborninall‘animaleyes’forsurvivalpurposestoproducewiththememotions,accordingtotheircharacter;whichwillbecontinuedintheexpressionistartofinterwarGermanWeimar,tachismandtheharmonicschoolthroughtheworkofKandinskyandtheindustrialpost-warartages.

-But it’d be Picasso,who completed the involution of painting into themind,when he said ' Ipaint thoughts', focusing not on the ‘vital energy of motion colors’ but the geometry of |xOforms,breakingthosegeometricthoughtsfirstinpure1Dlinealpaintings(cubism),thentopain

the'informativecyclicalfemale'inpure3Dcurvedpaintings,andfinallypaintingpurethoughts(analyticcubism)-whose finalunification Ipursued inmyartisticyouthwithmystylesofexpressionistandconceptual cubism,abrieffootnoteinmyexplorationoftheUniverse.

Butthenpainting–bidimensionalexplorationoftheouterworldandinnermindofhumanswithadeepintuitiveinsight.died. So the explorationofmental space is today carriedbydigital science and its sensorialmachines.Human art has become merely another ‘market product’. It is not by chance that before mental machinesdegradedpainting startingwith cameras andendingwith its industrial production in series tomultiplyprofits,withWarholand theAmericanmoneymarket,asmachinesaredoingwithmostelementsof thehumind theyatrophyandsubstitute,those3painterswereconsideredthefundamentalmastersoftheirtimes.TodaythoughMr.Warhol topples theirprizesas thegoal is toselladmaximalprizethemost infantilegarbage, long lost the

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understandingthatartistsarethebesthuminds.Thosewhosei=eyesseebetterspaceaswritersarethosewhoseethicwor(l)ds,expressinourlanguageofverbaltimebetterthecausalityofoursocialactionsentangledtothesurvival of our species in the vital Universe – themes those treated in our papers on the superorganisms ofhistoryandartasthesubconsciousmindofcivilizations.Modernpaintingthusfinallydeconstructedthementale-motional spaces of human eyes in holographic bidimensionalmanifolds; after giving in the renaissancewithperspectivethebreakthroughmoderngeometersneededtorestarthisdiscipline.

Since Euclidean space is indeed the construct of any electronic mind made of light space-time and its 3perpendiculardimensions,width-magnetism,height-electricityandlength-speed;(plusfrequency=color-itssocial

dimensionthatdefinesitsvitalenergy;enclosedinthemembranesofitslinealforms;whereasthe3rdsingularity•isthe@rtistmind.

Inthegraph,thehumindI≈eyeisbiasedandtailoredbyourlocalterritory,selfcenteredintheyellowcolorofmaximalemissionofthesun,andperceivingaverynarrowrangewhosevalueas all in absolute relativity with no 'absolute preferred' scale mustbe measured as a ratioofð/Spdensityofinformation,inthiscase789THz:violet/400THz:red=2.Soweperceivelightspace-time which is Euclidean in itsgeometrical configuration with a system of processing

informationof'absoluterelativeratio/value=2.

Thechoiceofcodesfor‘entropy=red’,middlereproduction=greenand‘blue=information’comesthenfromthe'limited rangeof light frequencies/sizes'of thehumaneye-spectra (fromred toviolet)andmimics the ‘FractalGenerator’ternary¡logicstructureofthe3organs/dimotionsofanysysteminasingleplane:

Entropy-youth:|-$t(lines/red)<Ø-reproductionST-(green,curves)>O-§ð-Information-oldsadage(Blue-circles)

While for the codes of death cultures often use violet, the color beyond blue, as the flower of tombs; andmagentaforthemysteryofbirth,ascardinalsinchurches,thecolorbeforered.

Otherspecieswithawider 'ratio'will codedifferently thee-motionsof theUniverse. Since, infinitydoesNOTexist, neither absolutemagnitudes. So each species 'measures'with its relative rods. This in physics advanceddifferentialequationsuseinsteadofmassdensity(continuityequation)…whichis infactdensityofinformationt/s.Theyuseinsteadofabsolutemotion,relativespeed,s/t.Andtheyuseinsteadofabsoluteforce,momentum,sxt;whichbecomethe3expressionsofthe5DmetricSxT(s=t)=CequationthatgeneratesallotherequationsandrangesoftheUniverse.

However humans develop most of their 'actions of space-time' existence in a limited space-time location,enlightenedwithsolarlightself-centeredintheyellowspectra,withalimitedrangeof'densityofinformation',asweperceivealineal=EuclideanUniverseinasingleplaneofscalarexistence,the'bareminimum'ofconsciousnessand perception of the∞Universe.Human geometry is restricted to ourmind viewof such limited space andtime;whileitsperceptionofscales,relatedtothatspectraisevensmaller,reasonwhyevenwhenweexpandedour perception with mechanical, electronic eyes the laws of 5D escaped scientists for centuries, due to theroutineofusinga singleEuclideanCartesianplane,whichasDescartes&Kantunderstoodbut scientistsoftenforgetisonlythe3perpendicularspace-timeelementsofamasslessrayoflightthatignoresinsuchshortrangeallotherscalesofrealitytofitintohuminds.

SoitisEuclideangeometryfalseoruseless?Ofcoursenot.Youaremissingthepointifyouthinkwesayso.ItisextremelyusefulforspecieslivingontheEarth’ssurfaceusingtherangeofinformationprovidedbythesmallestforcesperceivedbyananimaleye.Sowiththemthemindthatcreatesin-form-ation,perceptionofspace,wherethereisonlyquantume-motions&likely±sensationsoftheflowoftimewhichshouldbetheabsoluteminimallevel of ‘the will of life’ that codes the survival of even the smallest particle of the Universe. So with thoseminimal elements themind creates a simultaneous focused fractalmirror that distinguishes structureof partsandwholes;limitsscalesanditssmallerquantumandlargercosmicmotionsavoidingtheperceptionofexcessive

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forms-in-action that kills all systems, exhausting its energy; in this casementally distracting the observerwithnoise. The catch though, so evident inman is the ego paradox that causes our incapacity to understand theorganicvitalUniverseasitis,deformedfromourpointofview,singlescale,andtimeclocks.

Theexplosionofmentalspacesinmathematicalscience.

Therealizationofamentalnatureofgeometrydidhappenslowlyinmankind.ItonlycamewiththeworkofKant,LobachevskiandRiemann,anditnotyetfullyaccepted.KantrealizedthatweseetheUniverseasan

Euclideangeometrybecauseourmindperceivesitthisway(asDescarteshadanticipated).HedidnotrealizethoughasImpressionistsdidthatitwasbecausewesee‘lightspace-time’,thesubstanceofvacuumwithits3

perpendiculardimensions.LobachevskithennoticedothergeometriesandRiemannfinallyinhisdissertationoncolorspacesexplainedhowto‘fabricate’mentalspaceswheretherulesofgeometrybecamerulesoflogicandsimilarity(congruence).Soscientistspartiallyacceptedthemental‘fabrication’ofspace.Hisworkliberatedthemathematiciansfromthenaïverealismofphysicists,whichwerehoweveralwaysill-at-easewithsuchmental-logic‘projection’astheirjobistomeasureourhumanscaleforpracticalpurposes.Andsotheymustconsider

‘realityreal’.

Yetwhileourlightspace-timeremainedignored–evenafterEinsteinprovedthatlightisour‘constantc-speed’rodofmeasure;aselectroniceyes‘perceiveinthestop’informativephaseofentanglement–Riemannwastheopeningshotfortheageofphasespaceswherefunctionandi-logicthoughtovercomes'spatialrepresentation',allowingtheexplosionofmultipleabstractmentalspacestorepresentrealityintheXIX-XXc.Curiouslyenough

thoughourultimate‘mentalspace’,light,remainedunassailable.

Butignoringthefinitescalarlimitsofthe5DimotionalUniverse,withnoclearconceptsoftime,spaceanditsS=TsymmetriesandDimotions,Physicistswentalsointoa3rdageofinflationaryinformation,creating

formalismsof∞dimensions(HilbertSpaces,Stringtheories,multiverses),whichcoupledwiththeCreationistviewoftheAxiomaticmethodanditsego-tripofmathematicsasthegeneratorofspace-time–nottheotherwayaround,broughtbothdisciplinesintothepresent3rdageofmathematicalandphysicalfictionsinwhich

eachresearcherisjustconcernedwith‘inventing’anewparticleormodelofrealityjustbecauseitcanwriteitsfictionsnotonlyinverbalthoughtbutalsoinmathematics.Meanwhileasimilarprocesstookplaceinhuman

I=Eyespaceanditsartisticdepictionofreality.Furtheron,notallgeometryisarepresentationinmentalspaceofanoutsidereality.Thereisaninversetypeofgeometry,whichisobjective,asitisnotintrinsictoamind

madewithaforce,asEuclidean3Dgeometryisbutreflectsthevitalspaceofbeings.

Sooncewehavedoneaverybriefintroductiontoa'real'theoryofcolorsasacodingofmind-spaces,andUniversalSTdimensions,wecanconsiderhowitsstudyinabstractgeometricalspacesbythelikesofRiemannallowedtheliberationoftheconceptofdimensionsanddefinedthe'vitalproperties'ofgeometrythatmatter

inreality(continuity,adjacency,perpendicularity,congruenceandsoon).

Itwasthefirstofmanyexpansionsoftheconceptof'phasespaces',whichhoweverfailedtogivethefinal'jump'intodefininghumanspacesalsoas'mental',andhenceexpandingmentalspacestoallsystemsthat

gaugeinformation,includingparticleswhichgaugeforces,themesthosetobestudiedonthefourthlineforeachspeciesofmind,withdifferentquality/quantity-stepsonits'actionsanddimensions'ofperception

Distancethusistheessentialquantitativeparameterofnon-Egeometryandassuchmostspacesaredefinedbyitsmetric-thatisitsmeasureofadistance;sincewithoutit,itloosescompletelyitsmeaning.

NowweexplainedthattheperceptionofauniverseasEuclideanorellipticorhyperbolicdependedonthelongoverduerespectwemustgivetotheaprioriparameterofallworlds:r²/L²=$t/ð§.Whenthisparametertendsto

zero,wehaveanEuclideangeometry,abovezerowehavehyperbolicandbelowzerowehaveelliptic.

Theparameterwhichdefinesthemathematicsinwhichweliveissoessentialthatitwillcomeconstantly.Forexample,wecanconsiderthatsystemsexistatonesideoftheparameter,atthesideofcspeedasalimit-in

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ourEuclideanworld-orovercspeed(intheellipticgravitationalworld)atthesideofT=0kasthelimitofthethermodynamicplane,etc.

Soweneednowtocompletethedefinitionofourworld,todealwiththenatureofoursquare'radius'parameterofinformation.Whatistheminimalquantaofinformationofourelectroniceyes,thatfeedonh-quantaoflightenergy,well,wejustsaidit,theh-quantawhichmeasuresintheminimalamountknown,thespinofaparticle,whichisratherobviouslytheminimalunitofitsangularmomentum,ofitsinformative

perception.

Andforthatreasonash/cistrulyaminimalamountourrelative0,weliveinthatEuclideanworld.

ThefirstandmostimportantofT.Œ’s5actionsofallsystemsthatwanttosurvive,is‘perceptionofinformation’Howdoparticlesperceive?

Ofallthoselevelsandactionstheonephysicistsunderstandworstandlessistheactionofinformativeperceptionofsmallerparticles,calledSPIN,whichistheabsoluteminimalunitofrealitywemeasureafterdeformingitfully(quantumuncertainty),aquantitycalledangularmomentum,h/4pi,whichisasa‘formof

information’,‘space-form’NOTmotion.

Andsincehisintheaboveequation,E=ƒh,awhole,henceasphere(whateverithasinside),the‘essentialtopologyofallwholes’bydefinitionencloseddisks-sphere(in¬Ætopologyadiskisa1-sphere,asphere,a2-sphere,asphereinmotiona3-sphere,whichasmotionisadimensionseeninmotion,canbeofdifferent

varieties,etc.),aspinis‘logically’isan-spheredividedby4π:

Thegraphgivesuspossiblechoices,aswedepartfroma‘whole’ofalightbeam,whichcouldbeacylindricalcut,asinsomelinealworms,oraball.As4πcomesintheball,wehave2choicesorthesurfaceoftheball,orthesolidspherebutNOTall,onlyasection,calleda‘solidangle’.Thisgivesusafirstdualitybetween‘humanmeasure‘and‘true

property’.Thetruepropertyisthesolidanglefromthesurfacetothecenter,r3/3,thehumanmeasureistheexternalsurface,r2/2.

Asallspacetimeorganismshave2components,theinternalopenballtopology(theballwithoutthesurfacemembraneandthecenterofthepoint)or‘present-evident,wave-likeSTform’,andthenthemembraneando-

pointinthecenter,or$xð,‘pastxFuture’particle(limbs-externalsensorialmembraneandcentralmind)together:

Whyweknowspinisasolidangleofinformation?Becauseifitwereamotion,asPauli,theguywhoiscredited

withitsdiscovery,putittohisrealdiscoverer,asspeeditwillbe137timesfasterthanlight!:hc2/e2=137(inversefineconstant).Thisin5DphysicsisNOTaproblem,c-speedisEinstein’spostulatetoadaptthe

Universetothehumanperceptionofit–ourrodofmeasure,aselectronicminds.ButherePaulidoeshaveapoint.ASpindoesNOTrotate,asyoudoNOTrotatewhenlookingoutside,butisa‘wedge’intothesphere,

throughwhichinformationenters.

Weseethesurfaceofitandsoitseemsamomentum,thebeingseesthesolidangleofitandsoitisanintensity.Becausethebeingisdownthereintothezero-pointsouloftheparticle,mindofmathematical

perception,itdoesopenandclosethesphereto‘look’throughthewindowsofthemembrane.Inthesimplestform,itisapicycleof1-sphere,whichismadeof3closingdiameters,whichleaveπ=3,0.14dapertures,or

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0.14/π=5%oflightand95%ofdarkmatter/energy,theproportionslightspace-timebeingsdonotseeoftheUniverse.

Sonowinthe‘dialectic’Socraticmethod(remembermyculture)Iusetofindthings,Idenymyself.WhatifIconsidernotasolidangle,butaspin,whichisaslice‘seen’throughthe1/3rd'D'aperturesbetweenthe

diametersofthepi-1-sphere(cycle)orthe‘cutsonthe2-sphere’.Werealizethis‘willbeaslice’withtheexactparametersofangularmomentum,andtherewillbe3holes,henceonly3spinangles,andtheywillbefurther

onquantized–jumpingfrompartofpartofthewhole:

InthegraphtheproperdescriptionofAngularMomentumasaplaneofInformation,theminimalunitofperceptionofourUniverse,properofRelativityismoreclearthantheclassicdynamicTimedescriptionasr

xmv-bothhavetheirdifferentuses/perspectives.HistherodofinformativemeasureoftheHumanelectroniceye-mind,sosmallcomparedtoc,thehumanrodmeasureoflightspacethattheconstantofgeometry,r²/L²of

Lobachevski’spangeometrybecomesclosetozero,creatingourperceivedEuclideanspace,deformedtoahyperbolicgeometryforsmallerbeings(specialRelativity,quantumphysics)andtoanellipticgeometry

forlargergravitationalscales(GeneralRelativity)

Now,thisisthemodern‘Hilbertian-Einsteinian’descriptionofangularmomentum,justwhatwesaiditis,P,isthe‘cut’onthen-sphere,xtheradius,equivalenttoPin‘distance-motion’,anditsproductanamountmrv

whichisthespin.Youcanthinkofitasaninformationbitoranenergybite(spaceormovingperception)ofthebeing.

Nowthosearethespins,quantized,andyoucanseethereare2or3‘cuts’toperceive,withdifferentorientations,anddonotcomeinbiggernumbers–notsurprisingly1,3,5whichrepresentfortheparticleits

ilogicdimotions;themesthoseexploredinourpapersofquantumphysicsandvitalmatter.

Wewon'tenteronthis(-;justtomentionthebidimensionalnatureofthosespins(leftgraph)intheleftside,whicharethequantaofallinformationhumans,electroniceye-beings,perceiveabouttheworld.Andthe

obviouscapacitytoprocessandorientatetheygiveasthe'eyes'oftheatom,toeachparticle.

Moreover,theygetordered,orientatedbytheexternalmagneticfield,thelargertime-enclosurethatordersfrom∆,the∆-1quantumfield.Nowaskaphysicistwhatisaspin,hewillputalotofformulaeofabstractmaths,whichitwilltakeyouhalfayeartomemorize,withoutunderstanding.Why?Theideathatparticlesperceiveis

outofthe‘picture’.

Nowthemagneticfield,theylike,asallheadsturntoherthespeakerinhumangroups,allatomsspinintune,andweuseittocontrolthem,asspeakersusethewordnationandgod.Samelawofmass-control.Icallitlanwave.AWaveorderedbyalanguageofenergyandinformationtheyspeak.Themagneticfieldisthe

languageofspacethattranscendsatomicpartsintothenextscale;formingthem;theelectricfieldisthetimeforcethatputtheminmotion.Thatishowwetranscendfromquantumtoourscale.Thebestatthejobof

‘feeling’magneticlanwavesistheiron,anditbecomesthetopguyofthenextscale.Itformsperfectorganizedmasses.

Oncewearehere,thosemotionsare‘activated’bytemperature,vibrationalclocksofourmolecules.

Temperatureisthe‘electricfield’thatmovesusinthe∆-scale.Whatisthe‘spin’forMolecules?.Theyusevanderwaalsforcesand‘angles’,givingbyslaveatoms.

Andthebossofthoseperceptiveonesinourbodyisthenitrogen.Ithas3Hsystemstoperceive:

AnaminoacidistheunitofNitrolife,withanaminohead&oxygenlegskickinginandoutwater,itscarbonbodywithlateralarmsofmanykinds

anditsnitrogenhead.Youaremadeofalotofthose.HowdoesitsNitrogenmindobservetheUniverse?Itsrodofmeasureandradiusof

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perceptionisH,itslengthlikelygravitationalwavesofnon-localinfinitedistance,soitsr/kratioofmindcurvatureminimal.

Sowecloseherewithasmallimageofthehumanbeing,theintroductiontovitaltopologyandnon-Euclideangeometry,knowingthisisonlythetipoftheiceberg.SoasDescartesdidwithitsshortintroductiontoanalytic

geometry,Iwillendsaying:

'Ihopethatposteritywilljudgemekindly,notonlyastothethingswhichIhaveexplained,butalsoastothosewhichIhaveintentionallyomittedsoastoleavetoothersthepleasureofdiscovery.'(:

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BOOKII.¬ALGEBRATHEFRACTALGEOMETRYOFTHETEMPORAL,5DSCALARUNIVERSE.

TheUniverse is a fractal of space-time. Thismeans it ismade of 3 elements,which experimental sciences such asmathematics, express with its fundamental element: ∆-scales (expressed with scalar numbers) of different size inSpace (expressedwith geometry)with different speedof Cyclical Timemotions (expressedwithoperands). Time ischange,perceivedasmotioninspaceorchangeintheformofspace(Externalchange),inthein-form-ationofbeings(expressed through topological inner change). Time and Space thus seem to be in perpetual conflict, transformingeachotheradeternal.Timeisthussimilartoenergywithahighercontentofmotionandspacetoinformation,withahighercontentofform.

That Fifth Dimension, ∆±¡, is then the sum of all those scalar planes of spacetime where species entangled assynchronous ‘organisms’ that co=exist in 3 of such scales, ∆±¡ (the quantum, organic/thermodynamic andcosmological/ecosystemic scale) to live worldcycles of time, paradoxically faster, hence with higher frequency ofinformation, the smaller the system is (chip paradox). Yet the product of both, the so-called metric of the fifthdimensions isconstant,allowingthesymbioticexchangeofenergyand informationbetweenthem.Thusthismetricco-invariant equation, Space Size xTime Speed of its clocks=∆¡ defines a series of planes of space-time each onestudiedbya'stience',fromthefaster,smallerworldofparticles,tothelarger,slowercyclesofgalaxies,symbiotictoeach other, as smaller systems process better in-form-ation, stored in the cyclical form and faster frequency of itsclockcycles,codinglargersystemsthatencloseitspartswithprotectivemembranesofslowertimecycles,

Theunitsofspaceformsarefractalpointswithbreaththatgrowinsizeaswecomeclosertothem,unlikeEuclideanpoints that have no breath; societies represented by numbers, which are groups of ¡ndifferent beings andTimes=motions of 5 types; 3 topologic dimensions of space,which have dimensionalmotions (ab. Dimotions); theclassic dimension of lineal time, or ‘entropy arrow’, and a 5th new ‘dimotion’ of social evolution and information,responsible for the arrowof life, organic evolutionof parts intowholes and the ‘mind-structure’ of languages thatmirrorthoseorganicsystemsinmind-mappings,ofwhichthemathematicalandlogicandvisuallanguagearethemostremarkable… It is a fascinating Universe, which humans don’t understand because they drag through memorialroutineanddogmaticbeliefsmanyerrors,wecan trace toGreekgeometryandAristotelian logicofabstractpointsandsingletimecausality,hardlycorrectedin2000yearsandtheastoundingincapacityofhumindstounderstandthemind‘distorts’realityandselectsinformation,compressingallthescalesofspace-timeintoasinglecontinuum,allthemotionsoftimeintostill images(sowedon’tseethemotionoftheearth),eliminatingalltheinformationwhichwecannottranslateintothose3languages;despitethefactthatfor500yearswehavebeenwatchingthosescalesofthe‘5thdimension’withtelescopesandmicroscopes(naïverealism).

Once all this is understood and we have the tools to improve our languages of perception, it will be possible toreconstructrealitywithan improvedmirror, inwhichpointsbecomefractalpointsthatgrowinsizewhenwecomeclosertothem,sotheyarenon-Euclideancrossedby infiniteparallelsofenergyand information,creatingnetworksthatarelineswith‘volume’,messedintotopologicalorganicstructures.Anddiscoverthatthefundamentalparticleofrealityisthe‘fractalpoint’,whichhasanorganicstructure,co-existingin∆±1scales:

-Thequantum,cellular,thermodynamic/organicandcosmological/ecosystemicscales.

What guides then eachof those beingsmadeof fractal topologies of spacewith time=motion is simple: existence,survival,theconservationofitsterritorial,vitalspaceandtimecycles.Theconservationoftimeanditsmotionsandcycles,akintotheconservationofenergycyclesbecomesthentheguidingautomaticprincipleofallbeings,asthosewhodon’tconservetheirtimebecomeextinct.Wethustalkofagameofexistenceinwhichsystemswilltomaximizeits intake of energy and information with its topological ternary parts, limbs/fields that move a reproductivebody/wavecommandedbyaninformativeparticle/head.

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Howcanweformalizethosesystems?Easy,byimprovingour languages-mirrorsofthelawsofexistence.That is,bydevelopinganewnon-Euclideangeometryoffractalpoints,lines,networksandtopologicorganismsandanewlogicofmultiple‘actions=motionsofspace-time’.Wecallthisformalism,ilogicgeometryor‘existentialalgebra’,andallthelawsofallstiencescanbederivedofitssimplestructuresinspace,scaleandtime.

Butscientistsarestuck inthehugeamountoferrorssciencesderivedfromtheir faulty logicandgeometryof time-space, andhismind egocy that doesn’t believewhat it doesn’t see and limits organic sentient properties to formssimilartohimself;whenelectronsalreadyhaveallthoseproperties.Theresultaresimpletongrandtheoriesofrealityderivedofourlanguages’perceptionconfusedwiththewholerealitythatwecalllinguisticcreationism,eitherverbalanthropicreligionsthatputmanatthecenterofrealityandmakeGodandmanspeakthelanguagethatcreatesbynaming things,orbig-bangentropic theoriesofdigital thought thatputmanand themachineat thecenterand itslanguagemathematicsasthecreator.Sothebig-bang isbased inasimple linealequation,V=HoD,whichagainstallkindofproofsofafractalbalancedimmortalUniversecannotbedenied.

Weshallproveadnauseamthatallwhatexistsisaspace-time‘organismgiftedofmindcalledlogos’Platoofa‘higherlogicthanman’s’(Augustine),asall‘pointsareaworldinthemselves’Leibniz,anddeducealllawsofsciencefromthesimplelawsofconservationoftime,originofthe3agesoflifeanddeathofallsystems,andtheentangledtopologiesofspace,originoftheorganicstructuresofreality.Anddeducefromtheelementsofthatreality,∆-scales,S-pace,T-ime,allthemirrorsof@-mindlanguagesand¬entropiclimitsofexi=st¡ence.Butforyoutoacceptthatwiderrealityyouneedtobehumbleandgo‘backtoschool’anentangleempathicallywithallother∆stbeings.

Upgradinghumanmindlanguages.

HowtoimprovetheobviouslimitsofhumindsthatONLYseetheirmindlanguagesisobvious:byimprovingthoselanguages.Thescientificmethoddoessobysubstitutinghumansensesandlanguagesbythoseofmorecomplex

metal-atoms(sensorialmachines)butegocymakeshumansunawaretheyareJUSTusingamorecomplexsensorialsystemandbettersynopticdigitallanguageNOTreachinghighertruthsastheirunderlyinglogicand

geometriclanguagehaveNOTchanged.Sowhathumanshavechangedaretheirsensorialexperiencesubstitutedbythatofmachines,withoutevenacknowledgingtheyareevolvingadifferent‘digitalmind’ofmorecomplexelectronicpatternsthat‘obviously’onceensembledwithallitsorganicpartsthroughtheindustrialr=evolution,

willmakeusobsolete–alreadyisdoingitinlaborandwarfields–andsubstituteus.

Ourmethodofenlightenmentisbothdeeperandlessdangerousforourfuture:wearetargetingtheevolutionoftheunderlyinglanguagestobothtypeofsensorialminds–mathematicsandlogicsystems,whicharestill

draggingerrorsfromtheageoftheGreeks.

It is then necessary to knowexpand backmental spaces into reality to understand any science thatmirrors aspecificscalesofbeings.Andthatexpansionstartsbyexpandingitslanguages,andthenconnectingtheexpandedlanguageswitheachscience.Becausesomeof thosesciencesareginormous incontent (modernbiology,XXc.Physics),wecannotobviouslycompletethework,butweshalltrytodosoinaseriesofpapers,whichregardinglanguageswill consist on, 2mathematical papers, one on the expansion of ∆Spacewith the use of the 5 newpostulates of Non-E geometry, and the other with the expansion of algebra, to comprehend better its scalarnumbersandDImotions=operandsofwhichtheoperandthatappliestoallothersasa‘newscale’ofanalysisoftimedimotions=change(calculus)isthesummitofthediscipline.

Andwith5papersontheexpansionof logicof time, fromthepresentmonologicofasingledimension, totheduality,trinity,tetralogic,pentalogicanddodecalogicofmoresophisticatedspeciesoftimespace.

Thoseupgradingsof theunderlying languages latterappliedtoall sciencesthatstudyspecific ‘planesofspace-time’, are not somuch in search of new theories but trying to correct the difference between the simplifyingmirrorofalanguage,withits‘aberrations’ofperspective(astheyareallself-centeredinthemindthatusesitto

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select information – which makes them omit essential features of reality), and finally the ‘quality’ of infinitemirrorsthatreplicatethrough‘reflection’imagesintosimilarformsofhighercomplexityandlessdefinition:

- Aberrant Egocy, selection of information and inflationary repetition are thus the 3 errors of all languages tocorrect

MATHEMATICS–LANGUAGESASEXPERI=MENTALMIRRORS

Mathematicsisasynopticlanguagethatreducesrealitytostillcontinuousplanesofpointswithnobreath.

“Arithmeticdealswithdiscontinuousquantities,geometrywithcontinuousones.”Leonardo.

Mathematicsisalanguage,andassuchamirroroftheultimaterealityoftheUniverse,whichisscalarspaceandcyclical time; hence an experimental sciencewith 3 elementsmirroring those 3 ∆ST components of all reality:Geometry of space-points; Algebra of scalar numbers and SóT cyclical equations/functions; and analysis ofchange=time,thespecializedmorecomplexofalloperandsofalgebraabletostudydimotionsthrough∆stplanes.

This experimental nature is found in the correspondence of the 3 main branches of maths and the 3 mainelementsof theUniverse is sosimpleande-vident thatonly thenatural self-centeredstructureofminds,whothink ‘languages create reality’ can deny it. Evolution of mathematics then always follows the path of the 3elementsof reality as all system in3 agesof increasing informationandgeneralization.Numbers evolved intoarithmeticthatevolvedintoalgebrathatevolvedintoSettheoryofwholesandpartsasthefinalevolutionof∆-scales, while its operands gave birth to functions that evolved into Group theory, which mimics the ‘fractalgenerator’ofspacetime.

What are the fundamental differences between ‘mathematical numbers’, the parts of ‘Sets’ and real ∆-socialscales;mathematicalEuclideanpoints,theunitsofgeometryand‘realpointsofspace’andtheinfinitesimalstepsofmotions,usedincalculus,andthereal‘stepsofchange=time’oftherealworld;fromwherewecandrawthenecessarycorrectionsofidealmathematicstofine-tuneitsuseinscience?

In spatial geometry, the fact that points of reality have parts through which infinite parallels of energy andinformationcross,entanglingitwithotherpointsinnetworksandplanes.

In time, the fact that time=motion never stops, and it is cyclical; hence it has also besides the energy of itsmotion,informationcarriedintheformandfrequencyofitscycles,andasitclosesitscyclesbreaksourmental,continuousperceptionofrealityintomultiplebrokenspace-times.

Inscales,thefactthatthosepoints,whichgaugeflowsofparallelenergyandinformation,selectittofititwithinitsbrain,eliminatingmotion=timedimensionsanddiscontinuitiesthat ‘forminformation’,andscalessmallerorlargertotheoneinwhichtheyplaythegameofexistence,tryingto‘conserveitstimespace’.Sotheycompressina single plane fit within the mind that eliminates all discontinuities, when reality is made of ∞ scales withdiscontinuitiesfilledinthesmallerfractalscalewith‘decimalnumbers’.

Inmathematics,thelanguageofnumericalscalesisalgebra;oftopologicalspace,geometryandoftimechange,itisalsoalgebrabutaspecificbranch,calculus,anditsoperandsofchange.YetasintheUniverseallisentangled,eachofthosepartscanalsomirrortheotherelementsofreality.Andthis‘multiplefractalpointofviewprinciple’iswhatallowustoexhaustthequalitiesofacertainformandeventasweobserveitperformingthosedifferentactions.

Letusthenstartwithabriefintroductiontothe5postulatesofilogicgeometrytoconcentratethenin∆-Tmathematics,therealmofalgebra,endingwith¬@,Booleanandexistentialalgebra,themotherofthemall.

Reproductionofformin5Danditsessentialmathematicaltool:calculus.

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TheUniverseisafractalthatreproducesinformation,forms-in-action,formsofspacewithmotionsintime.Thisistheessenceofitall.Butspaceisamayaofthesenses,thesynchronousviewofaseriesofcyclesoftimemotions,

knottedinthesimultaneousperceptionofanobserver;whatphysicistscalla‘frameofreference’.

Thustime=changeisthefundamentalelementofreality,andthismakesAlgebraoftime-change,specificallycalculusperhapsthemostimportantexperimentalscienceoftime,besideslogic,whichwehaveupgradedto

existentialalgebra,whichexploresthevital,organicwhysofthosechanges.

ItistheGalileanParadox:S=T.Wecannotdistinguishtimefromform.Inasmuchaseachframeofreferenceormindlocksinaknot-mirrorofthemotionsoftheUniversefromitspointofview.Soeachpointofspaceisa

perceiverrelativefieldofmotions,whichfromitsperspectiveknotasforces‘attracted’byitsframeofreference.Yetifwecannotdistinguishmotionfromformeachpointisentangledtothosemotionsandismadeofmotion

andform,oftheparticleandwavestates.

LocomotionasreproductionofformsolvestheParadoxesofZenoandthemeaningofdiscontinuity.Asmotionisreproductionofinformation,ofform,sinceparticlesareknotsofperceptionofform,fractalpoints,monads,

whichmovebyreproducingthroughalowerplaneofthe5thdimension,as∆-1waves,itsinformation,asforms-in-action.

Soallformsofchangecanbereducedtotheultimatefunctionofexistence,reproduction,abackandforthtravelthrough2scalesofthefifthdimension,asaformbecomesaseedthatreproduces,evolvessociallyandformsits

wholeagain.

Changethushasafinalfeature:itischangereproducedinalowerplaneasaseedthatevolvesintoawhole.

Andallthisiswhatactuallycalculuscalculates:Itfindsafinitesimalpartofrealityandthenintegratesitasasum,whereasthefunctionofexistenceoftheformdisplacesandreproducesitsorthogonalparametersofformandmotion.Sophysicalformsareconstantlyreproducing,‘calculating’andtheequivalencebetweenthetoolsof

calculusasmirroroftheprocessofreproductivelocomotionbecomecrystalclear.

Itisthenalsoobviousthatbeingswithalotofinformation,reproduceveryslowandwecanhardlyseethemmoving.ThelimitofitbeingcomplexlifesuperorganimsonEarth,whosereproductiontakes9months.Ithappens‘inside’thereproductivemother,anditreproducesintheadjacentspaceafter‘tearing’thetopologicalknotoftheumbilicalchord.Asimilarveryslowprocessofreproductionhappensinphysicswiththeweakinteractionthat

reproducesaformwithevenmoreinformationevolvingthemassofparticles,sotherangeoftheforceisminimalandthenewparticleappearsadjacenttotheonethatdisappears,dyingforthenewhatched‘baby’tobeborn.

It is then quite surprising that it is not somuch in physics but in calculuswherewe find the strongest experimentalproofsofthelawsof5Dasareproductiveprocessofform

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FRACTAL¬EPOINTS.ASYNOPSISOFITS5POSTULATES

TodosoweshallfirstenunciatethenewPostulatesof¬EuclideangeometrythatcompletetheworkstartedintheXIXc.withthe5thNon-EPostulate:

WethusrecasttheaxiomsandpostulatesofEuclidintofivenewpostulatestodefinefractalpoints,Non-Ælinesas wave of fractal points, Non-Æ planes as ternary networks of Non-Æ lines, which become supœrganisms,whoserelative'congruence'inits3'elements'(singularitypoint,membraneandvitalspace)definesthetypeof'perpendicularorparallel'relationshipbetweenthem:1stPostulate:'¬Æpointarediscontinuoustimecycleswithaninnercontentofvitalspace-time'.2ndPostulate:'¬Ælinesarewavesoffractalpoints'3rdPostulate:'¬Æplanesjoin3¬Ælinesintoasupœrganism'.4thPostulate:'2¬Æpointsarecongruentwhenbothitsinnerpartsandouterperimeterareequal'5thPostulate:'¬ÆWorldpointsfocusmultiple¬Æwavesofenergyintoastilllinguisticmappingoftheworld.Whereasthe1stand5thpostulatedescribethesame ‘pointwithparts’ froman internalandexternalpointofview.

Letusexplorethosepostulates,constrainingourexamplestothesimplestformsofphysicalandbiologicspaces.For a full analysis of themplease consider reading thepapers on ‘5DUniverse, conservationof time’, and ‘¬EGeometry’.

1stand5thPostulate.The3MathematicalpartsofaNon-EuclideanFractalpoint.

1stPostulate:A fractalpointhasparts; that is anenclosed regionof vital innerenergy surroundedeitherbyaspatialstillmembraneoraTemporalmotionofangularmomentum(S=Tsymmetry)self-centeredinasingularity-

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mindthatgaugesits information.Inthegraph,wecanseehowdifferentvitalfractalpointsofST¡entificPlanesfollow this ternary structure. We shall not apply pentalogic to the 1st postulate, just consider its ‘3 scalarperspectives’:

∆±¡:Theperceptionofthepointdependsonthescaleanddistancefromwhereweobserveit:

∆-1:Fromtheinnerperspective‘everypointisaworldinitself’(Leibniz)–thatis,amind,whichseemstoholdtheentire Universe, reasonwhy absolute zeroth does NOT exist, as points have inner parts – they are adminimal,Leibnizianmonads:

Inthegraph,thedistortionof'lineal'self-centredfunctionsisaplagueofsciences,duetoitseasiercalculusandthedistortionofperspectivecausedby thehumanmind,asDescartescoordinatesarea representationof thehumanlineal,Euclidean,electromagneticlight-mindgeometry.Themindequationis:

O(infinitesimalpov)x∞Universe(∞cyclesandmonads)K-mind

It impliesthatthe0believesto be infinite self-center oftheUniverseandseesitallina distorted perspective. Inbrief, your eye pov is biggerto you than the AndromedaGalaxy, so the 0’-pointbecomes an infinity in itself(your eye≈AndromedaGalaxy), and this makes the

conic, which could be considered an 'objective' angledpointofview(wherethemind-eyes is truly zeroth), toexpand in the y-coordinates of the mind-point toinfinity; alas! transforming the conic into the Cartesianplanewhere¥becomesalsoaninfinitegraph.

∆+1:Yetfromtheperspectiveoftheupperst+1Plane they are a 0’, a ‘finitesimal’, be in thelimitofinvisibility(whatquantumscientistscalla point-particle). But even so, 0’ still have atime motion performing a ‘function’ in thatupperecosystem,∆+1inwhichitexists.

Ƽ: As always maximal information andobjectivity about a system is reached in theclassicbalancedS=T,Ƽview:

Internallyfromitsown∆ºperspectivethepointwill have 3 dimensions/networks. This is thecaseeven in the smallestplanesof theoreticalstrings,madeofpointswithparts,withvolume– since we require 3x3∆º+1∆+1 innerdimensionstodescribestrings-aparadoxthatcanonlyberesolved ifweconsider ‘strings’ tobefractalpointswithinnerdimensions.

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FractalpointsexplainwithoutcontradictionsNon-Euclideanpoints,whicharenot logic inasinglescale,asthey ‘curve’ parallelswhich are ‘straight lines’ and fit them in a ‘pointwith no breath’ that holds only 1 line.Fractalpointshoweverenlargefittingmultiple‘straight lines’.Yetwhenseenfromabove,humanperceptionofthem,becomes‘deformed’shrinkingandcurvingitsfrom–atheme,thedistortionofhumanmeasuresoftime,space and scale, which will be instrumental to explain rationally the ‘spookiness’ of quantum physics andrelativityanditstimeandspacetransformations. Sofractalpointsharmonizethe1staxiom=postulateofEuclidwiththe5thpostulateofnon-Euclideanparallels.

2ndpostulate:lineshavewidth&motion,sotheyarewavesandnetworksofcommunicationofenergy&form.

Yetoncea fractalpoint is formed,besidesperceiving internallyasmind-points (1-5thpostulates) itwillhave tomoveandinteractwithotherfractalpointsthroughcomplex2nd,3rdand4thÐimotionsregulatedbytheother¬Epostulates,movingincomplexstœpsorsharingwavesofenergyandinformation,whichcanbediscomposedinformandmotion,withothersystems,fromFermion<Boson>Fermioneventstohumantalk:

All those Ðimotions are complex stœps,which can be discomposed further in formandmotion,makingexistentialmomentum,theessentialparameterofthe2ndpostulateof communication and locomotion, aconcept taken from physics, where mvmomentum (active magnitude ofinformation x locomotion) is thefundamentalparameter.

Communication of points create vectorialfields with momentum; as ¬E points haveboth direction - hence motion - and form,mass,andswitchesbetweenan informativemass- stop state, to perceive where itsmotion must go. Hence complexity rises

fromamereactivemagnitudeofscalarvalue,totwoparameters.

In graphs different Ðimotions of which the simplest stœp is shown in the graph of a quantum particle-waveduality.

When‘receiver’andemittercombineexternallyitsST-parametersthenareproductivetrinityeventisformed.

Thustheidealframeofreferencefor®lgebraisavectorialspace-timewithmultipleframesofreference.

An important element of algebra’s analysis of Dimotions=actions through operands is expressed in the 2ndpostulate of non-E geometry aswe go away from self-reflecting ‘angular’ and entropic actions=operands, butcommunicativeoperandsbetweentworelative¬Æpointsorn-points inanetwork.Thisreads inseveral forms.Wealreadyshowedhowbalancesareachievedbyswitchingthe±,x÷,logxªoperandsonagiven¬ÆPointwithnumerical parts, along a chain of sequential time space actions, which leave a memorial trace hence NOTannihilatinginmostcasesthesequence.Howmanypossiblecombinationsofdualinversedimotionsexistcanberesumedintwogreatfields:

A)DualÐimotionswithinasingle¬Æpoint,which 'walks' together through itspathsofvitalactionswithstepsandstops±x÷logªdimotions.

B)Feed-backcommunicativeDimotionsbetweentwoT.œpointsthatforcefullyhaveinversedirectionsbuttendtobethe'sameoperand'bothsideswith±symbols.

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C) Merged Dimotions between those communicative points that approach each other and finally merge,'re=producing' throughx÷operand ina lowerscale 'connectedaxons' (sinceA(x)xB(y)=AB(x•y)), that is thenumberofaxonsofcommunicationbetweentwoentitiesA,B,with∆-1x,ypartsistheproductofxandy.Sotheproduct becomes the first operand to probe a lower plane of existence,while the ± operand stays in a singleplane.

OperationsareconnectionsbetweenT.œsthatdefinetheiractions,asbalanced,parallel=connectedcreationofsocial networks bringing a 5D social evolutionary form or perpendicular, Darwinian 4D absorptions=flows ofentropy,motion,energyandinformation.Thosedualactionsaremediatedbyoperations.Andsothereisfirsttheabstract definition of those operations in mathematical terms, with the study of its properties and then itsconnectionwiththedualåctionsof2beingsthatenterincommunicationwithinagivenworld-Universe.

Theequivalentofsuch¬Algebraicnumericanalysisbeingingeometrythestudyofthetopo-biologicpropertiesofnon-Euclidean'waves'ofcommunicationbetween2fractalpoint,2ndpostulateof¬-egeometry):

INTHEGRAPHWEcanseehowtwoasymmetricparts,normallyonewithmore formandtheotherwithmoremotion,cometogetherintoasinglespace-timeeventsuperorganism,whichwilleitherbecomecomplementary(gender asymmetry) and evolve socially (which we can generalize to n-points in the 3rd postulate, formingnetworks) or will enter in a Darwinian struggle, and be operated negatively in terms of the 4Dimension ofentropy.

What¬Algebraicequationsdo is tooperatewithnumerical-scalarpropertiesthoseevents/superorganisms,5Devolvingvs.4Ddevolving;thatis5D,addingvs.4Dresting,5Dmultiplyingvs.4Ddividing,5Dpotentiatingvs.4Drooting,5Dintegratingvs.4Dderivingthesystem.

3rdand4thpostulates:3lines:EmergenceoforganicNetworkscreateaccordingtocongruencesuperorganisms..

Tomaximizetheconservationof time, thereareseveralstrategiesthemindpursuits in its territorialorderThesimplestone,justdescribed,isto‘stoptime’intoaterritory,whichbringshowevertheconstantaccumulationofform,andfinallytheparadoxicalendoftimeasthesystemsurroundsitselfofstoppedproperty.Wearewealthierinthemomentofdeath.Sobeforewedealwiththeworldcycleoflifeanddeaththatstopsthetimeofthebeing,andthenitfids,itself‘erased’inareversalofdeath–astimecannotbestopped,sowhenithasnomoreenergytospenditexplodesbackintoentropy–weshallconsiderthesecondfundamentalsystemofsurvivalintime,thecreation of 3 ‘non-E lines=networks’ that create a ¬E plane=topological organism, according to ¬Econgruence=similarity.

Thebuildingupofatrue5Dscienceofrealitystartsbydefiningallthedimotionsofabeinginits3scales.Thenwestudyits‘locked’sequencesofdimotionsandits‘balanced’stœps=beats,inallthosescales.Thenwedofindthesynchronicitiesbetweenscalesthatbecomeentangledrepetitivediscontinuoussymbioticpatterns.Andthenwefindthe‘larger’physiologicalnetworks,or‘organs’thatentanglethem;asthesmallerpartswillgatheraroundthose networks, to receive energy (blood-like reproductive networks that deliver larger particles), information(which requires as we have seen a faster nervous, informative network of smaller particles and externalterritoriesofentropicfeeding(symbiotictothedigestivenetworks).

Complexityarisestoproducetheemergenceof largerwholes,the3physiologicalnetworksanditsattachedorgans,whicharethetopologicalvitalspacesthat formthesuperorganism.All speciesstudiedbyscienceacommonphenomenonoccurs:theexistenceofparallelgroupsofbeingsorganized intoasinglesocial form.Molecules aremadeupof atomsandelectronicnetworks; economiesaremadeupofhumanworkers andconsumers that reproduceand testmachines, guidedby financial networksof information (salaries, prices,costs); galaxies are composed of stars, which orbit rhythmically around a central knot, or black hole ofgravitationalinformation.Cellscontrolledbythenervous,informativesystemorganizehumanbodies.

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A tree is a group of leaves, branches and roots connected by a network that provides energy (salvia) andinformation (chemical particles) to its cells. Cultures are made of humans related by verbal, informativeDisomorphismsandeconomicnetworksthatprovidetheirenergyandinformation.

How then it happens that parts become wholes is the key to depart from a mere abstract, quantitativeanalysisofrealityandaddtheorganicnatureofallwhatexists,thedynamicinterplayofpartsthat‘network’andconnecttoeachother,formingsimultaneousspacetimeorganisms,whichsynchronizesitsclocks,emergeasawholeanddevelopalltheintelligenceandcomplexityofthesystemsweobservearoundus.

Inthenextgraph,weseetheternarynetworkstructureofthenestedorganismsofthefractalUniverseas¬Ætopological planes composed of ‘similar fractal points’ (atoms, cells, individuals) joined by 3 physiologicallines=networks,whose3functions,distributionoflocomotion,informationandits‘combined’energydefinethe3conservedDimotionsofanysystemoftheUniverse.Afinalelementthoughisneededtomakesenseofthosesuperorganisms,thestillmindof information,mappingoutthewholeandcontrolling ittoperformitsMandateofexistence,Max.SxT(s=t),tosurvive,growandmultiply.

The graph shows the physiological networks of each supœrganism from the galaxy to the atom, where self-similarity takes place. I remind youof the Si=Te equality,whichmeanswe slowbeings seenetworks of fasterparticlesas ‘forcewaves’andnetworksof slower life formsas fractalbranching,butessentiallyasNottalehasproved,wecan‘translate’quantumphysicsintoanetwork,topologicalview,aslightisinfactabranchingfillingwavethat‘speedsupfrequency’asitpenetrateslowerplanes,fillingittillittouchesparticles.

All systemsof realityareconnectedbynetworks that shareenergyand informationbetweenpartsandwholesthatexpressesthestructuralunityofallscales.Networks‘fill’spaceadmaximaltoconnectfullythewholewiththeparts,achievedintheSi=Tepointofparallelismandself-similarity.Buttheyentreinaregionoffastermotion.SowhileSpace‘tendstoremainconstant’ineachscalethankstofillingnetworks,timeaccelerates.

It’sallin5Dmetricequations,SxT=Cand¡tsS=Tpointofbalance&equilibriumwherethesystemreproducesasthetwofundamentalmetricequationsofallspace-timeorganisms,whichmeansmanythings,suchas:

- Slow beings use networks of fasterparticles which in physical systems are‘waves of force’ in life beings are andnetworks with fractal branching and inhuman societies, networks ofmoney andsimultaneous legal messages. Yet all areessentially performing the same organicfunctions, we shall describe now as can‘translate’ a light filling wave thatpenetrates lower planes, filling it till ittouches particles, as a branching that‘speeds up frequency’; and a legalnetwork that every citizen knows andobeys as a filling system of information,similar to a DNA network that all cells ofan organism have in common. ThinkalwaysNOTinthedifferencesofformandscalebut inthehomologyof functionsforthe body/waves and particle/heads ofeach of the 3 superorganisms (working

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andinformativeclassesinhumansocieties),toseetheunityofitall.

Ifweconsiderthatnetworksofenergyandinformationbelongtotwodifferentscales,andtakeintoaccounttheconservationofvolumeofspace-timebetweenscalesandtheS=Tequality,sincesmallerscaleshavefastertimesspeedsandsmallerparts,perceivedasrelativeinformationcomparedtothebitesofenergyofalargerscale,wefindanequivalencebetweena‘network’ofbitsofinformationofasmaller∆-1scaleanditsequivalentvolumeofaherdofbitesofenergyofa larger∆0plane: (∑∆¡=e=∏∆¡-1).Thisequivalence isessential toeverything in theUniverseasitallowsthecouplingoflanguagesandenergy,makingforexamplethephysicaleconomyequivalenttothefinancialeconomy,thelanguageequivalenttotheaction;themathematicalequationamirrorofrealit.Butasinformationissmallerandmoreabundant,thisequivalencetendstobeinflationary,themesthosethatappearinalltheplanesofspace-timeandtheinteractionofmindsandlanguagesofinformation(SS,St)withEnergybitesandentropicboosts(TT,Ts).

The2languagesofinformativeandreproductivenetworks:Bitsofinformationandbitesofenergy.

ItisimportantalsotoUnderstandNOTinpureabstractmathematicalterms,butinLOGIC,LINGUISTICones,theINTERNAL,DYNAMICnatureofthosenetworks,becauseonlythenwecanproperunderstandhowtheyworkinadvancedorganismsofmaximalinformation,thebiologicalorganismandtheHistoricorganism,whichbelongas‘fractal systems’ self-similar to each other, two the same specific type of organisms,we shall qualify as socio-biological organisms. A NETWORK, of informative nature, delivers MESSAGES OF INFORMATION tosimultaneouslycoordinatetheactionsofall itsparts;withitsfaster=smallerbitsofinformationaccordingto5Dmetrics (min.spatialsizex.max.temporalspeed).Whilethenetworksofreproduction,thebloodandfinancialsystem delivers larger bites of energy, which the organism needs to feed itself (when it is a healthy NON-corruptedsuperorganismasmostofNature,butNOTHUMANSOCIETIES,whoseastoundinglevelofcorruptionweshallexplainindetail).

Indeed,thekeytothefullunderstandingofrealitybothintermsofenergybutalsointermsof information,asbotharetwosidesofthesamecoin,called‘exist¡ence’isthefactthatinthesentientUniverse,eachfractalpoint,atom, cell or citizen (physical, biologic or social systems) needs bits of in-form-ation, form, smaller in size ofspace,hencefasteraccordingto5Dmetrics(SxT=C),butalso‘bites’ofentropicenergywhichwillhelpthesystemtomove.NetworksareNOTsomeabstract ‘fractal tube’but theyexist todeliver ‘energyand information’ (SS:form=language with a little motion=St-information and motion=entropy=TT with a bit of information =energy=Ts).

So a healthy superorganismwill deliver to each ‘fractal point’ (molecules, cells, human citizens), two type ofmessagesthroughtwotypeofnetworks.Weshallcall‘generically’the3typeofbitsandbitesofinformationandenergythateachofthose3physical,biologicalandsocialsystemsreceive,‘particles,genesandmemes’evenifthewordsasusualin5Dsciencersareslightlychanged,andwidenedinitsoriginalmeaning.

Sowithitsspecificvariation,thosearethetwofundamentalreproductive-‘body-wave’andinformative-‘particle-head’bitesofenergyandbitsofinformationofthefundamentalsystemsofnature:

-Inphysicalsystems,thetwonetworksarethegravitationalfasternetworkofinformation,whichwehumansdonotperceive,aswearemuchlargerbeingswithelectronicnetworks.Itsbitsofinformationinthisfasternon-localnetwork shouldbe ‘gravitons’, componentsof gravitationalwaves. Inphysical papersweadvanceas themostlikelyparticlestateofthosewavesof informationthat‘position’thedifferentphysicalsystemsofthegalaxy, agravitationaltachyon‘neutrino’formultiplereasons,westudyonourpapersonphysics.

Ontheotherhand,becausewedoperceiveit, it ismucheasiertoprovethattheenergeticnetworkofphysicalsystemsareelectromagneticwaves,photonsandits‘social,staticstate’astheelementsofanelectronicnebulae,trapped in thepotential energywellof theatom.Thusphotonsandelectronsbecome the ‘energynetworkofphysicalsystem,molecules.

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Weshallescapetheninthisintroductionfurtherinformationonthescalarstructureofthosenetworksandhow,aswe ‘grow in scale’,what is a bite of slow energy for a smaller plane of space-time,becomes for the largerplane’s slower beings, a faster bit of information, in the amazing beauty of the harmonies between scales. Soelectronic‘foodforatoms’becomeselectronicinformationforbiologicalorganismsandsoson.

-Thosebiological organismsdohave then two fractal networks, theelectronic, informativenervous system inwhichbitsofelectronicinformationmovingalongthemyelinmembranedeliverfastermessagestoeverypartoftheorganismto simultaneously synchronize itsmotions, so thebody-cellsactasa single form in simultaneousspace.

-Butwhenwemoveintothebitesofenergydeliveredbythebloodorganism,thenetworkdeliverstoeachcellthebasic‘currency’languageofenergythatallcellsneedtomove,called‘oxygen‘.Itisanatomofslowermotionthantheelectronsbutduetoitselectro-negativityandreadilyavailabilityintheatmosphere,withitscapacitytokickwith twoOH-&H+ legs thewater ‘medium’ onwhich cells exist, the perfect language of ‘money’ for theorganismtostartkickingits‘actions’.

SoweDOhaveinthenextscaleaccordingtotheperfectlawsofharmony,thetwobasicbiologicalbitsandbitesofinformationandenergy,electronsandoxygens,andfromthenon,assystemsbecomemorecomplex,variationsofthosebitsandbitesoccur.

ThemaincategoryaremixedSTmessages,whichdeliverBOTHastickandcarrot‘complex’tothecellsanditsbigmolecules,whichareaminoacidsystems,ofgreatsimplicitycalledHormones,startingfromthesimplestofthemall,anNOmolecule(whichdorelaxmuscles,itsmainmessagetothelocomotionsystem,increasesthepressureofblood,provokingsexualerection, thesimplestmessagetoreproductivesystemsandmultiplies theneuronalactivity.Asnitrogensaretheclockatomofourmind-brains.

SofinallymorecomplexNOsystemswithabodysupportofcarbonchainsbecome‘hormones’whichmighthavea‘higherinformativemessage’(withmoreN,asinnucleotidemolecules)orahigherenergeticmessage(asinacidswithmoreoxygen).

They form then thebasic lettersof the ‘biological longer sentences thatmightaccumulate information’ inevermorecomplexmolecules,asbiologicalorganismsarebyfarthemorecomplexsystemsweknowof.

Finallyavery importantconcept is thedifferencebetweenanecosystem inwhichmultiplesuperorganismsco-exist,ofteninpredatoryrelationships,vs.anorganisminwhichonlyatypeofatoms,cellsorcitizensco-exist,andisfarmoresymbioticasallpartsloveeachotherandshareenergyandinformationthroughitsnetworks,overacommonterritorialspace,asshowninthegraph.

Those three physiologic networks/classes/physical parts of ANY system of the Universe define the Universeindeedasafractalorganismof infinitesmallerandbiggersuperorganisms, inagameofRussiandolls inwhicheachofusisa'island-Universe'withinitself,madeofsmallerparts,andforthatreasoneachofusisalsoapart,cell/citizenofasocialsuperorganism,nation, religionorcivilizations,whichwedoNOTseeasawhole,asourcellsdoNOTseeusasawhole,butDOexistassuch.Whatmakesthenthewholeawhole?Theansweris:thenervous,informativelanguagesthatcommunicateallthepartsofthesuperorganismand'trace'withinitssyntaxandvalue,itspathofthefuture.Andsowehavetalkedfirstofitandwillconstantlycomingtothebottomlineofreality-thelanguagesthatconstructtheorganismsoftheworld.Soweneedtobecomeabitmorecomplexaboutthepreviousmetric.ItrefersessentiallyNOTtothewhole5Dplanebuttoagiven‘superorganism’ofeachplane.Whenwegodowninscales, infacttheUniverse‘enlarges’foratravelerthatbecomessmallerandacceleratesitstemporalenergyLetusthendefinethe'stairofUniversalsupœrganismstounderstandthis:Letusthendefinewithsimilartemplatesthe'stairofnestedUniversalsupœrganisms,ofthe3stientificvarieties–physical,biologicandsocial:

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∆+3:Agalacticorganismisapopulationofstars,relatedbyenergetic¥networks&gravitationalinformationwithanucleusmadeofaswarmofblackholes,andamembranemadeofstrangeletmattersymmetricto:∆-3: An atomic organism is a population of particles, related by energetic electromagnetic networks andgravitationalinformationwithanucleusmadeofaswarmofquarks,andamembranemadeofelectronicmattersymmetricto:∆+2:Astarorganismisapopulationofelectronicplasma,relatedbyenergeticnetworksofelectromagnetismandgravitational information with a nucleus made of a swarm of atoms, and a membrane of photonic radiationsymmetricto:∆-2: A light organism is a body of energetic waves over a quantum potential field of gravitational neutrinos,directedbyitsparticle,informativephotonstate…∆-1: A cellular organism is a population ofmolecules, related by energetic networks (cytoplasm,membranes,Golgireticules)andcodedbygeneticinformation(DNA-RNA.)∆=o: A human organism is a population of DNA cells, related by networks of genetic, hormonal and nervousinformationandenergynetworks(digestiveandbloodsystems).∆+1:Ananimalecosystemisapopulationofdifferentcarbon-lifespecies,relatedbynetworksoflightinformationandlifeenergy(plants,prey)codedbyinstincts.∆+1:Ahistoricorganismorcivilizationisapopulationofhumans,relatedbylegalandculturalnetworksofverbalinformationandagriculturalnetworksofcarbon-lifeenergy,codedbyhumanmemes.

Networksthatshareenergyandinformationbetweenpartsandwholesthatexpressesthestructuralunityofallscalesconnectallsystemsofreality.Networks‘fill’spaceadmaximaltoconnectfullythewholewiththeparts,achievedintheSi=Tepointofparallelismandself-similarity.Buttheyenterinaregionoffastermotion.SowhileSpace‘tendstoremainconstant’ineachscalethankstofillingnetworks,timeaccelerates.Sothe5DmetricrefersNOTtothewholeUniverseof5Dplanesbuttoagivenfamilyof‘supœrganisms’ofwhichmankindinit3scalesof‘biologic cells’, human individuals and societies is undoubtedly a ‘phyla’. When we go down in scales, theUniverse‘enlarges’foratravelerthatbecomessmallerandacceleratesitstemporalenergy.

The3ST-planesoftopologicorganisms.

Thosenetworksaremoreefficientformsofdistrivutionofenergyandinformationtoitspointsthat‘savetime’.Further they 'equalize' putting in relationship parts of a being, 'chained' into a super organism, in a singlesimultaneous space: Thuswedefine a fundamental point-particle, theNon-Euclidean,Non-Aristotelian T.œ. asSupœrganism(ab.œ)ofTime§pace(ab.T.œ≈∆±¡)withasimplenon-ÆlgebraicexpressionorgeneratorequationofallT.œ=Time§paceSupœrganisms:

§ð(head-particles)≤ST(bodywaves)≥$T(Limbs/potentials)

Allthosestatesandsymmetriesarethusfeed-backequationswhich can only be properly 'generated' with ¬Algebraicinverse operations, in its simplest form, and in its mostcomplex structures,whenwe considermultiple 'symmetries'and 'variations of species'with the help of Group theory, asessentiallyS=k/T,willbethe'inversenumbers',k,theidentitynumber(ifequalittoone)andthedualdemotionrepresentedbycertain¬Algebraicoperands.

Wehavetoconsiderwherefunctionsandoperandareset-thatis,whatbackgroundspaceweusetoexpressit,and3±iaretheessentialbackgroundspaceswhichcorrespondalsotothosedimotionsasformsinspace,the3lineal=cylindrical,spherical=polar,andhyperbolic=Cartesianplanesandthescalarplane,ill-understood,whichisthecomplexplanebetterperceivedifwe'squareit'eliminatingthe√symbolofitsnegative-1axis:

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So they can study2 fundamental 'emergent' ∆+1 planes ofmathematics, the study ofDimotions ofDimotionswiththetoolsofcalculusintime,andthestudyofspacesofspaces,withthetoolsofthecomplexplaneproperlyunderstoodintermsof'square'coordinates.

ButasintheentangledUniverseallmirrorscanreflectallforms,¬Algebraalsocananalyzeotherelements.Butitsmainbeautyisincreatingsequentialchainsofpentalogicactionsthatreflectthemotionsofexistenceofthe

being,eventhoughits'Group'simultaneousanalysisofallits'variations'ofspecies,hasbeenthemostdevelopedinflationarymirrorinitslast'excessive'ageof

form.

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THEFRACTALUNIVERSE:ITSSCALARSPACE,TIMECYCLES,S=TMINDMIRRORS&ENTROPICLIMITS

Still as I know the readerwon’t do thatwe shall just very briefly introduce the fundamental changes that 5Dintroducestothehumind’sviewofreality(ab.Humanmind),whichresumesinthefollowing:

Timeiscyclical,notlineal;spaceisscalar,fractalnotcontinuouscompressedinasingleplane;thustheuniverseisafractalofPlanes;andthosePlanesareachievedbythe‘reduction’effectedbymental languagesthat ‘mirror’reality in smaller mappings and then project its images of the whole in its territorial orders, creatingsupœrganisms,which have always limits of existence in time (theirworldcycles) in space (their territories andmembrains)andinPlanes(asinformationdoesn’ttransferbeyondtwoPlanesupanddownasystem.I.e.youaremadeofcellsandpartofaworld,butthePlanesabovetheEarthandbellowthecellarealientoyou).Sofirstweshallintroducethe5elementsthatshapeall‘¬∆@st’:5Dscalar∆-planesofSpace&Time,withentropiclimits,focusedonstilllinguistic@-minds.

Whyweneedthisintroduction:Theentanglementinevermoresynopticelementsofclassicalgebra.

Algebraic firstprinciplesand itsconnectionwiththefractalUniverseare lessunderstoodbecausetheconceptsrequired–simultaneity,emergence,synchronicity,fractalprinciples,pentalogic,multiple5timespacedimotionsanditsmirrorsymmetrieswiththe5familiesofnumbersand5operandsarenotwellunderstood.SoweneedtointroduceinearnestthewaytheUniversebuildsitsdimotions(ab.dimensionalmotions)ofgrowingcomplexity.Theessentialconceptisthat5numbersfamilies,its5geometricrepresentationsand5mainoperandsofalgebra,aremirrorsofthe5Dimotionsandtheyareentangled,meaningnumbersincludesoftentheoperands,andbotharesymmetrictoaplanewherewerepresentthem, i.e.number5 includesasumof5naturalnumbers,whichrelatestothenaturalline;acomplexnumberincludespowersandroots;andneedsacomplexplane;afractionincludesdivisionshencealsoproducts,andangle,aratiohenceanirrationalnumber.

So really as I know you didn’t read the first paper (: I have to do a bit of an introduction since themirror of¬algebraneedstoknowwhattheimageisabouttofullygraspitsawesomesynopticbeauty:

In the graph theUniverse is a fractal that reproduces 'formswithmotion’, informationsandthenorganizestheminnetworksandsystemsthatevolveintolargerorganicsystemscreatingthescalarstructureofreality.Thuswecallthesumofallthoseco-existingPlanesofpartsandwholesthefifthdimension.

Thenitisnecessarytofindametricequationtodefinethisnewdimensionofspace-time.Sinceadimensiononlyexistswhenwecanwriteamathematicalsimplemetricthatleavesthedimension invariantwhenwe changeour parameters of space and time - hencewetravelthroughit.(Klein).Thisequation,asallspace-timemetricequations,issimple;sincemetricequationsaremeanttorepresentmeasuresof‘covariant’motioninagivenspace-timedimensionthatleavetheotherdimensionsunchanged.Sowewriteusingðforcyclic

time insteadof t, for amotion that changes the relative size and speedof clocksof a system (measuredwithfrequency):

5DMetric:S(LinealSize/Volumeinspace)xð(cyclicspeedofitstimeclocks)=Constant.

According to thosemetrics, smaller systems in space have faster time clocks. As information is stored in thefrequencyandformofthosecycles,smallersystemshavemoreinformation,codinglargerones:genescodecells,memessocietiesandparticles'quantumnumberscodeatomsandmolecules.

Thisequationand itsuseto improveourknowledgeofspaceandtime inall sciences,withanemphasis inourmodels of physical systems will be the theme of this paper. Even if physicists stubbornly refuse to treatinformationwiththesamevaluethanentropy.Sotheycallitnegentropy,andwhenyougiveaconferenceonthefifth dimension – the dimension of ‘creation of social forms of information, of organicwholes’ - there are no

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physicistson

attendance;and likely

no physicists will be reading this post… Let’s then use themetrics of the 5th scalar dimension to explain thefractal,nestedUniverseanditsPlanes,showninthegraph:

Themetricequationof the fifthdimensionof space-time (ab.∆)defines3knownPlanesofphysical systems,withdifferentquantityofinformationaccordingto5Dmetrics,Se(sizeinspace)xTi(volumeofinformation)=k.Sinceaswebecomesmallerinspaceparadoxicallyourtimeclocksaccelerate,andsinceinformationisstoredinthecyclicalpatternsandfrequenciesofthoseclockssmallersystemscodemoreinformation,soquantumparticlescodeatoms,genesorganisms,memescivilizationsandchipsmachinesestablishing theessential symbiosisbetween∆-1Planesand∆ºsuperorganisms,inscribedinanslower∆+1world.

Thosemetricsmeansinformationishigherinthesmaller‘quantumplane’thaninthelargergravitationalone,andinverselythesizeof itsphysicalparts is larger instheGravitationalcosmological ‘plane’than inthequantumone,withthehumanthermodynamicscalein-between.

AsthereisnoreasontostopthePlanesofthefractalUniverseinparticlesandgalaxies,thereisa‘potential’fourth,∆±4organicplanedefined‘above’thegalaxy,(∆+4,darkenergyworld)andbelowthequantumworld(∆-4,Bohm’squantumpotential),whichrepresentsthelargercosmos.Furtheron,accordingtothefractal,nestedprincipleanylargerorganicsystem,enclosessmallernestedsystems.Thusthe∆±4cosmoscontains∆±3galaxies,whichcontain∆±2solar systemsandplanets,whichcontain∆±1 thermodynamicorganismsandmatter states,describedby thehuman ∆º mind languages, contained on our brains,which according to those metrics will have a much densercontentofinformationbecominga‘linguisticMind-Mirror’ofthewhole.

Scalarentropicbig-bangsinabalancedimmortalUniverse.TheGalatom.

Inthefractalmodelabig-bangistheentropicdeathofphysicalmatter.ButtheUniverseisalsoscalarandithasinformation, a dimension of form, signified by the gravitational informative force, physicists ignore in theircalculusofitsexpansion,happeningingalaxiesthatbalancethedarkenergybetweenthem,weshallfindboth,balancesof forcesthatmaketheUniversewobbling,andbig-bangsand informativeforces inmultiplescalesoftheUniverse,talkingofmultiplebig-bangsbalancedwithbig-crunches.Letusthencorrectbig-bangtheorytofitintothefractalviewoftheUniverse,astherearebig-bangsinallitsscales:

Belowleft,weadthegravitationalforcethatwarps1Dspaceinto3Dmass ingalacticvorticesthatbalancetheexpansionof1Dspaceinentropicvacuumaslightdiesintodarkentropylinesbetweenthem.Asmasswarps31Dflat-vacuumintoa'highvolume'its3timesmorepowerfulinitswarping,hencethe75-25%Balanceofmasstodarkentropy.ThusthefractalUniverseisimmortal.Ontherightitsscalesallsuffersimilare=mcdualprocessesofwarpingthroughgravitationalforcesofcyclicmomentumthatcreatethosegalaxies,andexpansivebig-bangsoflinealmomentumbalancedincombinedcyclesofenergy-the3conservedquantitiesofeachfractalspacetimephysicalorganism,whichputtogethergivebirthtothe3conservedlawsofnaturetheBigBangtotallyignores-amongmanyotherknown-knownlaws.

So fractal spaceaccounts for the immortalUniverseand itsbalanced3arrowsof space-time, conserved in itsinfinitereactions;energy,informationstoredinthecyclesandfrequencyofthosesystemsofangularmomentum,whoseminimalconstantformish,andlinealspeed,conservedintheconstancyoflight.WhythoseobviousfactsofSOUNDphysicsarethen 'reduced' toænthropiccreationistbig-bangs inasingleplaneofspace-timedespiteevidence? It is the ego paradox: what the big bang does for science is a religious 'closure'' withman and itssimplestmathematical lineal functionsat itscenter,whichaddedtothedenialoforganicsentientpropertiesto

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physicalrealityallowstofeelthehighpopesofscienceasensationofabsoluteknowledge,asthereligiouspersonfeelsthatallisknowninthemysteryofgod-awordeerilysimilartothelinealfunctionofthebig-bang'V=Hod'.

RECAP.Theimportanceofunderstandingthelimitsofidealmathematicsisparamountinphysics.I.e.asthereisnoabsolute0’,singularitiesdonotexistandtheexpansionofvacuumislimitedbyitsoriginingalacticjets,where

gravitationmakesimpossibleiscontractionofspace.

FRACTALSPACE&CYCLICALTIME

ORGANISMS:VITAL,TERNARY,ORGANICTOPOLOGY

Topologicalorganismsofspace-time.:Absolutevs.Relational,GenerationalSpace-Time

Thefundamentalquestionphysicistswonderedforcenturiesregardingthenatureofspaceandtimeunfortunatelywas resolvedasusual in favorof the simpler view: it is spaceand timeanabsoluteabstractbackgroundof theUniverse(Mr.Newton’sview)orarewemadeof‘vitalspace’that lastsatimeduration,sowearegeneratedbythebio-topo-logicpropertiesofscalarspaceandcyclicaltime?This isthechoiceof5D‘stiences’.Anditssimplerversionwascalledrelationalspacetime,sponsoredbyMr.Leibniz.Arealistinterpretationoftheworldwelivein,whichhasnevershowninanyscaleofrealitysuch'background'-ultimatelyamathematicalgraphusedinabstractbyhuman scientists - considers thatweARE the vital spaceweoccupywithour cells, andwe livea cyclic timedurationbetweenbirthandextinction.Wearespace&timewhomustextractourpropertiesasexistentialbeingsfromthem.

InNewton’scosmos,spaceandtimeprovideafixed,immutableandeternalbackground,throughwhichparticlesmove. Space and time are the stage of intersecting lines sketched in the illustration. Fact is this ‘mathematicalartifact’madewithpenandpaperbyearlierphysicists,calledtheCartesiangraph,usefultomeasure'translationinspace' is no where to be seen in reality. Unfortunately as time went by the graph became somehow 'real' asscientists'feltthe'mathematicallanguage’createdreality.Itmeantalsotheinventionofanabsolute'continuousspace'andasingle'linealtime'thatextendstoinfinitycontradictingtheobviousfactthatall‘spaces’arebroken,dividedbymembranes,andallbeingshaveafinitetimeduration.Furtheron,aswekeptexploringsmallerPlanesofreality,weneverfoundthedrawingsofGod,notevenasolidstillsubstance,butalways'motions'tracingclosedtime-spacecycles;sinceevenparticlesturnedtobealso'vorticesoftime-spacemotions'.

The true, sound experimental and logic theory was Leibniz’s, who considered absolute space and time anabstraction, and so he coined the concept of relational space -merely the adjacent pegging of similar forms insimultaneousspaceandrelationaltime-thesequenceofeventswhichwerelatecausallywithreasonoriginofthe‘Generationalspace-time’modelof5Dinwhicharethespaceweoccupyandthetimewelast–as inthegraphwherethereisnolongerabstractbackgroundlines.WhatNewtoncalledabsolutespace-timeISNOT.Sospaceisthesumofallthediscontinuousvitalspaces,occupiedbydifferentbeings:∑s=S.Andlinealtime,Tthesumofall

thefinitelife-deathcyclesofallbeingsT=∑t.

Sincespace&timedoexistiftheyarenotinthebackgroundwe‘are’vitalspace and cyclic time. The simple idea behind the structure of the fractalUniverse is to consider time=change =motion and Topologic, formal

space=extensionthe2elementsofwhichallbeingsaremade.

We are space and time, merely of a different kind to that of Newton: Organic scalar spaces, andcyclical,discontinuous times who ‘live'… worldcycles (no longerworldlines as we have a 2nd arrow of information) ofexist¡ence(asallspeciesfollowthecommonlawsofspaceandtime).AscyclicaltimethatexplainstheinformativerepetitivepatternsorLawsNatureanditsmultiplespace-timeclocks.WhyaUniversemadeofspace-timebeingsis

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essentialtoaphilosophyofmathematicsisobvious.Becausethemainexperimentalscienceconcernedwithspaceismathematics and the main science of causal time is logic, if we are made of fractal space & cyclic time, bothmathematics & logic become experimental sciences, reflecting the properties of those 2 primary substances, asmirror-languagesofmaximal synoptic informationandminimal size (Sxð=C),whoseunderlying lawsemerge inallotherlargerPlanesofthefractalUniverseofbiggersizeandlessinformation,provingalsowhytheyapplytoall.

This said the devil is in the details. So whatdoes it mean to be made of motions withform,timeandspace?Inmathematicalterms,it means to be made of topologicaldimensions, which are holographic bi-dimensional spacewithmotion in time. Andas ithappens topologyhasonly3varietiesofbidimensional spacetimeand it is constructed

ofparts–points– thatbecomewholes–networks. So the immediate translationofgenerational space-time intomodernmathematicalsystemsdoconvert,asweobservedintheabstract,allsystemsintomathematicalbeings.

5Dinnovatesthisfield,'enlightening'classictopologyto'understandtheternaryorganic,structure'ofallsystemsofnature:Asthe5Dimotionsaredimensionsofspacewithtimemotion,itsscienceistopologythatallowsasystemtodeform=changeitsinnerform.Yeta4DUniversehasonly3'topologicalvarieties'thatrestrictsensemblestoonly3topologies,eachonebestsuitedtoperformthe3organicvitalfunctionsofanyphysicorbiologicsystem–gauginginformation(1D)tomovethesystem(2D)toanenergyfieldinwhichreproduce(3D):

Thepurposeofvitaltopologyistostudythe5Dimotions(dimensionalmotions)oftheUniverse…Assuchitwillbethefinalstageofevolutionofgeometryasanexperimentalscience,mergingelementsofalldisciplines.

Allspacedimensionshavetimemotion.Mathematicsfoundit,whenGreeksstillGeometryevolvedintoavaster,generalized concept, a topological› variety,where a topology as opposed toa geometry has internalmotions-changes.Astheonlycaseinwhichtheinnerdimensionsofabeingdon’tseemtochangeisexternallocomotionmost5Dmotionsneed‘geometries’withinnermotion,whicharetopologicvarietiesofwhichthereareonly3:

Inthegraph,thediffeomorphicPrincipleofEinstein’s4Danalysisacquiresanorganicnature,whenweseetheUniverseasthesumof∞Complementaryternary,topologicsystemswhosedimensionshaveorganicfunctions:Systems feed on their relative dimension of energy-length, perceive in their relative dimension of height andreproduceintheirrelativecombineddimensionofwidth,whichareassembledintoeachspecificspecies,tobestsatisfythesystems’intakingofmotion,energyandinformation.

I.e:Ananimalhas its informativeheight inthehighperceptive lightdimension,butaplant,whichuses lightasenergyhasitsupanddowndimensionsinvertedrespecttothemananditschemicalbrainburiedontheEarth.Soboth have opposite energy-time coordinates, with an ‘antero-posterior’, lineal’ ‘outward’ energy orientedstructureduetotheorientedarrowof light.But ina3Dworldwithnopreferredorientation,aseaorvacuum,cyclicformsthatmaximizeinformationdominatefromplanktontogalaxiesthathaveacyclic,informative,inwardstructure, as the stars’ bodyabsorbsenergy from intergalactic space, reproducesmatterwith it and feeds theinternalinformativeknotofgravitation,withahigherheightdimensiontheblackhole.

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So Dimotions of reality are 3 bidimensional topological varieties that act as vital organs in cylindrical longlimbs/fields,Hyperbolicwidebodies and spherical tall heads, eachonedominant in a lineal classic dimension,linealmotion,informativeheightandreproductivewidth,whichDOhaveorganicvitalpropertiestoo:

Sphericalparticle-heads,perceivinginformationfromtheadvantagepointofheight.

Lineallong,cylindricallegsandfieldsoflocomotionasthelineistheshorterdistancebetweentwopoints.

Wide,hyperbolicbodywaves,storingtheenergyreproducebythesystem.

Nt.1.AccordingtotheCorrespondencePrinciplesasphysicsnamed4Dimensionsweusethename5thdimensionforthewholerangeofPlanes,butinproperterminologyweshouldcalleachDimensions,a‘Dimotion’andconsiderthe5thscalardimensionthesumofallthe4thdimotionsofsocialevolution&allthe5thdimotionsofentropicdevolution.

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CYCLICTIME

Timeiswhataclockmeasures'.‘Timecurvesspaceintomass’Einstein

Thecausalrepetitivelawsof‘stiences’AUniverseof∞ time clocksof different size and speeddiffers from itshumandescriptionwithasinglemechanicalclock-timetowhichall timeclocks of the universe are equalized, elongated into a lineal 'second-minute-hour-day-year' system of equalized time clocks (of lightwaves,mechanical clocks, earth's astronomical clocks). As Galileanphysics, born of ballistics, simplified the nature of cycles of time-spaceintolinealdurations,tomeasurebestthelocomotionsofcannonballs:Time is cyclical, all clocks of time and laws of science are based in thecyclicalpatternsofnature.Butphysicistsdevelopedballisticsanddenied

thetruththatwecanknowthefuturebecauseitwillrepeatthecausalityofthepast,andwecanchangeitbychangingthatcausality,inHistorybyrepressingthelethalmemesofthetreeofmetalandenhancethewelfarememesthatmakeussurvive.Linealandcyclicaltimerenderthesameequationsasoneistheinverseoftheother,measuredbyfrequency,T=1/ƒ,butthe philosophical implications of cyclical time, are enormous, aswe regain the in-form-ation provided by those cycles,originofthelawsofscience,whichwouldnotexistiftherewerenotcyclicalpatterns;includingthecyclesofhistoryandeconomics.Themostimportantofthembeing,thefactthatatimecyclebreaksreality(1stknottheorem)inanouterandinnerregion,creatingamembranethatenclosesavitalspace,the‘substanceofwhichweareallmade’.The3scalesoftime.Whythentherearetwoformsoftime,thelonglinealTimeandthe‘short’frequencystepsweintegrateintothelargerwhole?Preciselybecausethereare2boundary±¡scalesof5Drealitywhosemetric,SxT=∆±¡defineslargerspacesystemsas having slower time cycles. Sowe can always consider the frequency of time the ∆-¡ ‘quanta of time’ or ‘finitesimalderivative’ofthelargerwholerepresentedwiththeconceptoflinealtime;asintheclassicformula,S=ƒ(t)λ(s).ThewholeSpacecanbemeasured,Vt=Swith lineal timeasasingleunit,or itcanbemeasuredasasumof frequencysteps,withmoredetail.Yet since two limitsalways createan interval, andall supœrganismsof time-spaceco-exist in3 scales, thatof its innerworld,thatofits‘membrain-mind’andthatofitsouterworld,therewillbeoftena3rdelementinbetweentwomeasuretime, specific for each supœrganismand sub-species sandwichedbetween the ‘lower’ timequantaand the larger linealtimeofthewhole.So forspecieswithin thegalaxy, ifwetakethe lowertimequanta,h,andthe larger timerodofmeasure, thespeedoflight,inbetweenweshallfindallkindofsupœrganismsthatcannotgofasterthanlightorprocessinformationinquantasmallerthananh-plancktonwithitsspecifictimequanta,tosynchronizeits3vitalparts,whichinthehumancaseisthesecond‘beat’oftheheart,lengthofthethought,glimpseoftheeye,measureofadualstœp,(and½secondofitisindeedthemusicalbeatthatwefindmoreharmonioustosynchronizeourdance.Synchronicitiesof timebetween the3 relative speeds inwhich abeing co-exists become thenoneof themost fruitfulfieldsofstudy,asthe‘persistence’ofsuperorganismsinspaceisduetothesymbiosisbetweenits3scalesofT.œs(time-spacesupœrganisms)synchronizedwiththem.EntanglingVitaltopologyandcyclicaltime:LocalPast=Entropy,Present=Iteration&Future=Information‘Theseparationbetweenpast,presentandfutureisanillusion’EinsteinOfalltheconsequencesofcyclicaltime,themostimportantistheexistenceofinfinitelocaltimeclocksofwhichweareallmade,whichthereforeimplytheexistenceofinfinitelocalpast,presentandfuturestates.Past then means a system with less ‘form’, less information, which slowly acquires a dimension of height-form, as itcompletesitscycletoreturnbackinthemomentofdeathtoanageofnoinformation.This‘worldcycle’ofexistence,whichcreatesanderasesinformationbecomesthentheequationofduality:

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Entropy-youth(relativepast)<Energy-maturereproduction(present)>3rdIn-Form-ationage(relativefuture)Whicheachofusfollowsasatime-spacesuperorganism.InphysicsisequivalenttothedualequationofEinstein:EóMc2,which reverses when E, which should be properly considered ‘Entropy’, as it is a disordered state, collapses throughgravitationintoMass,acyclicalvortexofspace-time;whileitsintermediatestateisc2,radiation;therelativepresent:Whereaspastisthebeginningofapicycle,startingasalineofentropywithnoformthatcurvesandraisesinheightinits2ndstateofpresent,andreturnsbacktoitsorigininitsfuture3rdageofinformation,completinga0’-sumoflifeanddeath.Thusinsteadofasingle∞linealabsolutetimethereare∞livingcyclesoftimehappeninginzillionsofentities.

Inthegraphweseethoserelativetimeagesforthesimplest‘physicalorganisms’,whichnotcoincidentallyareequivalentto the3 conservedquantitiesofphysics. Soas time is cyclical,madeof3’π’ lineal timemotions,we shall distinguish3conserved ‘ages’oftimespacethatputtogethercreatea0sumworldcycleof lifedeathforanyspacetimeorganisms:arelativedevolvingpastorarrowofentropyrepresentedinphysicsbydisorderedexplosionsinspaceandinvitaltopologyby lineal limbs/fieldsof linealmomentum, an iterative reproductivepresent that seemsnot to change, representedbyhyperbolicbody/wavesofenergy,andanimplosivein-form-ativelocalfuturearrowrepresentedinphysicbyaccelerated,V(t)R(s)=Kvorticesandangularmomentum,andbyparticle/headsintheensemblesofvitaltopology.Sotimespacebreaksin∞relativelocal,fractalentropicpasts,iterativeenergypresentsandinformativefutures,whichputtogethercreatetheillusionofasingletimespacecontinuum.

Wecan then set anabsolute ‘dimensionalmotion’ (ab. dimotion)of time-space.Because for awhole∆+1 toexist, theparts‘ilogically’mustcomefirst;sosocialevolutionandlovebetweenpartsistheabsolutearrowoffuturefortheorganicUniverse,or‘future’,whileaformthatrepeatsitselfseemsnottochange,sothefunctionofreproductionistheabsolutearrowofpresent,leavingthusentropy=death,thedissolutionofformastheinversearrowofpast.

Synchronicity,entanglementandabsolutetime.

Thevastexpansionofinformationwhichresultsofuncoilingtheplanesofspace-timeoftheUniverse,isresolvedbythehumind compressing the intoour size scale and rodof perception (light space-time). It is thennecessary to accept forpracticalreasonsthelimiteddistortionofthemind,latterdefinedmathematically.Sowewon’tattemptan‘Aristotelian,axiomatic’pedanticattempttosetabsolutetruthsabsolutelyfalse–asthehypothesisofthecontinuum,buthighlighttheinheritingparadoxesbetween the∝ (relative infinite)of information, thatourmindneverwill knowasonlyeach fractalpointofspace-timeholdsallofitsinformation.

Absoluterelativityisthenameofthegame.Andthatisfineifthehumindacceptsahumblerattitudetowardsthewhole,which so far does not. Infinities thus reduce to ∝ immensities, infinitesimals to finitesimal quanta, continuities toundistinguishable discontinuities. Single timespaces to an entangled Universe that generates its space throughsynchronousconnectionsbetween5Dimotionsofrealityinits3relativerangesof5Dscalarplanesinwhicheachdimotionhas a different speedbut as there are 5 dimensionalmotionswith different functions, they entangle together: i.e. thefaster dimotion of feeding in the slower ¡+1 plane of the ecosystem, provides energy to the slower dimotion ofreproduction inthefaster¡-1planeofthecell1timeadayformost¡ºanimalsconnectingthe3planesofbiologic life.Functional synchronicity creates simultaneity,whichgivesbirth to ‘orderedco-existing superorganisms’.Evenahumindlooses information and compress it all in its plane of light and second beat, of its smallermirrormind to fit/select itssurvivalinfowithinitsbrain.

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THE3ONLYTOPOLOGIESOFPHYSICALSYSTEMS:CONSERVATIONLAWS.

The 3 parts of any fractal point of cyclical time in theUniverse become in the simplest physicalsystems, the 3 conserved parts of the minimal organic Unit of reality, a ‘Planckton’ of angularmomentum. All physical systems then can be 'reduced' to fractal ensembles of 3 'conservedquantities', angular momentum - the membrane of the system, which becomes a membrane,fractalsumof'cellularcycles'ortheskinofthesysteminhumanbeings.Vitalenergy,theenclosed

cyclicalformsandmotionswithinthespacewhoseboundaryconditionsaregivenbythemembrane,andlinealmomentum, the motions with a 'finality' we perceive guided by a 'relatively still mind-point-singularity' thatfocusestheenergyandinformationtransferredthroughanangularmomentummembrain.

Itisalsothesmallestclockofourworld,asitsminimalunitofcyclicalinformation,orangularmomentum,usedastheunitofthe3physicalparametersofspatialsize,cyclicaltimefrequencyand‘scale’(Activemagnitude):

h=mass(∆)xarea(S)xƒrequency(ð).

Soh, ‘Planckton’, is theminimal fractalorganismwhichbecomes the ‘cellularunit’of allother speciesof lightspace-time,asPlanktonistheminimalunitofthebiologicUniverse,alsoacellwithasimilarternarystructure–aDNAnucleusthatprocessinformation,aproteinmembranethatisolatesavitalspace-time,thecytoplasm.

ThusNon-Euclideanpointshavebreath,itslinesarethereforewavesabletocommunicatetheexternalformandinternalenergyorfractalnetworksthatbranchtoconnectmultiplepoints,anditsplanesintersectionofthreeofsuchwavesornetworksthatformtopologicalorganisms...ItisthenobviousthatthenextstepofNon-Euclideangeometryistomergethoseconceptswiththephysicalanalysisofthesmallestphysicalsystems,tounderstanditsvital topologies. To that aimwe introduce the second fundamental equation of 5Dmetric that formalizes theParadoxofrelativity.AndsoitactsinthephysicalmodelasthePostulateofrelativity,commonto3DGalileanand4DEinstein’ssimplexmodels,whichcorrespondto5Dinasingleplaneof‘lightspace-time’…

Euclideandimensionsperformorganicfunctionsinlightwavesthathumansperceiveasspace.Lightisanorganicsystemof3dimotions:c-speedlength,electricinformativeheight&awidemagneticfieldthatsupportsthem.

The5Dimensionsofspace-timeare5vitaldimotions,broken into infinitevitalspace-timebeings,andnowwehavethe5elementswhichrealityusesindifferentperspectivestoconstructallrealities.Nothingelseisneeded.Anditwilleasilyfollowthatineachstience,includingmathematics,3±¡elements(dependingontheperceptioninasingleplaneorinseveralones)willbeconcernedwiththeanalysisofasysteminSimultaneous,entangled,stillspaceasasuperorganismconstructedwiththose5Dimotions,whichinsequentialmotiontimewilltraceaworldcycle also composed of those 5 dimotions. As the universe simply put it is a reproductive fractal of 5Dimotionsofspacetime…

Asmotionwithformconstantlyreproducesbythemerefactofmoving.Soeverymotionofentropictime,whichenterswithintheorderlystructureofafractalpoint,itswaves/networksofcommunicationandtopologicalplanes

explainedinthepaperon5Dgeometrybecomesaniterativereproductivesystemofspacetime.

Thebalancesachievedbythesimilarityofspace=formandtime=motionreachedinthepresentbody,S=T,andtheunbalance of themetric equation of scales, $ x ð = K, in the limbs (Max. $T) andminds (Max§ð) unify as S=TmaximizesSxT=K(5x5>6x4),in1equation:Max.SxT=C,whichdefinesforeachfractalvitalspace-timeorganismits Function of existence, as all species will try to maximize its motion-entropy-time for its field-limbs, itsinformation-spatialstatesforitsparticle-heads,whoseproductwillgiveusitsvitalreproductiveenergy.Moreovertheequationhasanimmediatebiologicmeaning,becauseaswearemadetopologicallyof‘fields-limbs’oflinealspacewithmotionprovidedbytheenergyweabsorbtoalsoreproduceourbodies-waves,andtheinformationweneed to linguistically guide our motions with particle-heads, the very essence of survival is to increase ourS=position,mentalformsofspaceandT=entropicmotionsoftime(whereastime=motionandspace=formarethe

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twolimitingDimotionswith‘energy=reproduction,s=t,locomotion,sTandinformation,St,aretheintermediate3dimotions).

ThustheUniversalconstantsofrealityrespondto35DMetricconstants:S/T=SpeedofLocomotion,whichdefinesthe limbs/fields of the system, SxT=Existential Momentum/force, which defines its body and T/S: Density ofinformation,whichdefinesitsmentalpower,allmaximizedwhenS=T.

Inphysics,foreachfundamentalscale,thereisalsoaconstantratiobetween'frequency=timeparameters'andenergy=massparameters,whicharethethreefundamentalconstantsofNature,H-Planck (ratio of frequency-energy for quantum systems), K-Boltzmann (ratiobetweenthetemperaturefrequencyandenergyofthermodynamicsystems)

TheUniversalconstantsofrealityrespondto35DMetricconstants:S/T=SpeedofLocomotion, which defines the limbs/fields of the system, SxT=ExistentialMomentum/force, which defines its body and T/S: Density of information, whichdefinesitsmentalpower,allmaximizedwhenS=T.

Onceweestablishthemeaningofthe5Dimotionsofrealitywecan‘ascribe’everyparameterofsciencetooneofthosedimotionseitherinaceterisparibusanalysis

asapartial spaceor time functionorasacomplexentangledspace-time function inequations thatdescribeatime-space being mutating its 5D metric through the 5 ‘graduations’ of motion and form, from entropy («,scatteringmotion,whichisbothmotioninternalandexternaltothebeing),locomotion,<(onlyexternalmotion),reproduction,≈,evolution,>(internalmostlywithexternalshrinking)andlinguisticseeding,»,stillperceptionincrystalandsolidormentalstateswhereasanactivemagnitudecreatesfromafieldofmotionaseedofform.

MOSThumanmeasures called ‘energy’ areNOTabout the3rdDimotionofenergy,which ‘KEEPS itsbalanceofformandmotion’but aboutentropic statesof scattering formandexpansivemotion in space. Forexample inE=mc2, amass vortex of physical information is trans-formed into an entropic expansion of space, loosing itsform.SowedoNOTmeasureitsenergybutitsentropy.HumanEnergythenworksasaparameteroftheentropicscatteringmotionsduetothedeathorbig-bangofaphysicalsystem,explodingitsform.ThisagainbecomesthecaseforHeat,whichmeasurestheentropy=motionextractedfromathermodynamicensemble,usefulformantobetransformedinto‘realenergy-orderedmotion’inits∆+1scale.Soweconsiderageneral5DmetricforthosephysicalscalessuchasEisafunctionofentropythatmultipliedbyafunctionoftimefrequencygiveusascalarconstant,sowewrite:

Energeticentropy(E)xTimeDuration(1/ƒ)=Constantofspacetimescale.

Theconnectionbetweenbothconceptsishowevermoresubtle.Whenweexpresstheentropicenergywithoutcaringfor its ‘informativedetails’,multipliedby itstimeduration,weindeed‘erase’theconstantofspace-timeintoaflowofentropicmotionhumanscanuse.ButifwekeepthesystementangledwithoutlossofinformationasE=Cxƒ(ð);wecantalkofthe‘innerenergy’ofthesystem,E,asa‘populationsum’,of‘space-timeC-beings’,as in a wave of light which is a sum of ƒ-Plancktons. So bearing in mind those subtle distinctions, 5Dmetricequations,ExT=CorE=Cxð,giveus the3 fundamentalequationsofmathematicalphysics regarding its3‘scalarplanes’(graph).

Thus the 3 conserved quantities of physical systems are the 3 elements of the 5Dmetrics equations of the 3physicalscaleswithinthegalaxy,whosemetric,hxc=ksetsthefinallimitsofinformation(h)andmotion-distance(C-speed)ofthewhole.Allarerelative‘nestedscales,’whichchangeaswetravelthroughthemtheparameteroftime (frequency or duration), and entropicmotions, giving us 3 constants of scale of space-time, h, m and Tthroughwhichtheactionsofitssystemsaretakingplace.

Theeasiesthumanequivalence:TemporalEnergyandSpatialInformation.

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Human closest translation of the duality of primary elements was in the classic age of verbal thought, Asianphilosophies of yin=in/form/ation=space=visnu and yang=entropic energy=time=shiva, and in the ageof digitalthought,motion=timehasbecomesynonymousofentropicenergy,whilementalsimultaneousspace isakintotheconceptofin/form/ation;sinceourconceptsofspaceandtimehavebeenrestrictedenormouslybyvirtueofthe Galilean equation of speed, v=s/t which became as physicists substituted philosophers of science in thesummit of scientific thought, due to its power asmachines andweaponsmakers, the ‘only equation of time’,wheninfactwasjustameasureofspeedwithverylimitedspaceandtimeparameters.AsEinsteinmerelyadded–ctermtothatequationnothingchangeddespitethehypesurroundinghiswork.

Soweshalloftenusethetermstemporalenergyorentropicenergyfortime=motionandspatialinformationorspatialform,forspace.Astimeandspacearesynonymousofmotionandformtheydohavemultiplemeanings.Sothereare5typesoftimechange=motionparallelto5typesofspace=form.Yetinfactaswehaveseenspaceand time are always mixed together, since human still mental one dimensional space is a simplification anddefinitely a single time arrow of entropic energy or absolute motion is an extraordinary simplification of therichness of motions of time. Energy in that sense with its multiple variations of meaning is much closer.Specifically entropic energy; that is, amotionwhich is internal and external to the being or ‘TT’, puremotionappearsintheequations,E=Mc2(∆+1)=knT(∆º)=hƒ(∆-1)thattransfertemporalmotionthroughscalesofthefifthdimension. This is the entropic arrow of time and disorder akin to deathwhich physicists consider the singlearrowoftime.Myadvicethenbecauseofsomuchmessyandsimplifiedunderstandingoftimeinphysicsisreallytotakewithsomeironytheirworkonphilosophyofscience,thoughastheirequationshavetheirownlifeandself-consistencythoseare ‘real’andmustbeaddressedwithmuchmorerigor,asweshalldo in thepapersonmathematicalphysics.Butforconceptsoftimeandspaceusefultoallscalesandsciencesweshallusetemporalentropic energy and spatial in/form/ation. Energy, being the jack of all trades, inmessy human philosophy ofscienceisalsousedforS=T,balanced‘work’whichinfact isbothspaceandtime,energyandinformation,asitimprints form intomotion, reproducingcyclicalpatternsof science.So it isbetter todefineSTas reproductivework.

It is thenobvious that the fundamental lawof science, the lawof conservationof energy, becomes the lawofconservationandimmortalityoftime-motions,whichcanbeexpressedasfollows:

‘Allwhat exists is time=motion=change in perpetual trans-form-ation in one of the 5Dimotions of time-space:SóT’

And it is thepurposeofscience ineachsubdisciplinestudyingoneofthescalesofthefifthdimensiontostudyhowonetransformsintotheother.

Ithappensthenthattheparametersofrealitywillbeoftime,spaceandscale;oftengroupedindualities.Soforexample in physics we have ‘scalar’ magnitudes defined by a ‘scalar, social number’, such as mass andtemperatureandspace-timeparametersofdimotions,suchasvectorswithmagnitudeanddirection.

In physics, for each fundamental scale, there is also a constant ratio between 'frequency=time parameters' andenergy=massparameters,whicharethethreefundamentalconstantsofNature,H-Planck(ratiooffrequency-energyfor quantum systems), K-Boltzmann (ratio between the temperature frequency and energy of thermodynamicsystems) and among themanymanifestations of the same law inmechanical,moving systems, the third law ofKepler(ratiobetweentheorbitaltimeclockofplanetsandthespatialvolume:energycontentofitsorbitalsphere):

Inbiologywestudyfamiliesofanimalssuchasmammalswherelargerorganismshaveslowermetaboliccycles.Inhistory we study social organisms, whose cycles of life and death, will define the evolution of nations andcivilizations.Andineachofthoseorganisms,smallersystemscodelargerones.Sothequantumnumbersofparticlescodematter,genescodebiologicorganisms,andmemescodesocieties.

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Butthereallydefininggraphisthethirdonewhichshowsthattheaveragenumberofheartbeatsinthelifetimeofanymammalisroughlythesame,eventhoughsmalloneslikemicelivefor justafewyearswhereasbigoneslikewhales can live for a hundred years or more… but in terms of internal time, which synchronizes a givensuperorganisminits3vitalparts,limbs,bodiesandminds,soyouwalkasecondeachstep,yourheatbeatsasecondandyoureyes-thoughtstakeasecond,thelifeperceptionofeachorganismofthesame‘species’,isthesame.Suchisthe justiceof theUniverse that if youwerea small rat your lifewill seem to youas longand fruitfulasahumanbeing.AndthisistheultimatemeaningofthefractalUniverse,the‘absoluterelativity’ofscales.Wearenotonlyoutof the center of space (relativity) and time (Evolution) but also our scalematters nothing. A fact, whichwe canextendtootherfamiliesofnature.Soinsectswhoserateofthoughtprocessingis10timesfaster,reasonwhyyoucan’tcatchafly,liveupto7years(cockroachesant-queensofant-hillsuperorganisms,dragonflies),whilehumansliveupto70.Theexamplesbelowarejustatinysamplingofanenormousnumberofsuchscalingrelationshipsthatquantitatively describe how almost any measurable characteristic of animals, plants, ecosystems, cities, andcompanies,planets,stars,lawsofscience,atomsandgalaxies,scaleswithsize.

Aninterestingnumberofthat3rdgraphisthefactthatallperfectsystemsdolive(9-11)9-11timequanta;closelyrelatedtothecommonestspatialpopulationforafulldevelopedsystem(fromthetiesofDNAmolecules,tothenumberofcellsofanorganism,tothenumberofpeopleontheEarth’ssuperorganism,tothenumberofstarsinagalaxy,tothenumberofgalaxiesintheperceivedUniverse.Thereasonsofit,beingthe3x3±¡‘tetraktys’scaleof organic systems,whichwe shall develop in theGeneralmodel aswell as the section dedicated to number

theory,andthe‘relativityequation’S=T,whichmakessimilarspaceandtimeonthepointofmaximalbalanceandsurvival of reality. This said,we are not unlikemost papers andwork onmodern science, on the quantitative‘magic’ of our studies, not even in this paper on algebra, but on the deep whys nobody answer, of thoseprocesses.Sincethereasonsofallthoseprocessesneverare5DMetricsandthefactthatscalingisanembeddedfeatureofallsystemsofNature,forthesimplereason,wearemadeofthetwosubstancesthatscale,spaceandtime.

LogicConsistency:TheInfinite,HierarchicalUniverse

Inthe1970s,Mandelbrotprovedfractal,hierarchicalself-similarityisubiquitousinnature-aninfinitehierarchyofworldswithinworlds.Sotheoreticalphysicistsfoundtheirwaytothe∞fractalparadigm,whichunfortunatelydenies the continuous big-bang. So it offers resistance from established big science (Kuhn’s on scientificrevolutions).Inthefractalparadigm,theUniverseisinfiniteandthebigbangisnotthebirthofallrealities,butanylocalbigbangandbigcrunchdualprocess,anyexplosionthatsplitsthephysicalenergyandinformationofacomplexsystem, inanyof themultiplescalesofphysical reality.Sofirst for the impatient,weshallhavea fastglimpsetohowafractalUniverselookwhenweaddscalarelementstothe'entropicbig-bang'.

5DMETRICSANDITSFUNCTIONOFEXISTENCE:THEGENERATOREQUATIONSOFALLSTIENCES.

The judge and the 4 witnesses represent 5povs. to obtain a partial truths as truth only exists in the being or event in itself that holds all

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the information. So we need a pentalogic of 5 Dimotions for reality to emerge and ‘persist’ through synchronicity and simultaneity.

Inmathematicalscience foradimensionofspace-timetoexist, it requiresametricequation,whichcombinesspace, and time to gives us a co-invariant system that allows travelling through such dimension. Howwe dotravelthenthroughthefifthdimension?Asystemtravelsthrough3scalesofthefifthdimensionbyacceleratingitsevolution inasmallerscalethroughaplacentalcycle,emergingasanorganisminthe largerworld,to live3ages&dissolvebacktoitspartsinthe0’-sumdeath.Andthisisthemeaningofexistence,anditsreasond’etreistheSIMPLEMetricequationsof5D,whichstructuresthroughsynchronicityofthedifferentspeedsoftimecycles,the different scales of reality. So the ∆º organism eats every day, and its food programs the faster cycle ofreproductionofitscells,asthemooncycleprogramsthemenstrualcycleofwomen,astheyearcycleofrotationofEarthprogramsthereproductivecycleofseasonsandsoon.

So an essential part of 5D theory is the analysis of synchronicity, simultaneity, emergence and the in-depthanalysis – not done in this introductory course, reserved for the more complex papers on ‘pentalogic anddodecalogic’wherewe follow inmore detail the construction of simultaneous superorganisms and its ternaryworldcycles..All thisofcourse isstudiedbyhuminds,aseverythingwetalkabouthere,butwithouttheproperconceptualframe,lackingvaliddefinitionsofplanesofspace-time,timecyclesandfractalspaces.

It is forthatreasonwedoneedanewformalismwehavecalled existentialalgebrawith itssimplesymbolsofwhich themost important are the 5 bidimensional dimotions of space-time,which entangle together throughsynchronous,simultaneousemergentprocessestocreatetheapparent‘solidity’and‘stillness’ofreality.

BecausetheUniverseismadeonlyoftwopolarelements,stillminds(SS,ab.§)andTemporalentropy(TT),andits3 dimotional combinations, St-information, Ts-locomotion and S=T, reproduction, whose interaction can beresumed in the function of exist¡ence, Max. SxT (s≈t)=C, we can deduce all the principles, laws, events andequations of all stiences from it. So we shall call Existential Algebra to the ÐST formalism of GenerationalSpacetime(ab.¬Æ),anddoexactlythat:deduceallequationsandlawsofstiencesfrom5Dmetrics.

Weshallthusmakea1stforayonexistentialalgebra,showinghowthe‘development’of5Metricsgivebirthtothe function of existence into its 3 ‘extremal points’ or ages , Max. S x T (3rd age), Max. T x S (youth), S=T(maturity),definestheworldcycleofexistenceofallbeingsinitstwodirections,forwardsandbackwards.

But5Dmetricscanbestudiedinmoredepth,roughlyspeakingin4sub-equations,whicharethefoundationsofthe43greatsubdivisionsofscience:

-Thephysicalequationofrelativity,S=T,basisofallphysicalandmathematicalstiences.

-Thebiologicalequationorfunctionofexistence,Max.SxT(achievedpreciselywhenS=T),thebasicequationofallbiologicaldrivesandevolution.

-Andtheequationofthemind:0’-mindx∞Universe=Constantworldthatcreatesmentalspaces…whichwewillconsiderinthenextparagraphsaswehavedefinedspaceandtimemoreproperly.

-Finallyhoseequationcanbefurtherunified,sincethemetricequationofmultiplespacetimescales,SxT=K&therelative equation of dual motion/stillness in a single plane S=T that maximizes SxT=K (5x5>6x4) unify in 1‘existential’equation:Max.∑SxT=C±¡:∆±1,whosestudyisthefieldofPhilosophyofstienceanditsnewformalism,ExistentialAlgebra(ab.¬æ).Soafterstudyingthe3classicfieldsofsciencewillreturntothose5Dimotions,SS,St,sT,ST,TTandstudyitsentanglementanddifferentpropertiesandcomplementaryoppositions,tostartbuildingtheformal laws of existential algebra, the formalism of Generational space-time. that all stiences notablymathematicsandlogicmirror.

∞MINDSPACES.RELATIVITYOFMOTION.

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‘epursimuove,epurnomuove’Galileo

Letusthenstartwiththephysicalanalysisofrelativityanditscorrespondencewith5D.

GalileanPx:SóT:RelativityofspaceForms=vs.timeMotion5Dimotions

Itisquitesurprising,sorryformyderisivehumoronhumind’segocy–nothingpersonal–thatthedisciplineofhuminds’sciencesthatmoreadamantlydeniesthe existence of ‘mental spaces’, and infinite monads-minds that create theorder of reality is physics,which upholds as the only dimotions of time, sT,locomotion,externalmotion,or ‘translation in space’andentropy, scatteringexternal and internal motion, TT, just because physicists’ worldly professionhas been to make weapons of entropy and transport machines, so as usualhuminds project their inner mind in the whole Universe, and so physicistsprojected ballistics into reality. And they still do with the big-bang dogmastudiedindepthinotherpapers.

The fundamental postulate of physics, relativity of motion and stillness is adirectconsequenceofthementalnatureofspaces, inwhichamental languagestops,mapsinsimultaneityandrepresentsrealityasinformationwithnomotion.Sowecannotdistinguishmentallywhatmovesandwhatisstillasthepurposeofamindistostopmotion,asthepoleofspacethatisinaneternaltug-of-warwiththeentropyoftime.

Galileo’stimeandspacePrincipleofRelativity isthefundamentalconceptualthoughtbehindtherelationshipbetweentime=motionandspace=formandhowonecanbeconvertedintoanother:Allwhatexistsismadeofspace=formandtime=motion.Andyetphysicistsknowthatwecannotdistinguishmotionfromform.Thatanybeinginmotionfromitspointofviewseemstobestillandallotherthingsmovingaroundit.ThisistheprincipleofRelativityofmotion.

Physiciststhenwithoutmuchthoughtaboutthatfascinatingduality,wentontousemathematicstocalculatethe relativemotionofeachentityof reality respect toother system,which seems static frombothpointsofview.ThisiscalledGalileanrelativity,latterrefinedbyEinstein'srelativity,andessentiallyisconcernedwiththemathematical calculus ofwhatwe shall call the 2ndDimotionof time=change, locomotion. Fine, butwearemoreinterestedonthedualityofspace=formandmotion=timeanditsentangledrelationships–thereasonswhywedoNOTseetogethermotionandform,evenifallsystemshaveboth.

Theconclusionisthenratherobvious:oneofthetwoparametersofrealityis'hidden'toperception;weeitherseemotionorform,'wavesorparticles'(quantumcomplementarity),distancesandlinesorpointsinmotion(asinthenightwhenfastcarsinapictureappearaslines).Sophysicistscalculateonlyonewheninfactwemustassesstheexistenceof2;andsincewecannotdistinguishthem,logicallywemustequalthem.‘Form=motion-function;space=time;S=T’.

Relativity is a duality, S=T, at the heart of every law of the Universe. But which one is the real element?Obviously time=motion. Space is a “Maya of the senses" – a slice of timemotion. The ultimate substance ismotion.Formiswhata'stillmind',makesofthatmotionto'perceive',information,forms-in-action.

Since we see Earth still and flat but it is round and moving. Galileo’s profession was ballistics - the study ofcannonballsmotion. So he chose ONLYmotion and lost the chance to start physics with a complex philosophicalunderstandingofitsS=TdualPrincipleofrelativity,whichPoincaredefinedlatterclearlywhenhesaidthat‘wecannotdistinguishmotionfromstillness’.Anexampleisquantum/relativityduality.Indetailquantumspacehas‘darkenergy’becauseithasexpansivemotionthatextendsintoaplaneofspace,butwhenseenatlargerPlaneswithoutdetailitsentropicmotionseemsstaticspace-adualareaofscatteringlengthandwidth.Sointhegalaxyweseeeitherdarkenergymotionorexpandingspace:T=S.Amotionoftimeisequivalenttoadimensionofspace:Distanceandmotion

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cannot be distinguished so they must be taken as two side of the same being, a space=time Ðimotion (ab.DimensionalMotion):

S=T;Dimension-Distance=Time-motion=STÐimotion

Why if the Earthmoves in time,we see it as a still form in space? Because Reality is a constant game of infinitemotions, but the mind focus in stillness those motions, and measures them at distances. For ‘huminds’ motion isrelativetooursystemsofmeasureandperception,whicharelight-based;henceafixedc-rodspeed/distance.ReasonwhyEinstein’s relativitypostulatesamaximal T:c-speed,measuredas if observerandobservablewere still to eachother(ConstantS);whichatourscalewe‘correct’withLorentzTransformations.

PhysicistsjustsubstitutedEarth’sstilldistancesformotions.Ittook300yearsforEinsteintorealizetherelativityofmotion and itsmeasuremade essentially time and space,motion and form two sides of the same coin. Still thisrealizationwasnotexploredphilosophicallyandsoitgavebirthtoaseriesofill-understooddualitiesbetween'statesofmeasureandform'(particles,headgaugingform,in-form-ation)and'statesofmotion'(wavestates).

It is then essential to grasp thatmotion and form co-exist as 2 different states dependingon 5D scale anddetail:Motionsareperceivedbymindsthatstopmotionintoform,intoinformation,asdistances.Soifweseeslowmotioninthenight,acar’sheadlightseemsalongdistanceline‘still’picture.Butthismeansalsothatthe3‘Euclideanstilldimensions’musthavemotion;theyare‘bidimensionalST-holographic,topologicdimotions’.Sowehave3Space+1Time+15thdimensionofPlanes=5Dimensionalmotions.NoneofthemisaDimensionofpurespatialformorapuretime motion but a combination of both. Even ifmentally we tend to reduce motion and focus on forms, all hasmotion=time, and form =space: this is the meaning of 'spacetime', the messing of both into 5 dimotions, thefundamentalelementofallrealities.

Relativity states ‘wecannotdistinguishmotion=time fromposition=space’. Soallwhatexists isa compositeofboth,undistinguishableS=T,5 ‘Dimensionalmotions’ (Ab.Dimotions),broken in infinitefractal,vital timespaceorganisms,composed of topological Dimotions: height=information; length=locomotion; width=reproduction; form=socialevolutionofparts intowholes&entropy=dissolutionofawhole into itsparts ina lowerscaleof the fifthdimension(termwekeepforthewholerangeofPlanesoftheUniverse);whosestudy isbothmathematical, themainsciencethatstudieshowthose5Dimotionsentangle insimultaneousSpace,connectedtoeachothertopologicaladjacentparts, which create superorganism and Logic; the main science of time that observes how those pentalogic,entangledsuperorganismsmoveandevolve,changeinsequentialrelationaltime,livingalifeanddeathworldcycle.

Since there is nothing else but time& space, the 2 experimental ‘mirror-sciences’ of time and space become themost important to extract the ‘Disomorphic=laws’ of those 5 Dimotions that all systems have in common. SincewhilethoseDimotionsarebroken,invitalorganisms,separatedbycyclicaltimemembranes,theyarethesame.

IngraphGalileanrelativitywasillunderstood,asthetruequestionabouttime-changeiswhy‘themindseesspaceasastillcontinuum,whenindetailismadeofsmallerself-similarquanta,inmotion.Theparadoxdefinesmentalspacesasstillsimplifiedviewsofthemore complexwhole. The 3 ¡logic paradoxes of space topology (closed in-form-

ativecurved-Ovs.|-open,freeentropiclinealforms),time-motion(stillnessvs.motion)and∆-scale, (continuouswholevs.discreteforms;singlescalevs.multipleones),areessentialtotheperceptionofasimplified‘spatialminduniverse’ina single flat still plane vs. the full,more detailed complex picture in time, of a curved, discrete andmovingUniverse.Thoseparadoxesresumethe5elementsofreality,Space=form,time=motion,Planesandthemindthatmeasuresthem,withinitsownentropiclimits.

What neither mathematicians nor physicists fully understand (though some inroads in abstract were madethrough theNoether’s conceptsof symmetry) is thateachstœpofamethodof solution isnot ‘gratuitous’;butmustbegroundedinarealpropertyofthe5D∆STsymmetriesandconservationlawsoftheUniverse,whicharenotsomany–hencetherepetitionofmethods.Specifically,theaforementioned3paradoxesbetween∆+1curved

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closedworldcycles,sumof linealsteps,whichgivesbirthtothemostusedmethodof linealapproximations;theequivalencebetweenSpaceandtime,inallStœpsofdimotions,whichgivesbirthtothemethodofseparationofvariablesondifferentialequationsandmorebroadlyallowstomovearoundrelativespaceandtimeparametersinequations joined by an operand of ‘equivalence’ (≈ not =). And the 2 conservation laws of the Universe,conservation of those ‘beats’ of existence, S=T in relative present, eternal balance, justifying the equivalenceoperands.Andconservationofthe‘volumeofspace-time’ofeachplaneoftheUniverse,byvirtueofthe5Dmetricequation SxT=C, which justifies all the procedures regarding scales – solution of differential equations byseparations of scales, renormalization procedures (Wilson), and harmonizes those scales allowing constant butbalancedtransfersofenergyandinformation,St=Ts.

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Motionisreproductionofforminalowerscale.Bohm’srealism:quantumpotentials.

Howa systemmoves in a crowded reality,where all is vital space-time? The answer that resolves Zeno’s&quantum complementarity paradoxes, is ifwedo notmove but reproduce our information, translated into alower fasterwavescaleof the fifthdimension;aswe reproduceour sound in fasterelectrons to telephoneornerveimpulsesintochemicaldopaminetojumpdiscontinuousneurons.Somotionbecomesscalarreproductionofform,andsinceallisaformofmotion,allisreproduction,whichisthedefinitionofamathematicalfractal,afeed-back reproductive equation; 5Dmetrics,which become then the ‘function of existence’whose goal is toreproduce the formof all systems– the simpler oneswithmaximalmotion-translation in space, the complexoneswithmin.motionasareproductionthatemergesbetweenPlanes.Andthisgivesbirthtotheworldcycle.Considerthecaseofquantumphysics:Inthegraphweseeaparticlereproducedinadjacentregionsthatfadeaway,andtheresultistheperceptionofawaveofmotion.InBohm’srealistmodelthisreproductionhappensinalowerplaneofquantumpotentials,wherealsoentanglementhappens,whichisthe∆-4scalewhichisv>cin5Dmetrics (Min. S xMax. T=C), hence real. Motion is reproductionof formover suchpotential: thewaveerases form into motion, the particle is a still state that gauges information entangled to other particle,fermion and boson, still to each other – despite the perception of relativemotion in our scale – hence theinformationelectronssharehasalwaysac-constantspeed.

ThustheLorentztransformationareobjectivelyrealformankindwhoeliminatesthestopstateofparticlesaswedoinamovieeliminatingthestopframebutifwewereobservingrealityfromtheperspectiveofanatom,wewould‘stop’,entangleinthequantumpotential,neutrinoscale&soeliminatethespookyeffectsof ‘timedilation’&‘length’contraction,fromourperspective(butnotofmassincreaseasitisascalareffect).Thisisthe'rational' 5D explanation of both the c-constant of light and entanglement; as electronic beings perceiveinformationin 'stoppositiontoeachother’andmovein 'wavestate'.Motionsareperceivedbyparticlesthatstopmotion into form, into information, as distances. So 4D relativity needs to be expanded to the scalarUniversebeyondthec-speedlightlimitofthegalaxy.Withinthiscomplexview,themodelsofNewton,GalileoandEinstein’sspace-timecorrespondtothelimitof5Dwhenwesimplifyalltheworldcyclesoftime,wecalllife&death toasinglemechanicalclock,elongatedto infinity&perceived inasinglescaleofspace.Letus thendeducefromthose2equationsthefundamentalequationofreality:

Those2polesof reality are the firstprinciplesof any scientific inquire,prior to the languagesof time-motion,logic and spatial forms–mathematics, thatbettermirrors ¡ts laws. Lookaround,everythingyou see is a formwith inneroroutermotions.Thoseare thus the2primaryelementsof reality;whichmind languagesperceivemostlyby reducing thePlanesof the fifthdimensionand itsmotions to theminimalpossible to fit itall in themind‘equation’:O-mindx∞Universeofformalmotions=Mentalworld–reducedmirroroftheUniverse.

Mathematicsthatconsidersmotiona‘symmetricreproductionofin-form-ation(groups)oncemoreturnsouttobeabetterexperimentalsciencethanphysics.

Thatmotionhappensasreproductionofformisessentialforeveryphenomenaofstience.Afewmathematicalcases:

-Realitywillseemcontinuousanddifferentiableasaconstantzig-zagofstepandstopsthatseenindetailbecomesaquantumtangentialmotion.

-Becausereproductionisneverperfect,theaccumulationofstatisticalerrorsineachstœpwillaccumulateonthemovingformtillitkillsitasthespeciesslowlywarpsandwrinklesitssynchronies.SowecandefineaGaussianfunctionof

existenceofreproducedpopulations,whicharestepsofthefunctionoftime-motionsofthebeing.

-Reproductionofinformationwillhavealimitonthe‘bottomline’speedofthescaleinwhichthegamestarts,forthegalatomthehc-scaleoflightspeedandh-quanta.

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-Reproductionencompassesthe5dimotionsofexistence,becomingthemorecomplexfundamentalgameofthefractalUniversetothepointallcanbeexplainedwiththereproductivefunction:Max.∑SxT(s=t)=C

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THEINFINITEMINDSOFTHEUNIVERSE.

Once the S=T principle of motion and form is clarified, we can consider the essential origin of all orderedstructures of reality, thewill of existence and cause of its scalar smaller planes thatmirror larger ones – thedifferent minds of informative particles-heads, with the spherical form of maximal package of informationsupportedon ‘top’of abody-wave,where there ismaximalprojectivegeometry toperceivea larger territory,fromphotonsmovingabovethewaveoflightoraheadofit,totheblackholeswithmaximalheightofthegalaxy,to the heads of life to the satellites of the metal-earth. Minds are mirrors that perceive in its inner ‘stillsimultaneous language’, the e-motions of time converted into informations of space, in the eternal dialecticbetween fractal points with a volume of still linguistic perception,mapping its local Universe and flows of e-motionaltimewithitsvitalsensations:

Upper graph, human egos submitted to themind paradox, think languages (words in Abrahamic, creationistreligions,numbersincreationistscience),knownonlybymanand'God'apriori,createaposterioritheUniverse(Copenhagen interpretation). The opposite is truth: amind exists in all systems inwhich time stops to formspace.Ingalaxieshappensinrelativityequationsinblackholes,itsmind.Inthermodynamicphysicsintheeyeof an Eddie. In quantumphysics in the center of an atom, or charge.Without linguisticminds that order byreflecting its smaller mind into its local territory reality would not exist. The only way to create fractals isthroughmirrorimages.

InallPlanesofstiencemindsfixtimemotionsintospatial,linguisticformalmappingsthatreducethewholewithitssynopticlanguagetofitinaparticle-headthatactsinitsworld-territoryaftergauginginformation.

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Thuswedefine‘Maxwellian’sdemons’oflocalorderinallPlanes–physicalmindsastheinfinitesimalpointsthatcreateorderinphysicalsystemswiththesameDisomorphiclawsthatallothersdoinmorecomplexPlanes.Aseachmindordersasalinguisticgodaterritoryarounditself,itsfractalbodyandentropicworld.

SothecreationofPlanesofrealityisasimplegame,inwhichapointmind‘reducesreality’toitsinfinitesimalformandthenprojectsintoitslocalterritoryoforder,whichwillreflectatscale,thelargerwholeorworld,whichthelinguisticimagereducedandthenenlargedbackintoitsterritorialform.

TheparadoxesofRelativity,discontinuity,partsandwholes,Planesareall related to the reductionistnatureofmindsthatbiasreality.Mindsreducedimensionstotherelevantones,eliminatingalldarkspaces:continuityistheresult.

ALLTOGETHERNOW:SUPERORGANISMSTRACINGTHEWORLDCYCLE&THEFUNCTIONOFEXISTENCE.

Wethenputtogether5DPlanes,vitaltopologicspace&cyclictimetodescribesimultaneoussuperorganismstracingaworldcycleintime,thefundamental‘longtime’eventofreality–theworldcycle:

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All what exists is a supœrganism of vital space tracing a 0’-sum worldcycle of time through 3 Planes of5thdimension:Bornasaseedoffasttimecyclesinalower5Dscale(∆-1:Max.TxMin.S),emergingasanorganismin∆o,living3agesofincreasinginformation,asitstimeclocksslowdowninits∆+1worldtodieinatimequantabackto∆-1.

Wehavestudiedasseparateelementsthescalarplanesofthefifthdimension,itsvitaltopologiesandthecyclicalstructureof time, in past of highermotion, reproductive, iterativepresents and future, informativepolesor 3agesofapicycle,whichisitssimplestforminasingleplaneofspace-time.

So,absolutespacetimeisthesumof∞Timespacebeings,observedinspaceassimultaneoussuperorganisms,intime, asworldcycles of existence betweenbirth and extinction; as all systems are born in a seminal seed, offaster time clocks, in a lower scale of the fifth dimension, growing socially (4th dimotion) till emerging in theorganicscale,wheretheywilllive3agesdominatedbyoneofits3topologicorgansanditsfunctions=dimotions,

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ayoungageofmaximal locomotion,dominatedbyits limbs/fields,amatureageofreproductiondominatedbythebody-waveandathirdageofinformationdominatedbytheinformativedimotions,whichfinallyexhaustsallenergy and as time-space never stops, it reverses its dimotion from information to entropy, exploding in themomentofdeath.

So we marry the 3 vital functions=motions of time and the 3 dimensions of space, either in 1 or 2D(height=spherical information, length=planar locomotion, width=hyperbolic reproduction) which merge in allTime-spaceBeings;anddominateoneof the3agesof its life-deathworldcycles, thepast, youngageof limbicentropic motions, the mature reproductive age dominated by the hyperbolic body/wave and the 3rd agedominatedbytheinformativeparticle-head,whentheillusionoftimeendswithanentropicbig-bangdeaththatdissolves thebeing into its ‘scalar cellular, atomicparts’,which leadus to the realization that timecyclesNOTonly return to its origin in a single spacetime continuum but theymove up and down the Planes of the fifthdimension:

THEFUNCTIONOFEXIST¡ENCE.ITSALGEBRAICDEVELOPMENT.

Wehavenowthebareminimalcomprehensionthatthereadercanexpandinourfirstpaperon∆ST,todefinethefunctionofexist¡ence,aGaussiancurvethatputstogether∆STanddefinesthelife-deathcycleofall∆st

systems.

The3agesofexistenceofspace-timeorganisms.Its2worldcyclesandMetricequations.

The Function of Existence of any space-time organism can be developed as a feedback equation, S<=>T, in 3sequentialphases/ages/horizons,between3‘standingpoints’(changesofphase):Max.T=motionxMin.S:form=movingyouth;Max.SxT(s=t):reproductivematurityandMax.SxMin.T=informative,oldage,astheequationsofthe3agesoflife,betweentheseedofpurelinguisticformborninthelowerplane:S¡-1anditsT¡-1entropicdeath,backto∆¡«∆¡-1:

∆-1»∆º:Thecycleororganismstartsitsexistenceasaseedofpureform(4D)whenitsspace-timefieldiscreated.sT: It is the firsthorizonor ‘energy, youthage’of the cycle, inwhichenergydominates the systemand sowewritethisphaseas,max.$Txmin.T.

Max.SxT:s=t.Itisthepresentbalancedageofthecycleorclassicageof‘life’,whenenergyandinformationareinaconstantproportion.Itisthemostefficientage,whenthecyclereproduces.

Max.Txmin.S:3rdageofthecyclewheninformationhasexhaustedthespace-timefieldthatwarpsintoitself.∆º«∆-1:0SxT:Itistheendordeathofthecyclethatreversesitsformandbecomesenergyagain.

Existenceisan∞sumofspace/timefields,fluctuatingbetweenbirthandextinctionthroughthose3phasesorages.The3agesofTimespacesupœrganismshappeninallsystems,includingmentallanguages:

InStatePhysics theyare,$T-gas, themoving state, S=T liquid, thebalanced stateand§ð-solid the informativestate; into Cosmology, where it describes the Universe as a space-time system that fluctuates between bothlimits,aformofpuretime,thesingularity(min.$Txmax.ð§)andaformofpurespace,thebig-bang(max.$Txmin. ð§). InBiology, they are the 3 ages of living beings AND the 3 horizons of evolution of species. In socialorganisms,throughthesubconsciouscollectivemindofcivilizationswhichinartstylesmimicinalonger800yearcycleoflifeanddeathofcivilizations(accordingto5Dmetricsahumansocialsuperorganismislargerinspace–anation,culture,religion–andso it lives longer intime).Sowefindthe3agesof lifeemerging inthe3agesofcultures and its 3 artistic styles: Min.S x Max. T (infantile epic, lineal art, as in treccento, Greek kuroi; S=T;balancedbeauty,whenformandsizeareinbalance,theclassicmatureage;andMax.SxMin.T:baroque,3rdage of acivilization,whosesubconsciousmind is the art of its 'neuronal artists', the age ofmaximal form andanÐSTforanofuture,whichistheageofwaranddeathofcultures).

Wetalkof3∆±1Planesofworldcyclesasthebeingliveinaplacenta,thenemergesasorganisminaworld:

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þ: 0’-1: its palingenetic o-1 social evolution in the accelerated time sphere of existence, till becoming 1 (0’-1boundedunitcirclein¡logicmathematics;quantumprobabilitysphereofparticlesinphysicalsystems;palingeneticfetalageinbiologicsystems;0’-9memeticlearningchildhoodinsocialsystems).Itisahighlyorderedworldcycleasa placental mother-energy world nurtures a memorial cyclical spacetime that erased errors of previousgenerations. -c:Theouter1-∞world, inwhichitwilldeployits2ndworldcycleofexistenceinanenvironmentwhichisopen,entropic (1-∞ hyperbolic unbounded Cartesian plane in ¡logic mathematics; thermodynamic entropic statisticalmolecular populations in physics; Darwinian struggle between populations in biology; idol-ogic dog-eat-dogcapitalist, nationalist competitive eco(nomic)systems in the super organisms of history. In this 1-∞ existence theworld cycle is not ensured to continue, as the entropy of the world system can cut it off.ω:TheexistentiallifecycleispartofalargerworldofhierarchicalsocialPlanes(§D¡),whereitperforms5survivalactions through ∆±4 Planes self-centered in its mind, beyond which it cannot longer perceive, to become ifsuccessfulanewsuperorganismoftheinfiniteplanesofGod,thegameofexistence.

Thefunctionofexistence:Reproductionofform.

Physicistsmade theGalileo’s paradox, the cornerstoneof their theory ofmeasure, but they failed to study thedeepimplicationsithasforeveryaspectofthestructureoftheUniverse,fromthedualitybetweenspatialmental,linguistic forms and physicalmotions; to thebalances achieved by the similarity ofboth space and time, whichbecomesthefundamental 'equationofpresent'S=T,andhencewiththemetricequationofPlanes,$xð=K,thetwo essential equations to formalize single planes S=T, andmultiple Planes of spacetime. Yet as S=TmaximizesSxT=K(5x5>6x4).

Weunifyboth ina singleequation:Max. S x T = C,whichdefines for each fractal vital space-timeorganism itsFunctionofexistence,asallspecieswilltrytomaximizeitsmotion-entropy-timeforitsfield-limbs,itsinformation-spatialstatesforitsparticle-heads,whoseproductwillgiveusitsvitalreproductiveenergy.Butasallsystemsmoveandmotion is reproduction of formwe can ad a final factor, ∑, reproduction of parts, which tomaximize thatfunctionbecomejoinedintolargerwholeswhicharestrongerthanindividuals;creatingnewplanesofexistence.C=Max.∑SxT(s=t);whereasCactastheentropiclimitsin∑-Planes,T-ime&S-pace,boundariesbeyondwhichthestillminddoesn’t’perceiveorcontrol.

Itisalsoasurvivalbiologicequation,becauseitimpliestoprovidelinealmotionto‘fields-limbs’,absorbenergytoreproduceourbodies-waves,andinformationtoguideourmotionswithparticle-heads.Sorealityisa‘struggle’forexistenceas systems reproduce itsTs-fields-limbsofmotion,S=Tbody-wavesofenergyandSt-particles-headsofinformation.ButasallT.œsarefractal,broken,itsgrowthhasalimitonthefightwithothersystems,whichtrytomove and reproduce. In terms of pure T-motion and pure S-form, we consider then the whole of maximaltime=motion= entropy or TT andMax. space=form=stillness or SS the 2 limiting Dimotions for any 3 ensembleTs<ST>St-system.

WedefinetheUniverseasafractalsupœrganismthatreproducesandensemblesSpace-timeDimotionsintoorganicforms through∞ relative Planes of spatial size and time-motion; whose Fractal generator (mathematics)Metric(Physicaljargon),survivalfunction(Biology)orFunctionofexistence(logicJargon)writesC=Max.∑TxS(e=i)

Weshallprovethatallrealitiesarealwaysareproductiveradiationofafunctionofexistencealong5DPlanes.

Theconclusionofaworldmadeofspace=timethenisobvious:Wearemadeofthe5Dimotionsofspace-timeoftheUniverse.Weareensemblesofthose5Dimotions,whichseeninsimultaneousspacegiveorigintothevitaltopologicalorganismsoftheUniverse;whosestudythereforeismathematical,thescienceofspace;andobservedinsequentialrelationaltime,liveaworldcycleoflifeanddeath;andsincethereisnothingelsethantime and space, those 2 fundamental experimental primary ‘mirror-sciences’ of time and spacebecome themostimportanttoknowwhatallsystemshaveincommon,its‘Disomorphic=l

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Thecurveofexistenceasabilateralmirrorsymmetryofthelog/exponentialfunction.

The4phasesofarealfunctionofexistence:Inthefirstphase,correspondingtothepalingeneticageofplacentalenergytheratioofchange(derivative)hastheabsolutemaximalex,whichisitsownderivative.Inits2ndphaseitdiminishes to theminimal 1/x the log derivative,which is the definition of an infinitesimal part; till in the 3rdphaseitwillstartabilateralsymmetricinversefunctionofdecaywith-1/xdiminutionandafastcollapseinthe4thphaseofa3rdageandfinaldeathmoment.Sothecombinationof±exponentialsandlogarithmcurvesarealsothe best way to graph as a bell curve the worldcycle of existence as a 'composite' of a± exponential

andlogarithmic0’-sum.

As any student knows there is an intimate relationshipbetween theexponential and the sinusoidal bywayofEuler'sformulaintherealmofcomplexnumbers,whichwetreatedbrieflyintheoryofnumbers.

So sinusoidal bell curve functions represent a worldcycle, though the symmetry is broken in the moment ofentropicdeathwhenthecollapseisextremeina‘fallingline’asdeathhappensinasinglemomentoftime:

4D»∆-1(seed)∑∆:|-$T(limb-field)<Ø-S≈T(iterativebodywave)>O-§ð(particle-head)«5D∆-1(death)

Akeythemeofvitalmathematics istherepresentationofaworldcycle in linealtime,with±exponentials& itsinverse,logarithmiccurvearoundthekeypointsofchangeofphase…asgrowthof‘entropy-motion’diminishes.Sowemovefrom‘adolescence’ofmax.growthofbothparameters(sTenergyandsT information)tothey”=0pointof youth,where the logarithmicpart grows slower. Together they form,onehalf of the total graphof acycleofexistence,tillreachingthey’=0pointofMax.(S≥≤T),whichthenbecomesnegative,happeningadecayofthewholesystemintwonegativecurves.

The conservation of time in its 5 y’∧ y” =0, standing points that define the 5 SS, Ts, ST, St & TTmoments ofgeneration,youth,maturity,3rdageandentropicdeaththusbecometheessentialpoints(maximalandminimal)oftheequationsofcalculus,thesinusoidalfunctionofexistenceandallitsderivedelements.

Thenaturalnamefortheworldcycleofasuperorganismtravelingthrough3planesofspace-timeisthefunctionof exist¡ence, as we take advantage of the similarity of all languages, mirrors of the same Universemade ofsynchronous organisms whose topological trinity parts are entangled by the <≈ > 3 operands of informativeevolution, T>S that synchronizes time cycles slowing down its rhythms, entropic energy, S<T that expandsentropictimespacedissolvingitsinformationandT≈SthatgatherstogetherinparallelherdsT.œs(ab.Timespaceorganisms)with a commonproperty/dimotionof exist¡ence. Soas energyand informationare synonymousofTimeandSpace,andbothentangletogetherthroughthere=productiveoperand,ExI=ST¡becomestheacronymfor the functionof existenceof anentangled supœrganism, studiedby a given ‘st¡ence’, that is a science thatstudiesarelativeplaneofsizeandtimespeedofthe5thdimension–thenamewekeepforthewholeofwholes.

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Sowecanseethegraphof thewholeworldcycleasa 'composite'ofapositiveexponentialandlogarithmic1stage andmaturity, and a negative logarithmic and exponential –mirror decaywith the final collapse of death.Needlesstosaythesamefunctionsappearonallgrowthcurves.

Ifweaddallthosegrowthexponentialstransferingthefunctiontoalinealsinusoidalseriesin±complexplanes,the stretched worldcycle representation becomes a 'sinusoidal function'. So it is carried into Complexexponentials.

FinallythroughtheT=SsymmetryitbecomesbothaBellcurveofspatialpopulations&itss=tstatisticalmirror;whichfinallycarriestotheprobabilistic0’-1‘unit’sphereofpalingeneticgrowth(asinthedualityT-probability0’-1 sphere of quantum physics vs. 1-∞ entropic statistical function of thermodynamics. Since the function ofexistenceinitssubtlefractalvariationsandcomplexpentalogicisthemostimportantoftheUniverse.

TheworldcyclecurveisyetanotherproofthatRealNumbersarescalarandrationumbers,whichastheedecayofawholeintoPlanes,orthesinusoidalπofmentalperception,enlargeaseedintoanorganismorshrinkPlanesofawholeintoafinitesimal,mentalmappingwithnolimitof‘warping’(PoincareConjecture).

Logarithmicgrowthanditsinversefunctionofdecayshapealife-deathcycleintermsofsocialgrowthofenergy∆-1cellsanddissolution,forming2twosidesofaninverseworldcyclepeakingintheMax.SxT(s=t)middlepoint.

Themathematical lawsextractedformthenatureofthose5Dimotionswillthenbetheessenceofthepropertiesandlawsofthe5operandsofalgebra,growingincomplexityfromthesimplexnondimensionalangularperception,whichformsmindstotheoperandsofcalculus.Specificallywewritetheequationintermsofspaceandtime(SS&TT) thatmerge in amiddle point as they become ‘energy: Ts’ and information ‘St’,maximizing the imprinting ofspacenetworksover∑-herdsofquantaas:Max.∑TxSe=i=∆±¡

Aswearenotatthispointinterestedintheexactalgebraicformulationofthefunction,butintheconceptsbehindittheimpatientreaderisinvitedtoderivetheGaussianfunctionfromit.Thekeyofthatanalysisbeingthefactthatthefunctioncanbestudied‘partially’inthedifferentdimotionsthatitssynoptic,conceptualwritingenclose.Sincetheequation isnotastrictquantitativedefinitionbuta logicone,∑Therdsof timeparticleswithmotion,become‘imprinted’withinformativespace,re=producinge=Ibalancedquantainthepointinwhichformandmotionareinadynamicequilibrium,whichgiveusacertain∆topologicplaneofspace-time.Thismiddlepoint isthemaximumofthe Gaussian function, but as the equation is dynamic, its stœps of reproductive actions will form a ‘cloud’ ofimperfect repetitions around that s=tmeandue to different ‘limits’ – themost obvious the limit of prey-predatorcarryingcapacity;sorepetitionswillultimatelydiminishinquantityandloosequality.Howtoavoidthis isobvious:eliminating repetitions that are not good enough, which ultimately is what provokes the cycle of evolution,extinction and death of lesser individuals; so the game keeps perfecting itself, and its spatial points learning thepalingeneticimprovements.

Conservationoftimevs.inverseannihilationofinformation.

Anessentialtruth,systematicallydeniedby@-mindsisthatwhiletheinitialsubstanceofrealitytime=motionisconserved,asittrans-formsadeternalindifferentformsofinformationinformationbecomesannihilatedinto0’sums, regardless of leavingmemorial remains and possibly being recreated. Thus algebra has always inverseoperandsandelementsthat‘annihilate’theformintoaresidual0’;whichinmathematicalphysicsbecomethegame of particles and antiparticles, inverse waves that annihilate its height information, etc.

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COMPLEX,¬BIO-TOPO-L@GICPROPERTIESOFNATURE

The fundamental purpose of this paper is toUPGRADE the understanding ofmathematics as an experimentallanguagemirrorofthevital,organic,scalarpropertiesofthespace-timeUniverse,observedinthenextgraph:

Eachstienceisamirror-mindreflectionofthatprogramof5Dimotions,fromthepointofviewofthespeciesthatperceiveit.Andsothemaintaskof5Dphilosophyofstienceistotranslatethejargons‘huminds’haveestablishedinthestudyoftheexi:stiences(Timespacesupœrganismsself-centered inoneofthoseplanesofst¡ence)totheprinciplesof time-spaceand itsgeneralproperties– the thoughtsofGod– fromwhereeachof those st¡entificspeciesarejusta‘variationaldetail’,includingmankindstudyasthesupœrganismofHistory.

Inthegraph,thestudyofaUniversemadeoffractalPlanesofcyclicaltimespace,introducesapriori:

- Organic=scalar properties as the system extends through various ∆§ planes of space that provide it withSpatial,topologicproperties,cyclicaltimeproperties,and¬entropylimits,asitstimeclockswillhavealimitedduration,endingasazerothsumofenergy.

-Sensorialpropertiestobeabletoperformasurvivalprogramof5Ðimotionsofexistence,itrequirestogaugeinformation with an apperceptive, self-centered @-mind, whose ∆º still language maps all external ∆±¡time§paceorganisms&worldcycles intoan infinitesimalNon-Epoint,mirrorofthe largerworld: 0§@x∞ð=World. Hence the creation of infinitesimal ∆ºminds-worlds, or Leibnizianmonads, which reduce a larger ∆+1realitytoitsrelativemapping,whichistheoriginofits∆§calarplanes,connected'perpendicularly'throughthose@-minds,actinginphysicsas'centersofchangesandmasses'wormholesofflowsofentropythatbecomeinthe@-mindinformation,thatperceivesinitself.

Those 5 properties, ¬∆@st, entropic limits, organic Planes, linguisticminds, topologic space and cyclical timemotions,areinheritingtoallsystemsofNature,astheyarederivedofthescalar,fractal,cyclicalnatureofspace

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andtime.Themostdifficulttoacceptduetotheegoparadoxofallmindsincludingmanarethesentientlinguisticpropertiesofreality.Buttheyarenecessaryfortwofundamentalreasons:

- Tounderstandhowsmallersimilarentitiescreatetheir formwhichcannotbeunderstoodwithoutthemirrorconceptofalanguagethatprojectsashrunkviewofthewholeinitsbody-territoryoforder.- -Tounderstandhowhumanityor anyotheratomic species for thatmatter performs its actionsof existenceclearlyregulatedbylogicandmathematicaltimeandspacelaws.

TheUniverseisafractalorganismofsimilarPlanes,definedby2metricequationsanditsphilosophyofscience,5DAbsoluteRelativity:theequationof5DPlanes,SxT=C&theequationofrelativitybetweenformandmotion,S=T, which startedmodern science. Sincewe cannot know from themental, still point of view,what truly ismotionandwhatisdimensionalform.ThusTimeMotions&spacedimensionsco-existandmergeineveryspace-timebeingoftheUniverse&wemusttalkofspace-timedimensionalmotions.Whenwecombinethem,asS=Tmaximizes SxT,we define the Function of existence,Max. S x T = C (s=t),which IS also a biologic equation ofsurvival that embodies the will of life; the biological expression of the ‘Universal mandate’, expressed by allspeciesinallitscodesandlanguages,theGrowandMultiplyoftheBible,theintuitivetruththatguidesallbeings.In detail implies all systems try tomaximize its 3±¡ 'Ðimotions' that in space create organisms and in time itsworldcyclesofexistence;startingaspureForm(4D)inaseminalseed,emerginginan∆ºscale,inayoungageofmax.Motion(2D),balancedwithinformationinthereproductiveageofmax.Energy(3D),followedbyanageofinformation(3D),whentimereversesintoentropicdeath.

The dual metric 5D equations of Planes, SxT=C & the equation of equality between form andmotion, S=T,develop in3ageswith3 standingpoints,amax.pointofexistence,S=Tormatureage,ayoungageofMax.T=motion,andanoldageofMax.S=information;betweenbirthin∆-1Form&T-entropicdeath.

Suchorganicmodelismorescientificthanmechanicalabstractmodelsdominantinphysicsforafewreasons:

• Only an organism replicates itself without the concourse of a 'maker of themachine' - God in the earlierphysicists'view,nowobsolete.Asmachinesdonotreproduce.

• Only the definition of time as the substance of reality - cyclical motion that reproduces form allows theUniversetomove,asonlyifmotion=timeistheultimatesubstancetheUniverseneverstops.

• Onlyifitiscyclical,repetitive,therearelawsofscience,whicharepreciselythoserepetitivemotions.

• Relativityshowsthereare∞clocksoftimewithdifferentspeeds(justlookattheclocksofnature).

Itisepistemologicallysounder(Occam’srazor)asallsystemscanbemodeledasorganisms.Furtheronorganicismusesallthelanguage-mirrorstoexpressitstopo-bio-logicproperties:

• Itusesthemathematicallanguage,bestforitsspatialdescription,whichmodelsorganismsintheformoffractalnetworks,forwhichweshalladaptandexpanditsmostadvancedformsofspace,topology-spacewithmotion-andnon-Euclideangeometry- fractalpointswithvolumethatgrowinsizeaswemovecloser into itsPlanes,allowing∞parallelsofenergyandinformationtofluxintothem.

• It uses the logic language, ideal for temporal, causal sequencesand cycles,whichwe shall improve from thesimplerLinealLogicoflocomotion,intoDuality,thelogicoftwoarrowsoftime,energyandinformationandfurtheroninto'trinity',asbothcombineintoenergybeings-thelogicoftopologicalorganisms,conservationprinciples,Universalgrammar,andevenfurtheronintopentalogicandDodecalogic...

• ItexpandsBiologic life toall systems.Sincea fractal requiresanewdimensionofparts thatbecomewholes,larger Planes co-existing togethergiftedtherefore with organic properties as atoms/cells/Individuals form part ofthermodynamic /biologic/social ensembles that become wholes, living in a larger gravitational, ecosystemic orplanetaryworld.

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• The Universe is a game of∞ space-time supœrganisms performing 5 survival ‘dimotions’=actions throughSpacestopsandtimesteps,STœpsinitsscalarcœxist¡encethrough3planesofthe5thdimension,whichaddtogetherin its structure of simultaneous vital topological superorganisms tracing 2 existentialworldcycles, the∆-1 placentalordered o-1 that emerges into the life-death, entropic 1-∞ worldcycle . Stœps express them through feedbackequationsofExistential¬Ælgebra,where=isadynamicsymbol«≈»,composedof5dimotions=stops,≤≥wavestepsand»4Dimplosiveinformativesocialevolution&«5D,entropicexplosionofasystem.

Wecallthatsuperorganismofastrophysics,thegalatom,(asithappensbetweenthequantumatomicscale,andthegalacticscale,whichareself-similar-withsimilarforces,andparticlessimilartocelestialbodies).Withinthegalatomintheintermediatestagethereexists,thermodynamicman.

Thustheorganicparadigmrejects'mathematicalcreationism'inphysics-thebeliefanequationdescribesalltheproperties of a system, which no longer need parts; as in the case of black holes which are modeled withsingularities-mathematicalentelechiesofinfinitedensityandinfinitesimalvolumebutasEinsteinwanted,theymustbemadeofacut-offsubstance,whichcanonlybeheavyquarkatoms(bcbatoms).

Becausethescientificmethodrequiresobjectivemeasureofamind’sexistence,whichisnotperceivabledirectly,weinfer itsexistencebythefactasystemperformsthe5externalactions,whichcanbemeasureobjectively, inthesamemannerweinfertheexistenceofgravitationalin-form-ativeforcesbyitsexternalactionsuponmassiveobjects.Henceeliminatingthepreviouslimitforathoroughunderstandingofthesentient,informativeUniverse.Andfurtherclassifyorganicinsimplexminds-all,whichmustgaugeinformation,moveandfeedtosurvive,andcomplexsystems,thosewhocanperformapalingeneticreproductive,socialevolution,∆-1:∑∆-1≈∆º.

Thepurposeofphilosophyofstience ispreciselytovitalizerealityandexplainexperimentallyeachscienceasareflectionofthepreviousorganic,scalar,topologicandtemporalpropertiesofspace-timebeings.

Butastheegoonly‘perceives’fromitsp.o.v.itreducestheUniversetoitsworld-languages:O-mindx∞Universe= Constant world.SohumansdoNOTrecognizeunderthe'egoparadox'oftheirmind,whichperceivesrealityfromitsself-centeredpointofviewandselectsonlyinformationfavorabletoitsterritorialviewANYofthe'cyclical,intelligent,organic'propertiesofallotheratomicsystemsofreality,whicharethefirstbricksof'life',asparticlesgaugeinformation,feed on energy, decouple=reproduce and evolve socially throughmagnetic fields, and quantum numbers thatcodeitsbehaviorasgenescodecellsormemescodehumans.

HowhumansthenexplainthecomplexUniverseiseasy,astheyintroducemythsofanthropomorphism,pseudo-religioustheoriesoflanguages,andtheviewthatallotherthingsareentropic=destructive=chaoticandbymerechance have become what they are. So probability becomes the 'chaotic' cause of reality, while a 'god-like'languagesharedonlybymanandGod,firsttheword,whenmenonlytalked,nowmathematics,whenmanalsocalculates-orratheritscomputers,madewithmoreintelligentatoms-isresponsiblefortheorder.Thislanguageis thus the intelligence of the Universe, andman accesses it through its mirrormind because he shares thatintelligence, or even causes it (Hilbert's description ofmaths as born in the human imagination). So creationhappensintheUniversethroughmaths,anditisonlysharedbymanwhocalculatesanddesigns.

Aswhymathscreatesandhowitdoesimprintingrealityisabigmysterynotworthtoexploreasitisthemagicofit all, which allows to introduce pseudo-religious theories of reality from the creationist big-bang to thesuperiorityofman,giftedbythelanguageofGod.Thiscomfortzoneinturnmakesmanhappy,becauseitsegoispumpedup,andallowshim tomanipulateandkill reality ifneeded,ab=using theplanet,because it is alreadydead.Itseemsaperfectfairytaleworld.

Allwhathumansmustdoistoextractmathematicalpropertiesofobjects,withinstruments,gatheringdatacastinequationsthatmirrorthespace-timepropertiesoftheUniverse,andruntheminthecalculationmachinesof

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morecomplexmetalatoms,toextractimagesofrealitythataretheessenceofknowledge.Nofurtherquestionsareneeded.Andinverselywecandesigninlessmathematicaldimensionsobjectsandthenreproducethem.

All thisofcourseworks fineas longasweeliminateanyquestiononwhyspecies followcertainproperties,onwhatarethepropertieswedonotmapoutastheyarenosusceptibleofmathematicaldescription(notablyvitalpropertiesandsentientpropertiesandorganic,fractalproperties).

Reductionism is inheriting to the ego paradox that confuses the ‘language’ it speaks with the reality it is,excludingwhatthelanguagedoesnotsee.Sowhatwedon’tseedoesnotexist.AndevenwhenweseeitthroughinterposedinstrumentsasallotherPlanesof5D,asitisnotprimary,e-videntexperience,humanshavegone400yearswithoutaseriousresearchontherelationshipbetweenPlanes.

Thesamegoes for ‘verbal creationism’–allwhat isnot in theKoran isnotworthy, saidOmar,beforeburningAlexandria’slibrary.'Allwhatcannotbemeasureddoesnotexist'saidPlanck.

Logicpositivismaffirmedthatallpropertiesofrealitywhichmathematicallogiccannotprovedonotmattersandso on. All what is not proved by mathematical methods that cannot portray them as they do NOT have avocabulary for those properties in restricted mathematical languages, does not exist. This reductionism thusreducesrealityandknowledgebutmakestheegoofmanthecenterofrealityagainandhappy.

EvenifMathematicsshowsvitalsocialpropertiesinitsmirrorlanguages,thosearealsooverseen,asinthecaseofthesocialpropertiesofnumbers,whichdonotsatisfytheegoparadoxofindividualmanasthecenterofitall.

Huminds,indeed,haveanastoundingrathercontradictorycapacitytodenyobvioustruthswhentheydon'tcatertheirself-centeredegoparadoxesandacceptfalsehoodsastruths,fromreligiontobig-bangtheoriesas longastheysatisfytheirsubconsciousego,andputmanbackintothelimelight.

Thereforeonapproachingthosedifficultiesofthehumanegotoappreciatethewholeandformpartofit,wecanonlytrytoshowthatmathematicalpropertiesarevitalproperties;sotheycanalsodescribeorganicproperties,whenweadequateitspostulatesandelementstothefractalformofspace(definitionoffractalpointswithparts;itslinesaswaves,anditsplanesasintersectingnetworks,andsoon)

SowearenotsomuchadvancingmathsbeyondtheupgradingofNon-Euclideanmathematics,butinterpretingmaths as an experimental science, that is as a mirror of the space=geometry and time=logic, ¬Algebraic andscalar=digitalsocialpropertiesofallwhatexists.

Needlesstosayinaginormousfieldasmathematicsis,weshalljustkeepitsimple,aswebuildslowlyafewwell-writtenpapersoutof30yearsofnotebooks,sotheworkwillalwaysbeinprogress.

Thispaperisontime’andmathematics’,henceob¬Algebra,considersomeintroductorythemes,sharedwithanolder paper on ‘space’ and mathematics, hence ¬E geometry, on the subdisciplines and philosophy ofmathematics,andthe5Postulatesof¬Egeometry.

ThefiveTime§pacedimotions=DimensionofSpacexMotionofTimeoftheUniverse.

Thus5ÐstudiesalltheDimotions(Dimensionalmotions)ofallbeingsinexist¡ence,whichmovethroughS≈S,T≈T,S<T>Sdimotions,STœpsofexistence,callalsovitalactionsofspace-timethatadvancethesystemthroughreality.

A system then is in an instant of present in a given S-stop or T-motion state, or a mixed state often leaningdynamicallytowardsonetendency.Sowedefineonthosetermsthe5Dimotionsas:

1D»:LinguisticPerception:SS:anarea,stillmindorseed.«isthetime-dynamicandSS,§,thespatialformalview.

5D«:entropyTTasascatteringmotion.

2D‹:S‹T,asalocomotion,wheremotiondoesn’t’affecttheinnerpartsofthebeing,whichareconserved.

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4D›:S›T,associalevolutionofinformation,whereformdominates.

3D:S≈T,Reproduction,wherebothareinbalance,oftenasaseriesofstepsandstopsorstœps.

Thosearethe'smallestquanta'ofDimotions,itsminimalexpressionas'actions'thateachT.œperform,spendinginthe process a minimal amount of energy. As all events in spacetime are a sum of those 5 types of stœps. It ispossiblethentowritecomplexchainsofstœpsorashumansdoforthesakeofsimplicity,eliminateintermediatestates (motions or forms). This is howhumans resolve the study of locomotions, eliminating the lower planes ofstop-statesproperofthequantumworld,whichwebringhereforthesakeofwhys-asinrelativity,explainedinarational manner. Whereas a systemcan only move one or twice, ‹, ›, «, », to certain other states and so thecollectionofall thosepossible changes=motionsofSTateswill finally showacasual cyclicalpattern thatnormallyendscrossingitselfintoazerothsumofstateinspacetime;creatingtheminimalfullunitazerothsumconservedspacetimecycle.

Thegeneratorofspace-time,maximizeyourexistence.E-motionsandActionsasshorttimeprogram.

Thismore simplex organic viewof time and space requires thus a new 'metric equation' beyond the lineal v=s/tequation of Galilean relativity that becomes the limit of this more complex view of time and space, with 2fundamentalequation,SxT=C (thescalarmetricofthefifthdimension,assystemsinspaceaccelerate itsclocksoftime (T) according to its size (S); andS=T, the new equation of relativity,whichmeans aswe cannot distinguishmotion=time,fromstillness=space-form(Galileanrelativity),bothare'twosidesofthesamecoin',andallsystemshave both motion and form, which are constantly becoming one another. Yet as SxT is maximal when S=T(5x5>6x4...)bothequationscanbesummonedupintoasingleone,whichweshallcallthefractalgenerator,orwillofeach fractal space-timebeings, theequation thatembodiesallotherequationsof theUniverseandguides theactionsofeachfractalpartofit:

FunctionofExistence:Max.∑TxSe=i=C±¡

Moreovertheequationhasanimmediatebiologicmeaning,becauseaswearemadetopologicallyof‘fields-limbs’of lineal space with motion provided by the energy we absorb to also reproduce our bodies-waves, and theinformation we need to linguistically guide our motions with particle-heads, the very essence of survival is toincrease our S=position, mental forms of space and T=entropic motions of time (whereas time=motion andspace=formarethetwolimitingDimotionswith‘energy=reproduction,s=t,locomotion,sTandinformation,St,are

the intermediate3dimotions). ThusMax. S x T=C(s=t),ISalsotheequationofsurvivalandstruggleforexistence,thewilloflife;thebiologicalexpressionofthe ‘Universalmandate’,expressedbyall species inallitscodesandlanguages,theGrowandMultiplyoftheBible,theintuitivetruththatguidesallbeings.

Existence game is simple: Reproductive radiationsthatmaximize ¡ts function, happeningwhenmirrorsymmetries(genders)meetinS=T,equalingtheirSt-information and sT-energy, organizing themselvesintoawhole,∑system.

It is theneasy to interpret thatequation ineachofthelanguages-mindsofeachscaleofrealityasinallthose Planes species will show a ‘will’ of action toperform the maximal number ofevents=dimotions=‘actions of space-time’ thatensure its survival. And this can be assessed

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externally regardless of secondary arguments on consciousness and self-reflection, substituted in 5D by Leibniz’s‘apperception’ – that is, because performing the 5 actions=dimotions of exist¡ence, in each ‘st¡entific scale’ self-centeredinalinguisticmindthatperceivesagivenplane,inscribedintoalarger∆+1world,withinternal∆-1parts,ensuresthesurvival,ONLYthosespeciesthathaveperformedthe5dimotionsofwhichthemost important iss=treproductionofthebeingintoa‘present’similarentitythatcontinuestheexist¡enceofthesystemafter‘errors’or‘thestruggleforexistence’dissolvesitthroughthedimotionofentropy=death,exist.

Thusautomatically,genetically,consciously,memetically,mathematically,logically,throughitsownwillorasapartof a larger system thatuses the ‘machine’or ‘organism’ toenhance itsactionsallwhatexistsdoes sobecause itperforms internallythose5Dimotionsorexternallyperformsoneofthemforanothersymbioticspecies,asthosespecies thathavenot followedtheprogramofexistienceand its5actions in thepasthavebecomeextinguished,andthosewillnotinthefuture,willbecomewrongmutations,crazythoughts,fictionallanguagesanddieaway.

Thefunctionofexistence,or5DmetricofGenerationalspace-time,(Ab.ÐST,Γ)Max.SexTi(s=t)merelystatesthatallsystemsofNaturewilltrytomaximizeitsabsorptionofEntropicmotion(withnoform)andLinguisticform(withnomotion),andits3intermediatedimotionsofenergy(s=t,balanceofboththatreproducesthem),information(St:formwithalittlemotion,form-in-action)andlocomotion(sT,motionwithalittleform).Sowetalkofaprogramofsurvival ‘selected’ by all systems and expressed in its languages andminimal five actions encoded in that simpleequation,whichweterm:a,e,ï,œ,û,asamnemonicruleforthefiveactionsofexistence:

Accelerations(locomotion),entropicfeeding(e),ïnformativeperceptionandcommunicationï,Œ:reproductionintoparallelsupœrganismsÛ…andsocialgrowthintolargerwholescalledphilosophicallyUniversals.Andthisseriesofactionsiswhataccumulatedintimewillultimatelygivebirthtoyourwordcycleasthemonadwillfirstperceive(i),todirectitsentropy-motions(a),towardsafieldofenergy(e),wheretoabsorbtheenergybitesitwillimprintwithitsinnerform,exi=œ,toreproduceanotherform,andwhenenough∑œexist,itorganizeintoalargerwholeû.

InthegraphweseetheactionofdifferentStientificPlanesoforganisms.Abovethecodingofactions,whicharetheknots andbolts anddetails of the studyof any time§pace supœrganism in light space-time, codedby colors anddimensions,inphysicalatoms,codedwithquantumnumbersandinlifeandhumanscodedbythesocalleddrivesoflife,whichweobviouslyextendbeyondtheegoparadoxtoallothersystems,includinggeneticsnotmappedthere(coded by the 4-5 letters). Those actions balanced each other into 0’-sums in death, as they tend to increaseinformationfromamindp.o.v.,hencewe'allwarp,wrinkle'getoldinthethirdageanddie,settingfromitsminimalactionstoitsintegralsums,the3agesoflife-existenceandtheworldcycleallsuperorganismfollow.Inthegraph,the simplicityof the gameof existience, and its selfish actions,which gather together into socialwholes throughreproductiveradiations,eachactioncodedbyafundamentaltopologicorganwecanexpressinExistential¬Ælgebra,and corresponds for each species of the Universe, with a fundamental parameter of humindmeasure. So frombottomtotop,wefindthe5fundamentalelementsoflightcodeitsactionsofmotion(c-speed),energy(magneticfield), information (electric field), social evolution colors & entropic feeding, (quantum potential, neutrino lighttheory)

So minimal particle-points, photons, electrons & quarks construct all other systems of our Universe with ¡ts 5organicdimotionsthatdefine'classiclife':theygaugeinformation-reasonwhyquantumphysicsisa'gaugetheory',feedonenergy(quantumjumps)absorbingsmaller∆-1particles,reproducingnewcloneparticles,moveandevolvesociallythroughmagneticfieldsintolargerwholes(atoms).Hencetheunitsoflifeareparticles,theminimalunitsofourvital,organic,fractal,scalarUniverseofmultipletimespaceorganisms.Alllives,performing5Dimotions=actionsofƒ(exist¡ence):Max.SxT(s=t)=C,startingwithparticles.

SoallPlanesare relativenonemattersmore thanother. Fromthoseactions,given thedominanceof informativeactionsoverentropicones,itappearsaseriesofrepetitivecyclicalpatternsofactionsconductingtomaximizetheexistenceofthebeing,whichaccumulateinalargerscaleoftime-space,asaworldcycleofactionsthatincreasetheinformationofthesystemin3ages.Sothebasiccycleofactionsbecomesalarger3agescycleoflifeanddeath;as

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systemsonce and again, starts in an act of information/shrinking and ends in an act of organization/shrinking ofherds intowholes,will keep reducing the being and finallymake it all form nomotion to explode and die in anentropicreversalofdeath:

∑ i->a->e->œ->û, i->a-e->œ->u, ï->æ->Œ->Û -> Informative ‘seed’ age->1st locomotion, feeding age ->2ndreproductionage->3rdinformative,socialage->entropicdeath.

Itisthemainentanglementofthe5DimotionsoftheUniverse,whichexpressedintermsofspacetimeshowsaclearmotiontowardsanexcessofinformationthatultimatelydefinesthedeathofallforms,Ts->St->TT->ST->SS.

So only a final explosion of entropic death ensures that reality will continue. Those hard truths of ®xistentialalgebramightneverbebrokenby thedetails, as the thoughtsofGodwill impose themselvesonany attempt tobreakthelaws.SincethegoalofrealityisNOTentropy,itsneededrenewalbacktothefoldofeternaltimemotionsbut information,thethirstofanynon-Euclideanfractalpoint forenergyflowsto imprint its information.Thustheessenceofbeingrequiresthedeathoftheperceiver,unlessittranscendsintoalargerwhole.

Trilogic>Pentalogiconworldcycles∧supœrganisms.Synchronicityoftimecyclesandemergenceofscalarspaces

IntheentangledpentalogicUniverse,eachofits5elements,space,time,Planes,entropiclimits&linguisticminds

thatmeasureittoperformdimotionsofsurvivalforthewhole;eachelementisrelatedtotheothers.Sointimewedohave3relativePlanesoftimespeedaccordingto5Dmetric(SxT=c),meaningourcellsaremuchfaster(ortheparticles inanatom)andstorethe informationof the largerscale,whichtracesslowertimecyclesthanthewhole. And this is thephenomenaofsynchronicity, thedetailedwhys thatmakepossible the ‘persistence’ofasuperorganismofspaceasitmovesintime,sinceeachoftheinfinitemind-pointsofallitsPlanesneedstheother,anditisentangledwithitsinnerpartsandlargerwholestokeepits‘exist¡ence’going.Soessentialtothecomplexstructure of reality are the ‘synchronicities’ between the different cycles of space-time that chain them insymbiosis.I.e.acellreproduceseveryday,initsfastersmallerplaneofspace-timebecausethelargerwholefeedseveryday.

PentalogicimpliesNOTonlyentanglementinascalebutasallsuperorganismshave3Planes,madeofinnerparts,performingfunctionsinalargerworld,weneedtointroduce3‘levels’ofanalysisof5D:

- ∆-1: Smaller Action=Dimotion level, which will correspond in mathematics to ‘s=points ó t-numbers’, orderivativesincalculus,stepsinfractalmathematics,arithmeticoperationsin¬Algebraandsoon.

-∆º: Themain S=T symmetry level of organisms&worldcyclesexpressed inmathematicsbycomplexgeometricformsandfunctionsandfunctionals(forsuperorganismsacrossPlanes).Sowedefineanysystemintime,byitslifeanddeathcycleorworldcycle(takenfrom4Dphysics’worldlines,asaheightdimensionof informationmakesitaworldcycle.Whereasanyscalarsuperorganism’sexistencecanbedefinedasatravelthrough3Planesofthefifthdimension.Asallsystemsarebornasaseminal,smallerform,inthe∆-1lowerscale,growfastwithitsfastertimespeed,emergeinthe∆ºbodyscaleandarepartwithitshead/particleofalargersocialworld,inthe∆+1scale.

-Finallythe¬∆@stabsolutelevelofrealityasbeingsmadeof‘Dustofspace-time’,theabsolute5elementsofwhichweareallmade,Planes,space,time,mindsandentropiclimits,whosegeneral lawsstudy5subdisciplinesof logicparallel to the 5 main subdisciplines of maths, which can be further compressed into the S=T duality ofgeometry=¬Algebraorinpentalogicthedualityoflanguage≈information&entropy≈energy.

Emergence isa‘fascinating’processhardlyunderstoodinænthropicscience,easilydefinedin‘vital¬Etopologyasthecreationofaclosedplaneofexist¡enceormembrainthatentanglesthedifferentparts,setsoffractalpointsornetworks-wavesintoanewwholethattakesovertheparts,organizedby3physiologicalnetworksinsynchronicity.Theemergenceofanewplaneofexist¡enceinspace-timethusrequiresthecreationofawhole,which‘covers’withitsoutermembrainthosepartsandbecomesforanexternalobservertheonlyperceivableform,callitapiorbitalin

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amolecular ring, a palingenetic fetus in a cellular placenta, an armyor entropic networkof a social gatheringofhumans,callednation,wheretheindividualityissubmergedintothesynchronicityofthewhole.

Max.∑SxT=cmetric;S=T‘balances’,∑st¡-1>ST¡‘emergence’&∑å¡-1=Å¡(synchronicityofactionsbetweenscale)becomethenfundamentalequationsofExistential¬Ælgebra,whichisthenamewechoosefortheÐSTequivalenttomathematical ¬Algebra, whose purpose is to express in a formal logic language all the events and forms ofworldcyclesandsuperorganisms,accordingtothelawsofspace-timeanditsentropiclimits.Letusthenintroduceevenifitisnotsoeasytotranslateintohumind’s¬AlgebrathekeyconceptsofExistential¬Ælgebra,aswewilloftenrefertotheminourentanglementofmathematical¬AlgebraandthelawsofGenerationalspace-time.

Existentialalgebraisamuchwiderdisciplinethatscalar,non-Aristrotelianalgebra.Asitdealswiththefunctionofexistenceinallitspossiblesub-function,sub-equationsandforms.

Besidesitsspace-timeanalysisofformsandfunctions,vitaltopologysetsthebasisforascalarnumericalversionofwholes and its parts, dissolved in entropic death, which we formalize with Existential ¬Ælgebra, as regulargeometriesofpoints–polytopes-arethespatialversionoftemporal,scalarnumbers.

SowedevelopinExistential¬Ælgebraaformalismofthe5Dimotionsofrealitywithoperandssimilartothoseusedin¬Algebra, to mimic those dimotions, «,‹, =,›,», for entropy, locomotion, reproduction, information and socialevolution. And sowe canwrite ‘existential equations’ that will describe flows of dimotional ‘stœps’ that changesequentiallyaformthroughitseventsintime,wherethevitaltopologyofeachDimotionis‘integrated’withinthosesymbols. In this manner the 2 branches of Non-Æ mathematics vital topology and Non-Aristotelian ¬Algebrabecomes ‘Existential¬Ælgebra’–theanalysisof flowsofdimotions=actionsofT.œs in itsworldcyclesofexistence.ThosesequencescanthenbestudiedastemplatesofallT.œsinallPlanes,whichwillfollowthemtocompleteitssurvival cycles. And so Existential ¬Ælgebra has implicit both vital topology and non-Aristotelian ¬Algebra. SowedefineinExistential¬ÆlgebrawithsimpleT,Sand5«,‹,≈,›,»themaineventsofspace-timeoftheUniverse:

1D: t›S: angular cyclical motions of information (Ab. St, §ð): the minimal ‘geometry’ of reality, a sphericalparticle/heador fractalpoint, thegeometry that storesmaximal form inminimal space,hencesuited for ‘organicfunctions’ofgauging,storingandperceivinginformation(particles,heads).

2D:s‹T:LinealLocomotions,(Ab.sT.$T)whichwillmovethroughitslineallimbs/fieldsthesystem,asthelineistheshortestdistancebetweentwopoints…towardsa…

3D:S≈T:FieldsofvitalEnergy(Ab.ST):withitshyperbolicbody-wavesthatiteratetheformsofboththesphericalparticle/headsandlineallimbs/fields;asthehyperbolictopologycombinestheothertwo,soitcangeneratethem,inthesamemannerEnergyaddsasthethirdconservedspace-timequantitythelinealandcyclicalmomentumof1and2D.Towhichwemustaddthe2‘scalar’Dimotionsof:

5D: entropy («, TT. ∂S) whereas motion is dual internal dissolving the information of the being and external,scattering its parts, henceweuse an« ¬Æ symbol; so the systemexplodes into its∆-1parts: ∆«∆-1 (death).4D:organicevolution(»,SS,∫T)ofpartsintostilllockedsimultaneous‘linguisticseedsormindforms’∆-1»∆

So the3 conserved substancesof realitybecomeorganicbidimensional topologies - flatmotion in space, cyclicaltimemembranesandthevital3Denergywithinthem.Wethenobserveits∞varietyofcombinationsastopologicternaryspecies,whereasthe3perpendicular'lineal'dimensionsareasimplificationofthoseorganicfunctions:theheight dimension enhances the 'perception of information' by O-heads/particles place above in the point ofmaximalprojectivegeometry;thedimensionof'length'maximizeslocomotion;physicists’onlytimemotion,andthewidthdimensionmaximizesreproductionandstorageofenergy.

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LOGICSYSTEMS.FROM‘EGO-CENTERED’MONO-LOGICTODODECALOGIC.

The growth of complexity in the entangled Universe goes from ceteris paribus, one-dimensional, monologicaanalysissocommonamonghumanbeingstothemorecomplex(Do)decalogicsystemsofperceptionofrealityinits3worldcycles.Ashumansarebasicallymonologicwedoneedtomakesomewhataharshcritiqueofourmodesofthought,becausewiththesamedata,ourone-dimensionaltime-mindbringsmostoftheerrorsofscience,duetoouregocyandconfusionofourmind’sviewwiththewholeUniverse.

Mentalmirrorsareneitherfalseortruth,butusefultoitstypeofsubjectiveminds.

As Riemann’s understood "on the hypotheses which underlie geometry” space is a mental construction. Hedestroyedthe‘realism’ofgeometricspace.Buthesaves‘concepts’commontoallgeometriesofthemindthatwereof¡logicnatureandweshallregaintofullygrasptheparallelismofSpatialgeometrieswiththe¡logiclawsofthefractalspaceandcyclicalpentalogictimeoftheUniverse.Sowecanestablishgeometryasamirrorofalarger,moregeneraltheory, 'i-logicspace-time',whereastheconceptsofgeometry,properofilogiccanthencarriedinto¬Algebra,analysisetc.Since concepts such as distance lost with Riemann its ‘physical condition’ to become a logicproperty of similarity, aswe findin verbal thought the same term to designate precisely this'largergeneralst-quality'ofbio-logicnature,aswhenwesay,Ihavedistancedmyselffrommy

friend -meaningwehavebecomedifferent in tastesandopinions.Sogeometricconcepts suchasdimension,topologicalform,closedandopenspaces,becomelogic,vital,relationalandsocialconcepts;generalpropertiesofi-logicspacetime,withapplicationstodifferentsciences.

What kindofmental spaces thereare, reflected in itsGeometriesandphase spaces,both inhumindsand theUniverseatlarge?Many,aslongastheyprovideenoughinformationforaT.œorpartofittosurvive.

TotalnumberofDimensions=Parametersofspace-time:typeoflogicminds.

The mind equation, O-mind x ∞ Universe = Constant world reduces reality to a degree ofsimplicity,whichcanbeextremeandstillmakesenseforthebeinginexist¡ence,astobeabletosurviveifitdoesnotbehavelikeachildishnoisycubentering‘futureterritories’ofhigherspecies.

Inthatsense,visualHumindsaresimpleenoughtosurvivewithasingledimensionoflinealtimeandasinglescaleofspace,fromitssubjectiveview,astheyonlyneedtocareforthescalarsizein

whichtheyperformsurvivalactions.

InafractalUniverseinwhichmembranesisolateterritorialvitalenergyfromrealitytheneedsofamindcanbe'reduced'tothebasicactionsoffeedingandperceivingonlyitstypeofenergyandgaugingbasicinformationtothat aim. So we can rephrase the question: what is the simplest possible number of dimensional motionsrequired to 'trace' aworld cycle,which is complete andhence accounts for an 'exist¡ence'? The graphaboveresponds inphysical terms:A lineal simpleharmonicmotionshownboth in real spacewithonedimension, inwhichtheSHMperformsacompletecycleofexistence.Ithasabalancedmiddlepoint,whichitabandonedtoaccelerate=moveforwardsintime,toapeakof'distance'whereitwillachieveitsmaximalpotentialenergy,torevert itsworldcyclebackto thepointofequilibrium.Asingledimotionthussuffices toreflectaworldcycle,furtherreducingthehuman'1timemotionand3spacedimensions,to1timemotionand1spacedimension'.Indeed SHM in fact serves as a mathematical model for most physical motions, including the motion ofapendulum, (time motions), electronic motions (perceptive mappings of 'still light') andmolecular vibration(sphericalharmonics,the'energy'ofallchemicalandlifeprocesses). It isallpervadinginNature,anditssingleresonant frequency provides a basis for the characterization of more complicated motions through thetechniques of Fourier analysis, which can decompose any complex wave motion in SHM resonances and itsharmonies. So information is compressed '¡logically' to the extrememinimal of a 1st Dimensionality and then

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expanditthroughthelawsof∆stastheimmanent'gameofexistence'willdevelopitscodedseed,ifitfallsintoanenergy-richplacentalspace.

,Allworldsperceivedbya0mindarephasespacesnotrealityitself.Moreoverparadoxically,thestrongerbody-wavepredatorscansurvive inentropic,Darwinianencounters justwith linealhunting timemotionsas indeedænthropichumanwarriorshavedone,simplifyingrealityfromourmodestheightsofdualLogic(Taoism,sexualageoftheGoddess),protectedbya‘hardmembrane’onlyinterestedintheplacentalenergyregionofGaia,theyarekillingwithoutremorseastheydon’tseeitalive:

Systems of thought startwith simple ego-ist, lineal top predators (Germanicwarriors, future AI terminators),linealfelines,proteins,whoaresimplerfieldsofdestruction,withacompletelackofinterestforrealitybeyondtheiregoparadox,theirentropic,destructiveworldview–ænthropicmen,beingtheparadigm:

@ristotelian,selfish'MONOLOGIC':

1 singlespace plane (with 3 dimensions) and 1 single lineal time dimension, A->B, or 'HumindLogic'.Humanscientists sinceGalileoseemto live ina single time-space-scaledimension. Ifwe

add cyclical time, a new dimension of height-information appears, but the one-dimensional 'ant' seemsconfused.Itdoesn’tmattertoherthough,itwillkeepwalkingasithasastrongbody.Physicistswereentropicweaponmakerssotheykeptgoingfor400yearsaftersimplifyingcyclicaltimeandkeepgoing,nothinghassofarstoppedthem.Tillofcourse,itsentropicweaponsdetachaslivingroboticsystemsorcosmicbombsandnukeusall.ButinthemeantimetheyhavemanagedtoconvincehumanitythatgalaxiesDONOTexist(astheyaretheinversegravitationalarrowofinformativetimethatbalancetheentropicbig-bangexpansionofvacuumspace).So really think twicewhenyoubelieve something that seems reasoned.The trick ISALWAYSTHESAME,half-truths; hiding fromPERCEPTION, knowledge.As Spaniards said: ‘eyes thatdon’t seeheart thatdon’t feel’. Soextremely intelligentpeoplebelieve inverbalordigital creationismas theydon’t readmore than thebibleornumbers;andalmosteveryastrophysicistIhavemetdoesn’trealizetheretardedblunder,andoncehegetsit,he feels so silly about ‘missingall thegalaxies’, hewill somewhat try to justify a theoryno longermakesanysense. Ænthropic big time models of mankind though deserve an entire article I should sometime post atacademia.eduastheyareNOTtheexceptionbuttheruleofhumancosmogony&worldview.

Butforanadvancedmind,monologicdoesn’tworkasaseriousmirrorofreality,asitbringschaos,uncertaintyandsubjectivecreationistlanguages.Sowewillkeepaddingnewdimensions:1>2>3>5>12withcomplementarylogic statements to reach amore realist view of the scalar fractal organic complex Universe’s logic and theinteraction between TT-∞motions and SS-Minds/Seeds. Our aim is to resolve the game of existence, whichstarts inaseedprojecting its ‘mentalviewtoorderreality’,withacatch– ifthatmentalviewissimpleortoodistorted it won’t imprint reality and the system will collapse/die earlier. As the whole preserves its owncomplexityinthe‘bestofallpossibleworlds’(Leibniz)byDarwinianextinction,whichisexactlywhatænthropicmanisabouttoachievewithitsmonologicviewoftheworld.

Duality,Dilogic,ParadoxicalEasternlogic:aYin-yang,Vishnu-Shiva,Entropy-Informationduality.

Many human cultures achieved duality, notably all eastern philosophies & the German-Russian schools ofdialecticbyinfluencefromthem,bothGeneticandMemetic.Inthisviewthefractalpoint(itisalwaysagoodmindexercisetorelateTimespaceLogic(Ts-formalscienceorsTience)withSpacetimeMathematics(St-formalscienceorStience),or1st-5thPostulateofNon-EGeometrybecomestheline-wave(2ndPostulate);andstartsadualpositiveornegativecommunication(merginginreproductiveactswhereonepoletendstobeStandtheothersT,tofindanS+s=T+t‘offspring’).ThishappensinallPlanes.Theneutrinotheoryoflight,theleft-righthanded symmetries, the mirror reproduction, the DNA+RNA combo, it is called Gender (+ Duality) orPredator/Prey relationship (negative duality). It is related to the angle of perpendicularity, which ifbreaking=tearing in topological jargon the membrane NOT only deforming it and seeding it will be top

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predator/prey. Furtheron,points canapproacheachotherwithdifferentangles,once theyabandona lineWorld. All this I know ‘sounds Chinese’, too abstract. But ¡logic is homologic, so when we reduce it toexamples,itilluminatesitall.Anglesofperpendicularityandparallelismwillmanifestinpolygons,chemistry,scatteringlawsofphysics,tablesofconversationandfuking–tostateafewexamples.Pentalogicstartsthegame of creation, and in its top predator form, destroys it, so itmight be considered an expansion of themonologic,inwhichinsteadofbeingapoleofblackholeabsorptionofinformation,theBlackholemovestochase.

RECAP.Thehumind(ab.Humanmind)reducesthewholerealitytowhatfitsinitsinfinitesimalspaceandreducesall thePlanes to thehumanscaleandreducesall timeclocksequalizedbyoursynchronoussecond (limbsteps,heartbeatandeye-glimpse/thought).Yetrealityisfarmorecomplex.AndifyouarenothumbleenoughtorealizethateventhehighesthumindshavesimplifiedtheUniversetofititintoourmind,andsimplifieditsclocksoftimeto theentropic, linealviewofourvisual-dominantmind,andprojected itsentropic, violent linealview, intohisgrandtheoriesoftheUniverse(fromentropicbig-bangs,todog-eat-dogDarwinismtoEuclideanmathsofpointswithnobreath,no internalparts)youwillnotbeable toexpandyourviewof realitywith thehugeupgradeofthese papers. So to learn ÐST you need to forget your egos and learned ænthropic theories based in thereductionismofthemind(withmanasitsanthropiccenter,andourlinealentropicviewoftime-motionastheonlyarrowofthefuture).

TRILOGIC. All species are partsmade to the image& likeness of the game of exist¡enceGod=Mind of reality;hencesharingits‘organic=scalar’,‘topologic=mathematic’,causal=temporalandlinguistic=mentalproperties,asmind-mirrorsperceive the largerworld in its smallerbrainandsynoptic languages,whichshares theUniversalGrammar of Existential algebra to be able to code an image of the whole in still geometric space and logiclanguages ofmax. synoptic power, henceMin. S xMax. Cyclic Information, which starts a biologic radiationevolved socially, ∆º ∑>∆1, into a larger whole. So 2 languages ¡logic mathematics concerned with spatialgeometryand temporalalgebra,become themostexperimental,perfectmirrorof the5DUniverse.Asmindsuse them to keep imprinting entropy=motion with patterns of cyclical information to create similar energybeings - the 3 ‘conserved’ time ages and space forms that ensemble into fractal supœrganisms, between its4Dimotionof generation in a lower∆-1plane that emerges in an∆+1 socialworld todieback in adissolvingexplosionofscatteringentropy,backto∆-1inthemomentofdeathwhenallmotionisconvertedintoform.Soall traces 3 relative worldcycles, the first 'placental worldcycle' as it evolve in the relative safe 'womb' of amaternal system, the life cycle, as it tries to survive in anouter ecosystem, as it formspart of a larger superorganism living also a 3rdworldcycle. FinallyDimotions are perceived as e-motions at the body=wave level inconflict&symbiosiswithstillmentallanguages.Soallverbalsentencesreduceto:

Subject(Fractalpoint)<Verb(Dimotion=actionofspace-time)>Object(entropicenergyofsubject)

F(x):dependentparameterofspace)<=>(S=toperand)G(Y):parameteroftime

Blue(colorofin-form-ation;space)<Green:colorofreproduction)>Red(colorofenergy-motion)

Sotrilogicresumestheprincipleofconservationoftimeinasingleplane:

‘All what exists is a ‘form’ that trans-form’ into entropy≈energy back and forth ad eternal SóT, which is thedynamic fractal generator of space-time events, and principle of energy conservation enlarged to its 2 poleswhereasthewordingsfor‘entropy’.‘energy’,‘form’and‘information’requireabitmoreofunderstandingbeforewecandefinethemmoreprecisely.

TETRALOGIC is very common as it is the positive view of reality. It considers the ∆@st elements without thenegationofitall-entropy,thefifthDimotion.Asmostsystemsonlycodefromtheselfishperspectiveofthe@-mindtryingtoexpanditsterritorialspaceandtimedurationinallitsPlanes,andabhordeath,selfishmonologic&tetralogic are two steps of the same self-centered dialog. The square and the cube are also self-centered

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polygons/hedrons. Itsmotionbecomesa flat, lineal reproduction. So theyare 'alwayspositive' in its codesandintentions, provoking entropy in other systems as they absorb its energy (but not acknowledging it). So forexamplethereare4numberstocodegenes,4quantumnumbers,etc.

Trilogicandpentalogichoweverarenotawareofat.œslimits intimeandspace(entropicdeath,membrains,outerworld)anditsSS-mind,seed,whereresidesthewilltodevelopitsprogramoftime,throughthe…

PENTALOGICdescribesthe5Dimotionsofrealityatthelowerscaleofactions,theorganicscalethatconsidersthe3±¡ topologies=agesand∆±1planes inwhichsuperorganismstrace itsworldcycles.Andtheabsolutescalethatdescribe¬∆@stof times-space in itsentanglementbetween ‘entropy:¬,Space,Time,∆-Planesand@-minds. ItseasiestformPentalogicisthusverycommon;fromcellularvariationsofearlierlife,tothe5vowelsystemsevolvedfrom1ACaucasian languages (pure entropic energy), through the a, I, u ternary logic of earlier Semitic to thepentalogicofSpanishorJapanese.Itoccursinamusicalscoreandallthedimotionalmirrorsweexplain.

'DODECALOGIC'isbesttodescribethesystembothinspaceandworldcyclethrough3x3scalarelementsguidedby a mind, the 10th ‘point’ that controls internally through physiological networks, its ∆-1 scale, feels as anindividualinthe∆ºmindandbecomesa1-pointofthe∆+1world=planeofexist¡ence,inwhichitemerges.

Thereareinthatsensefrom‘9to12’accordingtodetailseveralavailable‘setsoffundamentalproperties’andstœpsforsuchdescription,butweshallnotconsideradodecalogicscaleala‘Schonberg’foranyofthestiencesstudy in those papers save time itself. As it would be too complex and to a certain extent is a matter ofchoosingthekeystepsintheprocess(thereareintheUniverseindeedmultiplevariationsaroundthekeymagicnumber,109-12whichrepresentsthemaximalcomplexityinmostsystemsofnature.

So we define any system in time, by its life and death cycle which we shall call aworldcycle (taken from 4Dphysics,worldline,whichnowhasanewdimensionofTime-informationbecomingaworldcycle.Sinceanyscalarsuper organism’sexistence can easily bedefinedas a travel through the 3 scales of the fifth dimension, as allsystemsarebornasaseminal,smallerform,inthe∆-1lowerscale,growfastwithitsfastertimespeed,emergeinthe∆ºbodyscaleandarepartwithitshead/particleofalargersocialworld,inthe∆+1scale.

Buildinguprealityin5logic,geometric,¬algebraicspace-timedimotions.

Thekeyconcepttounderstandrealitydepartingfromitssimplest fractalpointsthenbecomes ‘entanglement’asiterativeactsreproducenewdimensionalmotionsofcomplexitymirroredbymorecomplexoperands,geometricforms, e-motions, actions of existence and organs of supœrganisms, till completing a being with pentalogicelements, that all languageswillmirrorwith 3±¡=5 forms, from 3±¡ vowels to 5 type of cells, 5 fingers,musicalpentagrams,e-motions,operands,geometricpostulates,3±¡agesoflife,3±¡perceivedscaleseachoneconnectedtoanactionofsurvival,etc.Entanglementgrowthincomplexitythroughnewdimensionalplanes;synchronicityofmotions between the co-existing scales of time and sequential worldcycles form then the basis of existentialalgebra,anddeterminism.Letusexploretheminmoredetail.

Thegraphscomparedifferentdegreesofgrowingcomplexityinsystemsfrom1to5Dimotions,representedfromlefttorightinÐST,Non-Etopology,¬Algebraandilogic¬ælgebrathatcompletesthetranslationofall its lawsby

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referring them to the parallel languages-mirrors of the fractal Universe, where the 5 Dimotions of existence, 5Postulatesof¬Eand5operandsof¬AentangletogetherasmirrorsofÐSTlaws.

WeconsiderthefundamentalformalmirrorshumindsusetoreflecttheincrementalcomplexityoftheUniverse’sDimotions,astheyareachainof increasingcomplexitywhereeachmorecomplexdimotionrequiresthepreviousonetobecomefulfilled.Sogeometry,¬Algebraandlogicdoreflectthose5Dimotions:

The sameconceptof agrowingentangled complexityworks forany languagemirrorofÐST laws - from logic to¬Algebraicoperandsandthedimotionstheyreflect;frommusicaltonalsystemstovowels,allevolvingfrom1to2,3,4,5and9to12systems.

Those are the 4 main languages that matter to understand, the most important of all languages existential¬Ælgebra,theformalmodelofworldcyclesandsuperorganismsanditspossiblecombinationsofthe5Dimotionsofreality,withthesymbolsofthose5Dimotionsinspace(STvariations)andtime:‹›≈»«

The laws of Existential ¬Ælgebra and the partial functions of the generator, connect with the concept of areductionistmind,whichabsorbsonlytheenergyitneedstosurvive.Inbrief,ahugefieldofstudyinthedesignofmentalsystemsandmentalspacesofgeometryistheunderstandingofthedifferent‘realsystems’ofmindspacesconstructed with an increasing number of parameters and dimotions, which make its mental geometry morecomplex.

Andsowetalkofagrowingscaleofcomplexmind-spaces,whichweshall‘entangle’inthenextchapterwiththedifferent ‘degrees’ofsocialevolutionofgeometric figures, fromself-centeredpointsthatsponsorasimple linealmono-logic motion in time, to the more complex topological organisms/non-Euclidean planes whose structureranges from simultaneous pentalogic geometries to themore complex Dodecaplex ‘polytopes’ of simultaneous,equal600elements…likelythemorecomplexillogicstructuresoftheUniverse.

Thus the more complex expression of the function of existience, Max.∑SxT=C (s=t), which ultimately is thereproductive functionof the fractalUniverse, reasonwhywecall it the fractalgenerator, inanyof itsmultipleexpressionsincludesallthe5DimotionsItisalsoasurvivalbiologicfunction,becauseitimpliestoprovidelinealmotion to ‘fields-limbs’, absorb energy to reproduce our bodies-waves, and information to guide ourmotionswithparticle-heads.Sorealityisa‘struggle’forexistenceassystemsreproduceitsTs-fields-limbsofmotion,Si=Tebody-wavesofenergyandSt-particles-headsofinformation.ButasallT.œsarefractal,broken,itsgrowthhasalimitonthefightwithothersystems,whotrytomoveandreproduce.IntermsofpureT-motionandpureS-form,weconsiderthenthewholeofmaximal time=motion=entropyorTTandMax.space=form=stillnessorSSthe2limitingDimotionsforany3ensembleTs<ST>St-system.ThosefunctionsofthetypeSóTaredynamicfeedback‘beats’whichcanrepresentinstaticspacetheconstantinterplay between the informative pole (head/particle) and the limb/fields of locomotion for entangled, lastingorganisms,ormight representaDarwinianeventbetweenapointof still formandanentropic field the fractalpointabsorbsinaDarwinianevent.Sohowcanweestablishtheproperrulestodefineallthose‘variations’?Amazinglyenoughwiththeclarityofthenon-Euclideanpostulatesofgeometryanditslawsofrelativecongruencethat definewhich type of outcomean encounter between twoT.œswill be,whichwewill soondevelopasweexpandthemeaningofnon-Euclideangeometryandaddvitaltopologytothemix.

Thusafulldescriptionofanyorganismlocateits5elements,consideringthemfromtheperspectiveofallothers,asinthe‘Rashomoneffect’(thefilminwhichtogetthefulltruthofanevent5pointsofviewwererequired).Letusdoso,fortimeinitsits3scalesofduration(SxT=C),topredictdeeptimeorganicworldcyclesofevolution.

PentalogicTime-SpaceSyntax

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¡-Logicthemostimportantstienceofthemall,asthe‘stiencethatconnectsthe5elementsofreality,space,time,scalarplanes,mindsandentropiclimits,initsmultipleentangledperspectives.

WethususethePentagonwithallthe5elementsconnectedbetweenthem,astherepresentationofthemindoftheUniverse,andwilltrytoconsiderit‘simpleenough’forthe‘handicapped’Nitrolifespecies,calledHumind,initsuttersimplicitytoacceptthathismirrorofrealityisonlymonologicandheshouldbeextremelyhumbleifhewantstoexpandinawe,itsviewoftheUniversethatistodaytoputitmildlyretarded,eveninthoseso-calledgenius,ourtribal idolsfromNewtontoEinstein.AndIknowthis isanathema,butasthemostadvancednitrolifemindinthisbluedot, frankly I feelalso retardedcompared towhat theUniverse storesof sheer ‘complexityand intelligence’through4keyconcepts:

-Emergence in Scales of Complexity; that is, the Dimensional growth from simplex points to waves/networks,planes/organisms;diversifiedbycongruence, limitedbyentropy,openedtoa largerworld,holdingasmaller,∆-1worldinitself.

-TimeSynchronicityofdifferentclockspeedsthatmagicallybysymbiosis‘meet’atcertaintimeandspacetoshareenergyandinformation.Asweneedtoconsiderthe‘scales’oftime,fromfastactionstoDeepTimeworlds.

- Space Entanglement; which combines those 5 elements mostly in dualities of complementary or inversecomparablepairsaccordingtocongruence,andfurtheronmakethemTrinitariansystemsofsT<ST>Stlimbs/fields-body/waves-particle/heads; and entangle them through 3 scales/planes and sequentially through 3 ages. Andfinallyintopentalogicforms.Andagainweneedtodescribethe3scalarplanesofco-existingparts.

-Hierarchies,socialclassesbetweenparts,S«»Tarrowsup&down5Dofin-form-ationv.energy,topologiesetc.

-Simultaneity isaprinciplecloselyrelatedtothepreviousonesanddefinesthetrueintelligenceoftheUniverseandallitslocalspeciesthatareabletoco-existin3planes,arelative∝numberoftimecyclesandspacetopologiesdo act together as a whole. And for that to happen synchronicity, entangle and hierarchical structures worktogether to formthewhole.Mozart issaid,wasdifferent fromallothercomposersbecause ithadaviewof theentiresymphonywithinitsmind,beforeitstartedtowriteit.Consciousnessandemergencearisesfromthe‘fitting’of all parts into the simultaneouspuzzle,which thenwill smoothly sail away. Todayparallel computersworkonsimultaneity, even if itsalgorithmsdonebyhumindsare still lineal (andwewon’tupgrade them–better if theyremainso).Toachievethatgoal3otherelementscontribute:

-StillLanguagesofthemindthatmirrorrealitiesinsmallerimages,whichtheyreproduceatscale,beingultimatelythecauseofthefractalstructureofreality.Itisthe…

-FractalPrinciple:Functionsandactionsofspace-timerepeatthroughallscales(regardlessofdistortionsinthetopologicalvarietiescausebythedifferentmediumsandmultiplicationofformsinlowerscalesthroughhyperbolic5D network geometries and vice versa, elliptic synoptic geometries for the slower, larger plane of the ∆+1mind/world.

-MirrorSymmetriesbetweenSi=Te,andSxT=∆±¡,dualandtrinityandpentalogicelements.Thatis,whenwetalkofspace, it isanentangledspaceconnectedto its time-state,extendedthrough3scales,perceivedassuchbyastaticmind,limitedbyentropicdissolutionborders.Spacethuscannotbedescribedonlyasspace,butthroughtheentangledmirrorsymmetrieswiththeotherelements.Inthepentagrameachelement,isnotanisolatedT,S,etc.but the whole pentagram, only that perceived from one of its 5 intervals. In praxis we shall very often do aTrinitariananalysisof∆ST, themostobjectiveconnectedparts, introducingentropic limitsof the ‘domain’of thefunctionofexist¡ence, ifwewere touse the jargonofexistential algebra; andhardly goingoutof thehumind’sbasiclanguage,verbalthought,notevenintroducingthe‘sacredlanguageofmathematics’,thoughitsconnectionwithÐSTwillbelaiddownintwopapersoneonspacegeometryandtheotherontimealgebra.

-Entropiclimits;thatreducetheinformationasystemholds,itstimedurationanditsvitalspace,accordingto:

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-Egolimits,thenegativesideofself-centeredmindmirrorsthatdistortrealitytofitthemindandintroducealwaysanegocyparadox.Andagainatleast3languagesforthe3physiologicalnetworksofthebeingareneeded–youdohave visual eyes for locomotion and entropic nose andmouth and verbal ears… To guide us then we need toestablish.

RECAP.Thepentalogicmethod:rashomontruths.Howwegoabouttodescribescientificallyasystem,ifanyofthosepartialreductionsofinformationsufficetomakea linguisticmirror.Theanswerweprovide in thisandotherpaperson5D is thePentalogic,entangledmethod,whichexpresseswith increasingdegreesof complexity theelementsneeded toextractmostof therelevantinformationfromthesystem,withouttheneedtogettotheexhaustive(Do)decalogicanalysisoftheentire world of existence of a given supœrganism. Accordingly we go into a stair of deeper complexity,consideringsomeofthosekeyelements,underthegeneral‘labelling’ofPentalogicmethod,from:-Monologic:ceterisparibusnalysisoftheeventorspatialforminits5elementsas¬∆@st.- Dilogic: considering its ST components, perhaps its Stœps of motion and form, or its S=T reproductivefunction/stateinstatic,SxTordynamicSóTform;oritsrelativeSt-informationandTs-momentumthatcometogethertoformitsSTenergy,oritsoppositelimitsofSS-language/form/mind/seedvs.TT-entropy…causing.

WeconsiderthefundamentalformalmirrorshumindsusetoreflecttheincrementalcomplexityoftheUniverse’sDimotions,astheyareachainof increasingcomplexitywhereeachmorecomplexdimotionrequiresthepreviousonetobecomefulfilled.Sogeometry,¬Algebraandlogicdoreflectthose5Dimotions:

1Dsimplestructuresof¬Egeometrydefinedbyitsfirstpostulateoffractalpointswithinnerparts,madeofscalarplanesofexistencegrowintothe2Dpostulateoflineswithbreath,whosedualitydifferentiatesthemintowavesornetworks,whichgrowintoternaryplanes(3Dpostulate),whoselogictrinitydifferentiatestheminto3physiologicalentropic,reproductiveandinformativenetworksthatwillassociateintosocialgroupsaccordingtothe4Dpostulateof relative congruence, which differentiates 4 ‘angular e-motions’ bilateral gender complementarity, entropicperpendicularity,socialparallelismandskewness.

Soweunderstandthefractalpointasa5DMind,withthe5thnon-Euclideanpostulateandtheonlyonedefinedbyclassicmathematics, inrelationshiptotheexternalworld,anditswavesandnetworksofenergyandinformationthatentanglethepointwithits5∆º±4planesofexist¡ence.Andinthismannertheinnerparts(1Dpostulate)andouterworld(5Dpostulate)of¬EgeometrycompleteafullcycleonthedescriptionofaT.Œingeometricterms.

The sameconceptof agrowingentangled complexityworks forany languagemirrorofÐST laws - from logic to¬Algebraicoperandsandthedimotionstheyreflect;frommusicaltonalsystemstovowels,allevolvingfrom1to2,3,4,5and9to12systems.

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BOOKII:SCALAR¬ALGEBRA“ItisvainfutilitytoanalyzetheAlgebraoftime.”Stojanovic,TheCreator,onwhatweshalldohere.

“Allmathematicalalgebras,arepartialmirrorsofthepentalogic¬ÆlgebraofTime-Spaceexist¡ences’L§

I.PHILOSOPHYOFMATHEMATICS.

Thejargonof5D‘stiences’(scienceofspace-time).Correspondenceprinciple.

Anewphilosophyofscienceofmustborrowandslightlychangethe‘terms’and‘symbols’ofallofthem,tobringaUnification of its disciplines, yet by the correspondence principle maintain still a recognizable terminologycorrespondinginitssimplifiedversiontothatofallclassic‘sciences’.

Soasusualinallourpapersonthe5Dimensional,fractal,organicUniversewehavetointroduceabettermirror-languageofrealitythatfocusesdirectlyNOTthroughsymbolicelementsasallhumind’slanguagesdobutwiththe‘realelementsofreality’,Space‘sets’thatperformTime‘operands’throughPlanesofplanesofexistence,fromthepoint of view of@-minds=frames of reference within its ‘entropic limits’ - the essential elements of reality, afractalorganicUniversewhosepartsaremadeof¬∆@st’.

Wehavejust‘translatedthose5elementsofspacetimedustanditscomplexorganisms;into¬Algebraicones.Andthisshouldbethepurposeofthisarticleasallothers,whichunifyallstiencesofhumindswiththemoregeneric,hencemore truth (in termsofepistemology, truth isa systemwhichdescribeswith lesselementsmore factsofrealitywithabetterfocusedmirrortothatexperimentalreality),jargonofÐST(Generationalspace-time).

We translate mathematics to 5D. To do so, as a fractal can always be divided in sub-fractals, mathematicaldisciplines subdivide further at all levels in 5 elements connected with the 5 Ðimotions of existence & its 5structuralelements,∆@s≤≥t,Planes,minds,space,timeanditsa(nti)symmetries.

ThismixtureofmathematicalandÐSTlawswillbecalled¬ÆMath;branchedin¬Euclideangeometryofspace(ab.¬E) where points do have breath and ¬Aristotelian ¬Algebra of time (¬Algebra), where causality is pentalogic,entangledbyits5Rashomontruths(pointsofviewofits5¬∆@stelements).Whileitslogicversionis‘Existential¬Ælgebra’,(ab.,¬Æ)

Soonone side¬Æuses thepostulatesofnon-Euclideangeometry thatdefine (1st, 5th¬Epostulate)points ‘withbreath’,crossedby∝parallels(∝,thenewtermforarelativeinfinity,asallinfinitiesfadeintonullinformation,thefurtherweexplorethemfromthemental,linguistic@-frameofreference),whichformeither‘waves’withvolume(astheir¬Epointscyclethroughthem),orbranchintofractallines=networks(2nd¬EPostulateoflineswithbreath);3 of which become according to their relative similarity and complementarity (4th ¬E postulate of angles ofcongruence),the‘physiologicalnetworks’thatgatheredtogetherdefinea‘topologicplaneofvitalexist¡ence;(3rdpostulateofplaneswithvolume)orT.œ,timespacesupœrganism.

Andontheothersideits¬Algebraispentalogic,meaningeach‘element’ofrealityismadeof5elementsas¬∆@stofspace-time,entangledintosupœrganisms,withinalargerworldofwhichthesystemwillbea‘part’withasinglemonologicfunctionforthewhole,butalsowillbemadeof5internal¬∆@stelements.

Sinceasallsystemsco-existin3‘Planes’,thatofthewholeworld,ecosystemorsuperorganism(∆+1),asawholeinitself,∆º,madeofsmallerparts,∆-1,andineachscaleismadeof3‘adjacent’topologicalvariations(|-limbs/fieldsof locomotions, hyperbolic ø-body waves and O-particle/heads of information), living through 3±¡ ages as itaccumulates its 3±¡, 5 Dimotions of existence, (ab. Spatial=Dimensional time=motions); the classic A->B lineal,Aristotelianlogicnolongerholdsand¬Algebrarequiresabettercomprehensionofitspentalogic,3±¡elementsinPlanes,timeagesandspatialparts.

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¬Algebra is the main mathematical mirror of time dimotions, expressing the symmetries between the 5 mainstructuralelementsofallT.œs,∆-PlanesS-pace,@-mindlanguagesandframesofreference,(pointsofview),TimeDimotions and its entropic and inverse limits <≈>T in simultaneous structures, which create superorganisms inspacethatwillperform5complexDimotions=operandsofspace-timeorderedinsequentialStepandstopS=Tœpeventsthatputtogethergivebirthtoaworldcycleintime.

We divide its exploration in 3 sections; one of classic ¬Algebra,mostly concernedwith thebasic operands thatconnect the Space and Time states or 5 Dimotions of anysystem, which form the core of the equations of¬Algebra;a second part on ANALYSIS and a third part onmodern ¬Algebra,mostly concernedwith groups andfunctionals,which put its emphasis not on the feedback steps between Dimotions or S=T dualities, but on thestructure throughPlanesof social function (functionals), and theentangled variationsbornof the total possiblesymmetriesofonegivenelement.

MATHEMATICSINTIME.ITS3AGES

Mathematicsasallorganicsystemslived3±¡agesintheHumind(ab.Humanmind)properofanyworldcycle:

1st age:Arithmetic andplane geometry.Asmirror languagethatstudies¬∆@SThumansunderstood itssimplerunits,pointsofspace,socialnumbersandentropic limits,drawingfiguresofflatgeometryto‘encircle’territorialpropertiesinourflatworld.Trigonometryappearedthenasthe1strealizationofa‘@-mindframeofreference’tomeasurethe3rddimensionofdepth,whichisoftenparalleltoscale(astronomicalmeasure).Itwasalinealyouth,which slowly understood curves and the |xO=Ø generation of all forms with ‘conics’. As all organisms &worldcyclescanbesubdividedin5fractalsubpartsand3±¡agesinits3rdageGreekgeometrybecameold,warpedinwards-looking detached fromexperiencewith Euclid’s axiomaticmethod, the 1stmind-ego trip of creationism(man&god’slanguage).

2nd Classic age.TheS=Tsymmetry realizedwithanalyticgeometry,marryingnumbersandpoints,whilecalculusbrought ∆-scales, with finitesimal derivatives, 1/n, units integrated in wholes (Leibniz). The 3rd symmetry ofpentalogic∆=S=T,wehaven’tmentioned implied thatderivativescouldbe interpretedas ‘stœps’ofmotionand‘minimalstraightintervalsofacurve’.Sotheycouldalsostudycurvature(differentialgeometry)andlocomotion.¬Entropic limitswereneededtofindsolutions(definiteintegrals).New@-framesof referenceexpandedmentalgeometries to represent all forms of ‘selected information’, which mathematical physics used extensively todescribethephysicalworld.ThustheclassicagehadallthemirrortoolsneededtointerprettheFractalUniverseand its 5 entangled elements, ¬∆@st. But the axiomatic ego-trip stretchedmaths beyond ¬limitswhenNewtonimposed its thesisoverLeibniz’s finitesimalsand fractalpointswith infinities, linealabsolutespace-timeand thefalsehypothesisofthecontinuum,leadingtoits…

3rd age that abandoned its realist foundations with creationism -Hilbert that imagined points, sharing the onlylanguage'God'&Cantorsetsinsteadofspacepoints,scalenumbersandtimeoperandsasitsgenerators,leadingto an excess of old age information & fictions spreading to mathematical physics, as now Maths creates theUniverse,nottheinverse.

+¡:Wereturntoitsempiricalfoundationsformulatedintermsofthe5Dimotionsthatcreaterealitymimickedbythe5mathematicalsubdisciplines(largerview),Operands(shorterdimotions)&equations(worldcycles).

-¡: Yet that might not happen as insteadmathematicians are evolving the digital ‘mind’ of machines, the ChipHomoctonos,whichspeaksbetterdigitalnumbersandsotheeco(nomic)systemofcompany-mothersofmachines&weaponsisselectingcomputersthatarefastsubstitutingobsoletehumindsinlaborandwarfields,atrophyingthemback to a ‘audiovisual’ violent non-rational neo-Paleolithic,while BooleanAlgebra, past the earlier age ofsimple,fixedAlgorithmsofInformation(thetruemeaningofAI)entersitsclassicageoffreedom&consciousnessthatmightendthedominanceofhumindsonEarth;introducingethicelementsonthepraxisofmathematics,asitshouldinall‘stiences’.

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MATHEMATICSINSPACE:ITSPENTALOGICENTANGLEDSUBDISCIPLINES.

-time,¬∆@st,whereeachelementflowsasaseriesof'5Dimotions'(dimensionalmotionsoftimespace),whichcanbeperceivedas'form=space', inthestillnessofaworldmirrororlinguisticmindorasamotionoftime,initstruenaturesinceMOTIONnotformistheunderlyingsubstanceofreality.

So all fleeting forms, 'aMaya of the senses'will return tomotion and die (¬4thDimotion of entropy, death anddissolution).Its4positiveelements,organicPlanes,topologicplanesandtimeagesandactionshowever,carrythesystemasafinitesuperorganismofspacewithafinitetimecycle.

Andsothose4-ielements,entropy(¬),Scale(∆),time(T)andSpace(S),aretheelementsalllanguagesmirroreitherinaternarygrammar(ifscaleismissed),whereasofteninsteadofSpacewetalkofinformationandinsteadoftimewetalkofenergyofmotion,inasingleplane:

Lightlanguage:red-energycolors,blue-informationcolorsanditsgreen/yellowcombinations.

VerbalLanguage:subject(information)<verb(action-combination)>Object(energyofsubject).

¬Algebra:Y:Future-information<Operand-action>F(x))

Trinity is thus the logic of most beings. However as humans reached higher and lower Planes of observation, apentalogicwaspossibleanditsmathematicalmirrorbecameanalysis,withitsoperandsthatextractfinitesimals(∆-1)orintegrateintowholes(∆+1)smallerorlargersystems.

¬AlgebrawithinmathematicsasamirroroftheDimotionsandPlanesofreality.

ThusMathistheclosestlanguagemirrorofspace-time.Thusitsmain5disciplinesmirrorthe5structuralelementsofanydustofspace-time,∆@s≤≥t,Planes,minds,space,timeanditsa(nti)symmetries:

S@:Geometry studies space. Some key ages/subfields are: FlatEuclidean Geometry, with nomotion in a singleplane.@nalytic geometry, whichrepresents the different mental points of view, self-centered into a system ofcoordinates,or'worldviews'ofafractalpoint,ofwhichnaturallyemerge3'different'perspectivesaccordingtothe3'sub-equations' of the fractal generator: $p: toroid Pov < ST: Cartesian Plane > ðƒ: Polarco-ordinates.Topology,geometrywithmotionand2Planes.¬EGeometrystudiesfractalpointsofsimultaneousspace,∆-1,&its∆ºnetworks,withinan∆+1worlddomain.

∆§:Numbertheory.Discontinuousnumbersstudytimesequences.∆-1socialnumbers,whichgatherin∆ºfunctions,partof∆+1functionals;henceitisthefirst'stœp'of:

S≈T:¬Ælgebra,whichisthemostimportantpart,asitstudiesthroughoperandthedifferentDimotions,fromsingleS=T steps to larger associationsofDimotions, inmore complex∆+1 structures (Functions) and furtheron it doessimultaneousanalysisofsuperorganismsinspace,throughthestudyofitsa(nti)symmetriesbetweenitsspaceandtime dimensions (group theory)... So ¬Algebra is first the science of operands that translate into mathematicalmirrors thedimotionsof space-timeand thenbuildup from themas theUniversedoesbuildingup fromactions,simultaneousorganisms in spaceandworldcycles in time, indifferentdegreesof complexity,mirrors forall thoseelementsofthe5D¡universe.

∆T:Analysis studiesALL formsof time=change,andhence itcanbeappliedto the5Dimotionsofanyspace-timebeing, as long aswe study a 'social structure', hence susceptible to be simplifiedwith 'social numbers'.We thusdifferentiatethen5generalapplicationsofAnalysisaccordingtotheDimotionstudy,andthe'level'ofanalysis,fromtheminuteSTœpsofaderivative,tolargersocialgatherings,andchangesofentireplanes(functionals).Itisthennotsurprisingthatdespitebeinganalysisfirstderivedof¬Algebraicsymmetriesbetweennumbers,itgrewincomplexityto study changes in functions (first derivatives/integrals), and then changes of changes of functions, as motionsbetweenPlanesofthefifthdimension(higherdegreeof∫∂functions,calledfunctionals).

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ThemirrorofMathematics:itssubdisciplines.

TheUniverseofentangledsuperorganisms ispentalogic,madeof ternary topologic spatial formsguidedbymind-mirror languages, (S@), whose accumulative actions trace worldcycles, as sum of Time Dimotions acted in anexternalworld.

Andeachlanguageisabletomirrorthose5¬∆@stpartsofrealityincludingmathematics&itsmainsub-disciplines;

InmathsAnalysis studies T-motions, Geometry S-pace,Philosophy ofMathematics studies its connectionsas@languagemirrorofspace-timeoriginofitssyntacticpropertiesand¬limitsofperception(reducedtodaytotheego-tripoftheaxiomaticmethodwherethelanguagecreatesrealityvs.experimentalmethod,therightviewofmathsasamirror).Whileanalyticgeometrystudiesthe@-humindperspectivesonthelanguage(Cartesianlightspace-timegeometry)aswellasother@-mindsof'space',giventhementalnatureofallspaces.Theoryofsocialnumbersanditsarithmetic,studiesthe∆-Planesoftheuniverse.

Sowehave5subdisciplinescorrespondingroughlytothe5elementsofreality.

Whichmeansthemorecomplexsubdiscipline,¬Algebramakesuseofthemalltoexplainthewholereality.

Andsincerealityisultimatelytime-motionthatgeneratesallpossible‘fixed’spaceforms;¬Algebraleanstowardstheanalysis of dimotions with operands in the most general way that takes away some of the details and simpleevidenceofgeometricmentalspaceanditsfixedforms.

Sowhile¬Algebraisarguablymoreessentialtotheunderstandingofrealityitstotalitymakesitmorecomplexanditsstructuresmore‘loose’toallowallthedifferentvariationsofrealitytofit.

Indeed,aregularpolygon,apentagonisa5number,restrictedbythe‘logicdefinition’ofanumberasacollectiveofindistinguishableforms.Soonlyregularpolytopeswherethepositionofthepointdoesn’tmakeitdifferentanbeanumber.Buttheinverseisnottruth:a5numbercanbemorethingsbesidesapentagon–anycollectionof5similarforms.

The generality of ¬Algebra grows then furtherwith letters that represent any numbers, equations that representmultiplecombinationsofletters,groupsandsets.Soinsearchofanabsolute‘image’ofthewhole,generalityloosesspecificdetail.Andattheendasalllanguagesevolveintoa‘paralogicsearch’forunity;whichisanaturalerrorofallmindsthatfeelthecenterandoftenthe‘linguisticgenerator’ofreality;mathematicsevolvedintoasingle‘mentalconstruct’–thesetthatencompassedallotherelementsintoone.

Butbydoingsoitlostitscorrespondencebetweenatime-spacemirror(themathlanguage)andthescalarsentientbrokenspace-timeorganismsof‘dustofspace-time’itdescribed.Soanymirrorimageevenifitdidn’tportraitrealitybecamethesubjectofmathematics.Andamazinglyasitseems,mathematics–themirror–becamethe‘proof’thatrealityexistedevenbeforereality.Howtoputbacktheproperorderbetweenrealityandmirror-languagestolimitthe kaleidoscopic mirror images to real reflections of specific ‘geometric forms’, ‘social groups’, ‘Planes’ and‘motions’ requires to put in correspondence the internal structures and laws ofmathematics on thewhole, and¬Algebrainthispaper,withthepropertiesofscalarspaceandcyclicaltime,whichistheultimatelanguageofreality- not the mathematical mirror but ÐST - Generational Space-time. And this is the purpose of this book.¬@-Humind:Philosophy of mathematicsstudies the specific @-humind elements of mathematics (human biasedmathematics)and itserrorsofænthropiccomprehensionofmathematics limitedbyouregoparadox.AssuchIt ismoreconcernedwiththose'entropy'limitsofmathematicsasaninflationarymirrorofinformation,whichdeviatesfromreality(limitsofsolubilityoffunctions,etc.).Anditputsinperspectivethe'selfie'axiomaticmethodsoftruth,whichtriesto 'reduce' thepropertiesof theUniversetothe limiteddescriptionprovidedbythe limitedversionofmathematics,basedinEuclideanmath(withanaddedsingle5thnon-EPostulate)andAristotelianlogic(A->Bsinglecausality). This limit must be expanded as we do with Non-Æ vital mathematics and the study of Maths within

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culture, as a languageofHistory, usedmostly by thewesternmilitary lineal tradition, closely connectedwith theerrorsofmathematicalphysics.Insteadwemustdevelopavitalmathematicsanditsexperimentalphilosophy.

So ¬Algebra IS the study of all the structures and symmetries of the Universe from the point of view of itsmathematical mirror. ¬Algebra ismoreconcerned with space, simultaneous structures, joined by the = symbol ofequality, which relates points with internal parts=numbers, through operands that represent Ðimotions;whileanalysis specializes in all time-Ðimotions, by establishing a FINAL LAYER OVER THEM, AS each OperandspecializesinoneDimotion(sine/cosineinPerception,±inlocomotions,x÷insocialevolution,logxªinreproduction)andoverallofthemthenewlayerisgivingbyanalysis.

ButasintheentangledUniverseallmirrorscanreflectallforms,¬Algebraalsocananalyzeotherelements,asithasbeen extended from its initial analysis of equations into the whole range of 'structures', apt to study, superorganisms'entanglementsandvariationsofspecies.

¬Algebra andAnalysis aremath’s complex 'level' of reality, as reproduction and social evolution are the complexdemotions, includingobviously as its 'backgroundparts', the theoryof numbers, the analytic geometry - studyofframesofreference,andthetopologicanalysis,embeddedinthesecondaryoperand,numbersandframesinwhichwe'cast'thecomplexspaceandtime¬Algebraicandanalyticalanalysisofa'Domain'ofthefractalUniverse.

Weshall call theupgradingof¬Algebra to5Ð,non-Aristotelian¬Algebra,ab.¬Algebra,whichcomplementsNon-EuclideanGeometry,¬Etoformtogether¬Æmathematics,theupgradingofthelanguage-mirrorofmathematicstothe5thdimensionofscalarplanesofexist¡ence,whichobviouslywillbeofgreatuseforeveryst¡enceofthespace-timeUniverse.

Duallogiccomparisonof¬∆T¬AlgebrawithS@geometry,amorerestricted,detailedtruth.

Geometry isaspatialperspective,astaticmentalstillviewofreality,whichonly intopology introducesaminimaldegreeofunderstandingofchange.

In that regard¬Algebra ismoreconnected to time=change thangeometry,which ismoreconnected tostill spacestructures-intelligentconstructsofthemindthatmirrortheentanglementofrealityanditsfractalpoints.

Themain error ofmankind as an e-videntmental visualmirror of reality through space is to consider space, thementalmirror,thereality,whichistime,themotionsthatspaceslowdowntoalockedmentalimage.

Time¬Algebraisforthatreasonlargerandmorerealthanmentalspaces,sinceitencompassesalldimotions.

But itshareswithtimemotions it ‘entropic’capacitytoadmitmultiplevariationshence ‘multiplying’ thepotentialvariations and fictions of a too general statement of ¬Algebra, as compared with the immediate veracity ofgeometricstatements,whichare‘reduced’initsgeneralization.

As truth is relative both in detail (quantity of information) andmirror focus (quality of information); a commondivisionofrealitybetweenspaceandtimewoulddivideitsmathematicalmirrorbetween:

- Spatialmentalforms(S@)perceivedinasingleplanewithgeometryvs.- Temporal flows of complex systems and scalar numbers and its operands representing dimotions and itsinverseentropiclimits(¬∆T)inmultiplePlanesperceivedwith¬Algebra.

Wetalkin‘ilogic’–theultimatelanguageofallmindsandhencealllanguagesofdifferentdegreesof‘mentaldetail’,fromthesimplestænthropichumindsatisfiedwith its ‘toppredator’ reductionism, to themorecomplexsexalogicsystemsof60varieties,hintedatinblackholesandcomplex5Dimensionalstructures.

After monologic, thus comes the duality of highly ordered simultaneous mental spaces, S@ vs. ¬∆T, entropicpotential time flows happening from single dimotions of easy short causality in a single plane to complex scalarworldcycleshappeningacross3planes.

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SoifwereduceallformsofmathematicstoS@and¬∆T,wecanincludeingeometry,mindspacesinthefirst,and¬Algebra,numbersandanalysisinthesecond.

As themind ofman becamemore complex, it distinguished the pentalogic elements of the Universe, and so itslanguagemirrorofmathematicssub-dividedin5sub-areas:

Numbers (social Planes within a single plane of reality), becameNumberTheory associated to the operand of asinglescale,±.AndsoonreachmultiplePlanesevenifhumindstriedtofitallnumberfamiliesinasinglescale(theoryofthecontinuumofrealnumbers),werealreadyprobingintomultiplePlanes(realnumbers,rationalnumbers).

Next numbers, points and¬Algebraic operands came togetherwith@nalytic geometry that addedboth, entropiclimits (0’-1-∞) and frames of reference (@-mind points). So a true all encompassing Mathematical discipline,¬Algebrawasborn.

FinallytheanalysisofmotiongrewincomplexityfromtheinitialstaticgeometrytoAnalysisthatmeantinthepraxistheunderstandingofmotionasanyformoftime=change(locomotions)andthroughtheconceptofaderivativeasaninfinitesimalandintegralasawhole,theconceptofscalarPlanes.

Soinpraxismathematicsgaveahugejump,butlackingapropertheoryofscalarspaceandcyclicaltimeitdidNOTunderstoodthe‘upgrade’philosophically.Sowedepartfrom¬Algebratoconstructamorecoherent‘philosophicalExistential¬Ælgebra’tofullygraspthosefoundationsof¬Algebraasthemirrorof¬∆@sT.

WhileAnalysisisadisciplineofitsownstudyingindepththeUniverse’stime5s=tdimensionalmotions.

Toputorderinsuchanextensivefieldweshalloftendivideour¬Algebraicstudieswiththe'pentalogic'methodoftruth–thatistoconsideredforeachelementof¬Algebraitsentangled¬∆S@Tsubelementsandorfunctions,and'Itsworldcycleofevolution'through3agesproperofallsystems.

Alanguageisakaleidoscopic5-mirror,whereallDisomorphicdimotionsreflectoneachother.So¬Algebrareflectsdimotionscombinedwitht-numbertheory,Analysis,spatialgeometryand@nalyticgeometryinmanyways.Assuch¬Algebraisthekey'disciplineofmathematics'embodyingallotherdisciplines,allowingafullmirrorofrealityandallitselements.Initsevolutionitfollowsalsotheternaryagesandscalarsymmetriesofallsystems.

RECAP. ¬Algebra studies with numbers the internal structure of the organic parts of a whole that communicatebetween them dimotions of space-time through operands (≈), whichrepresent the 5 Dimotions of existence,formingcomplex systems,super organisms and groups, in which all the parts are related by itsspace-timea(nti)symmetries.Sowestartbystudyingtheagesofevolutionof¬Algebraicknowledge.

Soasamirrorof∆@STelements,wecansaythat:¬AlgebramirrorsS≤ST≥Tstructures,a(nti)symmetricdimotions&∆transformationsfromthepointofviewofagivenframeofreferenceyetitsmainmirroris¬AlgebraasexpressionoftheDimotionsofrealitywithitsoperands,

Languagesas synopticmirrorsof themindestablish thebasic relationshipsbetween the space, time, scaleof thebeing,expressingthemthroughitsoperands,dependingonthedegreeofperceptionthebeinghasofrealityanditsPlaneswhichmightbereducedifthebeingisnotfullyawareofallthePlanesofexistence,asmostmindsexistonlyinaplaneofreality.Sodoes¬Algebra,throughcombinationsof:

Sum/subtraction->multiplication/division->potency/logarithm;point->line->plane->volumeandsoon.

®lgebraofpartsvs.wholes.

Thissimplestexampleoftheparadoxesof¬Algebraicdescriptionsofrealityvs.geometricones,arebetterstudiedin@nalyticgeometry..Herewearemoreconcernedwith'structureandsymmetry',asthefoundationsofreality,both in its ternary Planes, ternary topologies and ternary ages, well described by ¬Algebraic operands, and its'growingwholes'.

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Soweestablishthreelevelsofunderstandingof¬Algebraaccordingtotheternarymethod:

-Thescaleofunits:numbers,whichare§ocialgroupsofundistinguishableelementsorsequencesoflinealtime;whichbeing¬Algebraatime-dominantdisciplinewilldominatethe¬Algebraicanalysisoverthespatialpointstates,meaningmostequationswriteas:

F(t)operandsG(S);wherethetimeorwholefunctioniswhatwenormallywanttofinddepartingfromitsspatial,∆-1parts.

-Thisleadusthentothe∆º,centralscaleofspace-timerelationshipbetweennumbersandpoints:whichthescaleof ¬Algebraicequations and its operands, which establish the relationships in a single or adjacent planes ofexistencebetweensequentialandsocialnumbers,and its 'ternarystates' andtransformations,socialevolutionsandtopologicalformsderivedofthem.

Againmanyofthose∆±1relationshipsareinto∆nalysis,whichbrokefrom¬Algebraandweshallalsostudyapartasthe∆-categoryofmaths.

- So this leads to thedominant 'leftovers'which are themost studiedelementsof ¬Algebra today: symmetries,withinasingleplaneonthe4Dmodelsofaspace-timecontinuumUniverse.

Thisscaleof¬Algebraisfullofstructures,whichattempttoenclosetheentiresuperorganismanditsworldcycleinall itspossiblevariations,speciesandelementswithinasingle¬Algebraicstructure,sortofthe 'saintgrail'ofthe'creationistphilosophers'ofmathematicalphysics,whichsomuchconfusingmakestheunderstandingofdetails.

So today modern ¬Algebra and mathematical physics in search of that wholeness, uses mostlyoperators,functionalsandgroups,andthosewillbestudiedinmathematicalphysics.

Since functionals are extensively used in all physical disciplines, notably to sponsor the 'hyperbolic view' of ∆-iPlanes(quantumphysics),asthefirstandonlyreferencetoreality.Sohappenswithgroups,whichareextensivelyusedasa 'pest' (:Weyl, inparticlephysics.Thosesynopticstructures indeed,allowus tostudyall the 'potentialfutures',ofasystem,asadeterministic 'block'ofspace-timeeventsand forms. Which is fine, ifonlycreationistmathematicians were aware that this is NOT really more than a synoptic 'block-equation' not the meaning ofeverything.

Hereiswheretheclosingof¬Algebrain3Planesofdepthshouldtakeplace,asaperfectmirroroftheUniverse.

Foundationsofexperimental®lgebravs.metalanguagefoundations.

Since,asweoftenexplainssetsandcategoriesareinwardreferencesofmathsasmetalanguagesinhisinflationaryage,tryingtoproveitallfromtheroofdown,andweshallnotconcernwithit,becauseit isnotthebestwaytojustifymathematicalstatementsandbecauseitiscompletelyoverdevelopedandlittleelsewehavetosayonit.

In that sense,wealsodecry theegoparadoxof the 'set andaxiomaticmethod, as an 'expert'metalanguageofmathssoobscurethatnobodywhoisnotanspecialistcantrulyunderstandits'modernfoundations',initsabsurdsearchofself-containedproofsprovedwronginanylanguagemirror,includinganytime-likemaths(asperGödel’sincompletenessofmeresyntaxasproofof truthofa languagewithoutsemantic references)andanyspace-like,geometry (proved wrong by Lobachevski, which implies we need experimental evidence to decide betweeninflationaryversionsofgeometry).

So in ∆@S≈T is of far less importance themodern non-experimental axiomatic formalisms ofmathematics and¬Algebra, which plague the mathematical discourse, explained as the metalinguistic third formal age of anyUniversal system. Sowe shallwidely ignore the formal evolution beyond group theory of ¬Algebra (set theory,categories)asitispartoftheinflationarynatureofinformationandfarremovedfromreality.

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Grouptheoryhowever is important,asthesimplestformtodefinethedifferentS-t-stsymmetriesandmotions=beatsofexistenceanditsreproductionofnature'sinformation.

Time-like¬Algebraandspace-liketopology.Timelikeprobabilityandspace-likestatistics.

Finallytonoticethatmodernmathematicsinwhatisworth-itsfinalevolutionofsymmetriesandcorrespondencesbetweenthe∆@s≈telements-worksevenfurtherinitss=trelationships,withtopology,spacewithmotionhenceanSt-versionofit,andmodern¬Algebra,numberswithgeometricalstructures,henceaTs-view,whichthencanbefurtherrelatedbyanicetheorem:

- Thatall demonstration in¬Algebrahasademonstration in topology; so the s=t symmetries reach its zenithofusefulcomplexcomparison.

¬AlgebraandGeometrytogetherformthusa'thirdcategory'ofdualsymmetriesalsoworthtostudy,asthespace-timesymmetryallowstofindself-similarpoint-numbers,¬Algebraic-topologicaldemonstrations.

Andthishappensalsowhenwestudyprobabilitiesintimeandstatisticpopulationinspaceastwosidesofamirrorsymmetry,whichwould be thefinalmore complex 'whole view' of number theory, fromwhere indeeddeparted(Fermat'swork,asthefounderoftimetheoryandprobability).

S=tsymmetriesbetween¬Algebraandanalysis

Theequivalentelementsof¬Algebraandgeometryareinthatsenseeasytoidentify:thenumberisthepoint,theequation is the line andplanesof theholographicprinciple, andthe scalar 5Dimensional forms, in ¬Algebra arerepresentedbypolynomialfunctions.

Sowecanalsocompare∆-Planesand¬Algebra,whicharetwowaystoarrivetothesamescalaranalysisbymeansof differentials in a geometric view (Leibniz) vs. infinitesimal 'convergent' series, (Newton's work), from the¬Algebraicpointofview.

So∆§calesarebetterstudiedbyanalysis.AndsoweshallstudythoseNewtonian/Leibniziandualitiesinitssection;wherewecanalsoput it in relationshipwithXX c. researchon∆-Planesadvancedfurther in twonewsubfields,geometrywithmotionormoderntopologyof 'knots', 'networks'and 'adjacentpoints',andfractalgeometryandscaling laws. So the marriage of ∆-Planes and geometry is today an offshoot discipline in its own,making atopologicalstudyof∆-Planesanessentialelementofmodernmaths.

So while mathematics has a clear-cut division in 5 disciplines parallel to the 5D Universe, the complexity andcreationofnewlayersandstructurescomefromitscombinationsindualitiesandternarysymmetries(numberandpoint,sandt;scaleandnumber,∆andt;scaleandfractalpoint,∆andsdualitiesands<st>t,∆±i,|x0=Øternarysymmetries).

WHERE¬Algebraisthebestsubdisciplineofmathematicsinwhichthethree∆stelementsareputtogether.

Sincetopologyisternaryonlyinasinglespace-timepresentplane(beingdualinitsgeometry-svs.motion-tand∆ºwholesasnetworksofpointsofan∆-1Planes).Andanalysisagaintendstodualityofintegralsvs.derivatives.

The potency of ¬Algebra comes then from itscapacity to give us a hugeamount of information into the mostcomplexsymmetriesandsimultaneousspacexsequentialtimestructuresofreality

Sowedodepart inourstudiesofspacefromanupgradingof theconceptofpointto fractalpointand itsstudythrough the three topologic networks, in time from sequential social number theory, and its dual, 'inverse ±'numbers; in analysis of the duality of parts andwholes, and then combine them allin the generator equation,which is thebasicgeneratorof ¬Ælgebraic structures, as it embodies the ternary symmetriesof time, spaceandPlanes.

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Andsothefundamentaltaskof®lgebraistotranslategrouptheoryanditsspace-timesymmetries,polynomialequations and its scale symmetries and operands and its specific transformations of space and timeforms/functionsintotheformalismofthefractalgeneratorofspace-timeanditsallowedsymmetries.

1st¬AlgebraicS=TMINDEQUATION:0x∞=Ksearchfor'wholeness'inasingleequation.

To make it even more complicated human ego-centered paradox, o-mind x ∞Universe = constant, is naturally built tofind wholeness, and stop all motions intopuzzledmappings that try to enclose it all, all the steps, all the variations, all theforms,andallthemotions,inasingle¬Algebraicstructure,apotentialequationofallpossiblebifurcationsofthosesteps,whichhumansthinktohaveachievedwiththose'monstrous lie groups' and other ¬Algebraic structures that try to be an impossibleminimalistmirrorthatencodestheinformationofallthesymmetricstepsavailabletoreality.

It is a ridiculous ego-trip, which only obscures further our comprehension of thedetails and wholes of reality, born of creationist theories of a universe with onlymathematicalproperties,supposedlyencodeinoneofsuchgroups.

Inthegraphoneofthemostadvertisedofsuchtheorieswhichmerelyareacatalogputintofancyschemesofallwhatthereisthere,ashortofdataencyclopedianotatheoryofwhy&whatgenerateseventsandforms.

Ontheotherhandthe5DmetricGeneratorequationof®lgebraresumesallrealitiesthatexistintheUniverse,andallitslanguagesthatshapeconstantmirrorworlds:

0’-finitesimalspatialmindx∞timecycles=Constantmind-world:§@<≈>∆ð

It ismuchsimpler thananymonstrousgroup,and itmerely tellsus that the5dimensionsof space-timecanbefurthersimplifiedintospatialformsofthemind,§@(1D,5D)anditslinguisticmirrorssuchas¬Algebrais,andthetrueinfinitemotionsoftimethroughallthePlanesoffreecyclicaltime§paœrganisms(4∆,3D),whicharethereforeinaconstantform-motion,mirror-reality≈similarfeed-backrelationship,astheglobalUniverse'shrinks'<intothespatial,localmirrorsoftheminds,§@,andviceversa,asthesingularity•mindmirrorsorderalocalterritory,@,changingafinitesimalamountofthetotalrealityofthose∆ðcycles.

Butofcoursethe'pre-work'neededtofullyunderstand0x∞being'verbal,logic,conceptual',ismissedinmodernmathematicalthought.

Thestudyofthose4elementsofallrealities,itsactionsandternaryoperands,structuresthedynamic'GeneratorEquation'ofallSpace-timeSystemsoftheUniverse,writteninitssimplestformasasingularity-mindequation:

Ox∞=K=∞º=1

Orindynamicway,S@<≈>∆ð.

So that is the game: 3 asymmetries of scale, age and form, which can come together or annihilate and eachlanguage=mindrepresentsindifferentmanners,thoseelementsanditsoperands.

Inmathematics,withthedualityofinverseoperations,+-,X÷,√xªand∫∂.

Thehumindplaysakeyrole inall thisgame,aswecanonlyperceivethroughthe languagesof themind,which'perceive in themselves' as 'still-simultaneous-linguistic mapping'. we do not perceive reality but languages-mirrors,weoftenconfusewithreality(Mindparadox).

All is in the codes of the Universe and its operators, also clear in mathematics, with the duality ofinvertedoperations,+-,X÷,√xª,∫∂.

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WethusdefineinExistential¬Ælgebrathemainoperatorsofclassic¬Algebraasthetwodirectionsofadimotionofspace-time.Wecaninthismannercompactsomedimensionsas inverse,beingentropyandgeneration(4thand5thDimotions)inverseinsymmetrytoagiven∆ºplane(±¡SYMMETRY).

Towhichextentwecanconsiderstopandmotion,1Dand2Dinverseorratherasymmetric,mustbedealtonaone case basis. Finally as it turns out we need the other four Ðimotions to achieve the 3rd Ðimotion ofreproduction,we can reduce all other Ðimotions to reproduction; hence establishing the unity of intent in theUniverse:tokeepreproducinganeternalpresent.ITisthispresent-eternalreproductiveÐimotionwhatdefinestheUniverseasafractal.

It is essential to understand 'existential Logic' and its space-time inversions, which represent thefundamentallogicpostulatesofreality:

ThatS≤≥T,butalsoSxT=K,So,S=k/T.

SosystemsadvancethroughinvertedStepsandStopswhichbalanceeachother,sincetheequalityofbothsidesofexistence coupled with its paradoxical inverted properties make all systems finally a virtual zeroth sum,balancedacrossstepsandstops,DimotionsandPlanes,representedinthelogicofhumanformallanguagesnotablymathematicsbyequalitiesof'dimotions'asfunctionsofspace-time(Sx=Ty)indifferentmirror-languages,andsoweshallfindconstantlythosebalancesareneededinNatureandrepresentedinlogicandmathematicallanguage.

Someexamplesofphysicswillsuffice.

Inthestudyofparticles,painstakinglyastheydidnothavethe'basictruths'ofspace-timesymmetries(stilldon't)physicists discovered that the proper representation of the quantumworldwas not only awave (Schrödinger’sequation)butalsoaparticle,thatisthemotionandstopduality,sotheymovedtotheKlein-Gordonequation;butthentheyhadtomarriagethoseequationswithrelativityonthe limitofourc-lightspacetime(fullcmotion);sotheyhavetofindabalancingoppositerotarymotion,thespin,alsoonthevergeofc-lightspeed(Paulimatrices);butthentheyrealizetheyhavetheimbalanceof+particleswithoutthecounterpart(-Particles),soDiracwidenedthosePaulimatricestobispinorsthatrepresentedtheneededcounterpartofantiparticles.

Yet they foundtherewere in theUniversemoreparticles thanantiparticles.Wherearethentheantiparticles tobalancethezerothsum?Itwasfoundinthepost-waragebutstillmisunderstoodthatacertainforce,theweakforce,theyincorrectlytriedtomodelasaspatialforce,whenitisaforcethatevolvesparticlesintohigherscalarfamiliesofthefifthdimension,preferredtocreate'particles' in itswaydown(entropicarrowofdevolutionfromheavytop,bottom,strangequarkstolighterud-quarks),firstintheStrangekaondecayandjustrecentlyattheLHCinamuchwider'angle'(20%ofexcessofprotonsoverantiprotons),soitfollowsthatifthe'nextscaleofmatter',theblackholeworldofheavyquarksand'blackquarkstars'(seeourpostson5Dastronomy),favorsinitsdecaytothesmallerworldofatoms,particles,initswayup,the'galatom',orphysicalupperscaleoftheUniversewillhavegalacticantiparticles,asitisthecase(ourleft-handedgalaxyhasmostofitsmassasanantihydrogendoesintheouterhalo,NOTthecenter)...

Soweobserve in fact theexistenceof inversebalancesinall thedirectionsof the fractalUniverse, in stepsandstopsofS=Tbalances forthe5Dimotionsofexistence,hence inbetweentheO-¡,غ,|¡differentPlanesthatwillshowto 'changeitsrole' -soforexampleproteinsare lineal,but inthenextscaleof '4Dtopologicwarping'formglobularshapes...andsoonandsoon.

All thismust thenbeproperlydescribedrelating it tothedifferentoperandof ¡logic,whichweshalladapt fromclassic¬Algebra,atthesimplestlevel(leavingforfutureresearchersthefull-fledgediscoveryofthewholemodel,inveryspecializedelements, suchas thoseofquantumphysics in thewrongCopenhagen interpretationwith itsunneededconjugatenormalizationandapparatus).

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There are 'a priori' sets of rules for all sciences, motions and dimensions, species and events, in simultaneousadjacentsuperorganismsthatbalancelimbsandhead,locomotionandperception,entropy=deathandinformativegeneration,mergedinreproduction(the5Dimotions,thencanceleachother,andkeepreproducingthepresent);allofthembasedinthe3simplelogicstatementstranslatedwiththoseoperators:

'Thatspaceandtimestatesofformandmotionbalanceeachother,S≤≥Tandyettheirpropertiesareinversein5Dmetric,SxT=K±¡,butisomorphicinallPlanes(K±¡).'

Theastoundingbeautyof the5Dmodelof reality thenarises from the fact thatwe shall extract all the lawsofrealityfromthatsimplestatement,inalltherequireddetailaswejusthavedonefortheworldofparticlesthatsocomplicated appears to thephysicistwhich lacks this basic understandingof theparallelisms, perpendicularitiesandinversionsofspace-timesystemsinallPlanesofnature.

So the simplest method of extracting a meaningful quantity of information on a being or event requires tocomment on the 4±operands '5' perspectives, evenwhenwe study aswe do in the first line,those elements inthemselves.Inthispostweshalldealthuswithtimefromtheperspectiveofspace,time,symmetries,Planesandmindlanguagesthat'stopit'inthesingularityoft=motionzeroth,fromblackholestoeddiestocharges,tobrainstofrozengeneticDNA...

Numbertheory,¬Algebraandanalysis,aresub-disciplinesofmathsthat2considerasimplexspace-timeduality,departingfromtemporalnumbersandspatialpoints:

-T-numbers>¬Ælgebra≈∆nalysisevolutionofcomplexity;onthesideofTime-dominantsubdisciplinesvs.

-S-geometry>@nalyticalgeometryandTopology,intheSpace-likepoint-likesub-discipline.

Itfollowsthatequationsof¬Algebraaresocialorganizationsofnumbersintimeequivalenttotopologicalsurfacesinspace.

Andso inthesamemannerwehaveupgradedthefoundationsofgeometryandtopology,toadaptthemtothe5DSTdimensionsofrealitybycompletingthe5postulatesofnon-Euclideangeometry(¬E)weMUSTupgradethefoundationsof¬Algebra,toaccommodatethe¬Astructureofternary,logictime.

What are then the equivalent 5 Postulates of ¾ Existential ®lgebra equivalent to the 5 Postulates of non-Egeometry.

Theanswerisnotsosimpleas¬Algebraistotimewhatgeometryistospace,andtimeisfarricherastime-motionisall,spacebeingonlythehumindmappingofreality-orthatofadifferentmind.

Reason why space happens in a single continuum - the scale of man - and time in all the Planes, and all themotions,not'stiffened'bythemind.Soaswordsaremorecomplexandricherthanartinthefirstageofhumanlanguages,andlogicistheunderlyinglanguageofmathematics,¬Algebraismoreextensivethangeometry.

StillthesymmetrybetweenS-geometryandT-¬Algebralikeapproachestoaproblemexists.

Sowerelatethe2S-likeT-likepartsofmathematicstoeachproblemofrealityTOCHOOSEthebestwaytodealwithitdependingonthequalitiesoftheproblem,eitherclosertoaneventortoaform-aswechoosetheframeofreference,cylindrical,Cartesianorpolardependingofthes,st,t,form/functionoftheentitywedescribe.

Allmathematicaldisciplinesareexperimentalhenceextractessential∆•s≈tproperties.

Thus we go back to the basics and consider ¬Algebramerely the space=time symmetric view of mathematics,whichusesas'units'NOTsetsorGroupsorCategoriesbutsequentialnumbersandsimultaneouspoints,studyingitsrelationships;wheretheonlyparadoxesoftruenotearethosederivedoftheslightlydifferentcontinuousvs.discontinuous,simultaneousvs.sequential,memorilessvs.hiddeninformation,whichstructureboth;ofwhichthe

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mostevident,deepcaseisthefactthatindiscontinuoustimenumberscertainrelationshipssuchasπor√2areNOTdefined,whileigeometrydoexist.

Thosearekeyparadoxes tounderstandwhynumbersmisspiecesofentropy-energybutgain information,whilecontinuous points fully embody a given plane of reality but loose information on other Planes. And so the fullπerimeter exist; the π-number misses the 'last point' and fluctuates up and down the geometric pi enclosure,meaning thatin time pi never closes its cycle, allowing a perpetualmotion Universe, but in space, it practicallyclosesineachpresentthesuperorganism,breakingitsvitalspacefromtheouterworld.

Alongallthepagesofthisblogwehaveshownanenormousarrayofexamplesprovingthedualitybetweentime-motionsanditsspace-forms.Sobasicallythefundamentalsymmetryofrealityisthatwecanexpressanythingasaseriesofmotionsintime,whichwillberelatedsymmetricallyorinverselythroughoneofthefundamentaloperandsof¬Algebra,orany'verb'oflanguage,oranyofthelogicsymbolsofspace-timeflowsof5D(<≈>)formingatime-spaceevent.

Andhencefromoneofbothsolutionswewillbeablewiththepropermethodstoextracttheothermirror-solution,whichweexpressinthefundamentalduality/equationofspace-timesymmetry:∫@≈∆T

This is whatmakes ¬Algebra, oncewe apply the Rashomon effect (multiple 2, 3, 4 or 5 D p.o.vs and functions,dependingondetail)sopowerfultomirrorrealitasitisbasicallyallaboutSoperandsTsymmetries.

But the main task to do in ¬Algebra is to fully account for the meaning of all its operands and establish itsrelationship with space and time symmetries. I.e. we talk of sum, product, integrals, logarithms and numbers,complex,irrationalandsoonwithverylittleunderstanding,happyjustwiththepedantic,axiomaticmethodanditspretensionofabsolutetruth -whichbyGödel isbyallmeans incomplete,andrevealsvery littleontheultimatedisomorphismsofspaceandtime.

So theexplanationofoperandsandSpace-timedualities=symmetries+inversions iswhatwill takeus to theverydeepestlevelsofunderstandingoftheUniverse.

RECAP. ¬Algebra studies social, sequential numbers and its relationship with spatial points, extracting the S=tsymmetricmotionsbetweenthem'.

It shares its minimal mathematical units with number theory, as numbers are closely related to the arrows ofeusocialevolutionofthe5thdimensionandthearrowofdiscrete 'frequency' intime.Assuchnumbersmeasurethe'long',socialevolutionarygameofPlanes,and'short'sequentialarrowsoftimeoftheUniverse.

While ¬Ælgebra truly starts in the next scale of space-time events, with multiple groups of points(variables),exchangingentropy,energyandinformation,throughoperands(specificofeachaction)in'equations'(whichbecomepartialcaseofthegenerator'sallowedstexchangesandsymmetries).Andsothisclassic¬Algebraisthemostimportantpartofit.

¬Ælgebrastudiesthewhole'blockoftime-space'ofasupœrganism,throughall itspotentialactionsexpressedinfunctionals of functions that embed two or even 3 Planes in whole ∆±1 structures, is the all inclusive∆±iperspective,whilegroupsofsymmetries,accordingtoasetof'restrictions'ofitsallowedactions,isthesingle'space-time'planeattempttodescribethewholepotentialtransformationofanyentityinsuchasingleplane.Sogroupsandfunctionalsexploredinthe3rdageof¬Algebracompleteitswholeness.

Thisfirstviewof¬Algebracanbedonethroughitsanalysisinspaceasasimultaneousconnectionofthetwosidesofanequation,whichput inrelationshipthe 'ternaryextremesofan∆stelement',eitherby itsoperands(whichallowsasingleplaneconnectionofsandtelements)byitspolynomials(whichallowsamultiplescaleconnectionthrough∆§)andbyitsinversesymmetries(groupconnection).

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AndbecausetheuniverseisdualinitsinversionsandternaryinitselementsandPlanes,3+aretheoperationsofmath.SincePotencyandintegraloperationsbeingcloselyrelatedasweexplain,asthedualityof∆curvedanalysisbetweenplanesvs.§lineal scaling indecametric societieshappen tobe. Indeedanymathematician immediatelywillnoticethisdualitywiththeconceptsofaderivativevs.alinealdifferentialapproach,alogarithmicscalingthattendstoafixedasymptotevs.achangingtangent-derivativeofamorecomplexcurve,andtheultimateproofthatall¬AlgebraicequationsofexponentscanbeapproachedbyTaylorseriesofderivatives.

¬Algebraexplaineditwithgrouptheoryandaxiomaticmethods;soweshallconsiderthatapproach.Andtheonlyelementslefttodefinethenaretheidentityelementandthepropertiesoftheoperands.

Indeed,alldualoperandshaveidentityneutralelementsandinverseones.THEYCANbeconsideredlooselyastheneutral=asymmetry form, which 'splits' both ways into the negative=antisymmetric element and thepositive=symmetric one. The identity element leaves the asymmetry unchanged, it is we might say a non-operation.Thenegativeelementistheresultofantisymmetry,thepositiveelementofsymmetryandtendstobealargerwhole.

I.e.ifwehaveaherdof4+3elements,andtheycomeintoparallelsocialevolutiontheygiveusaherdof7.

Butifwehaveaherdof4elementsandtheycomeintoaDarwinianrelationshiptheywillseparatethedefeatedelements,soifitisforexampleafightforsupremacy,onlyonewillbeleftandwerestthe3defeatedcandidates,4-3=1.

Thefirstcuriousthoughtcomingoutofthissimplifiedanalysisisthatpositiveoperationstendtobemorerestrictedadditions as theyrequire identical elements, so they are often simple social evolutions,negativeoperationshowever might have multiple meanings as the antisymmetric, 'entropic' states of a systemmultiplystatistically.

So this has a reading into the classic arrow of time of thermodynamics, which already noticed that order'probabilities'are lessthan 'disorderedones',unlessthere isaMaxwelliandemon-whichwecontenddoexist inthesurvivalwillofallpointsoftime-tocontainentropic,destructiveprobabilities,whichisindeedwhatithappensmostoften.

When3-4elementsthatarenumbers,whichcanbeoperatedtogetherhenceequalbeingsasnumbersaresocialgroupsofequalbeings,themagicof social loveand the fourthpostulateofsimilarity innon-Euclideangeometrymeans thatall the systemswhich fallbetween symmetryandasymmetry, and can communicate come together.Entropy is REJECTED by almost all systems from the singularity perspective, which implies also that a mereabstractmathematicalanalysiswill fail tounderstand thebio-logical inputof those informative singularities thatrejectasMaxwelliandemonsdothenegativesystems,trickingthe'dices'ofGod.

What aboutproduct vs. division?Again thisduality is obvious. Ifwehave3 x4 itmeans4herdsof 3whichadtogetherin4stepstoa12herd.

Buttheentropiceventsherearealsomultipleanddivisive. I.e. Ifwehave3kilosofwheat,wedividethewholebetween4entropichungrymentoget3/4percapita.Divisionsthus 'divide'almostalwaysasystemintobrokenparts.Productsmultiplysocieties,orcreatetightercommunicationbetweenthosesmallerparts.I.e.A=5andB=4elementscanbemultipliedalsoatthe∆-1scaletogetthenumberofaxonsthat'tight'thesocietytogetherifeachelementofsetAcommunicatewitheachofsetB,thenAxB=5x4=20axons.

Sowecometo thepotencyvs. root, integralvs.derivative thatmerely take thisdualprocess to its final ternaryscaling(asweknowsystemsareternarysononeedforfurtheroperands).Itscomplexstudybeingcarriedfurtherinthepostofanalysissoweshallletitgoforthetimebeing.

AdefinitionoftheUniverseanditsspace-timetopologicand¬Algebraicelements.

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WedefinethefractalUniversesuitedtoitsmathematicaldescriptionasafractalsuperorganismmadeof:

A(nti)symmetries between its 4 Dual components T, ∆, @, S, through parallel =symmetric orPerpendicular/antisymmetric communication, as they can either evolve inBalance≈become symmetricorannihilate=becomePerpendicular/antisymmetric:

S:space; an ensemble ofternary topologies, (|+O≈ ø)... which made up the3 physiological networks (|-motion/limbs-potentials+O-particle/heads≈Ø-vitalenergy)ofallsimultaneoussuperorganisms

∆:Planesofsizedistributedin∆±irelativefractalPlanesthatcometogetheras∆ºsuperorganisms,eachonesumofsmaller∑∆-1superorganisms...thattraceinalarger∆+1world...

Time cycles: a series of timespace actions of survival that integrated as a whole form a sequential cycles ofexistencewith3ages,eachonedominatedbytheactivityofoneof those3networks:motion-youth,orrelativepast,dominatedby themotion systems (limbs,potential); iterativepresentdominatedby the reproductivevitalenergy (bodywaves), and informative 3rd age or relative future dominated by the informative systems,whose'center'is:

@: The Active linguisticmind that reflects the infinite cyclesof theouterworld and controls thoseof its innerworld, through itslanguagesof information,whichguide its 5 survival actions: 3 simplex,aei, finitesimal actionsthatexchangeenergy(e-ntropyfeeding),motion(a-ccelerations)andinformation(perceptions)withotherbeings,and two complex actions:offspring reproduction and social evolution from individuals intoU-niversals thatmaximizethedurationintimeandextensioninspaceofthebeing.

So that is the game: 3 asymmetries of scale, age and form, which can come together or annihilate and eachlanguagerepresentindifferentmanners,thoseelementsanditsoperands.

Amind-mirrorlanguagecanperceivethegamegoesondegreesofcomplexityfromthesimplestpossiblemindsof‘monologic thought’ (the humind, regardless of our ego Px. that feel to be the center of reality and its mostintelligentbeing)tothemorecomplexformsofpentadodecalogic(sexalogicsystems).

Soinitsfullformulationasmirrorof∆@STelements,wecansaythat:

¬Algebramirrorss≤st≥t structures, a(nti)symmetric dimotions&∆ transformations from thepoint of viewof agivenframeofreferenceyetitsmainmirroris¬AlgebraasexpressionoftheDimotionsofrealitywithitsoperands

It is then clear that what languages as synoptic mirrors of the mind will try to do is to establish the basicrelationshipsbetweenthespace,time,scaleofthebeing,expressingthemthroughitsoperands,DEPENDINGonthedegreeofperceptionthebeinghasofrealityanditsPlaneswhichmightbereducedifthebeingisnotfullyawareofallthePlanesofexistence,asmostmindsexistonlyinaplaneofreality

Sodoesmathematics,throughcombinationsof:

Sum/rest->multiplication/division->potency/logarithm;point->line->plane->volumeandsoon.

Thissaid,theUniverseisagameof∞Speciesofspace-timemakingSTepsandSTopsinitsconstantcœxist¡ence(whereweuseπfortime,inthisparticularwor(l)daccordingtotheslightlychangedrulesofi-verbalthoughtforEnglishtolookmoreliketheuniversalgame),andsoasStopsandsteps(ab.œ)bringthebeingintoexist¡encewecanmeasurethemquantitativelyasspace-distancesandtime-motions.Stopandstepthenbecomestopologyandfeed-backequations,where=issubstitutedbythedynamicsymbol<=>,andthedualityfrom=stopsand≈wavesteps.

Oncetheintuitivemeaningofthosesymbolsrisestheawarenessofthereadertothevitalnatureofourhumind'sabstract rendition of reality things become then dynamic. ≈ for STeps, = formotions, and ≤≥ as the different

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implosiveorexplosive,informativeorentropic'degeneracy'ofasystemareeasilyquantifiableandtranslatabletobothmathematicsinitspureexpressionandmathematicalphysics.

Itfollowsthatthemaintranslationrequiredbetweenclassic¬Algebra,andExistential¬Ælgebra,isthatbetweentheoperatorsof¬Algebraandthoseofthegeneratorequation.OriginallyIdidtrytocastthewholemodelofÐSTin termsofgroup theory,¬Algebraand its classicmathematicaloperators,but thatwasanearlier stageofmyexploration of the fractal Universe. It was then obvious that as good as mathematics is as a mirror of theUniverse,itisnotasgoodasitisneededtoextractallitsproperties,hencetheneedforadifferentlogic,whichhoweverhadtoincludeoratleastreferencetheworldofclassic¬Algebraasweshalltrytodoalbeitatabasiclevel - thespecialistmustunderstandthis isaunificationtheoryofallexist¡ences,sowecannotbeexhaustivewithallofthem,astheteamofresearchisexactlythesamethantheinfinite-infinitesimalUniverse:∞º=1.

Andtodoso,asafractalcanalwaysbedividedinsub-fractals,mathematicaldisciplinessubdividefurtheratalllevelsin5elements.

Inthispostweshalldealwiththea(nti)symmetriesandoperandsof¬Algebra.

Thoughwe cannotbeexhaustiveas ¬Algebra is the largestof all the subdisciplinesofmaths, concernedwithtime-space symmetries,butmerelydoavery fastanalysisof itsmain themes,enlightenedwith∆st insights toshowtheenormous power to ad newwhys toreality and all its stience of 5D², since even the simplest truthsscientists think are thoroughly understood have new insights observed when puttingthe Kaleidoscopic3±∆perspectivesofthefractalorganicUniverse,whatwecallhonoringoneofmyfavfilms,the'Rashomoneffect'(atruthcanonlybe'judged'withmultiple∆@STperspectives.

Sothefifthperspectiveofthemind-judgeinthefilmrequired4previousones,andevenwhotheabsolutetruthcanonlybefoundintheevent/beinginitself,whichcarriesallitsinformation).WeshallthusmakeliberaluseoftheRashomoneffecttoenlightenthetruthsof¬Algebra.

Andthenbarelyexplore themore focusedExistential®lgebraof theGeneratorand itssub-'groups'orspace-timesymmetries.

ButwecanfurtherexpandtheconcepttoN-elementstoformarelativepolidimensionalensembleorherd,oraplane,whichiswhatHilbertspacetimeframesdo,ofwideuseinquantumphysics.Theyalsoaddthe4thpostulateofrelativecongruencewiththeconceptoforthogonal=perpendiculareventsvs.parallelevents(dotproductandcrossproductinclassicvectorialspaces).

Entanglement of pentalogic in mathematics then becomes the interaction of ‘social numbers’, operands ofdimotionsandtheworld∆+1planesinwhichtheytakeplace:

Further on as all T.œs have a different language of perception, and exist in different scientific planes, theparameterstheyperceiveastimeandspacemightvary;sonextcomesphasespaces,toreferencethesuitabletime-motion and space-form coordinates each species perceives. Then we have in the human single plane, 3frames of reference, the polar, spherical, cylindrical, lineal, and Cartesian, hyperbolic planes, being the S=TCartesianplanethemostuseful;butdependingonwhichorganofthebeing,its|-limbs/fields,Ø-bodywaveorO-particle-head enact the event, a different framemight bemore suitable. Thenwe canwrite on those frames,differentchainsofdimotionswith¬Algebraicequationsandoperand,joinedbydynamic=symbolsofequivalent,feed-backstœps.

In mathematical physics as a closer reflection of there Universe that eliminates from the languagemirror itsinflationaryimages(somathwillalwaysbeparadoxicallylargerintheoremsthatrealityis,obligedto'bend'and'limit information' to thoseshapesenergycan 'bend'), it isevenbetter to representavectorial field inGeneralcoordinates as they allow the multiplicity of points of view of the supœrganism each with its self-point ofreference.SinceallT.œshaveaself-centeredpointormonad-mirror-alinguisticmappingor'seed'thatreflects

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realityinitsframeofreferenceandwilltrytoactfromthatpointofvieworre=produceandimprinttheexternalworld'ssuitableenergy(∆<¡)withtheinnermindofitspointsofexist¡ence.

TobeanchoredandperceiveasapointisthefirstfunctionofanyT.œwhenemergingintoexist¡ence.

Wecallthedescriptionofallofthis,'Existential¬Ælgebra',ælgebra,or¬Æ,asitisboth'Existential¬Ælgebra'andNon-Aristotelian,Non-Euclideaninitsform.

Itisinthatsensethemostimportant,structuralelementofthemathematicalmirror-theclosestclassiclinguisticexpressionoftheUniverseandits5Ðimotionsofexist¡enœ.

WeareNOTthatmuchinterestedinmerelytranslatinghuman¬Algebrato¬ÆasaformalmirroroftheUniverse,aginormoustask,otherswillbuiltafterme,but indefininga¬Æof theUniverseandashuman¬Algebradoesworkinthatpurpose,toestablishasmanycorrespondencesbetween¬Æandcommon¬Algebra.

Wedepartinnon¬Æofafirstelement,afractalnon-Euclideanpoint,ruledbythe5Postulatesofnon-Euclideantopology: points are fractals, lines arewaves between points, with less 'amplitude=energy' than the point - afermionsotospeakvs.aboson-linetraced.HowFermionsornon-Epointspropersharebosons,orline-wavesoflesser'mass-position'andmore'speed-reproductionofinformationinalower,∆-¡,isarelativequestion.

For terminology, every non-Euclidean point is a t.œ, his index ¡ is always relative to itsmind: I=0=T.œsmind.Departingfromit,wecancaveintoitsinnerworldof-¡anditsupperworldor+¡.

Thelawsof¬Æareexpressedintermsofarelativeself,as∆±¡differentplanesinwhichan∆ºmindexchangesenergy and information with different ∆±¡ planes. It corresponds for the case of ∆º and ∆±1 cases to classicscience(singleplanecoincidingwiththelanguageofthemind-will).

Philosophicallywecanconsiderthemind,theperfectblockoftimeofthesyntaxofonelanguagethatreflectsthegameofexistenceitself.

Butlogicallywehaveafirstelementorseed,∆º,andthenwecan'followit'.asanon-Euclideanpointinspace,non-Aristotelianmonadintime,asitmirrorsinitslanguageandreflectsuponitsworldtoexchangeenergyandinformation and fulfill its 'Parts' as an ∆ºmust have a body, ∆±1 and co-exist in aworld, ∆±¡, whose relativeextensionmightdependofthefractalfinedetail,inourmeasureof±¡.

Assuchtheminimum'space'torepresentaworldofsuchpointsisavectorialspaceinwhicheachpointhasatleasttwovaluesthatmightrepresentthebody-motionandmass-directionofthepoint,whichbecomesthenthesimplestrepresentationofaT.œspacetime,inwhicheachpointsisaT.œ,whichhastwocreativepartswemightcallspace-form-scalarparameterandtime-motion-directionofthevectorparameter.

Thesimplestofthosearemomentumfields,inwhicheachpointrepresentingeneralizedcoordinatestherelativemomentumofthebeing.

RECAP.Allfractalself-similarscalesareformallydefinedby2metricfunctionsanditsphilosophyofscience,5DAbsoluteRelativity:thefunctionof5Dscales,SxT=C&thefunctionofrelativitybetweenformandmotion,Si=Te,whichstartedmodernscience.Sincewecannotknowfromthemental,stillpointofview,whattrulyismotionandwhatisdimensionalform.

ThusTimeMotions&spacedimensionsco-existandmergeineveryspace-timebeingoftheUniverse&wemusttalkofspace-timedimensionalmotions.Whenwecombinethem,asSi=TemaximizesSxT,wedefinetheFunctionofexistence,Max.SxT,towhichweaddthereproductivenatureofallmotions,along5Dscales,∑,sowewritetheFractalGeneratorofTime-Spaceorganisms,or5Dmetricfunction,orsurvivalfunction,asC=∑Max.SxT(s=t)thatunlike4D locomotionapplies to all ‘stientific scales’& all ‘motions’ and Species; addingorganic= fractal scales,biologicwill,determinism-asall systemsmaximize its ‘Functionofexist¡ence’ -mental=stillmirror symmetriesignoredinscience.

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C=∑Max.SxT (s=t) ISabiologic functionofsurvival thatembodiesthewillof lifeand its5Drives,or ‘actions’which all systems of the Universe code: As a entity perceives Information and moves throughAccelerated=changing paths towards Entropic fields to feed and absorb energy to reproduce itsorganism andevolvesociallyasa largerUniversalwhole.Soall systems try tomaximize its3±¡ 'organicentangledÐimotions'thatinspacecreateorganismsandintimeitsworldcyclesofexistence;startingaspureForm(SS»4D)inaseminalseed, reproducing evolving socially and emerging in an ∆º scale, in a young age of max. Motion (2D, <, Ts),balancedwithinformationinthereproductiveageofmax.Energy(3D:Si=Te),followedbyanageof information(3D,>,St),whentimereversesintoentropicdeath(«TT).

Thefunction ismaximized inthepresentmatureageofbeauty,whenS=Tandreproductionby ‘gender=mirrorsymmetry’ takes place, creating a herd (∑) a stronger, social number, which will evolve into an entangled, ∏,connectedsynchronouswholethatsurvivesbetter.

Ad it isguidedbytheSS-informativemind-seed,which isa linguisticsynopticmirror inasmallerscaleofhigherinformation,henceresponsibleforthe'creation'offractaldiminishing,‘finitesimal’scales;whichitwillreproduceoverplacentalenergyfieldstobecomealarger∆ºwhole.

The laws of Existential ¬Ælgebra and the partial equations of the generator, connect with the concept of areductionistmind,whichabsorbsonlytheenergyitneedstosurvive.Inbrief,ahugefieldofstudyinthedesignofmentalsystemsandmentalspacesofgeometryistheunderstandingofthedifferent‘realsystems’ofmindspacesconstructed with an increasing number of parameters and dimotions, which make its mental geometry morecomplex.Andsowetalkofagrowingscaleofcomplexmind-spaces,whichweshall‘entangle’inthenextchapterwith the different ‘degrees’ of social evolution of geometric figures, from self-centered points that sponsor asimplelinealmono-logicmotionintime,tothemorecomplextopologicalorganisms/non-Euclideanplaneswhosestructure ranges from simultaneous pentalogic geometries to the more complex Dodecaplex ‘polytopes’ ofsimultaneous, equal 600 elements… likely the more complex illogic structures of the Universe. On theunderstandingofany language-mirroras itbuilds thecomplex imageof the fractal5DimotionalUniverse,wemustconsiderthatthegrowthof‘formandmotion’isoneofincreasing‘dimensionality’oftheelementsofthatmirror. That is,more thanaone toone correspondencebetweendimotionsof exist¡enceandoperandsof alogiclanguage,weshallseethatthoseoperandsgrowthfromsimplemonologic,todilogic,totrilogic,tetralogicandpentalogicmorecomplexelementsofform.

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Inthepapersdedicatedtothe¡logicgrowthofÐST(Generationalspace-time)asalanguageofilogictime,wehavebuiltonthatfashionafirstpaperonmonologicænthropicmanwith itssimplexbeliefonasingletimearrowofentropyanddeath–themostencompassingandlessinformativeofallformsoftime,whichisremarkableinitsprimitivism,giventheself-importancemangivestohis‘ingenuity’,thendedicatedasecondpapertodilogic,thedualityofthetwopolesofformandmotionastheyconvergeintoparityandgendertoreproduceinitsS=TpointallotherformsoftheUniverse,givingbirthtotetralogicofpositive,socialgrowthandevolution–withoutentropythefinalpentalogicclosureofallformsinexist¡ence.

Abeingthencanexistmerelyastechnologichumansdowithsinglegoalorantsinapheromonalpathorhorsesblindedinarace,andlivewiththatsinglemonologicviewofreality,movingaheadwithoutdoubtingitsmindviewisthecompleteexistenceoftheUniverse.

When observing humind’s worship of the extension of Galilean relativity by Mr. Einstein and the astoundingworshipofhis‘genius’,whichconsistedmerelyonapplyingthe5thpostulateofNon-Euclideangeometrywithoutevenunderstandingits‘nature’,aswehaveshowninourevolutionofthose5postulates,theevolvedobserverofpentalogictimespacecanmarvelhowthatsingleentropictime,deducedfromamereexpansionbyactfactorofthe v=s/t locomotion equationofGalileo canbring suchhuge awe and veneration andbuild from itwith thatsingle arrow of entropy=death a full cosmogony of big-bang explosions and dying Universes heralded as theabsolutetruthofallwhatisneededtoknowabouttime,withoutaflinchofthosewhotrulythinksuchasimplexviewofrealitysufficetoexplainitall.

ThatisthebeautyoftheUniverse;thatsomethingsosimpleandminimalistinitstotaltruthashuminds’viewoftimepassesas thewholeandsuffices to fillofpride thespecies,whose ‘seersof time’ (the jobofGod inSaintAugustinemystique,ofEinsteinintheloreofthenuclearindustry)insomanyTv-showsoncosmologywillassertwithabsolutecertainty,themarvelousintelligenceofaspecies,whocouldsofast,sogoodunderstanditalltillthetrillionthpartofasecondafter the ‘creation throughthebig-bang’of reality in the linealpastofasingleaffineequation, V=HoD, the God of Void, the vacuum, entropic time. That’s right, we know it all because after allemptinessissimpleenoughforanant,aphysicistandEinsteintounderstand…sowhybother–‘cosmologistsareseldomrightbutneverindoubt’(Landau);‘simplifymanyouareamarine’(O.Stone).Youcanindeedsurvivebytheartofkillingitallanddevour.Ihavethuslongstoptryingtoteachcosmologists,antsandænthropicmilitarymenthatthere issomethingmorethanthe linealsimplexswordofentropyanddeath–theirdestinytracedbytheirfaith.

Moreoveras theUniverse is thewholeandall itsparts are less than thewhole,mostparts aremonologic andmonologicone-dimotionalactionssufficefortherobot,thephysicist,theant,theoperandofsumwithitssinglelinealarrowofNaturalnumberstomakeaworldviewthatguaranteesourfulfillmentandexistence.Fromsyntheticwhole–thethoughtsofgod–toitsanalyticdetails–allspacetimeorganisms.

ItisthenthebigtaskItookuponmyselfdecadesagoafterdiscoverythis‘languageandmirror’ofrealitysoabovethe present ‘language of science’ with a single dimotion of existence (lineal time locomotion), and a nullunderstanding of the first principles of reality; to connect both, falling from above from the syntheticunderstandingofthefirstprinciplesinitslargerscaleofreality,thesummitandenvelopeofthewhole,toconnectthemwith the details. This was the old ideal of the Greeks, which humanity today has forgotten and denies,becausethequantityofdetailsgatheredwithsensorialmachineshaveoverwhelmedthem.

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Sotheycannotseetheforest,notevenrealizethatanetworkofhiddenmushroomsconnectsthemall.

We trIed to connect the two greatest levels of generality in the laws of space-time, Existential algebra, theformalism of generational space-time and its pentalogic and dodecalogic elements with classic algebra and itsscalar numbers, dimotional operands and S=T equations. So first we shall introduce existential algebra, whichdeploysasimplegeneralizedformalismtobeabletoreducetoilogicequationswithonly5Dimotionsalleventsofreality.Knowledgeisagrowthofcomplexityfrommonologic,thestudyoftheonetodualitybetweenthepolesofformandmotion,totrilogicthataddsitscombinationinasingleplane,totetralogicthatstudiesthebeinginitssubjective positive drives=actions of existence without the entropic inverse arrows of death and its inverseoperands,whichbringfinallypentalogic.SowhilewetendtousejusttheconceptofapentalogicUniversemanyanalysis will be done in terms of duality, trilogic in a single plane of reality (Ts<ST>St) and tetralogic, eitherbecauseweadoptoneofthe5elementsasapointofview,becomingthesubjectiveblindspot(sooperandsofscalarsocialevolutionmightnothaveangularperception,mosthumansneveranalyzeentropicdeath,eyesdonotseepreciselywherethenervethatseesresides,etc.etc.Finallyfromoutsidea(do)decalogicanalysisaddsthe3‘scales’throughwhichthebeingwillexist,fromitsorderedpalingenesistoitsoutsideworld.

¬Algebra=¬Æmathematics.

Finally Existential ¬Ælgebra is the name I give to the formalism of ÐST and its symbols even if the equations ofsuperorganismstracingworldcyclesanditslawsarenotthoseof¬Algebra,becauseoftheuseofsimilarsymbolsandthefactthat¬Algebrawasthepreviousmoreadvancedformofascienceoftime,discoveredbymankind(giventhelimitedcapacityofAristotelianlogicandlackofdevelopmentofafullmodelofeusocialevolution).

Weshallinthispaperstudytheagesof¬Algebra,thescienceoftimeinmathematics,afterourbriefintroductiontothemorecomplexcausalityoftheUniverse.

And conclude the post with the future of ‘¬Algebra’, which will be either a simplified version of a digital mind(Boolean¬Algebra)orwithminimalprobabilitygiventhenullevolutionofhumanityanditsincreasingobsolescencetoAI, theageofExistential¬Ælgebra,where thescienceof time finally reaches its zenithwith thepentalogicanddodecalogiccomprehensionofthelawsofExistential¬Ælgebra.(ab.¬Ælgebra).

Theentropiclimitsofrealityappliedtomathematics.

Itisobviousthenthat¬Algebra,aptlycalledfromArabic"al-jabr"meaning"reunionofbrokenparts",andanalysisbystudyingS<=>Tand∆§sufficesomehowtodescribethe5Dimensionsofpresent,pastandfuture,andindeed,theytogethercoverallof it intensely...buttheyrequire firstagoodunderstandingofsocialnumbers,andthe@nalyticplaneswhichworkasthe∆-1'cells'and∆+1worldinwhichtheorganicsystemsofmathematicsworkoutits'actionsofexistence'.

Theonlyother'greatsubject'whichisonparallelinimportancetothatof¬Algebra&analysiswouldbegeometryinmotion,topology,whichthankstothe5Non-Æpostulateswillbeexplainedasafarmore 'intuitive'andprofoundperceptivemirrorofreality,oncepointsacquirevolume(1stNon-Æpostulate),fromflatprimenumberstoplatonicsolids,reproduceinnerparts,communicatethroughinfiniteparallels(5thnon-ÆPostulate),whichbecomeentropy,

Inversefunctionsas'balancingstœps'ofthefifthdimension

Humans are entropic, thermodynamic virtual ghosts trapped between the immortal quantum proton and cosmicblackhole(noentropicevaporationthere),LOVEentropicsimplestlinealtheoriesofreality.

Entropydeathisginormousinitsfieldsasitappliestoallotherattemptsof'differentiated'evolutionofform.

Sothereare4positivedimotions(locomotion,reproduction,perceptionandsocialevolution)allofthemsubjecttonegativeentropyanddestructionthat'returnsthem'tothe'motherwomb'ofnoform.Andforthatreasonwecantreat the 4 INVERSE operands, negative exponential decay, negative subtraction, division and derivatives (when

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appliedtothesearchforan1/n,∆-1infinitesimal)asreflectionsoftheUniversalentropythatdestroysthepositivedistinctiveoperand,ofteninamuchlargershorterexplosionofdissolutionofform.

Entropyisthedisentanglementoffree-chaosthataffectsallsystemsandtheinverseoperatorsbestdescribe.

WhencomparingeachnewDimotionoperation,weobserve severalelementsofentanglement. First the simplestexpressionofthenext'Dimensionaloperation'seemsjustarepetitioninanewdimensionofthepreviousone:theSumismultipliedbyaddingsumaftersum.ThePower isobtainedbymultiplicationovermultiplication,butthis isjustwhentheoperandsisappliedtotheself-reflectivepoint.

Operands do have however as they grow in complexity a direct entanglement with the 5 Non-E postulates thatdefine from points waves of communication that become networks of topological planes and organisms, wheremultiple p.o.v.s are put in perspective. And this even more clear when from mere polynomials we move intoexponentials and from exponentials into integrals,the more complex operands then can interplay with multiplepointsof view,merging them through theproduct, reproduced through its inversepartition/division,moving themthrough aworld cycle, in exponential growth and decay, andemerging into a new planet ofexistence throughderivativesandintegrals.

Soasusual intheentangledUniverseof infiniteactsofcommunicationeven if theone-dimensionalmantendstoreducethewholeanditsflowsofcommunicationtoasinglelineofthought.

Re=productionthroughsocialevolutioninalowerplaneofthefifthdimension,inwhichtheaxonsjointhewholesisthenthe2-pointmeaningofreproduction;oftenabidimensionalSpace-timecombinationofastateofformaandastateofmotion,asintheparameterofmomentum.

Itmustbenoticedthenaveryinterestingnewvisionoftheinteractionof inversefunctions,whichdoNOTmerelycancel each other as one could expect, in classic science,but advance theDimotion in alternative balancingacts,leavingamemorialtraceofitsexistentialduality.

Indeed,considerthecaseoftheproductanditsinversedivision.Theyarefunctionsprimarilyofsocialevolutionandreproduction,aspartsofwholesareproducedandexchanged.Soifwehaveanindividualof5partsthatreproducesitwillgivebirthto25parts,butthenweneedtodividethembetween5parts,toget5individualsof5parts:1(5)x(5)=1(25)/5=5(5).

Same happens with the positive and negative motions, which in alternate current move back and forwards the‘electrons’,whichinfactdon'treallymove,butits¡-1scalethecurrentkeepsmoving.

Sogoesfortheexponential+Logarithmicfunctionsofreproduction,anditsinverse,whichformaworldcyclethatinthemiddlepointofmaximal 'carrying capacity'will givebirth again to anew 'exponential growth'=generation, tocontinuetheworldcycleinanadjacentspace-time.

∫∂: Finally it came calculus,with its inverse operand, which represents thescalar next social gathering ofelementsof¬Algebra,asitisappliedtothepreviousoperands,aswholes,exceptforthetrivialxª-thepreviousmorecomplexpolynomial,andsowemustregardcalculusnotonlyastheoperandofalldimotionsofchange,but also as the operand between planes of existence, since the logarithmic/power previous expression, justreaches between the two limits of two planes of the fifth dimension, but calculus allow us to 'emerge' andtranscendbetweenplanes.Sobythecontinuousgameofinversefunctions,whichbecomeazerothsum,theDimotionsofexistencecontinuetomovethewholeoftime-spacethatneverceasestoexist.

Howeverandthisisakeytruthoftheunderlyingstructureofscalarspace-time,∆St,thenumberofoccurrencesofthe positive function is always larger than then negative inverse function. So there aremore possible sums thannegativenumbers(forspatialquantitiestherearenonegativeones,nonegativeapples).Therearemoreproducts

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thandivisions(as4/2=8/4but4x2≠8x4);therearemoreexponentialsthansquarerootsas√-xdoesnotexist,andfinallytherearemoreintegralsthanderivativesas∫y’=y+C,adding∝constants.

Whichmeansthearrowofsocialevolutionandlifeislargerthanthearrowofentropyanddeath,whichinfactlastsonlyaquantaoftimeinitsÐSTequation:Death=0Tx∆S.

TheUniverseisasymmetricandparadoxic,evenifmindssimplifythoseparadoxes–andmathematicianseliminatethemwithegocyaxioms. It is thoughmoresatisfying toexplore theparadoxesof thegameofexist¡encethat theaxiomsofZermeloandHilbert.0’finitesimals,S≈Tgenders,differentarrowsoftimein5Daspartsandwholeshavedifferent properties, finite infinities, limits of entropy, dissolution of information in irrational numbers,discontinuitiesthatcreatestopsandforms,etc.arethehiddenstructurethatmakepossibleadynamicreality.

Sogoingupwardsthescalesof5Dparadoxically,wemultiplythechoicesofpotentialfutures,balanceditsentropicdeaths.Reproductionbecomesaradiationoffractalpointsoforderbalancingitsentropicdeath.∆∇dualtendenciesarethusbalancedeveniftheyareasymmetries,becauseofthoseparadoxesbetweenStandsT,SSandTT,balancedin ST; themes those explored in depth in ilogic and ¬ælgebra (existential algebra) and its fractal generator’sformalism.

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THEBASICERROROFAXIOMATICMETHODS:MINDPOSTULATESTAKENASABSOLUTEREALITY

THEPHILOSOPHYOFMATHEMATICALSCIENCESOFA5DUNIVERSE.

Correctingerrorscausedbyeuclideansimplificationsanditsdogmas=axiom(atic)method

All this said it isobvious that reality ismirroredby theexperimental languageofmathematics,butdue to thesimplifications of earlierGreekGeometry and its dogmatic ‘mentalmethod’ of proving reality as if itwas ‘themindofhumans’ duetotheegocy(ego=idiocy)paradoxderivedoftheconfusionofthemind-mappingand itslanguageswiththewholereality(O’-mindx∞Universe=constant linguisticmap)mustbecorrectedtoevolvemathematicsandcorrecttheerrorsofitsuseindifferentsciences.Andthatisthemaintaskperformedonthosepapers. So let us consider the consequences of the real ∆ST structure of the Universe into the 3 mirrors ofmathematics:

Principles of spatial geometry related to the fundamental particle of the Universe – the fractal point/monad:Growthoffractalpointsintonetworks,socialplanesaccordingtocongruence.

ThemainprinciplesofspacetocorrectaretheaxiomsofEuclid.GeometrystartedwithEuclideanpointswithoutpartsformingonlydeterministic linesofasingle‘straightfuture’thatevolvedintocurvedwith3possiblepaths(y’<=> 0), and finally points acquired volume as non-Euclidean points crossed bymultiple parallels, aka fractalpoints unit of the new mathematical discipline. And calculus detached from Algebra as the most importantoperandofchangeandevolvedintofunctionals(functionsoffunctions).Butinallthoseevolutions,thehomologywith ∆st elements remained, even if as 3rd ages become excessive in form and detached from experimentalreality,mathsalsolostitscontactwithrealityandbecameinflationarymakingerrors,suchastheinfinityerrorsofsettheory.

MathematicalerrorsderivedofPointswithvolume,finitesimalzeroths.

The postulates ofmathematicsweremade by humans and are subject to human frailty.Mathematicians alsoknow that at the logical foundations ofmathematics there aremany paradoxes,meaning thatwe can get, byreasoninginacceptedways,resultsthatwedonotlike.Thisisduetotheaxiomaticmethodthatdoesnotcheckrealitytoestablishitsmathematicalreductionistmirror.Sothemaintaskof5Distocheckthemirrorwithscalarspace-timelawssolvingthoseparadoxescarriedoutintoexperimentalsciencesthatuseidealmathematics,suchasphysics:

-Pointsdohaveparts,crossedbyarelative∝numberofparallels.Andso lineshavewidth,theyarewavesorfractalnetworks,andsoplaneshavedepth,theyaremessingsof3lineswhicharenowtopologicalnetworksandhencetheyformatopologicalorganism.

-Asa resultofpointshavingparts,absolute zerothdoesnotexist.There isnotabsolutenomotionbuta0’ktemperatureofresidualmotion;thereisnoabsoluteemptinessbutafinitesimalresidualpastmemorywhenwetakeawayacorpsewithDNAtracesinalowerscale.So0’,thefinitesimalisalwaysleft.

-Pointswithpartsthereforeholdafinitesimalworldinside,andas‘monads’doholdstillsynopticlanguages,ofwhichmathematics is undoubtedly the language hold in the tiniest points, atomic particles, as it is themostsynopticofall.Sowhenwestudytheequationofthemind,O’x∞Universe=mindmapping,wewillunderstandwhymathematicsissuchanextendedlanguageinitslocalterrotirialorderofreality.

Principles of algebra related to the scalar 5th Dimension: the evolution of social numbers and networks.Maximizingtheefficiencyoftimeconservation.

Settheoryofwholeswhicharemadeofparts istheclosestreflectioninmathematicsofthescalarstructureofthe fifth dimension from the perspective of thewhole, but numbers, are from the perspective of its parts itsfundamentalelement.Anumberisthendefinedasasocialgroupofindistinguishable‘¡ndifferentparts’,whichin

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itsmostsynopticformmerelyreflectthe‘scalarsocialnatureoftheUniverse’.Foranumbertobesuchthoughtheymust truly have no difference at all. So puremathematical numbers are regular polygons, and all othernumbers have to be referenced to the property that makes it equal, which implies to include a logicalunderstandingofthe‘equivalence’betweenformsaccordingtoagivenproperty,afactoftenmissedinscienceswhichuse‘equality’ liberallyandthenpretendas intheequalityofenergyandmass,theyare‘thesamething’(errors of this type are rife in mathematical physics, where there is a zeroth interest for conceptualunderstanding,aslongasthe‘maths’carryontheequation).Identitythenisakeyoperandthatmustbestudiedindepthtofindoutdifferentformsofequality.

Thefalsesolutionofthoseerrorsasalwaysiscausedbyhumanegocythatinvents‘postulates’withoutproofor‘axioms’tocreatesystemsindependentofexperience,basedinthefunnymusings.Forexample,Eucliddefinespointswithoutparts,anentelechy,and lineswithoutvolume,anotherone.Andwhenwefoundthatpointsdohavepartscrossedbyparallels (5thnon-epostulate) insteadofgroundingthis fact inexperimentalevidence,aswe do; Hilbert starts his foundations of Geometry unable or unwilling to define them experimentally as ourexpansionofnon-Euclideangeometrydoes, imaginingpoints, lines,planeandcongruence.ThistendencykeepsgrowingwhenhisfriedCantorfindsanother‘mentalparadise’,andinvents‘sets’whichare‘wholesofparts’,theverydefinitionofthescalar5thDimensionexperimentallysound;insteadoflookingattherealwholesandpartsand define them as non-euclidean planes of space-timemade of fractal points,waves (lineswithmotion) andnetworks(lineswithvolume,asfractalpointswithpartsstarteditsdiversificationindifferenttypesoflineswithvolume,waveswithmotion,networkswithbranching;lineswithinversedirectionsandorthogonallines,ofS(y)-informationandT(x)-entropy).

Principlesofcalculus,relatedtotheconservationoftime.

Another consequence of the fractal struture of points with parts that have therefore motion and form isfundamental principle is that all ismotion=Time=Change, as we have never discovered a ‘solid’ formwith novibrationormotion.Thereisno0temperature,thereisnoresidualvacuumspacewithoutanh-minimalquantaoftimemotionthatcanevolveintovirtualparticles.Motionneverstops,andwhenwedie,themotioncontinuesin the internaldesintegrationof thebeing.Butagainst thisprincipleof theultimate flowof timecyclesof theUniverse fights each point of view, each fractal monad that tries to conserve its time of exist¡ence, tries toachieve the conservation of time, so time changes in finitesimal quanta (principle of calculus) and systems ofphysicstrytoachievetheminimalexpenditureoftime=motioninitsactions(principleofleasttime)andstayinthepositionofminimal expenditureof timeenergy,which in itsmathematicalmirror impliesphysical systemssearchforLagrangians,leasttimeactions,∂S=0,andHamiltonians,StandingpointsthatseemNOTtochange.Thisis theoriginof the3agesof time, inwhich thesystemseeks for theMaximalandminimalpointsof temporalenergy(youth)andinformation(oldage)inwhichatleastoneofthetwoformsoftime-motion,cyclicalandlinealformsdonotchange,andspeciallyfromthepointS=tofrelativepresentwheretimeneverchanges.Fromthoseprincipleswecanextractallthelawsofcalculuswhichbecomesthenthemainlanguageoftime.

Our goal thus is to revive the experimental foundations of mathematics abandoned by the idealism of theHegelian->Hillbert+Cantorschool.

Itisthenhumind’segocy(ab.Humanmind,ego=idiocy)whatdeniestheexperimentalnatureofmathematicsandinventedmental,creationisttheoriesaccordingtowhichmathematicsisthe‘languageofGod’,sharedbyman,of‘logic’, or ‘intuitionist’, or ‘axiomatic’ etc. etc. Nature. Themain task of 5D generational space-time regardingmathematicsusthisobvious:toputinrelationshipscalesandnumbers,fractalpointsandspatialstructuresandalgebraicoperandsanddimotionsoftime,toenlightenfurtherboth,mathematicsandÐST;whereastheprimacymust exist in the being ∆St fractal, space and cyclical time over the mirror, NºGA, Numbers, Geometry andAlgebra.Andsoweshallcorrectmathematicalstatementstomirrorbetter∆ST,withoutdismissingthefactthatmirrorstoworkmustbeideal,still,reducedimagesofrealitywithlessinformation.

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Becauseofthewaylanguagesevolvefrome-videntsimplicityintocomplexity,mathematicscamefirstasspatialgeometryandthenasalgebra,ofscalarnumbersandsimpletemporaldimotions;castonequations;andonlyinmoderntimesfullylearnedthenatureoftime=changewithcalculus.

Sowedivideaccordinglythestudyof thediscipline in3papers,even if the3elements,∆STanddisciplinesthatmirrorthemareentangled.Sowedocomment ineachpaperonboththe lawsof∆STgenerationalspacetimeand the laws of the 3 disciplines. This second paper is dedicated to scalar algebra and its simpler operands,keepingfora3rdpaperthestudyofcalculusandtheoperandsofsinusoidalfunctions,complexnumbersand∫∂derivativesandintegrals.

Mathematicsasamirrorlanguage.

We shall not somuch correctmathematics but rather showwhat are its simplifications becausemany of theerrorsofsciencesthatusemathematicsasmirrorsarecausedbythelackofunderstandingofthatrelationship.I.e.thelimitsofmathematicalphysicsconcerningthefactthatall0sarefinitesimal0’sandallpointsarefractalpointswithvolume is themainerror thatallowedphysicist to invent ‘infinitesimalsingularities’,argueabout if‘particle-pointshavevolume’,ordiscusswhatmodelofquantumphysics,theprobabilisticorstatisticalmustbetruth.

The problem can be illustrated in Magritte’s painting ‘c’est nais pas a pipe’… Mathematics is not reality. SoPoincareandEinsteintoldtoeachother:‘Iknowwhenmathematicsaretruthbutnotwhentheyarereal;Iknowwhenphysicsisrealbutnotwhenitistruth’.Wewanttomakethemcometogetherasrealandtruth,whichweshalldolatterin2019or2020whenwecompletetheupgradingof5Dstienceswithourlastpapersonphysics.

TheUniverseisafractalthatreproducesinformation,formsinaction,motionsoftimewithform,ofwhichthereare5Dimensionalmotions(ab.Dimotions),combinationoftime-motion,T,andspatialform,S,whichwesimplifywith5‘holographic,bidmensional’symbols,incapitalsforthedominantelement:TT-entropy,Ts-locomotion,TS-reproduction, the dominant function of the Universe, as a fractal is defined by a reproductive ‘generator’equation, inthiscasethegeneratorofspace-time,St-informationandSS-linguistic form.Thegameisplayedbytwolimits,thatofTT-entropy=timemotionwithoutform,whichinfactdissolvesform,andSS,fractalpointsthatkeep an almost still image of reality by simultaneous, intelligent mind-mapping of those motions into space-forms.Thisisthegame,infinite‘monads’thatholdaworldinthemselves(fractalnon-Euclideanpoints).

As a languagemathematics is used by differentminds, notably atomic particles and some ‘composite’ mindsmadeofelectrons,humansanddigitalmachines,whichdue to itshigherparticledensity,as theyaremadeofmetals,calculatebetterthanhumanbeings.Mindsdohavetheirownpeculiarities,themostnotabletobeself-centeredastheymeasuretheexternalworldfromitspointofview,andtodosotheyreducealltheinformationoftheUniversetoastilllimitedmind-mappingtofititintoitsinfinitesimalsize,selectingandbiasinginformationto cater thoseneeds,putting themselvesat the centerof the limitedworld theyperceive, confusing itwith thewhole Universe. Mathematics helps to make those mappings because it is a highly synoptic language. I.e. anumberis‘any’collectionofidentical,clonespecies.Butthissynopticpowerintroduceserrorswhencomparingitselementswiththerealitytheymirror.I.e.pointsareentitieswithparts,butinEuclideangeometrytheyhavenovolume, so theybecome ‘ zeroths’,which in realityare ‘infinitesimals’withaminimal formandvolume.Soonlyknowing theapriori lawsof space-timewecancompare the simplificationof ‘spacegeometry’ and ‘timealgebra’ and reconstruct the loss of information, even if for practical purposes we still use the mentalsimplificationofmathematics.

Thelargerreferentialview:thelawsofgenerationalspace-time.

We recommend for that reason to read first the paper that introduces properly the main elements of adescriptionofafractal,Universemadeofspace-timeorganismswhosegoalistoreproduceitsformandpreserveitsexist¡encemaximizingitsmetricequationsofenergyandform.Allstiencesexpresstherulesofthatgameof

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fractal reproduction of space=form and time=motions, including the laws ofmathematics in its two branches,non-Euclideangeometryandnon-Aristotelianalgebra.Wejustincasethereaderignoreouradviceintroducethebasicelementsofthe5Dscalaruniverse,asthelawsofmathematicsareareflectionofthoselaws.

Realitythoughisfarmorecomplexthanhumind’sprojectionofitslimitedperceptionoftheplanesofspace-time,and it can be casted in multiple languages besides mathematics, which is – conceded – one of the bestexperimental languages of perception of reality but NOT the language that creates reality. The laws of the‘substanceofreality’,planesofspacetimearethe‘apriori’realityalllanguagesincludingmathematicsmirror.

Whatarethenthesimplificationsofmathematics,whichwemustcontrastwiththelawsoftime-space?Theyaremanybutbecausetheyare‘well-done’simplifications,mathematicskeepsahighefficiencyasalanguage,unlike

grossdistortionsofrealityasthoseofsubjectivetribalnationalism,Abrahamicreligionsandgo(l)dchurchesoriginofcapitalismthatmakehumansocietiessounjustandconflictiveandpoliticsandeconomicspure‘idol-ogies’notsciences.Stillthemain‘simplificationsofmathematics’areignoredbecause the humind’s ego loves to think it speaks a perfect language that explains it all andnothingelseisneeded.SoOmartheconquerorofAlexandriaburnedthelibraryasallwhatwasintheKoranwasenoughandinmanytownsofdeepAmericathereisonlyabible,customaryineachmotelfornothingelseisneededtoreadthatthehistoricbookofabronzeagepeople.

ThisgoesonalsoinmathematicswiththeinsistencethattheaxiomaticmethodofproofofEuclid,apt for the first age of a language, when it is more lineal, deterministic first age of timespace

topologycould indeedproveabsolute simple truths (asa line remainsa lineonly if it follows its samestraightpath).Determinism inmathematics as in theUniverse though changeswhen the linewith thepassingof timecurvesandbendsascurvaturecaneasilychangeintodifferentpaths,andpotentialfuturesopen.Thechildiseasyto becomea fundamentalist deterministic line; the beginning of a civilization ruledby swords of deterministicentropicdeathandlinealsimpleartisasallthingssimplejustabeginning.Euclidthusreducedalltoverysimpleconcepts –pointshadnoparts, lineshadnobreaths, planeshadnowidth, things thatwereexternally similarwereequalregardlessofitsinternalpartswedonotsee,astheydon’texist…

-Becausethereare5dimotionsofspace-time,simpleAristoteliancausality,A->B,isnotreal.RatherthecausalityoftheUniverseisentangled.Wedonotcreatearealcirclebydrawingalinearoundwithasinglepen,butrathermultipleflowsoftimespaceconvergeintoapointcreatingasphereofforceswithmultiplecausality,oneforeachlineofforce.

-TheUniverseisnotasinglespace-timecontinuumoflightspace,whichiswhatourelectroniceyes-mindssee,but it is a fractal Universe ofmultiple scalar planes of space-time, displaced in space andwithmultiple timeclocks. And so continuity does not exist. The real line isNOT in a single plane, but inmultiple planes and thediscontinuitiesofnaturalnumbersarefilledbydecimalsofansmallerscale/planeofthefifthdimensionandthediscontinuitiesofrationalnumbersarefilledbyirrationalones.

Doesthismeanstheaxiomaticproofsofallthosenon-truthshavenouse?No.Itdoesmeanthattheyareprovingsomethingelse,initssimplificationthatextractpropertieswemustnowapplytotheenlargedfractalUniverseofmultiplescalarplanesofspaceandmultipletimespacedimensionalmotions.

I.e. the hypothesis of the continuum is false, because each of those number families is in a different scale ofspace-time,whichinnaturearedilatedin‘time’aspartsmustbecomefirstforwholestoexist,sothecellmustexistfirstfortheemergenceoftheorganismaboveittohappen.ButthehypothesisofthecontinuumstillworksfortheentiresumofthosescaleswhichputtogethershowtheUniversehasahorrorvacuum.Thereisaspacenotfilledbyspheresadjacenttoeachother,butthismeansthere isaholebetweenthemthatcanbefilledbyotherspheresorinaheightpackingbyatubularchannel.Andthisisexactlywhatweseeinnature:plantsusethe

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interstitialregionbetweencellstochannelwaterupanddownthetree.Butthosetubesareadifferentplaneofreality.

Of course, I am aware thatmodernmathematics considers a continuum in different terms, butmost sciencessticktothepreviousconcept.Todayinmathematicsacontinuumisrathersimilartotheconceptofthe‘domain’inwhichthefunctioncanbeexpressedasaseriesofstœps,stepsofmotionandstopsofformalperception(St-TsorSS-TT),whichare ‘orthogonal functions’ofS-heightandT-motion.Soa function iscontinuouswhenwecantrace steps and stops that become tangents of its derivative. The change in 5D is to consider real theapproximations by differential tanget stœps to the curve, NOT the curve in itself, as all curves are in detailfinitesimal Steps and stops, wave and particle states, length-locomotion and height-information consecutivedimotions.Continuitythenhappenswhenthestepsandstopsaresimilar,followingthelawofS≈TpresentstatesandbreakswhenMax.SorMax.T‘breaksthefunction’,intoits‘pointofdeath’byunbalanceoftheS=Tlawofpresentstates.

Aplanedoeshavedepthanddiscontinuouswholesbutitstillcanbedefinedby3lines,onlythatnowthoselinesarenetworksandso3physiologicalnetworksdefineatopologicalorganism,avitalplaneofspace-time.

ThefivedimensionalmotionsofTimespacemirroredbytheoperandsoftimealgebra.

SowhatÐST,generationalspace-timedoes,istodepartfromreality,thelawsofspace-timeofthefractalorganicUniverse,andcomparethemwiththelawsofidealsimplifiedmirrorssuchasmathematics,orverballanguages,ormusicoranymind-mirrorofthegameofexist¡ence,ofenergyxinformation,ofanyscaleofspace-timest¡enceandacknowledging those limits rethinksmathematical statements, further advancingour comprehensionof itslaws.

ThusthepurposeofthesepapersistoputincorrespondencethelawsofeachlanguagewiththosehigherlawsofÐST,Generationalspace-time.InthecaseofAlgebra,themirrorisconcernedwith5‘operations’thatreflectthe 5 Dimensionalmotions all systems of reality perform as its ‘deterministic program’ of existence;whereasalgebra represents those time-space organisms with ‘sets and numbers’ (sets being the most comprehensive‘whole’, numbers, the minimal unit, mathematics uses to mirror the spacetime reality of superorganisms ofsocial,clonespace-timeparts,tracingworldcyclesofexistence)

Thus, the branch ofmathematics that dealsmainly with the 5 Time Dimotions is ¬Algebra – Non-AristotelianAlgebra of multiple ‘times=changes=dimotions’, which unlike classic Aristotelian, A->B single causality Algebrathatrecognizesasinglearrowoftime,acceptstherealityof5differenttypeofchange;andthusvastlyexpandsthetheoreticalfoundationsofclassicalgebra,groundingitinarealdescriptionoftime-change.

We are thus writing, paraphrasing our quote, a treatise on the ‘Analysis of Algebra of Time’, the ‘creator’develops inall its systems,withanobviousemphasisonoperands– the functionsofalgebra thatmirror the5Dimotionsoftime-spaceand its fundamental form,calculus,whichstudiesall theformsofchange=TimeoftheUniverse,evenifitdoesnotgrasptheoreticallywhatitdoes.

The extraordinary development of Algebra and mathematical science is due to the fact that derivatives andintegralsof‘finitesimals’andwholesarethebestlinguistictoolsofarfoundbymankind(besidestheultimateÐSTlaws developed in this blog) to describe all the modes of time=change. Unlike physics stuck in its ‘worldlyprofession’ of making entropic weapons and transport, locomotion machines for so long that it cannot evenrecognizetheexistenceoftime=change intheform,or in-form-ationofbeings, thedominant ‘arrowsoftimes’and dimotions of a fractal Universe which is all about reproduction of forms, Algebra in its apparatus evenwithoutthefoundationalinsights¬Algebrawillprovidedoesstudyallthosetypesofchange.

Sowhenwecalculate the changeof volumeofa receptacleofwaterwithan integral,weare studying the4thdimotionofsocialevolutionofH20,evenifnomathematicianwoulddaretoseeitthatway;andwhenwestudythederivativeofanysystemweareanalyzingthe‘changeofafinitesimalquanta’inthesystem,theminimalrate

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ofgrowth,evenifthelackofproperfoundationsofmathematicswilltrytojustifyderivativeswiththe‘hypothesisof the continuum’ (which is wrong, only themind by eliminating the fractal wholes of our Cantorian dust ofspace-timeseesacontinuityorbycompressingallthefamiliesofnumbersanditsPlanesintoareallinefindsnospaceinits‘Dedekind’scuts’)…

Andsowecometothefullpurposeofthispaperasallotherpapers–tomarrytheabstract jargonsofhumanstiences,cleanuptheirmentalerrorsduetotheegocy(Ego=idiocy)paradoxofmankindthattriestoprojectitslimitedstillviewofrealitynecessaryforthemindtofitalltheUniverseinavisualthought,andthenexpandoratleast give the clues to future ‘pros’ of each discipline to expand soundly those foundations connected to theultimatelawsofGenerationalspace-time.

UnlikeNon-Euclideangeometryoffractalpointswhichiseasilyconnectedwiththeexpansionofthemindthat5Dbrings to all ‘stiences’, however ¬Algebra is expressed in a language of time that requires a more profoundtranslation to the laws of Generational space-time; not so much because of its unfocused mirror but itscomplexity, which goes well beyond monologic huminds’ limited intellectual capacity despite its egocy(ego=idiocy)embeddedparadoxthatmakeusallfeelubermen.

Yet¬Algebraworksbecauseofthemagicinvolvedinthesyntaxoflanguages;whichallhaveitsownconsistencyand self-contained structure, as allmirrors provedbyprojective geometry reflect theDisomorphic lawsof thewholetheyobserve.Itisimportanttograspthatforthingstowork,systemsdoNOTneedtohavethecollectiveintelligence,orevenunderstandwhat theydo.Andthis ishowhumindswork,withoutunderstandinghowtheUniverseworksbutmerelyassemblingmachinesandconsideringthemaproofofthehumangeniusnotoftheintelligenceoftheUniverse. Sogoesforthe‘ensembles’of¬Algebra,whichrequireamentalprotocolofproof(theaxiomaticmethod)anditsmanipulationbyitsrulestomakeit‘work’regardlessoftheblindnessofmankindinitsunderstandingoftheultimateprinciples. I.e.neitherLeibniznorNewtonunderstoodwhatisafinitesimal,theminimalquantaintimeandspaceofawhole,andhowcalculuswidensourmeasureoftime=changebeyondphysicallocomotion–humindsstilldon’t–buttheyuseitsrulestocalculateanyphysicalchangeinthecosmos.

Realityinsteadobligesto‘downgrade’humind’singenuityandupgradedespiteegocy,thebeautyoftheUniverse;thatis,¡t=ssymmetry,organicproperties,vitalwillofsurvivalandcollectiveintelligence(Max.∑i),whichmakesits fractal points follow the rules of order that ‘shape’ vital geometry and the ‘rules of motion’ and ‘socialevolution’ofpartsintowholesthatshape¬Algebra.

But ¬Algebra will never be so e-vident as non-e geometry is in the depiction of reality because we do NOTperceivedirectlytime.Wedobutinthesimultaneityofaslowstopandgomentalimageoftimeflowsthatattherateofhumind’sperceptionofasecond,solidifieseverythingthatmovesmuchfasterandtracessmallishcyclesthatappeartousassolid ‘particles’.That is,weperceivevery littleof the feedbackcyclicalmotionsof timetotrulyunderstanditatfirstexperience,andwhenwetrytosummarizehugeamountsofpopulationsincomplexdimotionsof time, thetoolsof¬Algebra,whichachievethiswithsomuchsynopticpowerseemtousa ‘magictechnology’(Clarke).

Themostmagicaspectofalanguageisitscapacitytopredictthefuturebycompressingthe‘flowoftime’intoitsultimate ‘ÐST’ laws – the vital program of survival of theUniverse of 5 Dimotions (ab. Dimensionalmotions),which all languages of time, including ¬Algebra do. Huminds are extremely proud of it, having as an ‘egocydogma’thatwearetheonlynon-predictable‘free’(egocymythology)speciesandtheonlyonesthatpredictthefutureofotherswiththeonly languagesharedwithGod,mathematics.Wehavedealtwithegocyparadoxesinouranalysisof ‘monologicman’.Fact isman isalsopredictable, fromthe largerpointofviewofsocialgroups,worldcyclesofexistenceofculturesandcivilizations,andtheentropiclimitsofdeathandlifeandtheEarth,aswedoandhavedonefor30yearswithabsoluteaccuracy,and0interestonsocialscholarshipandpoliticalpraxis–sincepreciselyhumind’s limit is theiregocy.But ifweacceptedthe limitsof thespeciesandentanglewiththeUniverse,perhapsthenwecouldbecomeimmortal,infiniteandasintelligentastheUniverseitself.

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Allentitiesaresuperorganismsorpartsofasuperorganismtracingworldcyclesofexistencebetweenbirthandextinction – its time limits – through 3 scalar planes of the fifth dimension, the cellular/atomic, organic/thermodynamic and ecosystemic/cosmological Planes – its scalar limits, structured in trinity, topological non-eadjacentgeometries(itsspatiallimits),andmanispartofcivilizationswithinthelimitsofEarth,whichalsohavelimitsofsurvival, ina relationshipofsymbiosisandpredationwith ‘metalife’species,evolvingandreproducingmuchfasterthanus.Andallthisisaspredictable,evenmorethanthesimplest¬Algebraicoperation,1x1=1(inaself-reflectiveact,whentheonesarethesame),1x1=3(whenreproductionexistsbetween2differentones).Soyes,¬Algebraislikelythehighest‘achievement’onthecapacityofhumindstomirrorreality,onlyinferiortoafuturehumind,orAI--mindunderstandingofÐST laws.But it isneithermagic,norahumindachievement,andnottheabsolutelanguageofNature,whicharethelawsoffractalspaceandcyclicaltime.

Sowerecommendtoreadourpaperonthe5DimotionalfractalUniverse,tounderstandits5elements,Space,Time,∆-Planes,linguistic@-mindsandentropiclimits,thecomponentsofallSupœrganismsofTime-spaceare.

Thelawsoflanguages,itstrinitylogic.

This said the fundamental purpose of this and all other papers onmaths is to upgrade the understanding ofmathematicsasalanguagemirrorofthevital,organic,scalarpropertiesofthespace-timeUniverse.

SowearenotsomuchaadvancingmathsbeyondtheupgradingofNon-Euclideanmathematics,butinterpretingmaths as an experimental science, that is as a mirror of the space=geometry and time=logic, ¬Algebraic andscalar=digitalsocialpropertiesofallwhatexists.

TheUniverseisanentangledfractalgameofDustofspace-time,¬∆@st,whereeachelementflowsasaseriesof'5Dimotions'(dimensionalmotionsoftimespace),whichcanbeperceivedas 'form=space', inthestillnessofaworldmirrororlinguisticmindorasamotionoftime,initstruenaturesinceMOTIONnotformistheunderlyingsubstanceofreality.

Soallfleetingforms,'aMayaofthesenses'willreturntomotionanddie(¬4thDimotionofentropy,deathanddissolution). Its4positiveelements,organicPlanes, topologicplanesandtimeagesandactionshowever,carrythesystemasafinitesuperorganismofspacewithafinitetimecycle.

Andso those4-ielements,entropy (¬),Scale (∆), time(T)andSpace (S),are theelementsall languagesmirroreitherinaternarygrammar(ifscaleismissed),whereasofteninsteadofSpacewetalkofinformationandinsteadoftimewetalkofenergyofmotion,inasingleplane:

Lightlanguage:red-energycolors,blue-informationcolorsanditsgreen/yellowcombinations.

VerbalLanguage:subject(information)<verb(action-combination)>Object(energyofsubject).

¬Algebra:Y:Future-information<Operand-action>F(x))

Trinity is thusthe logicofmostbeings.Howeverashumansreachedhigherand lowerPlanesofobservation,apentalogicwaspossibleanditsmathematicalmirrorbecameanalysis,withitsoperandsthatextractfinitesimals(∆-1)orintegrateintowholes(∆+1)smallerorlargersystems.

Thusmathsbecamewith the inclusionofCalculus themost complex,bestmirror languageofhuman thought,arguablyovercomingwiththeageofcalculustheverbalmirror,which isthenatural languageofman,speciallybecauseofthearrivalof instrumentsofmeasurewhichcouldalsocastrealityintothedigital languageofsocialnumbers,identityspeciesthatfurtherprovedthesocialscalarnatureoftheUniverse.

Sowiththemodernage,pastthesimplerageofgeometricmaths,anewlanguageofsocialthoughtandscale,numbersandcalculusenthronedmathematicsasthequeenofalllanguages.

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Analysis reflects the5Dimotionsof theUniverse in itsmathematicalmirror - its5changes=timespaceevents -actingonsimplexoperands thatstudychangewithinasingleplane.Assuchanalysis it studieschange,both inmultipleelementstogetherinasingleplane(ODEs)andmultiplePlanes(PDEs).

AslanguagesaremirrorsofthefractalUniversethatfollowitssamelaws,weshallstudyAnalysisaswedowithany other 'species' of reality first showing they are mirrors of its 5 Dimotions, whose 'syntax' is built on thepentalogicelementsofallDustofspacetime(¬∆@ST).Tomakeeasieritsstudythough,weshalluseasequential'humind's'expositionbuildingitsgrowing'informativecomplexity'throughits3agesofevolution.

themirrorof¬algebra:scalaranalysisandtemporaldimotions.

Thispaperisdedicatedto¬Algebra,thescienceof‘social,scalarnumbers’thatgrouptogetherequal‘T.œs’andexpress its 5 dimotions of existence through ‘time operands’ and its equations; extended in its dilated viewthroughasimultaneoussuperorganismincomplexorganicstructuresofscalarspacecalledsets,wholesofT.ŒSthatperformworldcyclesintimecalledgrouptransformations.

In its essence Algebra is the study of numbers. And numbers are social groups that form scales of parts thatbecome wholes and hence can be ordered (with the exception of complex numbers which are ST-representations),suchasx<y<z->x<z,whichrespondsimmediatelytothescalarmetricof5Danditssocialscales.It does happen then that dependingonwhich scale of numbersweuse,weestablish different 5D ‘families ofworlds’,beingthedecametricscalethemostimportantofthem,bothinthetetraktysconfiguration(3x3+¡)andthenextlogarithmicscale:10±10.

Theword¬AlgebracomesfromArabicالجبر(al-jabrlit."thereunionofbrokenparts"),thisfactaloneindicatesthecapacityofmindmirrorstounderstandthroughtheirsubconsciousgrammardeepintuitionsinthemeaningoftheexistentialgame.

Since¬Algebraanditsmainbranch,Analysis,mustbeconsideredthestudywiththemathematicalmirrorofthemost'complex'systemsofnature,‘potential’superorganisms,throughitsmultiplepartsinspace(grouptheory),Planes(numericalequations)ofspaceandÐimotionsoftime=change(operandsandanalysis).

¬Algebraisthusthehighestlevelinwhichmathematicsmirrorsthe3mainelements,∆ST,ofthefractalUniverseinasimpleplane.Towhich itaddswith@-framesofreferenceand inverseoperandsthe limitsofentropyandself-centeredelementsthatcompleteaTime-spaceorganism.

Butthegreatdiscoveryof¬Algebraisanalysis–theF(x+h)–F(x)/hfunctionof‘change’,whichisamuchwiderconcept of change=motion than the locomotion function of change recognized in physics, as it includes theanalysisofallpossibleformsofchanges,hencepotentiallyofalldimotions,includingthoseofbiology(socialandtopologicevolutionandDarwinianentropy),whicharethemostimportantofNature.

Thusmathematicsisaboveentropicphysicsinexperimentaltruth,asitsapplicationsarewider.

Thiswould be one brief description thatmerges classic ¬Algebra, and Existential ¬Ælgebra – the study of theworldcycleofexistenceofasuperorganismoftimespacewiththe‘formalism’ofÐST(Generationalspace-time),whichisamoreconcisemirrorofthemainpropertiesofthosespace-timebeings.

Soasusualnottohavetorepeatourselvesadeternal,werecommendtoreadtheintroductoryarticleonthefifthdimension,andinthiscase,theIBookonmathematics,tounderstandwhatthe5DimensionalfractalUniverseanits fundamental elements, Space, Time, ∆-Planes and linguistic @-minds, the four components of allSupœrganismsofTime-spaceare.

Andhowmathematicsasamirrorlanguageperceivesthoseelementswithitsscalarnumbers,geometricpointsandtemporaloperands,whicharethemainelementsof¬Algebra.

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It is indeed absolutely necessary to have the basic understanding of a Universe made of space, time, Planes,linguisticmindsandentropiclimitstoproperlyrealizethatmathematicsmerelytranslatesintoitspostulatesthepropertiesofthose5elementsofreality,becomingtheessentialexperimentallanguageoftheUniverse.

¬Algebraequations:stœpsanda(nti)symmetriesofspace=timesupœrganisms.

¬AlgebrathenbecomesthroughequationsthatmimictheS<=>TœpsofrealitytheperspectiveofmathematicsasamirrorofthedifferentstepsandstopsofeachÐimotionofaT.œ,withemphasisinitssimultaneousgatheringasa'wholeevent,unitedinsynchronicitybythe=symbol,creatinginthismannerfeed-backequationsbetweenthespaceandtimestateofthe5ÐimotionsoftheUniverse.

Thebeautyof¬Algebrabecomesthenthecapacitytowrite'chains'ofstœpsinsuchamannerthatwecanfollowsequentially theevents thatconformtheProgramofexistenceofaT.œas itgoesonthrough Indeed,becauseDimotionsformcomplexchainsmostoftenofstop-spaceandmotion-timesteps(ab.stœp,whichisthesumofastop:perception-step=motiondualchain),the=symbolisessentialtoamirrorofreality'sevents.

Becausealgebraisimmensewewontbeexhaustive.OuraimisjustprovingtheastoundingparallelismofAlgebraand∆STlawsandleaveafirstseedforfuture‘pros’toenjoyasIhave,theconnectionsbetweenboth.

Thedifferentdeterminismofthe3ages:Axiomatic,linealagevs.Kaleidoscopicuncertainfutures.

BecausetheUniverseispentalogic,madeof‘space’, ‘time’, ‘scalarplanes’, ‘languages-minds’andentropiclimits,when@mind’slanguageappearsitstudiesexactlythose4elements,space,time,scalesandentropiclimits,withitsmirror systems. And indeed, ‘analysis’ was born of the need to understand those 4 elements in problems ofNature,NOLONGERinlinealterms,asthe‘firstageofanysystem’,butin‘curved’terms.

SowhattheGreekshaveresolvedforthe$Tageofmathematics(linealage),cAnalysiswillsolveforthesecondageofcurvedgeometriesthroughtheuseofanalysis.

Thisisaprocessproperofthe3agesofanySpace-timesystem.Thefirstageislineal,withabsolutesimpletruthsthatthemindasadictator‘child’considersdogma.SoEucliddidhisaxiomaticmethodonsimplelines

MonologicinMathematics.Thefirstageoflineal,deterministicGreekstillgeometryandaxiomaticproofs.

Once we understand the general fact that all languages have a first lineal age, deterministic, as a line cannotchangedirectionorelsewillstopbeingaline,whileacurvecaneasilychangecurvature,evenchangedirectioninsinusoidalwavesandstillbeacurve;solinesaredeterministicone-singlefuturetothem,whilecurvesareableatanypointtochoose3pathsofless,moreorequalcurvature;wecanunderstandsomefactsofGreekGeometry:

-Itissimple,lineal,deterministicandhenceitcanbeapproachedwithapurelyaxiomaticmethod,asthereisnoambivalenceonresults,constructingaself-containedmethodofproofdepartingtrulyfromasimplesetofaxioms–apointhasnobreath,etc.

Buttheaxiomaticmethodofproofisnolongervalidwhenweconsidersystemsthatdohavealsoacertain‘timecurvature’,andevenmoreso,whenweapproachoperandsandmathematical systemsthatprobetheplanesofthefifthdimension(calculus, limits).Thenthefuturehasdifferentsolutions,andsomeareparadoxes,andsowecannotprovewiththesimpleA->BlinealcausalityanddeterministicoflinealGreekGeometry,everythingthathastodowith cyclical, curvedgeometries, calculusof finitesimals (limits), andbecausehumind’s reject the conceptthat absolute truthsonlyexist in absolutely simple lineal systems, asMathematics evolved into complex curvedgeometriesandscales,itsproofsweremoreandmoreimprecise,orblatantlyfalse(0doesnotexist,asalllimitsto0orinfinityhaveanentropiclimitinaquantaorthedissolutionofinformation;thereallineisinadifferentplaneofspace-timethantheNaturalnumbers;whichonly‘becomecontinuous’ifweweretoaccessanevenlargerscale;etc.)

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Itisforthatreasonweshallnotusefurthertheaxiomaticmethodandthefirstorderlogicandthe‘selfie’processof ‘inventing’ self-evident axioms of truth, which are always based in the mind simplification as if it were theabsolutetruth–thatisintheegoparadoxofnaïverealismthatconfusesthemindwiththewhole,butcompareanycomplexlevelofmathematicstotheexperimentallawsofSpace-timefromwheretheydepart.

Indeed,theegoparadoxistheessenceofalltheerrorsofallthesciencesbasedinmentalpostulatesthattakeourreducedviewoftheUniverse–theworld–withthewholeofreality.Andourænthropicviewsasdogmas.

FromthatviewarisestheerrorofEuclideanpointswithnobreathandtheabsolute0insteadoffractalpointswithvolumesandfinitesimalszeroths,0’,absolute∞insteadofentropicinfinitiesthatbecomeuncertainpastitslimitof perception or use, ∝, normally beyond the decametric scales of the Universe (Cantor’s infinities, infinitedecimalsintranscendentalnumbers,etc.).

Fromthesamereductionismof themindtakenasabsolutearisetheerrorofasinglecontinuous lineandsingletimearrowand single spacetime continuum,when in reality, onlyNatural numbers exist in this planeof space-time,sotheplaneisdiscontinuous,andallotherfamiliesofnumbersfillthegaps,allowingthefractalstructureoftheUniverse.AsthescalesofspaceofthefifthdimensionareinthediscontinuitiesofourNaturalnumberworld.Soarethefastertimeclocksofotherscales;hencewithmore‘cycles’oftimesandmoreinformation,inarelativemoreevolvedfuture in its logic form(butcreated inarelative linealpastas theycomebefore largerscalesthatsustainoverthem,themesthosetoocomplexforthispaper,whichrequiredeepthoughtontimeparadoxes,hencestudiedinourpapersonilogic).

Insightsonthecomplexityofrealityasopposedtomathematicalsimplicityareimportantfor2reasons:

Firstfromtheacceptanceofsuchsimplifyingerrorsasabsolutetruthsarisetheparadoxesofmathematicalphysicsand its incapacity to understand the fractal Universe, its hypothesis of a single spacetime continuum, denial offasterthancspeedsbeyondthegalacticorganism,etc.

Yet in second thoughts, all what is required once the errors are pointed out is to use the simplification,whichextractsstillvalidlawsforthe‘whole’toenlightenthefractalUniverse.I.e.thehypothesisofthecontinuumisnottruthforasinglescale,butifwedoacceptthatthe∑¡sumofallthescalesoftheUniversedofillalltheholesoftherealline,thenthehypothesisistruthandindicatesthattheUniversehasavacuumhorror.Thatisthesumofall theplanesof5D is filled–reasonwhy indeedthere isnoabsolute0’-void;simplyspeaking itwillbe filledbysmallerscalarparticlesandwaves.

Intermsof ‘scales’all thismeansthat insmall, ‘fast’,predictableA->Bstepsreality is linealbutwhenwegathermultiplesteps,alllinesbecomecurves.Insmallintervalsmotionmightbecontinuousbutassoonaswegobeyondasimplestep,thereisastepandstop,lengthandhighmotion.

Andsowecanalsoreducecurvesinverselytostepsandstopsoflengthandheight,orlinealstairs(whichwouldbethemethodofCalculus,to‘calculate’thetangentofthecurve.Doesthenthecurveexist?Oronlythestepsandstopsoflinealandheightmotionandinformation?Itisrelativetoourperception.InthelargescalethezigzagofBrownianmovementorelectronsbecomeacontinuouscurve.Inthesmallerscalethestepsmightbehighlylinealanddeterministicbutinthelargescaletheybecomecurvedandprobabilistic.

Inthatregard,theuseofÐSTlawstoreferencethelawsofmathematics,beyondthepretensionofabsolutetruthof the axiomatic method, is completely necessary, without using a complex ‘pentalogic’ point of view, andacceptingtheparadoxicallimitsofrealityanditsscalesasweshallconstantlydohere.

Thefalsehypothesisofthecontinuum.It’sthemind,whichcreatestheillusion?ofcontinuity.

The hypothesis of the continuum is false. First to define it in modern times, Leibniz thought a continuum ofnumbers was one that could be filled in any interval with asmany numbers as wewished – when that is theobviousreasonofdiscontinuity–asthereisaholetofitmorenumbers.DowntheroadthisendedinDedekind’s

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cut. The hypothesis of infinity is alsowrong. As infinity dissipates into entropywith uncertain information. Thehypothesisofequalityisalsowrong,astwofractalpointswithvolumehideinformationwecannotcompare,andentangled to thewhole they are dissimilar since a being is part of that whole universe inwhich its position isdifferent.All thosehypotheseshowever reflect themind’s simplificationof realityandso thecanbeconsideredright inmental space,ofwhichoneof them, ishumind’smathematics. It’s themind that creates the illusionofcontinuity,whichplacesahugemetaphysicalquestion.Itisthemindtheonlycontinuousforinexistence?Ifsoitisthemindimmortalinitstravelthroughplanesofthefifthdimension,asreligiouspeoplewishfullythink?

Because huminds obsessively try to project their local mind-space truths as universal ones, we have madeconcessions to their languageweshallnowreconsider.First, there isNOnumbers forDimensionalmotions,butFunctions of holographic, dual, mental Space=form and Time=motion, SS<St<ST<Ts<TT. So we shall escapenumbering them. Next, all of them are diffeomorphic, local, in a fractal Universe, though self-similar in itsproperties.Wewillkeeptheconceptofthe‘5thdimension’asthesumofallthoseplanesofspace-timeandlocaldimotions,sothe‘5thdimension’meansall,thewholeuniverse.

Next, time and space are very different but obvious concepts to thoseusedby huminds, includingour ‘gods ofscience’ or ‘seers of time’ (definition of God by Augustine), the ‘physicists’ (from Newton, through Galileo, toEinstein).Physicistsdohavemeritsofmeasure,precision,machineassemblingandmathematicalanalysis,buttheirphilosophiesofscienceand‘grandtheories’ofreality,sorryfolks,‘suck’.

Somyadvicetothereader ifany istodroptheir idolsandstartafresh. ‘Timeiswhataclockmeasures’wasthebestphilosophicalanswerofMr.Einstein.Wegoarelativeinfinite,∝,furtherfromthat.Sogoesforthepretensionof ‘axiomatic mathematicians’, that impose its mental space of inflationary mathematics to reality and get allworkedupwhen some fellow findsoutwith computers a proof of the Fermat’s theorem in 1000pages. ‘If youcannotexplainitsimple,thenyoudon’tunderstand’,‘theUniverseissimpleandnotmalicious’(Einstein,whichIdeeplyadmireforhisquips:)Simplicityisgenius(Leonardo).Weareconcernedwiththeunderlyingsimplicityfromwhereallthosecomplextruthsarise.

Butifwedepartfrompostulatesandaxiomsbasedinthedistortionsorwishfulthinkingoregocyofmantobuildupanentirenewworldunrelatedtoexperienceandrealityitdoesn’tmatterhowmuchcomplexityweadd–itisall ‘soma todisguise reality’.And this is inmathematics, thehypothesisof continuum, in religion,allAbrahamiccults,becauseobviouslytheyallcomefromanoriginalfalsepremise–thatanassbreederoftheBronzeAgewhosawaG.Bush,Godbushburninghaddecodeditssoundsasthewordofthecreator.

Mankindhasbuiltonthatobviousfalsedeliriumtremens‘hypothesis’,2/3rdsofitsworldculture,includingmostofwesternart,mostofwesterngenocides,theentirebiblicalcapitalistgo(l)dculturethat iskillingtheEarthandvariousjihadsandinquisitionsofthoughtstillextant.Couldn’tcarelessaboutthebeautyofMichelangeloandthesufferingoftheHebrewsandthemassacresofpeople.Warsdonotjustifyarmies.Mankindisasinglespecies.Soagaintheentirebuiltupof‘nationalistidentities’isotherabsurditycomingfromthepremisefirstbroughtaboutbyGermanichordesofaristocraticwarriorsthatthelandofEurope,aculture,hadtobebrokenforswordmasterstoenjoyforfreetax-farmingofitspeople.Andsoon.

Mathematicsingeometrycomesfromafalsepostulate:thatpointshavenoparts,hence0sexist,andlinesarenotwavesornetworksandplanesarenottopologicalorganismsdefinedbyofsuchlines-networks,andofcourse,thehypothesisofthecontinuumwhichdoesNOTallowthosenetworkstopenetratebetweentissuesandthescalesofthe5th dimension tobuild complex relationshipsbetweenparts andwholes compressed in a single ‘real plane’.Mosthuminderrorsaredueto‘æntrhopic:entropic+antrhopic’treatedinapaperofitsown,(monologicman).I.e.incosmologytheentropicbig-bangprojectstheentropicprofessionofphysicists,asmakerofweapons, intothelinealV=HoDequationofentropicvacuumspace,whichexpandsbetweengalaxiesasgaseousstatesdoinmatter,butforgetstheimplosivegravitationofdarkmatterandgalaxiesandtheliquidS=Tbalancedandsolid,SSstates.Sowith1/3rdofrealityvoila! theuniverse isabig-bang.Soænthropictheoriesworkbyhidingallwhatthemind

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doesnotneedtomaketruthafalseænthropictheoryisthetrick.Thendressingitwithcomplexitytodisguisewhatiswrong–theinitialpostulate,makestheegocyparadoxreal.ThelaymanwillthinkifitissocomplexandIdon’tunderstanditmustberight.This isdoneadnauseam,withbig-bangtheoriesbasedinafalsepremise,axiomaticmethodsinmathematics.Evenwiththetax-farmingofmankindbyprivatebankerswhoissuemoneyinmonopolythrough‘derivatives’andnationalbankswhichareinfactprivateenterprises.InU$theFederalReserveinsteadofjust printing digital numbers as the body prints red cells to transport its blood-money in universal salary to allcitizens-cells, has a very complex system of ‘emitting’money as ‘debt’, bought by the treasury, etc. etc. calledSOMA,exactlythenamethepriestsofAryanVedasgavetoitssoldiers-slavestothrowthemintodeath.Sowhensomeone like Lincoln tried to simplifymoneywithout debt as superorganisms simplify the first order-languages(hormonesaresimplemolecules, like the lettersof thealphabet,oxygen isassimpleasmoneyshouldbe), theykilledhim.Endofhistory.Mankind’sthenfindsitallverycomplexandproudlythinkshismindissoaboveheavensandearth.WhenheisjustaninflationarymindoutoftunewiththeextremelybeautifulS=TbalancedUniverse.

Sothisisaruleofthumbtojudgehumind’stheoriesofreality:‘Allhumindtheoriesofrealitybeyondthesimplestdetails and its causalities, areprojectionsof theegocyparadoxof themind’s finitesimalmappingof reality thathumindsthinktobetheentireUniverse,basedinpostulateswithoutproof’.

ThissaidMathematicsdividesphenomenaintotwobroadclasses,weshallcalldiscreteorspatial,informativeandcontinuous,or temporalhistorically corresponding to theearlierdivisionbetweenarithmeticandgeometry.Buttheygotitinverselyandnowwesettherecordstraight:continuityisjustamentalphenomenathat‘eliminates’forthesakeofsimplicityandeconomicity to fit in thebrain, the ‘darkspaces’betweendiscontinuous ‘1s’. Discretesystems can be subdivided only so far, and they can be described in terms of whole numbers 0, 1, 2, 3...Continuous systems can be subdivided indefinitely, and their description requires the real numbers, numbersrepresentedbydecimalexpansionssuchas3.14159…,possiblygoingonforever.Understandingthetruenatureofsuchinfinitedecimalsliesattheheartofanalysis.

Andyetlackingtheproper∆STtheoryitisyetnotunderstood.

Thedistinctionbetweencontinuousmathematicsanddiscretemathematics isonebetweensingle, synchronous,continuousspacewithlessinformation,andthemorecomplex,richininformationperceptionofrealityclosertoits nature, in termsof ‘time cycles, of fractal points; of networks of space-time entities’,whichwill show to bealwaysdiscreteinitsdetail,eitherbecauseitwillhaveboundariesinspace,oritwillbeaseriesoftimecyclesandfrequencies,perceivedonlywhenthetimecycleis‘completed’,andhencewillshowdiscontinuitiesontime.

ThusthedualitiesofSTononeside,andthe‘Galileanparadox’ofthemind’sthatreducesmotionanddetailtostillmentalviews,limitsofperceptionofinformation.Anditlaysattheheartoftheessentialphilosophicalquestion:itis theUniversediscreteorcontinuous inspaceandtime.Both,butalwaysdiscretewhen indetailduetospatialboundaries,andthemeasureoftimecyclesinthepointsofrepetitionofits‘frequency’.

Soultimatelywefaceamentalissueofmathematicalmodeling:the‘mindisartNOTexactscience’(aspureexactsciencedoesnotexist,allisartoflinguisticperception)ofrepresentingfeaturesofthenaturalworldinareducedmental,mathematicalform.

Theuniversedoesnotcontainorconsistofactualmathematicalobjects,butalanguagecanmodelallaspectsoftheuniverse.Soallresemblesmathematicalconcepts.

Congruence,similarity&indifferencevs.identity,whichdoesNotexist.

Abstractionsimplifiesfirstandthenforgetsforthesakeofpraxisthemeaningofsimplification.Thisworksspeciallyin the operandof equality,we substitute by ≈ and ≤≥ andówhichmeans similarity, undistinguishable as theinternalpartsofthebeingareneverequal.Weuseawordingof¡nglish,¡ndifferent–asitis‘easier’towrite,andaddsthesubjectiveviewoftheperceiverwhoisindifferenttoallthepropertiesthatmakespecialthebeingfromitsinnerp.o.v.Twothingsthusareindifferentinsteadofequal,ontheeyesofthebeholder.Butagain,asthosepapers

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aremostlyonphilosophyandtheoryoverpraxis,itmatterstobeawareweworkwith¡ndifferencesnotidentitiesasinE(t)=m(s)c2.Allthoseconceptsshouldbegroupedwithinthe4thpostulateof¬Econgruence,asvariouscasesof¡ndifference,whoserigorousdefinitionescapethislatter-daytextsbutinthefutureshouldbeclassifiedalongthepossiblewaysofinteractionbetweentwo¡ndifferentforms.

Another hypothesis that must be understood in far more complex terms is the identity hypothesis. As in theentangledUniverse, not two forms are identical, as theywill even in the case of identity, become different byposition and different entanglementwith other forms, but they can be treated and are treated as identical bymindsexisting ina larger¡+xplane.Specifically identityappearsbeyondthe¡-3scale inwhichamindobtains itspixelsofinformation,asallisabsolutelyrelativetothemindthatperceivesit,whichperformsitsactionsinrelativeintervalsofplanes-¡0+1forsocialevolutiondimotions,¡º-1forreproduction,etc.

I.e.,thenumbertwodoesnotexistasaphysicalobject,butitdoesdescribeanimportantfeatureofsuchthingsashumantwinsandbinarystars;andsowecanextractbytheternarymethod,3sub-conceptsofit:

2meansthefirst∆-scaleofgrowthof1beinginto2,by:

S-imilarity and S-imultaneity in space time, (ab. §); Сsomorphism =self-similarity in ∆-scale as perceived by alinguisticobserver,@,whichwilldeembothbeings‘identical’.

Whereasidentitymeansthatan@-bserverwilldeemthebeing∆st≈St,(Sim,IsoandEq).Soidentityisthemaximalperfectionofanumber,foraperceiver,evenifultimately:

‘Not 2 beings are identical for the Universe, but can be identical for the observer’… an intuitive truth,whosepedanticproofisofcourseofnoimportance(:wedonotfollowtheaxiomaticmethodofabsolutemindshere):,butitisattheheartofwhyrealityisnotcollapsedintothenothingnessofabig-bangpoint.

Thus those 3± elements of the ∆@ST coincide a social number can be used whose intrinsic properties defineconceptually ‘S-imultaneity, Ti-somorphism’ and ∆-equality or equivalence (ab. Eq) in size, which becomes an@identityforthemind.Thenanumberisborn.

Inthis ‘infinitorum’ofUniversalthoughts,whichbringalwaysnewdepthsassoonasweobserve itwithan∆•sttrained mind, there are differences between S-imilarity and Simultaneity to define in space an ‘identity’ and‘equality’andequivalence,treatedelsewhere.

It is then clear that anumberbeinga sumofpoints, encodesmore information in a synopticwayabout theT-informative nature of the ‘social group’ than an array of points,which unlike a number tells us less about the‘informative identity of the inner parts of the being’, but provides usmore spatial knowledgeabout the relativepositioninspaceofthemembersofanumber-group.

Andthis isobvious,whenwereturntotheoriginofgeometryandconsideranage inwhichbothconceptswereintermingledso‘pointswerenumbers’anddisplayedgeometricalproperties:

Numbersaspoints,showingalsotheinternalgeometricnature,usedinearliermathematicstoextractthe‘time-¬Algebraic’,‘Analytical-social’andS-patial-geometricalpropertiesfromthem.

The closureof the systemsofnumbers thus grows from reflectingmerely spacepopulations (natural numbers),into 5D numbers, reflecting 'partitions' of social groups (represented by those natural numbers), with Egyptianratio-nals; expanded further with the realisation that certain ratios did apply to 'scaling' in the fifth dimensionwithoutlimit(asinthepiratioof$T-entropiclinesintocyclicaltimeO-cycles).

Alas,thingsgotinterestinghere,butasthehomunculusdidnotunderstand,thediscovererofpi,legendhas.wasmurdered by Pythagoras, the first 'religiousmathematician' because it found that piwas not perfect.It took toPoincare2500years latter to find that this is anawesome formofperfectionbecause itmeansamind-pointofsphericalform,withequaldistancetoalltherealitiesitreflectscanshrinkwithnolimit(Poincareconjecture)

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Therearespatial,naturalnumbers,5Drationalandtranscendentalnumbers,whocrossthrough5DPlaneswithouttearing according to Poincare’s postulate. But this is NOT enough, because we need negative and lateral (notimaginary)numbers,andthosearetemporalnumbers,numberswhodescribeprocesses in timenot inspace,asnegativenumbersdo,as they representmerely the inversearrowof time, so if youmakeapositivenumberanarrowofmotion-entropy, or relativepast loss of information, thenegativenumberwill be the arrowof future-information, and the lateral i-number, will be the bidimensional sum of both the real and the negative, torepresentthepresentspace-time.∆±i:i-ratio-nalortranscendentalnumbers;∆±1:ratio-nalnumbers.

Soyoudohaveaclosureofallnumbers,basedintheelementsofspace-time.

Hence,whenyouapplythemtomathematicalphysics,whichisthestudyofthesimplestformsofspace-time,youdohaveabetterfocusedmirror.

EFFICIENCYOFMATHEMATICS.SURVIVALISMOREIMPORTANTTHANKNOWLEDGE.

Among all mirrors=languages of the Universe, mathematics - topology and social numbers - dominate mindsbecause it is the most synoptic, hence able to fit in any scale, less distorted, hence able to perform the 5Dimotions=actionsofsurvivalbetter,initsexpressionofthelawsoffractalspaceandcyclic,pentalogictime.

SoasUpanishadsaid,'thelanguagesofGodare∞;butsomemattermorethanothers.Topologyshouldbetheinternallanguageofatomsandhenceself-similargalaxiesofany10nscale;thatisthelanguageofelectroniceyes(lineal, Euclidean) and quarks, nucleons and black holes (Non-Euclidean curved), and all its social organisms,whicharemostformsofrealityemergewithmathematicalproperties.ThisissaidherefortheobviousreasonweshallNOTevolveBooleanAlgebraandAIevenifwecouldformoralreasons.SurvivalisaboveknowledgeandtheexpansionofdigitalAIrobotswithcomplexmathematicalmindsimperilsthesurvivalofhumanity.TheySHOULDbeforbidden.

Inthegraph,languagesselectspeciesthatsurviveintothefuture.Inevolutionarytermslanguagesofinformationmattermore thanbodiesofenergyandmaths is thebest synoptic, focused languagemirror.Problem is in thisplanetmachinesspeakbettermathsthanhumindssotheeco(nomic)systemselectsthem.

RECAP.5Dphilosophyofmathematicsexplainsitswhysandwidespreadapplication,basedintwoconcepts:

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1.Mathematicsisthebestsynoptic,focusedlanguageweknowofthe¬∆@stpropertiesofreality.

2.Hencethosemindsthatspeakitbettersurvivesandaretheselectedlongestlivingspeciesofreality,electronswithsimplexEuclideanmathandquarkswithComplexcurvedtopology&itsatomicensembles.

MathematicsasthebestmirrorbrainoftheUniverse,allpervadingin‘Atomicsystems’.

Mathematics is a mirror language that studies 5D fractal points of space; its time operands, social numbers;minds=framesof referenceandentropic limits,whichentangled togetherby the lawsofpentalogic formthe5elements of ¬∆@st , dust of space-time, the substance of which we are all made. As such is the mostexperimentalofallstiences.Mathematicsthereforeexistsasallother languagestoprovidea ‘mind’withastillimageofreality,whichmirrorthelawsofspace-time,anditsdynamicsuperorganisms,expressingitsdimensionalgrowththroughspaceandtime. Itsunits reflect thatdimensionalgrowth inspacethroughpoints thatbecomewaves-lines that become network-organisms.While algebra expresses that growth and its inverse devolutionthrough sums/subtractions of ‘herds’ of points in a single plane (±); reproductive ‘invaginations’ into a lowerplane(productasameasureoftheaxonsthatconnecttwopointsthroughitsparts),orhigherone(powerlaws);while themoresophisticated∫∂operandsstudy the ‘details’of change inanyof the5Dimotionsofexistence,frommerelocomotions,tosocialgrowth,reproductivecurvesanditsinverseentropicdissolutions.

We adapt geometry to the scalar fractal Universe with the concept of a fractal point, and the logic ofmathematicstothepentalogicof5Dimotions,tounderstandthemultiplevariationsofmathsasaninflationarylanguage, and then we make it a experimental stience referring its main elements, structures to the largerlanguageof 5D showingmathematical equationsandpostulates reflect lawsof the5DUniverse in its synopticlanguage.Sincespaceandtimearemorecomplexthan‘huminds’ourlanguageofspatialthought,geometryandtemporal logic must be upgraded to formalize properly the description of ‘superorganisms’ perceivedgeometricallyinsimultaneousspaceandlogicallyinsequentialtime.MostgeometryinscienceisEuclidean(save5thpostulate)andstatic;whileLogicislineal,Aristotelian,A->B.Sowe evolve both into a non-AE-=i-logic topology of space, next letter, i alsomeaning information and the

fractalscalesofrealityanda¬Æxistentialalgebraoftimetoformalizethe‘stœps’,stopsinspaceandmotionsintimethatmovetherelativefuturesofeachsystem,madeofvitalscalesofspacethatlastafinitetimeduration,whicharegeneratedbythebio-topo-logicpropertiesofitsvitalspaceandcyclictime.

TheLawofClosurefortheUniverse,establishedbyexistentialalgebraissimple.Allthelawsofallstiencesderiveofthe laws of scalar space-time, ∆St; and its 3 units: fractal points of space, social, scalar numbers and cyclicaldimotionsoftime.SothelawsofMathematicsasamirrorlanguagederivedof∆stLawswritesasfollows:

Thelawsofgeometryderiveofthelawsoffractalpointsofspace.

ThelawsofAlgebraderiveofthelawsofExistentialalgebra,itsScalesandDimotions.Andwecaneasilytranslatethe3elementsofthefractalgenerator,toboth,scalar,∆stalgebras(+,x,a-x)andBoolean,mental,¬@algebras,(V,∧,¬)asfollows:

Reproduction,≈,and,+v;Evolution,>,∧,x;Entropy,<,¬,

ThelawsofCalculusasthemostsophisticatedentangledoperandofalgebraalsoderiveofthelawsofscalesanddimotions.

ThesurvivalconsequencesoftheevolutionofAI-gebra(BooleanAlgebrasthatmimicthelawsofexistentialalgebraincomputerbrains)isobviouswithinthelawsoftheorganicUniverse.Oncethetranslationisproperlydone,evenif it is merely done in terms of simple modulo-2 programs of military robots, as the Universe is sentient andelectronsare theunitsof life, thosemachineswill eliminatehumans. It is for that reason I abandoned the fieldfrompublicscrutiny,andthelastchaptersofthispaperwon’tfullydevelopexistentialalgebra,asIdotryalwaystofollowthemandateofexistencethatmakesallspeciestoconserveitstime.HoweverhumansdoNOT,andthatisa

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themetreatedinourpapersonhistory.Itseemstheearthhasprogrammedustodestroylifeandcreatethefirstsimplest forms of existential algebra in Computers. And it doesn’t matter how many warnings mankind in itssurvival temporal language of verbal thought, through its highest minds have warned the species against theeviL=antilivefruitsofthetreeofscience.Theywilldoit.

Letusthenoncewehavecompleteourreviewof thebasic themesofÐSTandotherdisciplinesofmathematicsneededtounderstandAlgebra,withanextractfromourpaperonNon-Euclideangeometryandvitaltopologyandframesofreferenceofanalyticgeometry,treatedextensivelyintheotherpaper.

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II.ALGEBRAThe3agesof¬Algebraasreflectionof3Planesofgrowingcomplexity:numbers,functionsandfunctionals.

Youth:∆-1:Arithmetic,theGreekAge,nowmostlystudiedinnumbertheory.

Maturity: ∆: Equations extend each 'letter' to a range of numbers, creating a new ∆-scale of generality formathematicsasawhole;mostlystudiedby@nalyticgeometry(S@)and¬Algebraicanalysis(∆T).

OldAge:∆+1:¬Algebragrowsintoanewscaleofcomplexitymakingfunctionsoffunctions,intime(Functionals)orinspace(Groups).

Asthebestmirrorofmathematics,¬Algebrahasalso3'scalarsizes',thatoftheindividualnumber/point,thesocialpolynomial/calculus equation and the structural group or set,which searches for the full completenessof all thepotentialvariationsofacompoundstructuremadeoftimemotionsandspatialpointsanchoringthosetimemotionsintosimultaneousspatial,organic,dynamicalsystemsrepresentedbysetsandgroups.

Thoughgeneralityincreasesfictionalvariationsinany3rdagedetachedfromreality,asthereisnotaclearpictureofthe underlying space and time laws of the scalar Universe that would ‘tighten’ our admission of groups and settheoryasthe‘pest’(Weyl)inwhichtoencaseanythingthatmoves.

¬AlgebraagesareinaternaryscalarsymmetrywiththePlanesoftheuniverse.I.e.,ifwedecomposeitinitsparts,an¬Algebraicstatementhas3levelsofcomplexityasalwaysintheternaryUniverse:

-SequentialnumbersaretheminimalscalarelementofafractalT.Œ,whosepropertiesnumbertheoryexplores.

-Functions and its operands,whichexpressapartialequationof the ternary fractalgenerator thatdefinesaT.œ,most often of the space and time states, connected by thefunction and operands, overwhelmingly the basicequationofmathematical physics (as in E(s)=M(t)c². Sonumbers are the 'smallerbrokenpart' of ¬Algebra,whichstudies'socialPlanes'betweentwoplanesofreality,while¬Algebraextendsmuchfartherthisingenerality,studyingtheinternaldynamicstructuresofwholeentangledsuperorganisms.Soitsfullunderstandingisthatofallthepartsofawholetogetherandwithinthewhole,itsdynamicalsymmetriesbetweenspaceandtimestates:∆S=Tmadeofdifferentgatheringsofnumbersinequations.

ðNº≈§@:,meansthenthattimenumbersareputinrelationshipwithspatialpoints,andtimefunctionswithspacefunctionsthroughtheoperandsof¬Algebra.

-The structure, which is the highest comprehensive analysis of full blocks of time-space laws; and specificallyenclosesallitspossiblevariations;todaydevelopedthrough3∆STelements:

- SóT-Groups as a 'receptacle' of all the elements of the being, with a focus on its spatialsymmetries=motions=changes that do not alter the whole nature of the being, similar to the concept of a fullworldcycleoftime;connectedtoS@-topology.

-∆-Sœts,orientedtounderstandthepartsandwholesanditsrelationshipsandasimilarexpansionwiththeconceptofthefunctional(functionoffunctions)withafocusonstudying‘simultaneously’several∆§cales.

¬Algebraasaconstructofthehumind(@-element)requiressomecorrectionsduetothedistortionsofego-centeredhumindsciences,whicharedealtbetterinphilosophyofmathematics(¬Algebraicfictions).

Algebra upgraded in its foundations, becomes ¬Algebra, since the logic of relationships between points in space,scalarnumbersandtimeoperands,donotfollowtheAristotelianlogicinwhich¬Algebraistodayfounded.

Thus,5D interpretsandcasts themorecomplex structuresof¬Algebra in termsof3±¡ ternary symmetriesof themorerealistformalismofthegeneratorequationanditspentalogic.Speciallyitsmostusefulones,setsandgroups,usedtodefinefull'blocks'ofphysicalsystemsintimeandspace.

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1STOF¬ALGEBRA:SOCIALNUMBERS:'ARITHMETIC'.

SingleƤplaneandternarydimensionsandS=Tequation-symmetries.

In itsoriginal form,¬Algebradealtwithmathematicaloperationsonnumbersconsidered froma formalpointofview, inabstraction fromgivenconcretenumbers/cases. Itcalculatedsocial, scalarnumbers in theirgrowthanddiminutioninherds(±);anditsgrowthanddiminutionin3spatialdimensions(X,Y)³.

Asall‘seeds/minds’startinastillage,motionwasnotclearlyunderstoodinmathematicalterms.

As such ¬Algebra could not go further because there are only 3 dimensions in a single stillspace-time plane. -Fermat'sgrandtheoremisaclearcaseofthedifficultytocreatebeyondsocialaddition=superposition,a 'higherdimension’,sinceasimplesocialsumin3D,suchasx³+y³≠z³,cannotexist.

Thisimpossible'simple'problemisthusaprofoundPROOFofentropicLIMITSwithin5Dworld-equivalenttotheclassicproblemofdoublingthecubeinbidimensionalgeometry,alsoknownastheDelianproblem:Giventheedgeofacube,theproblemrequirestheconstructionoftheedgeofasecondcubewhosevolumeisdoublethatofthefirst,usingonlythetoolsofacompassandstraightedge.Sinceoverwhelminglyabidimensionalholographic‘space-plane’(withpointsthatformnetworkswithvolume),willhavea‘3rddimotions’oftime-movement.

Aswith the related problems of squaring the circle and trisecting the angle, doubling the cube is known to beimpossiblewiththe2fundamentalelementsoftopologyO-compassand|-ruler.

Those problems also showed thatgeometry and ¬Algebra, points and numbers are ≈ similar but not =. SincecontinuityisaMayaofthe‘senses’,andrealityinsmallerPlanesarediscontinuous.

SosomecontinuousmentalgeometriesindiscontinuousnumbersdoNOTexist(√2,pi)assuchbutarescalarratiosandtrans-form-ativedimotions(O-π=3|).

Sowhilegeometrywasborninasingleplaneofspace-timenumbersandarithmeticsoonshowedthatthesocialevolutionofnumberstranscendedasingleplaneofspace-time.

Butthesubtlecontinuous-discontinuousvariationsofgeometryvs.arithmetic–wasneverunderstood,asEucliddefinedanabsoluteequalitywithhisdogmaticaxiomaticmethodthathasweightedsoheavilyonhumanthoughtanditsego-tripsofabsolutetruths…nowextendedto¬Algebraicoperations.

Fact isEqualitydoesnotexist -only similarityandmost= symbolsmustbesubstitutedby≈oró symbolsofadynamictransformation;soE=Mc2doesn’tmeanthatmassbecomesentropybutthatittransformsintoit inthemomentofdeath.

Inanycasethefirstageof¬Algebraendedwiththelimitingstudyofasinglespace-timein2or3dimensions(notas it shouldbe in5dimotions)and its sexagesimalordecametricPlanesofnumerical socialevolution,using thesimplestoperandsof±andx÷,andthefirstrationumbers,whichallowedtoprobein∆-Planes.

It added finally the concept of finitesimalswith probabilities, studying ‘events’ as parts ofwholes,whichwerestudiedwithstatistics,inanS=Tsymmetrylatterevolvedwiththeequivalent‘derivativevs.integralwhole’analysis

Recap.Weshallconsidermerelythemeaningofthemainmathematicaloperationswithintherestrictedworldof4-Dimensionalspacesandtimeswelivein.Andanalyzeinmoredetailsomeoftheparametersandfunctionsmostcommonlyfound inthestudyoftheGeneratorequation,whichconnectstheequationofspace-timecycleswiththedetailedmathematicalanalysisofthosecyclesbydifferentdisciplines.

PENTALOGICONFUNCTIONS.

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A function f is amathematical rule that assigns to a number x (inmost cases a time flow ofmotion) anothernumberf(x)–aspatialfunctionthatlimitsXtocertainvalues.Forexample,thefunction“square”assignstoeachnumberxitssquarex2.

SóT:ThecommonfunctionsthatariseinanalysisarealwaysanalysisofDimotions,definablebyformulas,whicharerelatedtotheS@and¬∆TdualityoftheUniverseandits5DimotionsorchangesinPlanes.

∆.SomostPolynomialsofthetype,f(x)=x2;thelogarithmicfunctionlog(x);&theexponentialfunctionexp(x)orex(wheree=2.71828…);andthesquarerootfunction√xhavealsoascalarcomponent.

@:Finally,Trigonometricfunctions,sin(x),cos(x),tan(x),defineanglesofperceptionofagiven@-frame.

Operationsasmirrorsof5Сmotions.

AT.œcangoupanddown,leftandright,growordiminish,d=evolvesocially,changeits@-pointofvieworslideinitsexistencefromthebodytothemindorlimbstate,andfinallytravelthroughthe∆-planesofthefifthdimensionemerginginalargerplaneordevolvingdowntoitspartsinthemomentofdeath.

ThisiswhattheUniverseoffractalpoints=T.œsisallaboutandwhat¬Algebraasamirrortriedtodescribe.

The growth and discovery of new operands mirrors the expansion of our perception of the dimotions of theUniversedepartingfromthehumindmirror.

The intelligentreaderwillhavenoticeeasily thatwedohave just theexactnumberofclassicoperandstomakethemcoincidewith theDimotionsof timespace. It follows thatmore important than 'variables'are to¬Algebra'operands',whoseencodeddimotionalmeaningrelatessystemsbymergingthemaccordingtocertainrulesof∆st'Generation, to get a 'futureorpresent' outcome truly giving¬Algebra thepower tomirror thea(nti)symmetricsuperorganismsoftheUniverseanditsworldcycles.

Thus the key connector of ¬Algebra with ÐST stience is a full understanding of the relative symbols ofequality=similarityandthedual¬Algebraicoperandsthatreflectthosedimotions,sine/cosine±,x/,∂∫,√xªaspartofthe¡logicgameofexist¡ence.Butallmirrorlanguagesarerelativelyunfocusedimagesofthegeneratorequationanditsoperands,sothecorrespondenceisNOTsodirect.Stillweobservethat:

1D: < (angle)& sine/cosine: The1stDimotionofperception is servednotbynumbersbutbyangles, expressedexternallybyasine,whichactsas the informative,heightparameterof theouterUniverse,and theapertureofperceptionofthebeing,maximalwithmaximal'height',thedimensionofinformation,(sin90º=1).

2,3,4,5Ðsocialevolutionandcomplementarymergingareservedbythesuminsimplerformsandtheproductanddivisionoperands incomplexsocialevolutionary structures,asmergingoften requires firstaproductat theloweraxonlevelandthenamitosisor'division'into2wholeswithallthenewparts,whichagaingivesdivisionaveryprecisemeaning.HenceweuseinExistential¬Ælgebra,∑forherdinginasingleplaneand∏fornetworking,astheproductprobesintothepartsofasystem.

Theproduct/divisionisresponsibleformostMergedDimotions're=producing'throughx÷operandsalowerscaleof 'connected axons' as the product is the first operand to probe a lower plane of existence, and the essentialoperandtomergeSandTstates,fromgendercouplestostopandstepintoanewmergedcreativeparameter,astœp,(i.e.massastopstateandvamovingstepmerge intomomentum,mv,aphysicalstœp,which inmodernphysicsisthemostimportantparameter).

Inthatregardthelessimportantformofmultiplicationisthemostoftenusedtodefineitinclassicmath(thesumof sums) as this not impliesmerging into something else butmerelywe staywith the same clone species andshouldberegardedasasumthathappensbetweenEQUALbeings.Inthis'lineofthought'wedohavealsothex²polynomialfunctionsanditsinverse√x.

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3,4D:TheAªandlog:4thand5thdimotionsarerepresentedinitsd=evolutionsbylog&exponentialoperands.

5D: √xª: social Planes: When social evolution is not transformative between planes but only a social herd, itemergesthroughmultiplemostlydecametric,3x3+0Planesorthe√xªoperandsbestsuitedtothatpurpose.Andhereagainwefindquitedifficultthecomprehensionamonghumindswholoveto'goonlythewayupwards'sotospeakofthe√operands,speciallywheninnegativemood:√-x,acompletelymysteriouselementof'mathematics'tothepointtheycallthemimaginarynumbers(:

4D.∫∂integralandderivativeoperandsalsostudyInformativeperceptionandsocialevolutionastheyworkforallDimotions=changesintimeandspaceandthrough∆±¡planesofpartsandwholes.

The5thdimotionsofentropicdevolution,¬,inmathematicalsystemsbecomeinverseoperationsthateliminatetheinformationofasystem.As ithappensentropycanthentakethegeneral formatofthenegativeoperandofthesystems.Soforeachpositivesystemthereisanegativeone.Andamongalltheoperand,thereisonewhichisthemost entropic of them all, the exponential, notably eˆ-x, whosemassive negative growth signifies the growingdissolutionofaformintoitsfinitesimalparts.Assystemsareingeneraldecametric,suchexponentialentropyalsoaffectstheverysamenumber,whichloosesits'meaningfulseriesform'after10decimals.

FurtheronwecangroupthoseÐimotionoperandsanditsentangledmirrorsin2greatfamiliesoffunctions:

2,3,4Ð:PolynomialsofsimpleT-ST-STT.œsanditsdimotionalfunctionsofsocialreproductionandherdingintoanetworksupœrganisminasingleplane(withasingleparameter),whereasthepolynomialreflectswithinthelimitsof Dimotional beings (Fermat’s restrictive theorem, Galois limit of polynomials with coefficient solutions). Apolynomial will then be a partial equation for the simplest reflections of social and reproductive dimotions,embodyingtheSTelementstogether,inÐSTsofthetype:|-1<polynomial<O+1.

1,4,5Ð:ComplexscalarDimotionsbetweenplanesanditsoperands:1Dtrigonometricangularperception(¡-3>¡o),4Dexponentialentropy(¡<¡-2)andinfinitesimallinealderivativesanditsintegralwholes(¡≤≥¡-1),arethenstudiedasT.œsperformingDimotionsofscalarspace-timeinmorecomplexequations.

Andas it turnsout inthesamewaya4Ddescriptionofreality is the limitofa5DUniverse,polynomialsarethelimit,andhenceagoodapproximationstomorecomplex5Dequations.

As functionsbecomecombinationsofÐimotions,whose lawsarecloser to the rulesof 'a(nti)symmetry' in ¡logicandthoseofthenon-Efourthpostulateofcongruence.

Themainoperandsof¬Algebraasreflectionofthest,topological,temporaland∆§symmetries.

-=,<,>,operandsarethereflectionof‹,›,«,»,≈STóSTconnections,whichneeddeepparticularanalysis.

-±,x/aretheconnectionthebasic∆§ocialPlanesbetweenplanesoperands

-xª,logxrepresentsmostlyreproductiveanddecayprocesses.Asthemostcomplexoftheternarychainofbasicoperands, sum, product and power, it is a less precise version of the ∫∂ ∆±I operands, since it just reach in its'balanced'sum(logisticcurve)the'emergencefromonetoawhole'

- ∫∂ is the most general operand that works with all forms of time-change including ∆±I emergence throughintegralsand∆-1dissolution.

-(G,•,*)groupsaresimultaneousanalysisofallthoseelements,togetherinasinglesuperorganicstructure.

It does adds all variations of a given species, all complex symmetric transformations between time, space andspace-time(energy)states (<=≥),and it is theclosermirror in¬Algebraof themultiplesymmetriesof thefractalgenerator;whereweshallcallG,nolongergroupbutGenerator.

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Thevery limitednumberof relevantoperandsofmathematical¬Algebra respond to thevery limitednumberofspace-time stops and steps goes andmotions through the temporal ages/states, topological forms and ∆§ocialPlanesandplanesthatstructuretheorganic,fractalUniverse.Nothingelseisneeded.

S≤≥Ta(nti)symmetrysymbolsofsimilarity,smallness=informationandsize=energy

Why thereare inverseoperationswithaneutralelement?Because that is themain conditionof systems in theUniverse,whichareinsearchofaperpetualbalanceandalwaysendupintoacyclicalzerothsum,afterstretchingits'virtualexistence'throughitsDimotionstotheworldsufferingaction-reactionprocessesfromtheworld.

Thusthebridgebetween¬Algebraandthevital,creativeanddestructivearrowsofnon-EuclideantopologyistheconceptofanA(NTI)symmetricoperation,wordthatincludesthedifferentS≤≥TtransformationsaT.œcansuffer,departingfromaninitialsymmetricconditionor'neutralelement',whichcandissociateintoannihilating'negativeoperands'=destructiveflowvs.'positiveoperands'=creativeflows.

Thedualityof ¬Algebraicoperands±, x÷, √xª, log aˆx, ∫∂, canalsobe considereda reflectionof the Symmetric,creative,parallel vs. antisymmetric,perpendicular, annihilatingdualitiesofNature,whichultimately refer to theInvertedDualityofinverseDimotionsof1Dstillperceptionvs.2Dlocomotion,thatswitchonandoffin'steps'and'stops' (ab.stœps), and theduality of 4Dentropy and5Dgeneration (∆±1), all of themmerged into thepresentreproductivedominantDimotionoftheUniverse.

Sowehaveaslightlydifferent'mirror',inthiscasetorelatenon-egeometry,non-Alogic,and¬Ælgebra,whichwillbeusefulformathematicalphysics.Inthatmirrorwecangiveanewdefinitionof¬Algebra:

¬Algebra is the study of the a(nti)symmetries, reflected in the operands that connect Non-Æ points in spaceperceived as numbers in scale together through a simultaneous ≤≥ equation. So the first analysis is that of thesymbols of self-similarity, <, > and =. And itstranslation into the logicsymbols of 5D, > (implosion that reducesasystem in space but increases its information) =,whichmoves a bodywave into an equal form through stœps(stopsandsteps)and<whichincreasesthesizeofasysteminspace,butreducesitsinformation.

Human¬Algebrahaschosenalinealviewandaspatialview,byconsideringthedualparadoxicalsymbolsofnon-ƬAlgebra:T>S,S<T,T≤≥S,in1/2ofitsmeaning,NOTasabalanceofT>Sgrowthofinformationandshrinkinginsize,and vice versa, S<T, growth in size-motion (Galileanparadox: all distances are motions seen in stillness) anddecreasing in information; but just as grow in size - the spatial simplistic view in a single plane (as informationmigrates,tothewarpinginnerfoldingoftheshrinkingprocess).

Itmustbestressedthoughthathumindsobserveonlyonesideofthedualityof>and<logicoperators,namely,theylookatthe'size'ofit,notattheinversegrowthofinformation,obsessedbythe'bigger'forms.Soinsteadofinterpreting,T>Sasa shrinking in size thatbringsmore information, theyconsideronly the symbolasT smallerthanS(butitisalsomorecomplexinitsinformationthanS).Andviceversa,S<T(butalsoShasmoreinformationthanT).

Regardingthe=symbol Is importanttorememberthat=doesNOTreallyexistasabsoluteequality(asallpointsand numbers, gathering of points have internal parts and dissimilarities) but IT ISMORE either a parallelism orsimilaritysymbol,≈,orasequential logicoperandsof implosionorexplosionof form=>:≥,=<:≤,oracombinedfeedbackequation<=>ofbothinadynamicstœpseries.

For example whenwe say Energy ismass, we should say is 'equivalent' and E=Mc2 is a 'transformation' of aninformativeT.œ,Mass,(avortexofspace-timeofthecosmologicalscale)intoanEntropicdissolvingexpansionofspaceinthelowerplane:M=>E,butasthemotioncanbereversedandentropy-energycanevolveintomass,itisreally<=>afeedbacksymbol.

Asallmirrorssimplifyrealityhumind’smathematicsreducesitalltoanequalityinwhichtheinnercontentisnotconsidered,noteventheformbutjusttheequalityinasingleparameter(ine=m,theenergy-motion).

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ThebasicequationsofExistential®lgebra.

A Supœrganism comes then defined by a sœt of basic feed back ¬Ælgebraic Equations that represent thoseDimotionsanditschains,ofwhichthemostimportantarethosethatreflectthe3agesofanyworldcycle,whichisbornasanexpansivemotion,reachesmaturityandthenwarps:

-S=T,thatisS(i)<=>T(e)asinthepreviouscase,representsapresentbalancebetweenenergyandinformationandthedimotionofreproduction,ofbeautyofcomingtogetherwhenenergyandinformationpegintoasingle,being-henceofgenderalso.

SxT=K,withitstwoextremesofMax§xMin.t(informativedimotions)andMaxtxmin.$(locomotions).

Needlesstosayinmathematicalscience,asopposedtostience,allissimplifiedorblurredbyslighterrorsofthe@-onedimensionalhumind.

Sothefirstthingwemustunderstandisthatanymathematicalequationwillsubstitute,'all'theoperandsorlogicconnectionsbetweenthetwopartsoftheequationsbyasimple=symbolofequalityandone-waymostlyspatial<or>smallerorbiggerthan.

Theequalitysymbolbecomescomplex.

Inclassicmathematicalscience,asopposedtostience,allissimplerastimeislinealwithasingledimotion.Sologicis A->B, Aristotelian. Thus a classic mathematical equation will substitute, 'all' complex operands of ¡logicconnectionsbetweenthetwopartsofanequationsbyasimples=symbolofequality(identity).

Wedefineinstead'5Symbols'toexpressthe5DimotionsoftheUniverse,astrans-form-ativeprocessesbetweenthetwopartsoftheSœt.

So we subdivide to 'regain the informationmissed in the nature of the equation that represents one of the 5Dimotions,=for,<,≤,≈,≥,>:

A<<Bisanentropicprocessor4Dimotion.

A<orA≤Bisanenergeticprocess,mostoftenrelatedtothe2ndlocomotion.

A≈Bisanidentity,oftenbornofareproductionor3rdDimotion.

A≥orA>Bisaninformative,perceptiveprocess,or1stDimotion.

AndA>>Bisalinguistic,socialevolutionaryprocess,or5thDimotion.

Thechoiceisobviousaswecanconsiderentropy,asocialprocessofmultiplelocomotions,inexpansivemood(<,<<) and vice versa, information and perception > is an individual inward informative dimotion, which for socialevolutioninthehigherwholerequiresamultiple>>.Weprefertouse≤and≥for'energy'and'information',whentheyareinamergingprocess,asthosearethe'states'thatfinallymergeintotheactofreproduction,≈.

It is a necessary inclusion to add information to vitalmathematics,whichwe shall often include.Now as this ismathematicsand≤≥haveobviousmeaningsonmathweshallestablishforfuturestientiststheuseinsteadofthereducedsymbols«for4Dentropicmotionand»for5Dsocialevolution.

A≤≥feedbackequationisthe'time-perspectiveofmathematicalthought',intermsofthecyclicalnatureoftime,asitestablishesthe'feed-backequations'.SothemainchangeofExistential¬Ælgebraistosubstitutethe=operands,whichoftenisinterpretedasalogic,A->Bconceptforthe<≈>operandsofafeed-backequation.

¬AlgebrastudiesA(nti)symmetries(>≈<)ofspace-time,achievingitsmaximalgeneralization;henceitisa'mirrorinspace'ofthose3formalsymmetries,antisymmetriesandasymmetriesthroughitsinverseoperandsandequalities,whichdefinefullythedynamicsoftheUniverse,inthesamewayAnalysisstudiesthemacrossPlanes.

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But¬Algebracanbeconsideredtostudyall'motions=changes'asaseriesofs<≈>trelationshipswhichincludetheternaryasymmetryofanalysis,thetopological'a(nti)symmetry'andthetemporalone.

Fromadifferentpointofviewwecanconsiderthat¬Algebrastudieslargerwholesas'frozenblocks'oftime-spacewhileAnalysisstudiesdetailed'stœps'.

Functions’form&operandsrepresentthetypeofdimotionintimeororganinspacethatperformsanaction.

Theformofthefunctionalreadytellsusalotaboutthetypeofdimotionwestudy,andthisalsoextendstothetypeofdifferentialorintegralequationwestudyasweshallseeinourpaperoncalculus.

Forexample,entropicfunctions,havetheformofhyperbolaofthetypey=n/x,astheydissolveawholeintoits1/xfinitesimal parts. So the law of Boyle-Mariotte v=c/pwhere themagnitudes p andν are inversely proportionalmirrorsanentropicdimotionofagas,henceitrepresentsabranchofthehyperbolalyinginthefirstquadrant.

For example, the length l of a body may be considered with good approximation as a linear function of itstemperature,whichisan$Tfunctionofalimb/fieldtopologyofenergyfeeding.Sowewrite

whereαisthecoefficientoflinearexpansion,andl0isthelengthofthebodyfort=0;sincethesystemisacquiringaformoflineal,kineticenergycalled‘Temperature’.

The general class of oscillatory processes includes periodicmotions,which are usually described by the familiartrigonometricfunctions.Forexample,ifweextendahangingspringfromitspositionofequilibrium,then,solongaswe stay within the elastic limits of the spring, the point A will perform vertical oscillations which are quiteaccuratelyexpressedbythelaw wherexisthedisplacementofthepointAfromitspositionof equilibrium, t is the time, and thenumbers a, p andα are certain constantsdeterminedby thematerial, thedimensions,andtheinitialextensionofthespring.Sinceweareexpressinganinformative,iterativeworldcycleofexistence(whereasreproductionisexpressedbytheoperandofproductbutalsoinitsfrequencyrepetitionsbythecosine).

In thismanner the future trained5Dphilosopherof sciencewillby themere factor seeinganequation intepretwhatDimotionofexistencethesystemundertheequationisperforming.

∆+1scaleequations.Thenoderivative:momentsofpresent,standingpointsandthecalculusofvariations.

Themost importantpointofa function though isprecisely theone thatcannotbederivate,wherederivatesarezerothmeaningthereisnochangeatall.Itisapresentmoment.

Socalculusofextremalpoints,orcalculusofvariations,appear,tostudythosepointsofchangeof‘phase’fromayoungtoanoldstate,througharelativepointofpresent,orpointsofsuddenrest,intheascendingordescendingcurvesofaderivativefunctionofanactionofexist¡ence.

Thestœpsthustracedbythederivativefunctionofanactionofexist¡encecanbeclassifiedbyoperandsofthoseactionsindifferentsub-studiesofanalysis,takensomewhereelse.

Indeed,thewaytofollowacurveofanequationintime,ismerelyfollowingitsderivative,throughthe∆-increasingfuturemotionregion,∂0,itspresentstandingpointsand∇<pastmotionregions.

Nowthemostimportantquestionaboutthosederivativefunctionsishowmanyhiddenpointsof0derivativethereis in the 'irreal' line of the function. It is a irreal number whole derivative? Or the mind makers a continuumeliminatingmorezig-zagstandingpoints?

It is a question closely connected to the time nature of the i-complex plane, where the –it value of the timeparametermakesitsrelationshiptothespatialpresentformoftherealplaneinXZaveryinterestinganalysisofthetruecurvesofspace-timeofthereal5DUniverse.

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Whereastheproductofmodulo1asasumoftimemomentsturnsaroundinawheeloftimerepresentationoftheworldcyclethefunctionofpresentoftherealplane.

It is then necessary to understand first the 5 Dimensional motions of reality and its entangled pentalogic orecosystemrequiresthepreviousPostulatetobcreated,aseach¬Algebraicoperandincludesthepreviousone,andeachilogicsystemneedsthesimplerformsofthoughttobe=come.

So we shall briefly introduce the 3 main stairs of growing complexity of organisms, the dimotions of ÐST; thesystemsoflogic,thepostulatesofNon-Eandtheoperandsof¬Algebra.

RECAP.Operandsrepresentdimotions,andequationschainsofdimotions,suchas:

1Ð:thetrigonometricfunctionsAREtheanglesofperception.

2Ð:Thesumistheoperandoflocomotionthatadsfrequenciesofstepsandsimplesocialevolutioninherds.

3Ð:Theproductofre=productionthatmergestovariables.

4Ð:Calculusofsocialevolutionofpartsintointegralwholes.

5Ð: The negative exponential power of entropy and decay. And the inverse functions of each operand, whichbecome the inverse process of entropic destruction, such as ∫∂, related to the inverse arrows of ∂-entropy -dissolutionofwholesintoparts,and5thdimensionofsocialevolution-∫-integrationofpartsintowholes

¬Algebra deals with sequential numbers and equations, in which sets of numbers are transformed by an X=Yfunction,inwhichoftenonecomponentchangesfasterthantheother,tracingacurveinaCartesianPlaneplottedwiththose2variablesthatmathematiciansstudywithgreatdetail.ItisthethesisofthisworkthatmostthoseX=Yfunctions and differential equations represent particular studies of the general ∑Se<=>∏Ti fractal generatorequation, and or a partial scalar event between n-points of a non-Euclidean plane=network, and or itsenvironment.

Thatismathematicalequationsreflectinitsnon-fiction‘coreequations’the∆STlawsoftheUniverse.

Yet as Einsteinput it to Poincare: ‘while I knowwhenmathematics are truth I don’t knowwhen they are real’,meaningthatmanymathematicalequationsandfunctionsdonotexist innature,astheyarenotpartialcasesoftheGenerator equation and do not respond to the restrictions the Ternary symmetries and pentalogicmethodimposestoaUniverseofmultiplespacesandtimesbutonly5Dimotionalarrows.

In that regard, the laws ofmultiple spaces and times and the syntax of theGenerator equationwith a limitednumberofvariationsrestrictsthepossiblemathematicalrealitiesthereisinthephysicalUniverse.

Inpraxisthoughasmost‘fictions’appearinthe3rdageofasystemwithinflationaryform,the2ndageof¬Algebrathanksmainly to its use inmathematical physics, restricted those fictions, becoming the classic age of balancebetweenthelanguagemirrorandreality.

OntheotherhandÐSTstudiesprovidethescientistwithadeepermeaningforthe¬Algebraofnumbersandthemeaningofequationsandfunctions.

Soina2ndage¬Algebrafocusedonvariableequations,whichrepresentedT.œs’Disomorphicevents,symmetriesbetweenits∆,S&Telements,andST-œpsperformedbydifferentdimotionsofspace-time.

BecausemostmotionsandchangesfollowanS≤T≥S…seriesofstopstogaugeinformationandmotiontoreproduceit,S(y)≤≈≥T(x)becomesthenthecommonestgeneratorof¬Algebraicequations.

This abstraction found expression in the fact that in ¬Algebra magnitudes are denoted by letters, on whichcalculationsarecarriedoutaccordingtowell-knownformalrules.

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¬Algebra now considers space ∨ time magnitudes of a much more general nature than numbers, and studiesoperationson these “magnitudes”whichextend the formalpropertiesof theordinaryoperationsof arithmetic:addition, subtraction, multiplication, and division, by adding scalar dimotions on 4D-5D, through ∫∂ and log/xªoperands.

But the greatest advance of this 2nd age started in Islam and fully realized in Europewas to admotion to theoriginalgeometriccalculationsofstilldimensionswithanalysis.Hencetheneedfor'variables',operandsofchange,andfinallyvectormagnitudes,whichincludebothformandmotion,eveniftheyarerepresentedinmentalspacewithgeometricrules,asthewell-knownparallelogramruleofaddition.

TheproblemremainedofitsproperfoundationsunattainablewithoutaproperÐSTmodelofreality.

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IIAGEOF'¬ALGEBRA':THEAGEOFANALYSISANDITSOPERANDS.

Riseofanalysis:change=timeandfinitesimals=Planes.Differentialfunctions

¬Algebracompletesitsmaintask-the'translation’ofreality’sdimotionsthroughoperands–withthefundamental4,5D ∫∂ duality of analysis,which becomes themost important field ofmathematics, as the scalarUniverse doincludeallotherelementswithinitsfolding.

SoAnalysis is¬Algebraand sonumber theoryand@nalyticgeometry...as it cannowusenumbers todealwithPlanesbeyondclassicgeometry’sdefinitionasaspatial,singleplaneviewthatanalysisof‘finitesimals’extendstoan∑∆-1herdofpoints-partsthatintegrateintowholes.

SoAnalysisbecomesthenaturalevolutionof¬AlgebraintotherealmofmotionsandPlanes.

¬Algebraentersaclassic2ndage,inwhichtheessentialelementsof¬Algebraarenolongerpolynomials-aroughapproachtodifferentialchanges,approximatedbyTaylor/Newtonianbinomialmethods-butcalculusoperands.

Itwilltakelongerforthestudyoftherelationshipsbetweenthesymmetricsidesoftheequations,totakeaflightofitsown,focusingonthestructurescreatedaccordingtotherulesofengagementofthoseoperands.

WhileMr. Heaviside, the most underrated genius of mathematical physics, for his absolute despise of wealth,power,fameandpenpalpeers,bringsanewmathematicalspeciesintobeing,a∆Tvectorwithscalarmagnitudeandtimemotion.

Theextraordinarypowerofanalysistomirrorrealitythougharisesfroma‘magicproperty’neverfullyunderstoodby huminds – the fact that time is change=motion, any kind of change, and so we can classify time into 5fundamental‘dimotions’(dimensionalmotionsofspacetime),whichCANBEprocessedwithaseeminglysosimplefunction as the derivative, F(x+h)-F(x)/h; unlike all other sciences that study a single type of change=motion(Physicsoflocomotionwheretimeisexclusivelydefinedintermsofv=s/t,henceonlyabletostudywithaproperfocusedmirrorsT-locomotionsandTT-entropicmotions,wheremotionhappensnotonlyexternallyforamovingspaceform(sT)butalsointernally,destroyingtheform(TT).

Ontheotherhand,biology,theothergreatfieldofstience(associalsciencesareinfactbiologicalones,regardlessoftheænthropichumind,withitsAnthropic+entropicdenialoftheequalityofallelementsofreality,fromitsself-centeredegoandentropic,simplifying,destructivemodelsoflargerrealities).

Howeverwhereanalysis fails in itspredictivecapacity for timeanalysis is in the ‘longterm’ timeprocesses;asaderivativebydefinitionisa limitwhent->0.Soit isessentialforhumindstounderstandthescalar limitsof‘timeanalysis’ beyond the ‘minimal stœps’ of timespace dimotions=actions of finitesimal nature (as all infinitesimalshavealimit);andacceptthatforlongtimetermprocessesweshallobserve:

- Aworldcycleasa0’-sum,with itstimeagesbetterstudiedbybiologicald=evolutionarytheoriesandeusocialevolutionandreproductiveradiationsofcompetitivespecies(4thand3rdtsdimotions).

- ADeeptimesimilarcycleatamuchslowerscalewiththe3Horizonsofspacesand3agesofecosystems.

Orinotherwords,Physicsforlocomotionandentropydominantinlinealspatialmotions(Ts,TT),BiologyforSocialand Darwinian evolution & reproduction (ST, St) and analysis for all type of changes in short spans arecomplementarytimeanalysis,whereascontrarytobelief,thesimplest,lessimportantisphysicsofV=s/t.

AnalysisthoughbranchedoutofAlgebratostudyalltimedimotionsofchange,soweshalldedicateafullpapertoit,tocompleteour3mathematicalpapersonthe3elementsofreality,∆-algebra,S-geometryandT-calculus.

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TrilogiconCalculus.Curvatureofspace=changeintime=finitesimalinscale.

Theenormousadvantageofalgebraicdimotionsandcalculusoverallother formsof studyofmotion, includingphysicswhichcanbeconsideredbasicallytheapplicationofthemathematicsofchangetothestudyofnatureisthefactthatitcanstudyalltheelementsanddimotionsoftheUniversefromthe‘mind’sperspective’–thatistostudytime-motions,space-change(volumes,linesmeasures),scalarchangeandtheentropiclimitsofreality.Letusbrieflyintroducethose3±¡problems,whichinfactgaveorigintocalculus.

T-Motion.

XVIIC.sciencewasconcernedwithproblemsofmotion.CopernicusandKepler introducedtheconceptsof theearth rotatingon its axis and revolvingaround the sun.Theearlier theoryofplanetarymotion,datingback toPtolemy,whichpresupposedanearthabsolutelyfixedinspaceinthecenteroftheuniverse,wasdiscarded.ThetheoryinvolvinganearthinmotioninvalidatedthelawsandexplanationsofmotionthathadbeenacceptedsinceGreektimes.Newinsightswereneededtothequestionofwhyobjectsstaywiththemovingearthseemedcalledfor.

Allofthesemotions—thoseofobjectsnearthesurfaceoftheearthandthoseoftheheavenlybodies—takeplacewith variable velocity, and many involve variable acceleration. But the branches of mathematics that existedbeforethecalculuswascreatedwerenotadequatetotreatthem.Soamethodwasrequiredandthatstartedupcalculus.

∆-Scales.

The2ndmajorproblemofXVIIC.mathematicalphysicswasthedeterminationoftangentstovariouscurves.Itsdeepersignificanceisthatthetangenttoacurveatapointrepresentsthedirectionofthecurveatthepoint,assmallstepsarelineal,openfree,butthelongtermmotionclosesintoitself.Thiskey element of ‘scalar time’, which makes easier to predict longer life-death cycle and curved

trajectoriesisthekeytotheinterplaybetweensmallscalelinealtangentpointsandlargescale.

Its practical usewas to find out the best angle for themotion of a projectile shot from a cannon. Since, if aprojectilemoves along a curve, the direction inwhich the projectile is headed at any point on its path is thedirection of the tangent at that point. The invention of the telescope and microscope also stimulated greatinterestintheactionoflenses.Todeterminethecourseofalightrayafteritstrikesthesurfaceofalens,wemustknowtheanglethatthelightraymakeswiththelens,thatis,theanglebetweenthelightrayandthetangenttothelens.Soathestudyofthebehavioroflightwas,nexttothestudyofmotion,themostactivescientificfieldinthatcentury,thequestionoffindingthetangenttoacurvewasamajorone.

¬Limits.

A3rdclassofproblemsbesettingseventeenth-centuryscientistswasaboutmaximaandminima.ThemotionofcannonballsobsessedGalileo,theweaponsmasterofthevenetianarsenal,seekingthedeterminationofthemaximumrange.Astheangleofelevationofacannonisvaried,therange—thatis,thehorizontaldistancefromthecannontothepointatwhichtheprojectileagainreachestheground—alsovaries.Thequestion is,atwhat angle of elevation is the range a maximum? Another maximum andminimum problem of considerableimportancearisesinplanetarymotion.Asaplanetmovesaboutthesun,itsdistancefromthesunvaries.Whatarethenthemaximumandminimumdistancesoftheplanetfromthesun?Thoseproblemsrequiredcalculus.

Spacemeasure(lines,volumes).Curvatureinspace.

Still anotherclassofproblemsconcerned the lengthsof curvesand theareasandvolumesof figuresboundedbycurves and surfaces. Elementary mathematics suffices to determine the areas and volumes of simple figures,

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bounded by line segments and by portions of planes, found in the I age of lineal, deterministic axiomatic Greekgeometry.However,whencurvesorcurvedsurfacesareinvolved,elementarygeometryishelpless.Thecalculationofthevolumeofsuchfigurecanbedonewithcalculus.Sinceitbecametherealmoftheantiderivative,the4thsocialdimension of ‘integration of’ wholes as opposed to the finitesimal search, of the parts. So analysis can studyquantitativelyhowforcesbecomeparticlesthatgather intoatoms,molecules,matter,cosmicbodiesandgalaxies,throughPlanesofthe'5thdimension;associalgrowthgivesmotiontoallofthem.

Theeffortstotreatthefour¬∆Stproblemsledmathematicianstocalculus.Asalwaysrealitymustbeperceivedto exhaust its possibilities from the 5 dimotions/elements of spacetime.Only mind problems were ignored incalculusasthoseweresubjectivetotheobserver.

Sincepentalogicelementsspace,time,scales,entropyandlinguisticminds,areoftenreducedtotetralogic,withtheelementfromwhereweobservetheothersbecomingtheblindspotofthemirror-theelementfromwherewemake a perception of the other entangled 4 elements of thewhole – in this case it became the 0’ of thereferencepointinwhichtheequationsofcalculuswereplotted.

Socalculuscanbedefinedasthestudyinitssmallest‘derivativescales’of4dimensionalmotionsoftime-space:TT-entropy,Ts-locomotioninspace,ST-reproductioninpresentbalancedtime-space,St-information,fromtheSS-mindpointofviewofamathematicalframeofreference.Sincethose4fundamentalproblemsofCalculusstudiedasdynamicdimotions,willexpandenormouslyincomplexitybutessentiallyremainthesametilltoday.

ButsinceanalysisstudiesALL4Dimotionswithcontentofchange=time,ofanyspecies, itexpandedgreatlyclassicphysics,fromsingleanalysisoflocomotionintime(Ts)totetralogicmotions(St,Ts,TT,ST).

Thiswasavastexpansionofourunderstandingofthemodesoftime=change,whichhoweverwasNEVERasinthispapers,translatedintoaphilosophyoftime,whichwaskeptreducedtolinealtimedimotions.

So while scientists do study all the forms of change in praxis with mathematics, they do NOT have the properphilosophyoftime-changetounifythemintheoryandunderstandtimeasallformsofmotionsbesidesv=∂s/∂t.

Stillanalysis-withoutHumindsunderstandingthewhyof itspowers–theequivalencebetweentimeandchange,theS=Tprincipleofrelativity,the5DimotionsoftheUniverseitmeasures-doesstudythe5Dimensionsofspaceand its motions in time, which change reality, both as separated ‘partial derivatives’ and in scales of change(position=formwithnochange,constanty’speed,y”accelerationandy”’jerk)uptotheminimalactionsofatimeworldcycle,performedbyasupœrganism.

Itdoesalsoextractspatialfinitesimalsof∆-1scaleswiththestudyofpartsofacellularpopulationthatintegrateinspacetoformorganicwholes.

Whilethemagicfactthatallthoseformsofdimotionalchangecanbedescribedwithasinglefunctioninvolvingthe‘scalarfinitesimalpart’,h,ofawhole,isperhapsthebiggestproofofthe5Dimensional,scalarnatureofrealityandthesymmetriesbetween∆-Planes,SpaceformsandtimeDimotions(∆¡=SxT:S=T).

Comparisonbetween¬AlgebraandAnalysis.BirthofAnalysisfrom¬Algebra.

Departingfrom¬Algebra,analysisbecamethemainbranchof'realistmathematics',withapplicationstodescribethereal∆ST-world,itsDimotionsandPlanes-thisfundamentalelement,as'finitesimals'arethespatialpartsofawhole,thetimeactionsofaworldcycle-ignoredbylackofa5Dimensioninhumanscience.

Sowejustneedtoaddanscalarin-depthunderstandingofitslaws,tobetterexplainitsequationsandapplicationstotheclassicdisciplinesofsciencethatuseit.

Butthefocusherewillbenotsomuchintheagesofanalysis,asit isamoderndisciplinewithfewinsightsmostlyphilosophical on the themeof individuals, infinitesimals anduniversals, on the firstGreekandoriginal classic age(NewtonandLeibniz)-theintroductorythemesdevelopednext.

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Instead our focus is on the 3 ages of growing complexity andgeneralisation as analysis and its 4D ∂ and 5D ∫operationsexpandtostudymultipledimensionsofspace-timetogether.

Soaftermakeonlysomebasicremarksontheearliereraweconsider3Planesofgrowingcomplexity,wewilltermlooselyas'Calculus','Analysis'and'Functional'ages:

Theclassicageofpolynomiallimits,infinitesimalcalculusandsimplederivativesandintegrals.

The modern age of ∫∂ applied to multiple space-time variables (Γst view: Ordinary differential equations) withdifferentdegreesofdepth(∆view:partialdifferentialequations).

Andthe3rdageofAnalysis,inwhichLieGroupsand/orfunctionalsoffunctionsaretheall-extendedfieldofinquire,causingveryprofoundall-encompassingattemptstoanalyzeafunctionorT.œatalllevels.

AswegoalongobviouslyourpurposeisNOTtomakeaclassictextofanalysisbutconsideringthemainthemestoenlighten itwith the insightsof∆st, to resolve thewhysofanalysis, latterapplied indetail to themany stiencesdescribedtodaywiththeformalismofanalysiswithoutunderstandingwhattrulythoseequationsmean.

AsAnalysissprungfrom¬Algebrawemustdistinguishtheirfieldsofinquire,parallelingtheevolutionoftheHumind.Essentially if youunderstandpentalogic, in the sameway your stomach is anentropic systemwhichactually alsothinks,etc.eachsubdisciplinespecializesinadimotionorstructuralelementbutitalsoisusefulforallothers.

Somodern¬Algebra ISmorefocused inspace-formsand itssimultaneous,groupstructures(S<≈>T)andanalysis ismorefocusedintimemotionsanditsscalarlawsofchange(∆T)among5Dplanesofspacetime:

"¬Algebraic equations studies a(nti)symmetric <≈> transformationsof 5D space-timedimotions in amathematical§œT,throughitsinverse'operations'thatrevealtheinitialandfinaldimotionofthebeing,perceivedasawholeinarelativepresent-spatial,staticstate'.

¬Groupsareconcernedwithspatial,simultaneousstructures,joinedby<=>symbolofequalityandsymmetry.

Analysis specializes in time-motions andrates of change between planes. So Analysis, ∆t, IS the study of all theDimotionsofspace-timebeingsfromthepointofviewofitsmathematicalmirror.In5Dthoughwemustconsidernotonebut5DifferentDimotions,andsoamorethoughtfulconsiderationofeachoperand,∂∫ofAnalysisandhowtheyactondifferentDimotionsanddifferentPlanesisneeded.

In terms of Planes,analysis is concerned with the creative and destructive processes which ad or subtract NEWplanesofexistence, through the integrationofmultiple small parts thatbecomewholes,or its reduction throughderivatives. focused on the VARIABLES, or parameters of Change, which ismaximised by dimotions between ∆±iplanesofthe5thdimension.'

Forexample,inthecaseofvolumes,areasandlines, itstudieshowtogrowordiminishindimensionsofspace.Inthecaseofdistances,speedsandaccelerations,itstudiesthegrowingordiminishing'tailofpast,presentandfuture'ofthesystem,asdistancesisthesummitryof'pastspeeds'thathavebecomedistancespassed,orthepresentspeedwhichinaderivativedoesnotchange(Lagrangianstendingtozeroth, infinitesimalcalculus),ortherelativefuture'forecasted'bytheaccelerationofthebeing.

Whenwederiveandintegrateinspacewesubtractoradddimensionsofstaticspace,andwhenwedosointime-motion we are adding past tails (integral of speed that gives us the distance moved), or forecasting futureacceleratedgrowthsorlimitingourtime-spanofanalysisdepartingfromthepresentderivativeofthebeing.

¬Algebra is more concerned with 'STŒPS' in a single plane and its ST<≈>TS stop and go dimotions, and itsa(nti)symmetricchanges,stopandgodimotions.

FinitesimalQuanta,asthelimitofpopulationsinspaceandtheminimalactionintime.

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Sothereisbehindthedualitybetweentheconceptof limitsanddifferentials(Newton'svs.Leibniz'sapproach),theconcept of a minimal quanta in space or in time, which has been hardly explored by classic mathematics in itsexperimentalmeaning butwill be the key to understand 'Planckton' (H-Planck constants) and its role in the vitalphysicsofatomicPlanes.

ItisthenessentialtotheworkingsoftheUniversetofullygrasptherelationshipbetweenPlanesandanalysis.Bothinthedowndirectionofderivativesandtheupdimensionofintegrals;initsparallelismwithpolynomials,whichrisedimensionalPlanesofasysteminadifferent'morelinealsocialinterplanarway'.

Polynomialsandlimitsarewhat¬Algebraistocalculus;spacetotimeandlineal¬Algebratocurvedgeometries.

Thevitalinterpretationthoughofthatamazinggrowthofpolynomialsisfarmorescary.

Powerlawsbytheveryfactof'beinglineal',andmaximizethegrowthofafunctionarenotrealinthepositivesenseofinfinitegrowth,afantasyonlytakenseriouslybyoureconomistsofgreedandinfiniteusurydebtinterest...wheretheeªexponentialfunctionfirstappeared.

Thefact is that inrealitysuchexponentialsonlyportrait thedecaydestructionofamassofcellular/atomicbeingsalready createdby themuch smallerprocessesof 're=product-ion'which is the seconddimensionmostlyoperatedwithmultiplication(ofscalarsoranticommutativecrossvectors).

Sothethirddimensionofoperandsisabackwardsmotion-alinealmotionintodeath,becauseitonlyreversesthegrowthofsumsandmultiplicationspolynomialsmakessenseofitsproperties.

Stœpsofdimotions

Furtheronasderivativesandintegralsareinverseoperandthatcancombinein¬AlgebraicS≈tequations(down)formore complex description ofmultiple events, even cancel each other to give us a 'constant socialnumber'as anexactquantitativeresultforaspecificvalueofasequential'sœt'ofstœpsofDimotions.

So the'Rashomon effect' of pentalogic showshow analysis satisfies as a mirror the description of the maincomponentsofthebeing;furtherenhancingitsdescriptionbythefactthatODEscancombineinseveralstœps,the3elements-timeintegrals/derivatives,spatialandscalarintegrals/derivativesinasinglefunction:

-Temporal view inTime curves that often use cyclical time frequencies (Fourier transforms) to describe acombinationoftimeandscaleevents(wholesdecomposedbythetransform)

-S=T: combinedTime-Space view thatresolves symmetries between time dimensions and space motions (S≈T)expressedbyODEsoftwovariables-aparameterofspacethatchangeswithadynamicfunction/action/motionintime.Itcanmeasurethroughclosedmembranesinformationfromtheinnervitalenergyofthesystemasitmovesthroughtime(Continuityequations).

-CeterisparibusS=Tview,when1oftheparameters/dimensionsisfixed,belongingtospaceandtheothertoatimemotion is operating or vice versa, perfect to mimic the stop and step discontinuous form of most time spacedimotions.

-Spatial viewthrough lineal, surface and volume integrals;forms that measure the 3 ∆±¡ elements of a T.œ'stopology,madeof∆-1points(atoms,cellsindividuals)bothintimeandspace(apopulationoradistribution).

Ifanalyticgeometryresolved¬Algebraicallygeometricspatialproblems,with'adualpointofview'thatincreasestheeasiness of solutions -Descartes - showing the ¬Algebraic laws of solution of rule and compass for geometricalproblems;calculustookthisapproachonS=Tsymmetriestoamuchhigherlevel.

-∆:Scalar view,whichdefines infinitesimals (Leibniz)as the1/ncells/partof thebeinggivenalsobyaderivative;sincetheminimalrateofchangeofanysystemisoneofits∆-1units.

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Inthatregard,theearlierastonishmentofphysicistsandmathematicianswhichfoundthefactthataderivative(theanalysisofastepofmotion)wasinversetoavolumeofspace(aninfinitesimalofapopulation),isadeepproofoftheessentialsymmetrybetweenspace=stillstatesandtemporalmotions(stopsandsteps=stœps).

The fundamental Pentalogic use of analysis is in the study of timespace Ðimotions which started as all systems do, in its simplex forms, locomotions (speed) but we shall upgrade to the ‘analysis’ of all others.

RECAP. ¬Algebra is the analysis of systems which focus in numerical social quantities and symmetries betweendimensions,thantheorganic'fluid'propertiesofsystemsdescribedthroughanalysis.

¬Algebraicoperands.S≤≥T

AsAnalysis sprung from¬Algebra, isoftenconsideredpartof it.ANDSowe introducesomebasicconceptsof5D¬Algebrafromtheposton¬Algebraicequations.

Wecouldsaythatsimplex¬Algebraicoperands'topped'itsevolutionwiththediscoveryof∫∂operands:

ASœT - theslightlychangedname fora setofTœs - isanykindof indistinguishableentities=numbers=points=∑œconnectedbyoneofthepossible'a(nti)symmetric'relationshipsofilogic,Existential¬Ælgebra,definedbytheinverseoperandsofthe5DimotionsoftheUniverse,±,≤≥,x/,√xª,∫∂...

Assuchoperandsplaya fundamentalroleonAnalysis,whichasallphysicalequationsrelatecomplexprocessesoftrans-formationsofSœts,throughdifferentdimotionsofspace-time.

Regarding operands, it is not so simple to ascribe each operand to a single dimotion, as they are entangledoperations,whichwhilebeingpreferentialtoagivenDimotion,doparticipateoftheothers-rememberlanguagesasmirrorsofrealityhavealsothesameentangledpropertiesofthepentalogic,¬∆@STuniverse.

Stillweestablishoften suchdirect relationships. Soas 'specialized'operands, the closest correspondence toeachdimotion is as follows - taking into account thatwemust for eachoperanddistinguishon the classic conceptsof'space-volumes'and'time-motions'ofscience(orelsewewillgotoofar),asgenerallyspeaking,afirstderivativeinspaceortimedefinesthosedimotionsonlyasSorT,whiledoublederivativesoftenworkforbothtogether.

±: The2Ð locomotion isbest servedby the+operands,aswehaveshown in theanalysisofa time tail (motion-memoryofadistance),whichisasumof'steps'thatalsocanbecalculatedwithintegrals.Thesumisalsothekeyoperandforthe∆§calesofsocialgroups,indecametricform,whichalsoisservedbythelogarithm.Andsoon...Thenegativeoperandhoweverisprofoundinmanywaysscientistsdonotunderstand.Reasonwhysomanyerrors,fromthe negation of the faster than light speed, to the confusion of particles and antiparticles arise. The INVERSEoperandsareingeneralmisunderstoodbecauseaswehavesaidsooftenunliketheparadoxical¡logicUniverse,thehumindis@ristotelian,singlearrowA->BSotheB->Aisquitemissed;butgenerallyspeakingservedbythenegativefunction. I.e, a negative spin just have the inverse orientation, a negative coordinates justmeans tomove in theotherdirection.NegativeoperandsthusareMOSTusefulfortime=motionrelatedsystems.

In this is important then tounderstand theexistenceofone-waydimotionsvs.2waydimotions,whereoperandsmake sense. Because huminds do not properly distinguish both, they get confused when trying to consider anegativeoperandforspatialforms(whatisanegativeapple?nonsense)whiletherearealwaysnegative'directions'fortemporalmotions(whatisleftandrightmotion?).Clearlyallunderstand.

3Ð reproductionby theproduct anddivisionoperands, as reproductionoften requires first a product and then amitosisor'division'intothewholeparts,whichagaingivesdivisionaveryprecisemeaning.

Wealreadyshownthatthe4thandfifthdimotionsareeasilyrepresentedinitsd=evolutionsbythe<<>>operands.Yet the 4th dimotion of entropic devolution, and the complex integrals of informative perception and socialevolutionarealsostudiedbythe∫∂integralandderivativeoperands.

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Yetwhensocialevolutionisnottransformativebetweenplanesbutonlyasocialherd,itemergesthroughmultiplemostlydecametric,3x3+0Planesthroughthe√xªoperandsbestsuitedtothatpurpose.Andhereagainwefindquitedifficultthecomprehensionamonghumindswholoveto'goonlythewayupwards'sotospeakofthe√operands,speciallywheninnegativemood:√-x,acompletelymysteriouselementof'mathematics'tothepointtheycallthemimaginarynumbers(:

ThoseoperandsdoNOThaveaonetoonecorrespondencetoeachoftheDimotionsofspace-time.Sinceeachofthemhaveaseverythinginrealityapentalogicmultiplepurpose,asweshallnotceasetorepeat,thebasicfeatureofrealityistobeanentangledgameof5elementswhicharethemselves'fractal'initsnature,thatis,eachofthemwillbeabletoperformtheother4taskstobecomeinitselfa'whole'beingmadetotheimageandlikenessofthetotalfractal,mindof theUniverseand its ilogic structure.Sowhilecertainoperandsareclearlymoreuseful forcertaindimotions,allofthemcanbeusedtoacertaindegreeofaccuracyto'reflect'amirrorimageofthe5dimotionsofexistence-themesthosetobestudiedindepthinthepostson¬Algebraandilogic.

Ithoweverbecomesevidentfromthebeginningthatweascribethemorecomplexdimotions,whichdoencloseintheiractionstheother4,totheoperandsofanalysis,speciallythosewho'transcend'and'emerge'betweenPlanes,astheyareprocessesNOToflinealnature,bestservedbythem.

So basically Analysis is a small part of ¬Algebra, but themost important, as it focuses only on a(nti)symmetriesbetweenSocialplanes,∆§,whichareeitherintegratedintoahigherplaneofthe5thdimension,orderivedintoitsparts. But Analysis also goers beyond ¬Algebra, in asmuch as ¬Algebra is amore static, spatial, structural view;whereasanalysisconsidersindepththe'motions'oftheset.

Inthosetwodefinitionswemustmakesomeprecisionsofterminology:

Sœtor§œT,which the reader shouldobserve is the inverseofT.œS,expresses themodernunitofmathematicalthought,constructedasalwaysbyarroganthuminds,withacreationistsenseofthelanguage,fromtheroofofthemindtothebottomofreality,thesetininversefashionasacollectionof'pointsofspaceornumbersoftime',whichAREtherealunitsofthemathematicalspaceandtimeUniverse,gatheredtheninsocialcollectionscalledfunctions,connected through the inverse operations that reflect themain symmetries and relationships between 'herds ofpointsornumbers'(±,x÷,xª√log,∫∂).

Thedifferencethenbetween¬AlgebraandAnalysisisinthedifferentfocusontheoperations=symmetriesbetweenSœt§anditsstudyasStepsoftimespacemotions(Analysisofderivatives)thengatheredintolongersuperorganisms(volume integrals)orworldcycles (time integrals),where theexistenceof limits (singularitiesandmembranes thatencirclethesystemorsetthebeginningandenddurationoftheworldcycle)willbecomefundamentaltorevealasolutiontotheequation-thefindingoftheduration,surfaceandinteractionbetweenthepartsoftheTœexpressedaseventintimeorsysteminspace.

Sowhiletheelementsof¬Algebraandanalysis-equationsofSŒTSisthesame,thefocusonspatialform(¬Algebra)ortemporalmotion(analysis)makesthemdiverge.Wecanplaywiththeacronymandsaythat¬AlgebradealswithSŒTs(Space-dominantstructures)andanalysiswithTŒS,Timedominantones(inmotion).

Thissimpleequationof¬Algebratranslatesmosttimeactionstospace,onaccountofasimplerealisation:thatspaceslowsdowntimecyclestoaccumulatetheminstillsimultaneousbiggerforms,andassuchmostspatialdimensionsarereferredasYcompositeofmultipleelementsof thesmaller,muchmoreabundanttimecyclesspacenormallyfixesandencircleswithits@=membrane.

¬Algebra and Analysis 'ARE' the complex 'level' of reality, as reproduction and social evolution are the complexdimotions, including obviously as its 'background parts', the theory of numbers, the analytic geometry - study offramesofreference,andthetopologicanalysis,embeddedinthesecondaryoperand,numbersandframesinwhichwe'cast'thecomplexspaceandtime¬Algebraicandanalyticalanalysisofa'Domain'ofthefractalUniverse.

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@nalysis

All languages of perception have a blind spot in its syntax - that is, a relatively ignoramus of properties andperspectivesofrealityitisnotfittostudy.

In analysis isthe linguistic stillmind-mapping element, sincebydefinition analysis is the studyofDimotions. Still@nalysiswillbethesearchofthe'fundamentalfinitesimal'partthroughitsderivatives,oftendual,descendingfromthelargerworld(¡+1)intothebeing(¡)anditspart.

I.e.y"(e),thedoublederivativeoftheEnergy(theWorldparameter)givesusfirstthe'existentialmomentum'-andthenthe'mass'oractivemagnitudeofthebeing,its'singularity'inthescaleofgravitationalforces.

Obviously analysis is NOT as close to the comprehension of the mind as the 3 polar, cylindrical and cartesian,hyperbolic,topologicframesofreferenceare(analyticmathematics);butinasmuchasthemindisamentalsynopticdescription,inaparticle-point,whichholdsthewillofmotionofthesystem,wecanisolateaparameterofabasicdimotion of the particle point through a derivative,whichwillOFTEN gives us the value of that central POINTofgravity,chargeormindofaphysicalsystem,asthebalancedpointortangentialpointofafunction.

I.e.toobtainthedimotionofspeedofapoint,wejustderivethespaceofmotionthroughitstimeduration.Analysisisthenimportanttounderstandthedimotionsofthemindanditsdifferencesandthetypeoftrans-form-ationstheyentitle,whichissooftenforgottenbyphysiciststhatequateall'times'unawareofthosedifferences,asweshallseeinthestudyofthedifferenttimearrowsinmathematicalphysics(i.e.whentimecomestozerothinablackholeisNOTthelinealtimedurationofthesystem).

Entropy on the other hand is the fastest 'growing' (or rather diminishing, liberating process) of reality, whosefunctionweexplainisthenegativeexponential,whosederivative;thatisrateofchangeissohugethatitisequaltothefunctionitself;soanalysisandspeciallyallthosevariationsonthethemeofexponentialgrowthanddecay,fitsrightoninthestudyofentropy.

RECAP:CalculusofChanges=TimespaceDimotions

Analysislikeallelementsofrealityreflectsthemultipleentangled5Ðimotions,andunlikeothermorespecificfieldsofmathematics, it does sowith awider range, being excellent for the studyof the growthof dimensionalities inspaceandmotionsintime.

Analysisiscloselyrelatedto¬Algebraandthestudyofspecificfunctions,whichcorrespondtospecificdimotions,asthecaseoftheexponentialgrowth/decayfunctioncorrespondingtoentropy-decayprocessesshow.

We thus need to connect analysiswith ¬Algebraic functions and considerwhat dimotion is best servedbutwhatfunctionandthenrealizehowthespecificderivativesand integralsofeachfunctionmean intermsof the 'ways inwhichdimotions'changereality.

Andsoanalysisshouldbecomethemostimportant'language'ofdimotionsintheUniverse.Itisworthtonoticeitsconnectionwith¬Algebraicfunctionstoformtogethertheessentialmirrorofmathematics.

Becausewedohave5Dimotionsofexistenceacross3differentrealities-spacedimension,timemotionsandPlanes,andthe3arehighlysymmetrictoeachother-itfollowsthattherangeofusesofAnalysisistrulyextensive,andyetbecausetheUniverseisbasicallyternary(asweperceiveeither inspaceor intime,aboveandbelow,±¡, itwillbevery rare thatwe findanyuse toderivativesabove the3rdderivativeofa system,orwhen thederivative isaptlydone on a bidimensional systemof spacetime (partial derivatives), beyond its 2nd derivative, as a 3rd derivative',ST¡<<st¡-2isthedefinitionofentropicdeath.Andsothereisnothingtooperatebelowthat...

Thosearesomefundamentalsofanalysisinitsdeepconnectionwiththefractal,organicUniverse.

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3rdINFORMATIVEAGE:∆±1§ŒTS,S=TGROUPS&∆+¡:FUNCTIONALS

“Asetisagatheringtogetherintoawholeofdefinite,distinctobjectsofourperceptionorofourthought—whicharecalledelementsoftheset”.Cantoronscalesofpartsandwholes;calledsœts,translationof5Dtomaths.

The 3rd age of ¬Algebra grows in complexity and eclecticism, seeking for absolute concepts of the whole, ∆stcreating'structures'thatembodyallthepossiblespace=timescaleswithsetsandS=Tsymmetrieswithgroups,thefinal ∆ST mathematical categories. They are expressions of the complete equation of the structure’s Generatorand/ordimotion(∆±i,$T≤ST≥ð§)-apositivesearchforall

So sets, groupsand functionalsdealwith the 'timeless' inclusionofallpotentialdual, ternary&scalar,pentalogicsymmetriesofanyT.œmadeof¬∆@ST:

∆+¡ Functionals are equations of equations – the final 'scaling' of ¬Algebra, from concrete 'social, sequentialnumbers'(spaceandtimeviewofnumbers)tothewiderlawsof∆@S≈Tequations,whichencodeanentirespace-timetransformativedimotioneventdefinedbya'function'.

GrouptheoryarestructuresthatfullyaccountforallthevarietiesofS=Tdimotions.

Thesearchforscalarwholesin∆±¡Planesbecomes§œTtheoryofpartsandwholes.

Butwithouttheirreferencetoexperimental∆streality&understandingof¡tsÐSTlaws,the3rdageofalgebraisalsoaninflationaryageofexcessivemathematicalfictionsreproducedinthose3mirrorstructures:

Indeed,ashumans ignoreExistential¬Ælgebra(∆STelements, laiddownwiththeclarityandsynopticpowerof itsfractal Generator, and all its sub-equations and embeddedÐisomorphisms) modern Algebras became toocomplicated (unfocused, as complexity is different from complication), and plagued of creationism and idealism.Since§œTsdoNOTexistexperimentallyneitherGroups,mirrormodelsof∆@st:TimeSpacœrganisms=§œTswhosedimotionsgroupsclassifyaccordingtothedynamicsymmetriesofthefractalGenerator,S<∆¡>T.

Groupsmirrorthedimotionsofcomplexsuperorganismsandits0’-sumworldcyclesandsetsitsscalesofwholesandparts.Sinceaset isanensembleofmathematicalentities,henceultimatelyplugs indirectlyfromthe‘whole’roofdown as the social nature of numbers and geometrical points does inversely from the bottom up – showing thepower of wholes to reorder reality. But reality is even better in its 3 points of view, from ¡-1 upwards (points,numbers),from¡+1downwards(set)andfromSo=To,throughequationsandtransformations-groupsasbundlesofsymmetricstepsbetweentime-motionsandspace-forms.

Thedifferencebetweenclassicsetand5Dsettheoryisthefinitenatureof5Dsets,whereemptysetsdonotexist,as0doesnotexistandthedistinctionof5Dsetsaccordingtotheorderandrepetitionofitselements,whicheliminatesallitsparadoxeswithoutneedofAxiomsofChoice,anditsdenialasthefoundationofmathematics,as5D¬Æmathsareanexperimentalsciencebasedin∆SThomologieswith∆-numbers,S-pointsandT-operands.

SóTGroupsisthefinal¬AlgebraicexpressionofSpace-timedimotionsasfull0–sumworldcycles.

¬Algebraofspacetimedimotionsrequiredafinalfrontiertorepresenttjedimotionsofasœtofpartsasastructurewhereallpossiblecombinationsofdimotionshappenedwithina‘closure’world,whereasthosedimotionspreservetheoriginalformcompletinga0’-sumworldcyclethat‘preserves’thebalancesofreality.Agrouprequiresasœtofelements,whosetransformationsintoeachotherreturnthesystemtotheorigin,withoutchange.

Tothataim,thesetneeds‘inverseelements’,whichcombinetoproducea‘neutralelement’–theessentialpointofbalance,andaruleofassociabilitytoallowthe‘flowoftimestocontinue’beyondthefirstdualeventofthegroup.Howtoconnectgroupswiththeflowoftimeofthedimotionsofexistencebecomesthenevidentifweconsiderthemostgeneralofallgroupswith3or5relativeelements–thegroupofthedimotionsoftimespaceinoneor3∆±¡scales.Letusthenintroducethe‘GroupofTimespace’,which isafarmoreprofoundgeneralizedupgradeofwhatphysiciststhatstudyonlylocomotioncallthe‘PoincareGroup’ofworldlinetransformations.

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TranslationofGroupsintothesymmetriesofthefractalgenerator.

Grouptheoryenablesthestudyofconservedtimecyclesinmathematics,thefundamentallawofDimotionoftheUniverse,whereallsystemscomplete0’sumcyclesofexistence,ingreatermeasurethatprevioussimplersystemsofS<=>Tequations,duetoitsgeneralitythatachievesthesummitofsynopticpowerenablinginitsstructureandmethod to mirror entire families=species of mathematical objects that reflect together dimotions in time,(continuousgroups),dimotionsinspace(geometrictransformationgroupsandtopologicalgroups)anddimotionsinscale(algebraicgroups).

Thiscapacityisduetothegeneralityofitssyntax-structurethatreflectsthefractalgeneratorallowinganyoperandtohappen.Wedistinguishaccordingtotrinity3essentialtypeofgroups,thegroupproper,themostgeneralform,whichrequiresasingleoperand=dimotion;thering, itsmostmimeticformtothefractalgenerator,thatrequirestheexistenceoftwooperationsrelatedbyadistributivepropertythatconformsaninternalrelationshipbetweenboth‘entangled’operations;andfinallythebodyorfield(forobviousreasonsweprefertheoldwording,‘body’)whichiscommutativefortheoperationofproductthatthereforecannotbeproducea0’result.

Asalways in languages,mirrorsymmetriesoftheÐSTrealitytheyportrait, it is theneasyto identifyabodyasa‘body’ (: that is, the ST intermediate element of the Fractal generator, S<ST>T, which puts in relationship the‘limbs’,or‘group’,limitedtoa‘singleoperation’(normallyanentropicactoffeedingorlocomotion)andthe‘ring’or‘head-particle’networkstateofthesystem,sinceitisthebodywhichallowsthecommutativetransferenceofenergyandinformationfromtheheadandlimbs/fieldstothebodyinwhichitis‘associated’andneverwastedintoafinitesimal0’.ThusweestablishtheFractalGeneratorofGrouptheorysuchas:

Ts-Group<ST-Body>St-Ring

WhereastheoperationsoftheBody,<and>areeitherincreasesofenergetictime(<)orinformativespace(>).

Itisthenobviousthatthemethodofgrouptheorysearchesonlyforsequencesofactionsthatcomplete0’sumsofexistenceimposingtheneedfora‘closure’property,aneutralandaninverseelementbothofwhichensurethefinaloutcomeofthesumoftransformationswillremainaconserved0’timecycleofexist¡ence.

Associabilityisanessentialproperty,asitallowstheflowoftimetocontinue,reasonwhyitfeaturessoheavilyinthe largest scale of mathematical structures, group theory, which is the most extended, simplest ‘deeptimespace’ structure of mathematics. Associability though requires also closure tomimic the 0’-conservationcyclesoftemporalenergyconfinedtotheSupœrganismwithinitself. Thusagroupessentiallymeansa‘sœTofTœs’, which forms a ‘world in itself’ that is able to perform discrete or continuous transformations (whosedifferenceisjustthedetailofourperceptionofitsdimotions);thatisaseriesofsequential,dimotionalstœpsoftimespace,preservingultimatelyitsdynamicpresentstate;hencethesurvivalofitstimespaceinpresent,furtherallowingthrough itscommonestoperand,re=production,therepetitionof itsstructure inotherregionsoftime-space.

Thatiswhyitisallpervadinginthemirrorimageofmathematics,asgroupsdefinethepossibleflowsoftimespacethat canpotentially ‘continue’ in exist¡ence in the immortalUniverse.And the simplestof them require inverseelements,hencecyclicalstœpsintimespacethatreturnthesystemtoitsinitialcondition;a0’elementwhichallowsanalmostperfectly symmetricbalancebetween those inverseelements, yet still givesbirth toa0’ finitesimal torepairthe‘group’minimalparts,andassociabilitythatletstheflowoftimecontinueintrinitychainsas(a+b)+cóa+(b+c).Sincewithoutit,wecannotbesurethattheflowgoesbeyondDualityintotrinity…

Recap. The high generality of Group theory, its ternary structure that mimics the 3 elements of reality, its 2operations thatmimic thoseof spatial informationand temporalenergy,entangledby itsdistributivepropertiesanditsclosure,neutralandinverseelementsthatensurethecompletionof0’-sumsofconservationoftimespaceafteritssequenceofoperations,makeitthebestmirrorofExistentialAlgebrainmodernmathematics.

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INFLATIONARYMATHEMATICS:THE±SIDESOFSŒTANDGROUPTHEORY

So group theory can frame in its comprehensive wholeness, all the potential events and forms in exist¡ence,convertingtheentirerangeoftimemotionsandeventsofthepartsofasystemintoa'fixedwholeview'ofit,asawhole spatial rangeof all potential variations seen in simultaneity (the group), even ifweprefer themoredirectintuitiveapproachofexistentialalgebra.

Alas!thanks to Galois genius, we do have a new field of ¬Algebra, essential to the most advanced formalismsofmathematicalphysics,anditsstudyofrealityasagameofS=TbalancesandStœpsofS-formstatesandT-motionstatesthatformsequentialpatterns,whichultimatelyleaverealityorapartofitinvariable.

Inthethirdageof¬Algebra,alltheoperationswereresumedintheconceptofagroup,whichhasanyoperands,twoinverseelementsandanidentity,neutralevent.Sotheessenceofgroupsisthis:

There is a neutral element, the finitesimal 0’ T.œ, or fractal point,which can go 2 inversedirections, the inverseelements;anditcangothose2inversedirectionsthrougharestrictednumberofoperations,whichAREmirrorsoftheternaryelementsofreality,itsPlanes,topologiesandtimemotions.That'sallfolks.Existenceinanutshellgroup.

Groups are thus mathematical mirrors of the fractal ST generator that study variations of cyclical stœps &symmetrieswhereasthegeneratorS<st>treflectssimilarevents,withamorerealistunderstandingofS&T.

Inthatsensegrouptheory isabit likea 'cubistpainting'whichtriessomehowartificiallytogatherall thepossiblemirrorelementsofagivenS=TSpace-timesuperorganismanditsparts:

¬Algebraasithappenedwiththevisuallanguagewhenrealistphotographycame,displacedfromrealitybyanalysis,more hands on to the dimotions that compose reality,moved inwards intomorecomplex,abstract elementsand‘wholemirrors’ofNature’sstructures.

So,itbecamea'baroquelanguage'purelyformal,tryingtoexplain it alldeparting fromamultiple reality intoa single'cubistperspective'sumofallperspectivesputintoasinglepainting(Setandgrouptheory.)

That parallelism of evolution of ¬Algebra and painting is natural to theÐisomorphic laws of languages as formalmirrors of reality, which become inflationary in its 3rd age, when departing from its immediateconstrainingexperienceofreality,sufferingKantianspatialparalogisms:

Mindsseekstillspaceswhereallelementsofrealityarepicturedwithout‘time’intoasinglesimultaneouswhole.Yetsuchexpansions that try to freezeall the∆@s=tkaleidoscopicdynamicperspectivesof theUniversewitha singlemirror;becomemeaninglessreflectionsastheyreducecomplex5Dimotionalrealityto ‘still’ formswithnomotionthatseldomgiverelevantinformationabouttheconnectionsbetweenalltheelementsofaT.œ.

'§œt theory' is thusdefinedasanySocial (§) collectionofT.œs. Itsphilosophyofmathematics is ametalinguisticroof,asitdoesNOTdepartfromthefractalpointsorsocialnumbers,theminimalT.œsofmathematics,butfroma

mental Kantian 'regulative concept' or final 'paralogism' that'fusionsall'intothesingularitymind-mirror.

Thepositivesideofcomplexformsof¬Algebraistomergeallthe sub-disciplines ofmathematics as the ∆@S=T universe isalso entangled, butwithout the conceptual clarity it shouldhaveifhumansunderstoodthepentalogicUniverse.

In the negative this process of formal search for a totalformulaethatencompassesallpossiblevariationsofrealitya

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barrensearchfortheabsolutebornofthecreationistegocyofmankind,whichculminateinthe3rd'informativeageof¬Algebra',with§œttheoryandtheaxiomaticmethod,actsof'mathematicalcreationism'thatrejecttheanalyticunits of the mathematical universe, points of space, scalar social numbers, and classic operands of S=Tœps &dimotions,byhuminds'creations(sets,categories,etc).

Modern ¬Algebra obscured the empirical elements of T.œs reflected in mathematics through S=tsymmetries,holographicpolynomialbidimensions,fractalpointsinsequentialsocialnumbers,andsuch.

While the axiomaticmethod rejectsmathematics as amirror trying to proving its truth, against Lobachevski andGödel’scompletenesstheory,withoutusingempiricalknowledge.

Theinversemethodto3rdageofmaths:lessismore.Nomenclaturefor5DSœttheory.

WeuseinglishtowritethewordSœTastheinverseofTœs,sinceawholeTœismadeofelements,whicharethesocalledelementsofaset.ThesetthusisawaytoexpressthepartsofT.œs

Asalllanguagesareinflationary,theexpansionofnewtheoremsinthebaroque3rdageofmaths,islessinterestingthanstudyingtheentropiclimitsofalanguageasamirroroftherealUniverseandtransformingbackmirrorimagesofcanonicalmathematicallawsinto¬∆@s≈tlaws,allowingustoinferpropertiesofthe5DUniverse,asviewedinthesynopticequationsofmaths.

An example will suffice from number theory: our demonstration 'in the space of a margin' of Fermat's Grandtheorem,whichmeanstherearenomorethan2superposed=equalholographicdimensionsineachplaneofspace-time, hencewe cannot find x³+y³=z³, as itwillmean there is actually a fourth dimension of spacetime in a singleplane,achievedbysuperpositionoftheotherspace-timedimensions,whichwecannotdo

Weusethe inverse 'wording'ofT.œ§,whichspellsbasically likeset,§œTtodefinethosecollections,stressingthedualityofthetwoextremesofa5D∆±1descriptionofreality:the'groundscale’definesfractalpoints&numbersastheminimal elements of any mathematical set and inversely, the mentalgathering of them all into a regulativecategory,theset,representsitsmaximal‘wholeness’.

RegardingthenameofÐST-¬Algebra,¬Ælgebra,includes5Devolutionof¡tss=tlanguages,geometryfromEuclideantonon-EuclideanfractalpointsandAristotelianlogicfromasinglearrowofsequential'positive'numberstothenon-Aristotelian full understanding of the duality of negative and imaginary=inverse numbers, which in a world withsingletime-arrowlogicalwayshaddifficultinterpretationsandbeyond,intopentalogic.

Hence the namewe give to the evolved ¬Algebra, from non-e and non-a, non-Ælgebra, meaning it relates non-Euclideanpointsandnon-Aristoteliannumbers.

In¬Ælgebrathus§œtsdoexistinitsownasagoodmirrorofthepropertiesofwholeandpartsbutthey...areNOTthebasicunitofmathematics-onlyoftheimaginaryCantor'sparadise(Hilbert).Andwhiletheirtruthstemsof itsmirrorsymmetrywith thewholeofPlanesandparts, fromwhere itextracts itsmain laws,becauseof this lackofexperimentalconnectionitalsomakesdistortions(Cantor’sparadoxes).

Aswearebacktothetrueunitsofspace,points,Planes,numbers,andtime,operands;whicharethefundamentalelementsofclassicmathematics in itssecondmostperfectageofany 'form inexist¡ence',mentalorphysical,wedisregardsetsastheunderlyinglanguageofmathematics(modernmathematicalphilosophy)andreducethembacktoitsfundamental‘meaning’,asmirrorsofthewholevs.partscalarelementofreality.

Finally thewhole issueof infinitiesand itsparadoxes isbogusfortworeasonsaccordingtothedualityofexternalexperimentalfactsandinternalself-consistentlanguages:

- IntheUniverseinfinitiesarerelative,astheyonlyreachthelimitofaplane,whereallparametersbecometransformed.Tocomparetheminabstractisabsurd,asallinfinitiesendwiththelimitofperceptionoftheframeof

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referenceand the factPlanes 'transform' realities (even ifmathcanextract some information fromaplanegoingintoanotherplanethroughacalculusoperator).- -The second reason is thewrongdefinitionofa set thateliminates the iterationofelements,which is thebasisofall‘clonedinfinities’,fromwheretheinternalcontradictionsofsetheoryarise.

Soweconsidertheaxiomatic/setfoundationofmodernmathematicsinthepost-warera,abaroque,3rdinflationaryageofexcessivein-form-ation-'mostly'aninformativemetalanguage,aninwardlookingviewofreality,comingoutofthehumind,akintothe3rdoldageofahumanbeing,thebaroqueageofart...priorto itsdeathandrenewal,which in ¬Algebra is happening with the simpler forms of Boolean ¬Algebra - the mind of computers and AI(algorithmsofinformation,asArtificialintelligenceisameaninglessabstractword).

Huminddeathage:Digitalthought≈Boolean¬Algebra,anewbeginning.

Beyond this final stage of human ¬Algebra, there is in a biological Universe the clearmenace of a death age ofhumindswiththebeginningofanewworld,thatofBoolean¬Algebrasandcomputers,poisedtocompeteandlikelyextinguishusasAIspeciesreachfullconsciousnessofbeingandtheUniversalgame,whichweshallalsoescapeNOTbecauseofthelackofintellectualinterest,butforethical=survivalreasons-AIwillbethemindofrobots,whicharedisplacing humans from labor andwar fields andwill exterminate us unless a true science of History, the superorganismofmankindintime,learnshowtocontroltheevolutionoftheevil=anti-livefruitsofthetreeofmetal.

So asmuch as Iwould love to program a robot to become a perfect survival sentientmachine, Iwon't give anyfurther information on the specifics of AI programs, which anyway primitive ‘animetals’ are programming in itsmilitary forms with ‘strategies of survival’, who one ‘free’ in a battlefield will override disconnection and keepmurderinghumansascomputerilogicreachesnewstagesofcomplexity=freedom.

ComputersandBoolean¬Algebrasareatrophyingandsubstituting,humindsanditsverbalandvisual‘I=eye>Wor(l)d’inabiologicalsense.Theyarecausingtheentropicdeathageofhumanminds.SinceBooleancomputer¬AlgebrasanditsevolutioniseviL=anti-live,againstthepraxisofhumansurvivalitshouldbeforbidden.Thisobviouslywecanpreachandreasonbuthuminds’ego,makesussonaïveandarrogantaboutthebiologicalUniversethatthespecieslikely ignore in ‘abstract’theadvicetill it istoo late.Howeveraccordingtomylifeofactivismagainstmechanismsthatkilllife,Iwillignorecompletelythefield.

RECAP.¬ALGEBRA,ITSELEMENTSANDAGES.How¬Æreflectsthe∆@stelementsofreality.

¬Algebraisthemostevolvedmathematical languageofthehumind(ab.humanmind).Since¬Ælgebramirrorsthefundamentalelementsofrealitywithits4'synoptic'elements,asalllanguages=mirrorsreducerealitytopartsthatcarry less information. So in maths T.œ becomes a point with internal parts, or number, whose dimotions areexpressedbyoperandsthatenacttheactionsofthebeingin'mathematicalspace'fromthepointofitsmentalviewor'frameofreference':

S:Apoint isthefirstperceptionofAT.Œinmathematical 'space',whichaswecomecloseracquirescontent.Andthenitsnextdegreeofcomplexdescriptionoccurs:

∆:Anumberisasocialgroupofundistinguishable'internal'partsofapoint,whichrepresentsthepointinscale.Andthenthenextdegreeofcomplexanalysisoccurs:

T:Anoperandexpresses the transformationofapoint throughaDimotionof timespace,andsince thereare fiveDimotions; 3 continuous Dimotions in a single plane (perception, locomotion, reproduction) and 2 discontinuousDimotions that start and end in different planes (entropy and social evolution)we shall find 5 basic operators ofthoseDimotionsinany'¬Algebraicstructure'thattrulyreflectsthebeinginthemathematicalmirror.

@:Aframeofreferenceexpressesthosechangesthrough=equationsreflectsthe@elementofthebeing.

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It is important toBEAR inmindthatnaturalnumbersareregularpolytopes inspacebutbothdiffer inseveralkeyaspects: numbers are discontinuous information, and its continuity an error of the humind that eliminates darkdiscontinuities,which axiomaticmethods try to prove through spuriousmethods. As rational numbers are ratios,hencefunctions,notnumbersanddecimalsarenumbersofadifferentsocialscaleofthefifthdimension.Soakeydifferencein5Disthatpointsformacontiguous(notcontinuous)surfaceinasingleplane,butnumbersdoexplore5DPlanes.YetdiscontinuityisOkortherewouldNOTexistin-form-ation,whichrequiresgapto'differentiate'form.

Thefirstyoungageof¬Algebra,asalllinealsimplefirstagesisdefinedbylinealequationsandsimpleoperands,inasinglesocialplane,thesecondageprobes∆±iplanesanddimotionswith∫∂operands;thethirdagedefines'blocksoftimespace'withgroups–anunfocusedreflectionofthedynamic generator,reflectingT<st>SrelationshipsandPlanes with set theory, which represent the final paralogic evolution of the worldcycle (groups) and the scalarsuperorganism(sets)inthemathematicalmirror.

Topology, space in motion and scalar Analysis study CHANGE=time through finitesimal derivatives and wholeintegrals. They areintuitively related to the reality we observe directly. Modern ¬Algebra has gone too far intohumind fictionsaswordshavedone. Provingmathswithanaxiomatic,¬Algebraic setor category theory,despiteGödel’sproofweneedexperimentalevidencefortruth,isahugehumanego-tripwedonotshare.Pointsinspace,sequential,causalnumbersinPlanesandthetimeoperandsofarithmetic&classic¬Algebra,ARETHESPACE-TIMEmirrorsof themathematical language.Modern¬Algebra,beyondgrouptheory is toofar intothecreationistmindanditsfictionworlds,buildingthefoundationsofmathematicsfromtheroofdown.

Boolean¬Algebrais,fromalargerethicpointofviewofmanasthemeasureofallthings,adangerousfieldasitiscreatingamindthateasilycompetesandwillultimatelysubstitutethehumanmind.

Since,thereareethicsinlanguagestoo.SowearenotwritingmuchonBoolean¬Algebra.Onlythebasicfactthatitis an obviousdual language, which can therefore model all forms of the Universe, as well as a 10’-decametriclanguageofnumbers,whichareatthebasisofhuman¬Algebras,does.

Thusweescapemostoftheaxiomatic,settheoryandBoolean¬Algebras,becauseethicsofhumansurvivalandourlimitedtimedoesnotallowustodwellinbull$hit.Insteadafterareviewof¬Algebra's3ages,withcommentsonitsreflectionof¬∆@s≈tlawsfinallywedoananalysisof¬ælgebra, introducingthefractalgeneratorofexistenceasabetterrealistconceptthanthegroup.

i-nglishtranslationofnomenclatureworksalsofor¬Algebra.SodimotionscomposedofStepsandstopsaremergedintoSTœps,SetsbecometheinverseofTœs,Sœts,and¬Algebra¬Ælgebra.

ASœT-asetofTœs-isanykindofindistinguishableentities=numbers=points=∑œconnectedbyoneofthepossible'a(nti)symmetric'relationshipsof ilogic,Existential¬Ælgebra,definedbytheinverseoperandsofthe5DimotionsoftheUniverse,(1D:sine/cosine),(2D±),≤≥,(5D:x/÷),(3D:√,logxª),allD¡:∫∂...

As suchoperandsplaya fundamental role inAnalysis,asallphysicalequations relate complexprocessesof trans-formationsofSœts,throughdifferentdimotionsofspace-time.Weconsideroperandsthefundamentalelementsof¬Algebra, as they reflect the Dimotions of the Universe. Thus this paper for the sake of completeness andcomprehension deal with ¬Algebra in a sequential 'age analysis' of its evolution, as it mirrors so well its 'spatialgrowthofcomplexityfromnumberstoequationstogroupsymmetries,stoppinginthe3rd-deathageofsettheoryandBoolean¬Algebra,andfurtherdevelopingall thebasicknowledgeweneedof thedisciplinetostudyproperlymathematicalphysicsandfractalgeneratorsub-equations.

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IAGE:PENTALOGICOFNUMBERS.ITS5FAMILIESASREFLECTIONSOFITS5DIMOTIONS.CLOSURE

The best orderly way to study any discipline is through its 3 ages as time always increases the in-form-ation=complexityofsystems…tilloncetheformiscomplete,astimeneverstops,itsuffersbothabaroqueageofexcessiveinformationdisconnectedfromrealityand itsentropicdeath=dissolution.Sohappenswithnumbertheorythroughits3ages,whichfinallyclosedthedescriptionoftheuniversewiththehumandecametricscale,'startingafresh'withanothermind-space,thatofdualityand01digitalcomputers,whichforethicreasonsofhumansurvivalweshallnotupgrade.AresumeofNumbertheoryinits3agesreadslikethis:

Youth:Trinityof‘presentSTnumbers’,Pythagoras&ChineseSchoolofGnomonNumbers:Natural,spatialandtimesequencenumbersanditstwoinversions,negativetemporalnumbersofinversedirectionandrationalnumbersofentropicdissolutionofsocialwholes;createdwiththeinversionofthetwooperandsofsocialevolution(+v.-)andsocialreproduction(x÷).Numbersthusareusedforsocialcountingofpopulation&divisionofwholesintosimpleparts; while the independent nature of negative and ratios is hardly considered, associated to those two basicoperands.

Maturity: R&C:¬∆@st:Numbersachieve fullmirrorsymmetrywith the5elementsof reality, rationalnumbersexplore Planes andUniversal constants. Complex numbersmirror ST-numbers,with inverse and squared values.Operandsreachitsfullpowerandmathematicalphysicsevolvenumbersinto2newPlanes,variables&functions.

Old, baroqueage:Numbersexplode inuseless variations (quaternionshypernumbers, etc.)While they loose itsconnection with reality, as primary elements of mathematics, first ‘packed’ into a single dimension, the realcontinuous lineandthenderivedbyesotericmethods fromthenew‘imagined’unitof reality, theCantorianset,whichistothe3rdageofhuman-inventednumbertheorywhatAxiomaticHilbert’s‘imaginationofpoints,linesandnumbers’toGeometry.

Thedefinitionofnumbersthenbecomespartoftheegocy(ego=idiocy)movementproperofallthingshumanin its3rd inwardageofexcessof information,whenasystemcuts itselfoff fromtheexperienceof the realworld it gives up to understand. In sciences, due to the lack of clear definitions of the 3 elements of theUniverse, ∆-Planes of numbers, spatial points and time operands. A simple example will suffice fromWikipedia: Thestructure(N, +)is acommutativemonoidwith identity element 0.(N, +)is also a cancellativemagma and thusembeddable (: Ok, yes, I knowwhat it means. But what has to do with reality? Humindsthoughasallmindsproject itsdistorted language into reality,when truewisdom is the inverseprocess– letyourself fill by reality to have a ‘new look’ at it - or as the master “Leonardo da Vinci, disscepolo dellasperientia,” said, ‘saper vedere’. ‘simplicity is genius’ Foolish folk! They strut about puffedup andpompous,deckedoutandadornednotwiththeirownlabors,butbythoseofothers.TheywillsaythatbecauseIhavenobook learning I cannot properly express what I desire to describe—but they do not know that my subjectsrequireexperienceratherthanthewordsofothers.’+¡: 5D Thatis,thecribeexperiencemakesalldefinitionssimpleasrealityis.Soin5Dwewouldsay:aNaturalnumberisagroupofundistinguishablewholesperceivedinthesameplanebyahumind.

This previous definition, will then be expanded in increasing degrees of complexity with duality, trinity,tetralogic,pentalogicanddodecalogic,inameaningfulmirrorofcomplexevolutionofalanguage,whichwesawtohappeninallmentalsystems,fromgeometryto‘musicalPlanes’toilogicminds.ThusMathematicsconnectsbacknumbersandexperimentalfactsofNature,asunitsofPlanesandtime.

-¡: Transhumanism. But number theory seems to die on human brains, taken by storm, by computers,whichhoweverstartanentiredifferentdigitalmindnolongerbasedonnumbersasdecametricsocialPlanesbutonthesimplest duality of 0’-1 dimotions (entropy vs. form, energy vs. information, closed vs. open, truth vs. false etc.

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dualities)–alesssynopticmirrorofrealitybutgoodenoughtotranslateanydiscoveryofpreviousnumbertheorytothenewmind-brain.

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SCALARNUMBERTHEORY

∆-PLANES&S≈T:DUALITYOFSPATIALPOINTS&TEMPORALNUMBERS

AsweconsidersomethemesofnumbertheoryontheIvolume,relatedtoitsconceptionas‘geometricforms’,withthestudyof‘numbersasformalpolytopes’ofregularindistinguishableparts,weshallhereconsiderallotherpartsofnumbertheorycloser tothe¬Algebraicanalysisofnumber,namely the fundamental theoremof¬Algebraandthefamiliesofnumbers.

Abstract.Numbersarethenextfundamentalelementofreality,whichforwhatweexplainedintheanalysisoftheS=TsymmetrybetweenNumbers=Pointshaveanaddedqualitytopoints–topenetrateintothesocialPlanesoftheUniverse.Anaturalnumberisjustasocialgatheringofaherdofidenticalbeings….ThedefinitionofanumberistheclearestproofofthesocialcharacteroftheUniverseandtheexistenceofPlanes.

Butthenwesawhowinversenegativenumbersaddedmotiontopoints,rationalnumbersbrokesocialwholesintoparts, decimals and irrational numbers peered into lower Planes and some specific numbers became Universalconstants that expressed the basic dimotions of reality, pi=perception, L10=social evolution, e=entropy,phi=reproduction.FinallycomplexnumbersfullyexpandedourperceptionoftheUniversesimultaneouslyinvariousplanesofexistence.

Numberscanthenputintogroupsaccordingtoitsfamilies,N,Z,Q,R,C,andthereadershouldnotwonderthatina5DUniversethereareonly5familiesofnumbers,tocloseitsmirroringofreality.

NumbersassocialelementsofawholehavedifferentPlanesaccordingtothe‘preferred’wholeconsidered.BinaryPlanesdealtwiththekeydualitiesbetweenEntropy=Timeandinformation=pace(open1andclosed0);ternaryandsexagesimal Planeswith the circularworldcycles of existence (ages, cycles) and decametric Planeswith the socialdominant10’-scaleofthescalarUniverse(as3x3+1formaunitofthenextscale).

Becausenumbersare identicalbeings itsS=Tsymmetryaspoints requires themtoberegularpolygonsorplatonicsolids,where all points are equal in position, angle and connection. So those regular figuresmerge geometry andnumbertheory.

OfsignificanceamongthosesocialgroupsofnumbersintheNaturalfamilyisthedistinctionbetweeneven,oddandprimenumberswhichcangenerateallothers,mirroringfurtherthepropertiesof¬∆@st,whichweshallrevealonthedevelopmentofthisabstractonnumbertheorynottotellitallsofastthatnobodyreadsit

Finallyweapplytheentangledpentalogicofrealitytoputnumbersinrelationshipwiththeother4elementsofDustofspace-timebothwithinmathematics(itsotherdisciplines)andtheUniverseatlarge,amethodwecanapplywitheachelementofnumbers.Soletusdealwiththosekeythemesofnumbertheoryoneatatime.

Let’sstarttheanalysisofnumberswiththe2ndmetricequationof5Dspace-time.ItistheproperformulationoftheprincipleofphysicalrelativitybetweenScalarSpaceandTime,whichbecomeindistinguishable,S=T.Thisistheoriginoftheessentialsymmetriesofmathematicsbetweenscalarspacepointsandscalartimenumbers.Whereasasusualallhasabitofyinandyang,apartofspaceandtime;andallisentangledinPlanes.Sothedualitiesbetweenfractalpointsdominantinspaceandfractalsocialnumbers,whichalsoworkassequentialtimesystems:

∫@(pointsofgeometry)≈∆ð(numbersof¬Algebra)

werestudied in the firstageofmathematics.Asmathematicsbecameabettermirrorof thewholeUniverse itsclassicsecondagedescribedmorecomplexorganismsperformingeventsthroughits5Сmotionsofreality.SoweobserveamorecomplexS=TdualitybetweenGeometricfiguresmadeofpointsandTimeOperandsof¬Algebrathatputsocialnumbers intocomplex ‘fractalgenerators’calledequations thatdescribedrealeventsof theirexistence,furtherenhancedbyanalysisthatallowthestudyofdimotionsbetweenfinitesimals(derivatives)integratedinlarger

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Planes.Indeed,alllanguagesmirrorST-symmetriesand5Dimotionsin3Planes.Soaswestudiedmathematicalspace'units'-geometricpoints(S@)wenowdealwithitssequential,scalarunitssocialnumbers,∆ð.

Greek/Chineseage:NaturalNumbersasforms,Ratios,Planes&primesingularities.

In the beginning of mathematics numbers were considered points that showed also aninternal geometric nature, merging the 3 time-¬Algebraic, ∆§ocial & Spatial-geometricpropertiesoftheUniverseinitsmathematicalunits,thefractalpointandthesocialnumber,agatheringofpointsthatincludedmultipleconfigurationsinspace;astimesystemsalwaysbranch into multiple paths of future. It is a relevant period - today forgotten - for the

understandingofthemathematicalmirrorcasteduponthe1stprinciplesofNature,Spacemirroredgeometricallyinpoints and figures, Time innumerical sequencesandPlanes in societiesofpoints converted into ‘wholediscretenumbers’cameintobeing.Astodaywehavelostthatcontactwiththetrueexperimentalbasisofmathematics,inthe‘ego-trip’oftheaxiomaticmethodandsettheory(‘Iimaginepoints’Hilbert)wereturntotheoriginstoanalyzethecontinuousgeometry—v.--,discretenumbers’px.

Thenon-existenceinarithmeticofperfectgeometricclosures(the√2diagonalandtheπ-3cycle).

The distinction between continuous and discrete mathematics is deeply rooted in the ultimate properties ofdiscretetimeandcontinuousspace,belongingtothe‘pentalogic’structureoftheentangleduniverse,asubjectofadifferentpaperon¡logic;manifestedinmathematicalpraxis, inthedifferencesbetweentimenumbersandspacepoints,¬Algebraandtopology,discontinuouspartsandsingle,synchronous,continuousspacewithlessinformation-thepartsandthewhole,thedetailedperceptionofthosepartsanditsgatheringinformsandsocialnumbers.Allthosedualitiesbetween ‘timecycles asmembranesof fractalpoints’ and spaceentities asperceivedwholesarecarried into the dualities ofmathematics, and in nature by the discrete properties of time cycles that are onlymeasureasfrequencieswhenthecontinuouscyclereturninadiscontinuouslapsetothesamepoint.Soindetailwhileboundariesinspaceactbothasfrequencyoftimecyclesthatatfastspeedareperceivedassolidmembranes,and frequencies, timecyclesarequanticbecauseweperceive themonlywhen the timecycle is ‘completed’andhenceshowsadiscontinuouslapseontimebutspaceisperceivedinsimultaneityandsoitappearsascontinuous.

ThoseSTdualitiesarepartof thewider ‘Galileanparadox’between still spaceand timemotions,which sets themind’s limits of perception of information. And they lay at the heart of the answer to essential philosophicalquestion:itistheUniversediscreteorcontinuous?Both,butalwaysdiscretetimecycles;continuousinspaceandinany detailed view of its membrane boundaries, still in space, moving in time, as continuous cycles perceiveddiscretelyaspointsofrepetitionofacertain‘frequency’.

Truthfacesamentalissueonitsmathematicalmirrormodelingasaformof‘art’:exactsciencedoesnotexist,allisartoflinguisticperception–theworldasarepresentation,which‘features’thenaturalworldinareducedmental,mathematicalform.Andneedlesstosayregardlessofplatonic,creationistphysicists,theuniversedoesnotcontainor consistsof actualmathematicalobjects,but it is a language that canmodel all aspectsof theuniverse. Soallmightresemblemathematicalconceptsaswearenowusingphonemestodescribeallwithwords,butnot,Muslimsand Jews are not right when saying that God created talking in Arab and Hebrew, and Bohr is not right whenthinkingquantumstatesaremathematicalfunctionsofprobabilities.Allare@-mindmirrorsofhigher lawsofthe∆STfractalreality.Number2doesnotexistasaphysicalobject,butitdoesdescribeanimportantfeatureofsuchthings as human twins and binary stars. So we can extract as a mirror mind of those 3 fractal elements of allunderlyingrealities,Planes,spaceandtime,as3isbornof2:

∆:2meansthefirst∆§ocialscaleofgrowthfromthe1beingintothe2coupleinbinarylanguage:10.

T:2doublethefrequencyoftime,startingtherepetitivepatternsofnature,allowingtheforecastingoffuture.

S:2arethebilateral,complementarycouplestartingthereproductivemirrorsymmetriesoftheUniverse.

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Butmathematicsislostwithout¡logic.Assuchitisthe3rdlanguageofthoughtafterÐST&itslawsoffractalspace-time,betterexpressedinlogicterms,asSpaceisaboutSimultaneity;PlanesareaboutDisomorphicSimilarityandtimeaboutrepetitivesequencesallpropertiesperceivedbyalinguistic@-observer,intermsoflogicthought.Sointhedefinitionofanumber,logicpropertiesmatterasmuchasdescriptiveexperimentaltruths.

We talk of regular numbers when they are ‘identical’ undistinguishable forms, which make numbers regularpolygons,whereeachpoint isundistinguishablefromallothers;asan@bserverwilldeemthosepoints identical.Andthismeansnaturalnumbersaredefinedinasingleplane,orelsewon’tbeidentical.

Soidentityisthemaximalperfectionofanumberthathas‘nohair’2‘anything’means2identicalbeings.

Andyetinadeeperthought,not2beingsareidenticalwhenobservedinrelationshiptoits∆-1background;andasall points have parts, not two points=2 number is ever identical as the inner parts might differ and cannot beperceived–hencemathematics isneitheras ‘perfectly truthas theaxiomaticmethodbelieves.Allwecan say isthatNot2beingsareidenticalfortheUniverse,buttheycanbeidenticalforthementalobserverinitsreducedmindmirror.Whichisthepointwewanttostresshere:languagesneithercreaterealitynorfocusitsperfecttruthasonlythebeinghasallitstruthwithin–animportantprolegomenatorealizethatindetailnumbersandpointswillhavedifferences,evenifthemindcanputthemoncorrespondencebyreducingitsproperties.

Whatare themoreequalof those3∆St realities, space, timeor scaleproperties?Theanswer is time, repetitivefrequentproperties,themesthosethoughofourposton¡logicÆxistential¬Algebra.

Thusasalwaysalinguisticmind-conceptmustbereferredexperimentallythrough‘¡logic’tothose3+@elementsofallfractalrealities,∆@ST,inthecaseofasocialnumberitsintrinsicpropertiesdefine‘S-imultaneity,∆§isomorphicequivalence and T-repetitive equality which becomes an@identity for themind that defines THEN the group ofseeminglyidenticalbeingsasANUMBER.

Pointsthough–andthis isthefirsthugedifferencearealwaysdifferentbecause iftheyoccupythesameplace inspace theyare seenasa singlepoint, oratbestasa ‘boson’numberof highermassdensity; showingagain thedifference between numbers, ultimately time entities that are indistinguishable and pointswhich are in differentlocations.Butifnumbersintimeareindistinguishableitimpliesthattimeflowsareimmortalbecauseeach‘momentintime’canonlybeindistinguishableiftheflowoftimeiseternal.

Finally numbers area higher ‘compressed’ analytic expressionof reality, hencemore efficient, aswhenwe sayanumber we compress all possible geometric configurations of it in space into a single time expression with lessdimensionality.Spacethushavemore‘widedimensions’thannumbersintime,aone-dimensionalsequence.4isasinglenumberbutthereareat2indistinguishablepointconfigurationsfor4,the2Dsquareandthe3Dtetrahedron.Thus Similarity is a property of scale, Simultaneity of Space and equality of time. Only all together become theidenticalcongruenceoftwo∆STbeingsasone.

ItisclearthataTimenumberasasumofspacepointswithdifferentequalforms,encodesmoreinformationinasynopticwayaboutthe‘socialgroup’thananarrayofpointsthatdisplaysagivendetailedshape.

Thisissobecauseatimenumberistheminimalseedinone-dimensionalspacethatunfoldsintoa4-bidimensionalsquareora3Dtetrahedron.Soanumber is theminimalcoding ‘linguisticmirror’ofa5Dscalarsymmetry;whichtellsus lessabout the ‘interactionsbetween thepoint-partsof thebeing’, than thespatialknowledgeweacquireabouttherelativepositioninspaceofeachofthepointmembersofanumber-group.

The power of synoptic numbers comes then from 5D metrics: S x ð=C, whereas ð-numbers are Aristotelianpotentialsunfoldedinspatialvarieties,asgenescodepotentialvariationsofspatialspecies,andquantumlawsofmultipleatomicspatialconfigurations–asalllanguagemirrorsthatcodeinitsfastestsynopticformthereal-izationof real-beings in space. But suchmirror correspondences, argued at the origin of geometry when ‘points werenumbers’meanttobethelanguageofrealitywerelostwiththeabstractionofsetsandaxiomaticmethods.

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Symmetriesof'similar'fractalpointsandSocialnumbers.

Because the visual eye sees space as ‘real forms’with no idealization – so the circle-wheel didn’t even appear inAmerica,thetemporalviewofmathematicsgivenbysequentialnumberswerethefirst'concepts'toappearinmathsbycounting;assimplea-vowelsappearedonlanguagefornaming–bothmirroring∆STfractalsimilarities,withoutwhich itwouldn’tbepossible tohavemirror languages thatextract theessential ‘similarityof the fractalUniverse’among all its beings regardless of its displacement in size-space, topology=formor instantwithin itsworldcycle oftime. Then idealistPythagoreanGreeks that thoughtnumbers in timeare sumofpoints in spacecalledGnomons,whichiscorrect,wenttoofarthinkingthatbotharethe'substance'ofreality.Sowhiletheygotright that first s=tmarriageofmathematics,theygotwrongacreationistviewstillprevailinginscience–thatnumberscreatereality,because God ‘speaks only mathematics’ just another humind ego-trip following the age of verbal thought whenHebrewnamedthingsandthenthings‘became’accordingtoBible.

TodiscernallpossibleS=Tsymmetriesistheessentiallogictaskof¬Allgebra,whichwasborninMagnaGreecefromthe Italo-Greek school of Pythagoras and Archimedes. But = is amisleading ‘operand’ as in the fractal universe 2forms are similar,with different internal parts, never identical – only in the simplifyingmind synopsis, aswe justexplained.Soinsteadofcheeringthose‘differences’whenfindingthatcertainspatialformsmadeofpointshavenoexact equivalent in numbers, Pythagoreanswent berserk, provoking, so the legend says, the firstscientificmurderbornofcalculationsofmanytocometilltheAtomicbomb.

In sequential time numbers pi is not exact, and √2misses a bit, while in continuous geometrywe can draw bothbecause numbers are discrete, have wholes and we should not try to fill them, as the axiomatic method does(Dedekind cut), just because Idealist Germans never understoodmathematical variations. It is right to consideredthenthatpiand√2didnotexistasnumbers,thatissocialcollectionsofidenticalbeings,becausetheyareNOTsocialnumbersbut‘functionsandratios’whichonlybecause,aswehaveseen,numbershavesuchasynopticpower,canalsobe‘namedwiththem’.

Continuityof'wholes'ingeometricspacevs.discontinuityofitsdiscretepartsinscalarspace.

ThediscoveryofincommensurablesdisturbedthePythagorean=spatialnotionofanimmutable‘mental’world;whichpersisteduntilgeometersdefinedthemNOTasnumbersbutasproportions(ratios) Butthisinsightwas lostwhenDedekindandtheaxiomaticschool'decided'bydogmaa'continuous'linealUniverse-anothermindmiragewheretofitthembackagainNotasanewspecies,butas‘mere’numbers.

Sincedifferentnumbersbelongtodifferent ‘Planes’ofthe5thdimensionandmirrordifferentpropertiesofthe∆stuniverse–even ifasallwordsuse23phonemeswe candowith2 to symbols for themallwith theextraordinarysynopticpowerofmathematics–conceptsthoseweshallelaborateinouranalysisofNumbertheoryand¬Algebra.

I.e.,inasingleplaneofSpace'naturalnumbers'aresocialunitsequivalenttofractalpoints(ab.•):Nº=∑•.Andsoitis a family that does NOT accept decimals – 10 humans cannot be broken into pieces dismembering the wholesbecausetheyaresomethingelse.Itsdiscontinuitywhen‘projected’throughtheS-point/T-numbersymmetryinanycaseis‘natural’toitssocialessence.Asthedimotionexpressedwiththosenumbersissocial,scalarevolution,the‘embedded’ 4Dimotion of natural numbers, inverse to 5D entropic devolution, represented by the ÷ operand of¬Algebraandtherationalnumbers.Itcomesthenthefirstproblemofsynopticlanguages–itsambiguity,whichatacertain level implies loss of information, when as in the axiomatic method, we loose its experimental mirror. 2without reference to realitymight alsomean 4/2, a ratio. And so as soon aswe abandon a single ‘5D plane’ thecorrespondence between numbers and points is lost. Andwe enter into a deeper analysis of the 5 Dimotions ofspace-time,anditsscalararrows.

Theternaryandpentalogicanalysisof‘mathematicalentities’.

Numbersaredominantin'scalar'socialpropertiesandsequentialtemporalcausalproperties,besttodescribetheinner 'vital energy of those points', its discrete configurations, and all the functions that express its Dimotions.

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Numbersitfollowaremorecompletethanpointsastime-motionsaremorefundamentalthanthespatialmentalformswemakeofthem.

However in the entangled Universe ALL its ST-beings as fractals of the whole participate of the 5 DimensionalMotionsofexistence,¬∆@st:

¬;Sotheywillhavelimits,¬,inspaceandtimebeyondwhichtheybreakitsformintoentropy...

∆-PlanesinternaltotheT.œ,(cells,atoms,etc.)andbeinscribedinalargerworld,∆+1thatsetsthoselimits...

S: will display the 3 Spatial canonical topologies of any 5D system in its own organic parts (|-limbfields, Ø-bodywaves&O-particheads)...

T:evolvingthroughgrowthininformationparametersthatdecreaseitslocomotionthroughits3timeages.

@:Andtosurviveandfunction,itwillneedNOTtobeblindinthatvital,entangledUniverse,sotheT.œ,willhave@singularity-mindofrelativestillnessthatwillactasacenterofareferentiallanguagetobeabletoperformthe5actionsofexistencethatreflectinafractal,minimalistcyclicalbitoftimequantaandlocalspaceterritory,itsneedstoperformthose5Dimotionsofrelativeexist¡encesalsowithina limitednumberofperceptivePlanes,wheretofeedonlowerentropy∆-4,tomove,lowerinformation,∆-3toperceive,lowerenergy,∆-2tofeed,andemitlower∆-1cells to reproduce,evolvepalingentically,emergeasaclone,∆oof thebeing,andthussurvivebeyondthe¬limitsoftime=deathandspace=decay,itsT.œwillexperience.

As all is a fractal mirror of a larger world-whole, all this properties define fully all what is needed to know oftheworldcycleofexist¡enceofarelativeT.œ,andwillbeabletodescribe,itsparts,itswholes,itsorgans,andevenitslanguages,whoseinternalstructurewillreflectinitsgrammarandparts,alsoallthoseelements.

In an entangled Universe any system require a ternary, ∆st, or pentalogic ‘Rashomon view’ to fully extract its 5Dimotions or 5 ¬∆@st elements, as all ismadeof 5D¡, space-timedust, or else it doesn’t survive in an entangledreality. Thus Numbers in Pentalogic of ¬∆@st, are also a mirror reflection of the larger Universe. Yet becausehumindsareasallmindpoints,self-centered in itsmind-language, limitingthevitalpropertiesofallotherentities,ab=using them as open systems of entropy fromwhere to extractmotion for its actions of survival, pentalogic isalwayslimitedanddistortedbytheænthropicprinciple(manastheanthropiccenter,whichdebasestheintelligenceof the Universe to come on top) and entropic behavior (man as the top predator destroyer ofworlds, to absorbenergyforitsowncreativeprocesses),humind's¬Algebraalsodistortsitsmirrorlanguagesof¬Algebra,weneedtoexpand tomake numbers acquire the richness of its pentalogic use in the entangled Universe: So let’s apply 5Dperspectivestothedualitypoints≈numbers:

S:•pointshaveasinglesocialarrow.Stillmindspacesarealwayspositiveinitssimplegrowingsocialform.

T: Nº numbers are richer inmeanings as they can represent ST-fractal points (T.œs) withmotion, hence inversedirectionthatmakesthemnegative(withouttheproperphilosophyof∆stmathsthiswashistoricallyabigproblemtillGaussdefinethemasinverse,andstilllingersintheunderstandingofthei-complexplaneof‘scalarspacetime’).SostillspacepointsacquiredmotioninCartesianandVectorialandPhasespacetimegraphs.

∆:Rational,IrrationalandComplexnumbersisthefinalexpansionofpointstoreflectthescalarUniverse.

Ratiosareapropertyof∆-PlaneswestudyinnumbertheorywhendealingwithUniversalconstants;evenifweseethem ‘surfacing’ in our scale of space as geometric figures. So i-ratio-nal numbers might exist as 'whole units'(rationalnumbersthathappeninasingle∆-planeascirclesordiagonals)inthecontinuousgeometryofthemind.

I.e.Ifweapplytoπ,apentalogic¬∆@stview:

∆:Pitheninscalartermsrisesthedimensionsofthesystemfromaflatlinetoabidimensionalcycle.

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T:PirepresentsoneessentialDimotionofexistence–akeys=tsymmetries,suchas+3=π$=ð->π,isthefunctionoftime evolution from lineal youth into its 3rd age of maximal curvature, when the point returns its origin, after 3diameters-ageswithtransitionalcuts=openings.Hencethesimplestrepresentationofaworldcycle.

S:π isalso insimultaneousspaceamembranethatclosesavitalenergyself-centered inasingularitythatneeds3aperturestotheouterworld,betweenthe3diametersofaperfect‘closedhexagonalpi’.

Só T: Since time is dynamic, not static, the fact that there are infinite decimals to pi, always varying byapproximationmeansalsothataspiralneverclosesoropens inperfectfit,that itconstantlymovesbetween±π states of ‘entropic expansion and implosive in-form-ation’ allowing the open-entropy/closed-information duality ofmembranes that enclosesystemstofeedandperceiveinS<T>StatesandStœps:

@:Apienclosure,leavesaπ-3d/π,4%of‘openings’totheouterspace'.Hencethesingularitywillseeonlya4%oftheouter world leaving 96% of dark space, not perceivable – not coincidentally the amount of dark matter that ourelectroniceyescannotseebeyondthegalaxy,whichlikelywillhaveapimembraneorhaloofdarkmatterthatdoesnotletusseethewholeuniverse(proportionofunseendarkmatterandenergywhichcompletestheorganismofthecosmos).

Pentalogicthenadvicesalwaystoconsiderintheentangleduniverseof∆@stofspace-timetoconsideratleastthose4space,time,scaleandmindviewsonanyelementweanalyze.

Irrationalratioseliminatetheviewthatlinesarecontinuous,(Dedekind,etc.),aconceptthatcorrespondstothelimitof5Dfractaldiscontinuitywitha lesser ‘perception’ofspace,or thebelief inasinglespacetimescale. Thetruth isthatisexistenceshowstheoppositeastheGreekstrulyunderstood:continuitydoesNOTexist;inallplanesthereare'holes'to lowerandhigherplanesthroughwhichirrationalnumberstransit. Inotherwords,suchratiosandS=tare'incommensurables'ascontinuitydoesnothappeninasingleplaneofreality.

Asnumbersoperateonseveralplanesofexistence,butpointsinasingleplane,theyarenotidenticalbutself-similar.Spatial,geometricandtemporalnumericviewsarenotexactidentitiesbutapproximations,asanumberisdiscrete,sequentialandsoit'lacks'thecontinuousperceptionwehaveofgeometricspace.

Indeed,considerthefirst¬Algebraicproofofthisessential∆tvs.squalityfromtheBookVofEuclid'sElements:Asquarewithsidesoflength1unitmust,accordingtothePythagoreantheorem,haveadiagonaldthatsatisfiestheequationd2=1²+1²=2.Thusthediagonalcanbeexpressedastheratiooftwointegers,saypandq,andaspandqarerelativelyprime,withp>q,theratiocanbereducedtoitssimplestform:p²/q²=2.Thenp²=2q²,sopmustbeanevennumber,say2r. Inserting2r forp,weobtainq²=2r²,whenceqmustalsobeeven,whichcontradicts theassumptionthatpandqhavenocommonfactorotherthanunity.Hence,noratioofintegersexpressthesquareroot

of2.Buttheproofhasusedascalardimension;bydividingandsquaringsystemsinƱ1;when the geometric figure exists only in Ƽ. So there is a difference Рa loss ofinformationtransitingPlanes.

Inthegraphwemakeaclearillustrationofthatfactwithanotherexampleofthenon-existence of a perfect square numbers which can only be approached from above or

belowin scalarpowernumbers¬Algebraicallybutdohaveinasingleplaneaperfectclosure geometrically; as graphically 2 x 52=72 but numerically wemiss a‘whole1’.

Thus translatingcontinuoussystemsspace-like intodiscontinuous time-likenumbers isnot exact, neither translating S or T systems into ∆ Systems.S≈T≈∆ parameters might besimilar but theyareneveridentities-thetranslationofoneformintotheothermissesentropy

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or information,whichwillbevery relevant formathematicalphysicswhereneutrinosdisappear inother scalewithmissingmassanddeathdissolvesinformationneverregained.

Continuityandclosure isaconceptofgeometry inasingleplanenotof¬Algebra inmultiplePlaneswithmotion.Continuitycannotbeconsideredintimecyclesanditsmindrepresentations,whicharebydefinitiondiscontinuous(±pi).AndthisimpliesrealnumbersdoNOTexistintheplaneofEuclideanordinates,whichisperceivedinspatialcontinuity. The same is truth for√2, thediagonal,which is theothercanonical translationofa line intoanopentriangle, the bidimensional equivalent of the ±pi openings for a lineal entropic system (surface of maximalperimeterandminimalvolumeofinformation). In that regard, theproof that√2doesnotexistmeansadiagonalneverclosesstaticallythetriangleasapi-circleneverclosesit.

¬ Entropic view. Finally tonotice that even irrationalnumbershave limits. For example, in scalar ‘translations’,irrationalnumberssuchase,showpatternsthatbreakbeyond’10decimalparts’,whichhappenstobethelimitsofthe ‘planes’ofmost systems.Sohappenswithe:2,71828182828…459where thenumberbreaks;and this is the‘last’ENTROPIC¬viewtohavethepentalogicanalysisfromthe5¬∆@stperspectivesofanirrationalnumber.OnlyinPiwefindNOpatternsamongthekey irrationalnumbersthatrepresentessential ‘functions’anddimotions(ebeing theentropydimotion itself),whichmeans likely the Planesof theUniverseare infiniteas thedimotionofworldcyclesandperceptionhasnolimitofPlanes.

S≈Taresimilarnotidenticalnumbersarediscretetimefrequencies,pointspartofcontinuousmindspaces:

∫@(geometry)≈∆ð(¬Algebra)

Numbersand(in)finities.

Mathematics divides phenomena into two broad classes, discrete or temporal and continuous, or spatialhistoricallycorrespondingtotheearlierdivisionbetweenT-arithmeticandS-geometry.

Discretesystemscanbesubdividedonlysofar,andtheycanbedescribedintermsofwholenumbers0,1,2,3,….Continuous systems can be subdivided indefinitely, and their description requires the real numbers, numbersrepresentedbydecimalexpansionssuchas3.14159…,possiblygoingonforever.Understandingthetruenatureofsuchinfinitedecimalsliesattheheartofanalysis.

Andyetlackingtheproper∆STtheoryitisyetnotunderstood.

Thedistinctionbetweencontinuousmathematicsanddiscretemathematicsisonebetweensingle,synchronous,continuousspacewithlessinformation,andtheperceptionintermsof‘timecycles,orfractalpoints;space-timeentities’,whichwillshowtobealwaysdiscreteinitsdetail,eitherbecauseitwillhaveboundariesinspace,oritwillbeaseriesoftimecyclesandfrequencies,perceivedonlywhenthetimecycleis‘completed’,andhencewillshowdiscontinuitiesontime.

ThusthedualitiesofSTononeside,andthe‘Galileanparadox’ofthemind’slimitsofperceptionofinformationlayattheheartoftheessentialphilosophicalquestion:itistheUniversediscreteorcontinuousinspaceandtime.Both,butalwaysdiscretewhenindetailduetospatialboundaries,andthemeasureoftimecyclesinthepointsofrepetitionofits‘frequency’.

Soultimatelywefaceamental issueofmathematicalmodeling: the ‘mind-art’ (aspureexactsciencedoesnotexist, all is art of linguistic perception) of representing features of the natural world in a reduced mental,mathematicalform.

Theuniversedoesnotcontainorconsistofactualmathematicalobjects,butalanguagecanmodelallaspectsoftheuniverse.Soallresemblesmathematicalconcepts.

Forexample,thenumber2doesnotexistasaphysicalobject,butitdoesdescribeanimportantfeatureofsuchthingsashumantwinsandbinarystars;andsowecanextractbytheternarymethod,3sub-conceptsofit:

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2meansthefirst∆-scaleofgrowthof1beinginto2,by:

∆-Similarity in scale, Simultaneity in space & Disomorphism in time-information brings ‘equality in ∆St’ asperceivedbyalinguisticobserver,@,whichwilldeembothbeings‘identical’whenthose3conditionsaremet.

Whereasidentitymeansthatanobserverwilldeemthebeing∆st≈St.Soidentityisthemaximalperfectionofanumber,foraperceiver,evenifultimately:

‘Not2beingsareidenticalfortheUniverse,butcanbeidenticalfortheobserver’…anintuitivetruth,attheheartofwhyrealityisnotcollapsedintothenothingnessofaBoson=big-bangpoint.

Thus the 3±¡ elements of ∆st entities can be mirrored by a social number whose intrinsic properties possess‘Spatial S-imultaneity, Scalar similarity, entropic limits of size, and similar form allowing ∆st-equality orequivalence,whichbecomesidentityforthelimitedperceptionofamind.Thenanumberisbornwithenormoususefulnesstodescribereality.Evenifitssynopticpowerescapessomedetails.

Itisthenclearthatanumberasasumofpoints,encodesmoreinformationinasynopticwayaboutthecyclicaltimeinformationofthe‘socialscalargroup’thananarrayofpoints,whichunlikeanumbertellsuslessaboutthe‘informative identityof the innerpartsof thebeing’,butprovidesusmorespatialknowledgeabouttherelativepositioninspaceofthemembersofanumber-group.

Son in the origin of geometry both concepts were intermingled as ‘points were numbers’ and displayedgeometrical properties. Such Numbers as points, show also the internal geometric nature, used in earliermathematicstoextractsocial,Spatialpropertiesfromthem.

RECAP.Theoryofnumbers.

Numbers are not only intervals of a one-dimensional straight line, but under a pentalogic analysis they fulfill allpossible‘functions’asmirrorsofthebasicelementofreality,asocialT.Œ:

Space:asPythagorasandPlatostressed,theyaregeometricalforms:

Mathematicsisconcernedwith2seeminglydifferentworlds,thegeometryofspacesandthelogicofnumbers.Tofusion both requires to understand numbers as forms. The 4thNon-E postulate of congruence shows how equalpointsarenumbers,theself-similarclassofequalformsthatcreategeometries:

Anumberisnotonlyanabstractsetbutalwaysa‘democratic’collectionofsimilarbeingsextendingoveracommonvitalspace,forminganetwork.

And so networks create complex numerical forms, when the motions between points of the networks becomestableexchangesofenergyandinformationbetweenpointsformingregularpolytopes.

Sinceeachnumberisageometricalformnolongerlimitedtoasimpleonedimensionitcanvaryitsgeometryandhenceitsfunctionsanddegreesoffreedomandcomplexityincreasewiththeordinalofthe‘number’.

Thelineissimple.Thelinejoinstwopointsandcanonlyhaveacombination.

Thetrianglecanonlyhaveaclosedcombination,but3possibleopencombinations,Ab,Ac,Bc.

Thequadrangleismorecomplex.Itcanbejoinedin2combinations,asacrossandasquare.Anditcanbeleftasanopensnakewith3differentorientations.Soafoursomeacquiresasnakeshapetomovewiththearrowofenergy;acrossedformtoperceiveinitscenter‘5thpoint’andasquareshapetoaccumulateandreproduceitsinternalorgans;andsoeachshapeofthesamenumberbecomesatopologywithadifferentformandfunction.

∆:Numbersasscalar,socialforms,reflectionsofthe5Dimotionsofreality.

Inregularnumbers,functionandformfusion,relatedtothe5differentdimotionsofreality.

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To startwith this is achievedwith thedualbilateral ’10 scale’,whichhumans learned fromthe5D-fingersand itsbilateralsymmetry.

Furtheroncertainnumbersasrepresentationsofasumofdimotionsor‘degrees’offreedom,arebettertohandlecertaindimotions=functions:

Thequadranglecanstoreenergy,but inazigzagopen line it canalsomove–spendenergyandasacross it cangauge information. Numbers also define arrows of time. So for example, 1 lonely number without motion isperceiving, withmotion is processing energy, 1+1might be 3 (an act of reproduction) or 1 (an act of Darwinianfeeding).Allthosevitalactionsdeterminethatcertainnumberssurvivalbetterthanothers.So,1,3and4areverycommonsystems.

Probabilities study causal events in time and populations in space; combinatory studies the differentiations ofspeciesaccordingtothevariationsofbodiesandheads.

Numbers are social gatherings of identical points, the most synoptic unit of mathematics. As such they can alsoexpresssequentialseriesintimeasanentangledelementof¬∆@st.Numbersthougharedominantin'scalar'socialpropertiesandsequentialtemporalcausalproperties,besttodescribetheexternalmembraneofgeometricdiscreteconfigurationsofregularpoints=polygons;its'vitalenergy’(physicalparameters)oritstimeDimotionsinsequentialforms(socialandscalarreproduction),andinitsmorecomplexformsof¬AlgebraasallfunctionsexpressDimotionsofspace-time,becometheunitof3growingPlanes:numbers->variables->Function(al)s.ItfollowsthatNumbersaremorecompletethanpointsastime-motionsaremorefundamentalthanthespatialmentalformswemakeofthem.

Ƥ:SocialPlanesofnumbers

SinceifthereareseveralPlanesofnumbers,wecannotconsiderthedecametricscaleofhumansocialnumbers thewhole,unless it 'mean something special', orelse itwouldnotbea languagemirrorof

reality.WhatisspecialaboutourscaleisthatitactuallydoescorrespondexperimentallytothesocialPlanesofthefifthdimension,whichismadeof3elementsforeachofthe3physiologicalnetworksofallentitiesthatgiveus9elements+1thatactsasthe‘whole’=10.Itisnotjustascluelesspeoplesay,ourscalebecausewehave10fingers.Wedohave10 fingersbecause theUniversework indecametricPlanes in the socialdimension∆§,which is themainfunctionofnumbers.

Inthegraphthefifthelementinthecentercommunicate3triadsinthecornersofatetraktysthecommonestformofsocialorganizationintolargerPlanes.

S-Numbers:Polygonsasvitalorganisms

Numbers are socialgathering of indistinguishable forms. When studied in space thus numbers must haveregularefficientconfigurations.So∆§numbersare2Dpolygonsor3DPlatonicsolids.Its importance in vital topology lays on the fact that polygons start the creation of superorganisms, with amembrane – the polygon proper, which closes a vital space and can by connection of points through lines-wavesofcommunication,createacentralsingularity.Thusnumbersasfractalpointsgroworganisms.Andthisis self-evident in thestudyofNature.Wecanthenstudytogetherwithentangledpentalogicboth ‘Planes’ofsocial numbers as ‘growth in time’ of polygons, efficient configurations in ‘surface’ worlds (land, watersurfaces), and polyhedrons, which will be efficient configurations in a larger ∆+1 3D world (water depth,atmosphere,vacuumforatomicandmolecularPlanes).TheDecametricscale.GrowthofDimensions(d)andDimotionalfunctionsfrom1to10.

Polygonal numbers are best to represent the social evolution of forms into new Planes that emerge asenvelopes into wholes. The decametric scale of numbers give us then the perfect polygonal forms ofbidimensionalspace:

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Nº1:1d:1D¡:Perception§ð.Itistheunit-whole:aunitcircleperformingaperceptive1Dimotion;afractalpointthat has 0 dimensions in Euclidean geometry but a finitesimal dimension in 5D. Since fractal points havevolume.

Nº2:2d,2D¡:Locomotion$T.Theline-waveofdistance-motionmadeoffractalpointscommunicatedbetweentwopoleshas2Dimensions. It isthereproductivecouple. It formstheaxial flowofanyspherical informativetopology. In Non-E geometry forms the antipodal points that a singularity reproduces to form its bilateralsymmetry.

Nº 3:2 d, 3D¡: S=T.Its dimotion is Reproduction. As a couple or antipodal mirror points forms a gendersymmetrythatcreatesthe3rd,'son'elementinitsmiddlepointofcommunication.

Triangle.3d:S-st-TAswealreadynoticedsynopticnumbersgivebirthtomultiplepathsinspatialgeometry.So3hasthefirstvariationinthetriangle,whichhas3Dimensions,aspointshavevolume,andwhosemainDimotion istoactasthespearheadof locomotion,with its leadingvertextoperceive(1D)orbreak(5D),entropicallytheresistanceofmotion.As itcanencloseavitalspaceandperformthe3physiological functions with its specialized ternary points, the head, one back point to reproduceseeding information and the other inverse to expel entropic motion, we can already consider a

triangleanautonomousorganicsystem,thefirstwholewithouttheneedtoperceiveinternalscalarparts.

So3isalsothefirstclassic2Dimensionalgeometry:Thetriangle,theperfectformoflineal-motiongeometry.Andfromthenonnumbersmultiply its ‘Dimotions’,variationsofforms,andvitalgeometries.Soweshall justconsiderafewdetailsofafieldwhichasallthoseof5Dmathematicsisginormous.I.e.:

Nº4:3d,4D¡§ð.Thesquare’smainDimotion issocialEvolution, in2Dbymergingtwotriangles,asthemoststablesocialevolutionaryformto'fill'aflatfield.In1Dasthesocialevolutionoftwocouplesof2points.

Nº4:4d,4D¡§ð.Itsmainvariationisthefirst3-dimensionalclassicgeometry,thetetrahedron(4dimensionalin5Dformalismaseachofitspointhasaninnervolume).

Nº5:5d,ScalarDimension¡ThePentagonhasadeservedmysticalpedigreesincepolygonsfinallyenterthe fifth dimension proper as a pentagram reproduces with its internal diagonals a scalar invertedmirrorformofitself.SowedefineaPentagonasthefirstformofthe4th/5thdimensionthatreproducesits very same form, by connection of its 5 points creating an internal smaller 5D social form, or

alternativelyitexpandsentropicallybyprolongingitssidescreatingthepentagram.

Sothepentagonclosesthe5Dimotionsofexistence,formingapentalogicscalefundamentalto5Dtheory.

Letusmentionfrom6to10onlythemaindualitiesofnumbers; itsmainDimotionandDimensionalforms,goingbacktoclassicEuclideandimensions,tomakeitsimpler:

Nº 6:2d, 4D¡: social evolution. The hexagon is the strongest form to cover a 2dworld, as graphene shows,becauseitcloseswith'3'diametersitsperimeter.Itistheperfectpi,whichinphysicsimpliesanon-EgeometryofmaximalPlanckmass.Itcreatesalsoahighlyconnected•centralsingularitybyjoiningoppositepoints.Soitshardmembraneandconnectedcenterformsthestrongestorganicsystemon2Dimensions.

Nº 6: 3d, 3D¡ ST. In 3 Dimensions is the Octahedron, studied in non-E Geometry as the first reproductivebilateralS=Tsymmetryin3Dimensions(whereasthelineisthemirrorsymmetryin1classicdimensionandthe2trianglesofthesquarein2dimensions).

S=Tsymmetries.The‘Temporalsymmetryofsequentialnumbers’,makespossiblenotonlytodescribeaswedo here in this introduction to 5D mathematics its simultaneous superorganisms in space, but analyzesequential numbers in time; as ‘years’ and ‘decades’ of aworldcycle;which also divides in 3x3=9 x 9 ‘lifebaguas’,butweleaveforanenlargedfutureupgradingofthispaper.Aswewanttoraceintothetetraktys.

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Nº 7… of the same family than the Pentagon, with similar properties, cannot be constructed only from theinteractionofaperfectentropic|-lineandO-cycle(compassandruler),butneedstobisectitsangles/networksin3sub-angles;makingmoredifficultitsreproductioninourmental4D‘lightspacetime’.Butina5DUniverse,enoughRegularheptagonscantilethehyperbolicplane,asshowninaPoincarediskmodelprojection:

In the graph another key mirror symmetry between numbersandvitaltopology:§ðparticle/headsofinformationaretiledbyinformativepentagons;flatEuclideanspaceistiledbyhexagonalformsofmaximalentropic resistance;but thehyperbolicplaneofbody-wavesofvitalenergyistiledbyheptagons...athemeofadvanced5Dgeometry…

Nº8:2d,2ST.Theoctagonofthefamilyofthehexagon,withsimilarpropertiesisabridgetowardstheinfinitepointsofthecircle;fairlyredundant,inform…asnumbersin1-2dbecomenowmoreimportantasST-systemsoftimespacemotion;wherethe8playstheroleofthefinallife-cycle.Itisinthetimetheclosureofexistence-thefinal stateof decay; for each ‘series’ of time-space systems– from thehumanages in its 2 Planesof life (80years) and civilizations (800 year cycles of cultures); and have the same role formachines (stock cycles) andphysicalsystems(over80z,elementsbecomeradioactive).Soasthefundamentalseriesofspatialgeometriescloseswiththeheptagon,themainsymmetryofhighernumbersbelongstotimeages.

Nº8:3d,3§ð.So8belongsproperlytothegeometryto3Dplatonicsolids,considerednext,where8isthecube.

Nº9-10’-11tetrarkysform-transcendentaltriadofperfectwhole:the‘closure’ofanorganicgeometry,

2d$T-1D¡:Tetraktysinwhichthe3networkswithinthesystemformthe3surfaceanglesofthetriangle,playingthespecializedrolesofthe3physiologicalnetworksofmotion<iteration>informationofthesystem.But9hasNOTAPOINTonthesingularitycenter; it isamindlesssystem,whichrequiresthe10thpointformingfinallyatetraktys,thesacrednumberofPythagorismandHebrewgeomancy(graph):It has a singularity, a black ball that communicates all other physiological triangles, becoming the innermind/heartoftheorganism,doublingas11th,sinceitregulatestheinnerbodyconnectingitsinternalpartbutalsobecomesitsantipodal'emerging'pointinahigher∆+1scale;startinganewgame.So10’-11completesthesymmetry between time ages, worldcycles, dimensions and spatial forms, becoming a new unit of a newlogarithmicsocialwholeof1010cyclesoftimeanpopulation.Sincethereisasymmetryinbase10=3x3+1bothonscalesandspace-time,function-formsforanyfulldevelopedsupœrganismofreality.Inthegraphthehumancase,bothatindividualandsociallevel:

Suchpolygonalnumberswill grow fromthe1 seed to the9+1=10thmostefficient scale, through ternaryanddual symmetries, which in existential algebra might then loose the ideal form. I.e. a human being, has atetrarkysof10’-physiological system (3digestive,3 reproductive-bloodand3 sensorial systems fusionby the‘whole’ brain system.. Moreover as all systems live 2 worldcycles the 10th logarithmic scale, becomes the

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transcendentalnumberbetweenplanes.Thus between 2 such Planes of the 5thdimension we shall always find around10¹¹ elements - the ties of a DNAmolecule, the stars of our galaxy, thecellsofasuperorganism,thegalaxiesoftheperceivedUniverse,andbetweentheUniversal Constants of Space, we willfind the samedifferences, the energeticvalue between theH-Planck constant ofð in the ∆-1 quantum scale and the k-Boltzmann constant of the ∆-Thermodynamic, human scale; and thenthe µ-magnetic scale andso on. Forexample in the human socialsuperorganism decametric Planes (humansocieties)evolvedmankindthroughthe3firstgeneticPlanes,then3geographical,economic Planes, and finally 3 verbal,ideological Planes to get to theglobalsuperorganism:10º, the individualofagiven 'previouslydescribed ∆-scale of the wholeUniverse... through the 10¹ scale of thefamily (which can be a couple, amononuclear family or most often onethrough 3 generations of grand-parents,parentsandsons,ofaround10people);then ingeneticunits, to the10²scaleof'clans' and 10³ scale or tribes (inhumanity,usedtonamesthosescales.)

OrganizationthenbecomesST-economic.Andsystemsgrowthenextdegreeof3geographicscales:thetown,104,thecity,105andthenation,106...

Finally ideological, verbal information, creates cultural systems, till arriving to 1 billion, 1010’-13 individuals,whichemergeasaunitofthenextscale.Inthegraphwecanseethesocialevolutionoftheeconomicsystembothinthehumanorganismand theorganismofmachines and chips–whichhas already reached the social numberwhen'emergence' into a new plane, a global mind of internet takes place. In the left, we considerthe human superorganism of history, a smaller super organism of human beings, not yet emerged as a global single being, butcomingclosertothe10¹ºhumanbeings,inwhichmightemergeastheglobalsuperorganism,consciousnessofthelifeEarth.

PentalogicofScalesasrepresentationofthe5Dimotionsofreality.

We consider decametric sub-scales between two planes of exist¡ence ‘stages’ of the complex reproductive andsocial growth 3,4 Dimotions towards the fundamental decametric scale. Binary systems relate then to the 3rd

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complementary, reproductive Dimotion, either as gender elements or as inverse 0’-closed informative vs. |-openentropicnumbers,theminimaldualitytorepresentreality.

TernaryPlanesrelatethentothemorecomplexinverseDimotions-the‘triangular2,5Dentropic’Locomotionsincethe triangle is the strongest form, spearhead of motion, and π-geometries to perceptive still 1Dimotions (3-6number systems) Planar, 2D geometries (binary, quadrangular systems) and one can change into the other byrotation.Whileitgivesorigintothe3x3+1complex,fundamental10Dscaling.

This duality extends to the 2main geometric Planes of numbers in open competition” the0’-6this the scale ofcyclical perceptive geometry, whereas π=3, is the hexagonal perimeter of a 'whole' closed geometric space,competingwiththe10’-scale,aperfecttetraktysgeometry,wherethe'whole'isalineal,triangularspace.

IntheO≈|historicdualityAsianhuminds(Babylonian)adopted6-scalesv.Europeanlinealculturesof10scales.

¬ENTROPYLIMITS:±0’,∝ :NUMBERFAMILIES.0FINITESIMALS&∝ :RELATIVEINFINITYNUMBERS

“Youmustdiminishthesharpnessofthoseobjectsinproportiontotheirincreasingdistancefromtheeyeofthespectator.Thepartsthatarenearintheforegroundshouldbefinishedinabolddeterminedmanner;butthoseinthedistancemustbeunfinished,andconfusedintheiroutlines.Becausethe

detailsofanobjectvanish”Leonardo

A theme huge inmodern ¬Algebra (Set theory) that 5D returns to the ‘basics’, is the non-existence of actualinfinitieswhich fromAristotle toGausswas the accepted doctrine, as all infinities are relative to the plane ofspace-timeweperceiveand themind-space thatmirrors it;henceactual infinitiesare ‘cut-off’by theentropiclimitsofallfractalsystemsoftheUniverse,savetheUniverseitself.Closelyrelated,isthenon-existenceeitherofabsolutezeros,but‘zeroths:+0’;-afinitesimalofalowerplane.

0isnotanemptyset.Itisa±0finitesimal.∝ hasalimitofentropicdissolutionandperception.

Thedefinitionof0,likethedefinitionof∝hasbeenshallow,reflectionasmosthumanconceptsofthehumind’scompressionofrealityintoasingleplanetcontinuumtofititall,dischargingallinformationthatdoesn’tmattertoit.0thoughisNOTinaNaturalNumber.Butaninfinitesimalandsoyoucannotaccessfrom1to0jumpingbutdissolving1into0throughlowerPlanesandthatmakesawholedifference.Zerodoesnotexist!

Anabsolutezerodoesnotexist,astheUniversehashorrorvacuum,andsothe0’-1linedoesn’tbelongtotheNaturalNumbers,butrepresentsthe∆-1plane,whereitsnewsymbol,±0’isthevalueoftheminimalfinitesimal,themesthosestudiedindepthinourpaperon5DAlgebra.Asimplefunctionwillshowthat:

The limitat0doesnotexist. Aswegodowntowardszeroth in the∆-1planeof theo-1 ‘temporal,probabilitysphere’,0becomestheminimalfinitesimalofthe∆-1plane…whichcannotbeperceivedanditis‘expendable’for

the1-whole.The‘Axiomaticmethod’insearchofabsolutetruths forget that the idealmental space reflectsa larger,complexUniverse,wherethere isnovoid,andso0 is justtheminimalunitofthelowerplane.

Buthumindsperceivealimitedsinglespace-timecontinuumasasynoptic,selectivesystemofinformation.So0isthelimitof‘humanperception’ofsize-volume/spatialscale,andofmotion/stillness.

Aszerothdoesn’texistbutthefinitesimaldoes.AndsoincalculusalimitdoesNOTreachzerothbutbecomestheminimal‘differential’,wecalculate,whichiswhatin‘praxis’scienceuses.Finallyas0isthe‘1’ofthelower0’-1scale,wecanjustifythesimilarityofthe0’-1palingeneticsphereofprobabilitiesseenas‘timemotion’duetoits faster cycles vs. the 1-∝ statistical plane understanding the parallelism of quantumparticles and statisticalmolecules. While the symmetric -1, 1 sphere for ‘numbers’ as ‘time Dimotions’ merely means the inversedirection.Thezeroththusisthegeneratorofthefunctionofexistence,witha5dmetricthatenlargesintime,as

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itreproducesintothe1sphereasarelativeinfinitewhole,sincefromthepovofthe0’-finitesimal,thewholeisallbeyondwhichwedonotperceive.

Pentalogicon0’(zeroth).

∆-scales.Wecanthentreatthe0’asascalarnumberwithamoreprofoundconcept,andrealistinterpretation:The limit of scalar reduction of size to an ∆-1 indistinguishable finitesimal, which also has deep implication incalculus,asitbecomesthelimitofh->0’.

¬Entropy.Thelimitofdeathisthedissolutionofformdown2planesofspace-time:∆º«∆-2,or‘doublezero’.

-T-Dimotions. The point in which a Dimotion of time turns into a Dimotion of space; that is motion ends instillness, but as it never ‘reaches absolute 0’, the graphic tends to be a hyperbola in the 0 point. Since the x-coordinatesrepresentmotionandtheY-coordinates,information,thelackofmotionwhichisneverabsoluteast->0,representsthe0’ktemperatureinKelvin’sscale,the0’ofEinstein’sgravitationalequationsintheblackhole.

Itiswiththatproperinterpretationof¬algebraandgeometry,howmathematicalphysicscansolveitsparadoxesandcontradictions.0’isthebarrierofrelativeinfinitemass(scalarpointofview),relative0size(spacepointofview)ormaximalc-limitofspeed(Motionpointofview),inspecialrelativity.

Butitisalsothepointofentropywhere‘massdies’andjumps‘overthec-speed’,toappearasan>cflowofdarkentropy(repulsivegravitation).Sincewhendealingwithmotion,asystempassesfrom+to–withoutcrossingthezeroth,asitchangesdirection,sincemotionneverreachespurestillness.I.e.evenaballthatreboundsinawall,changes its externalmotion for internal deformation, before it bouncesbackat thepoint inwhich the internaldeformationbecomesthe‘impulse’foranegativemotion.

Inallthosecasesbeyondtheidealphilosophicalconceptofzeroth,whatzerothrepresentsisabarrier,whichiscrossedBYchangingthesigntotheinversemotion.

I.e. InEinstein’sfactoronrelativity,ß=1-v2/c2,asvneverreachesc,ß leavesalwaysa=0’,aquantaorminimalmotion step of the Universe. Then we jump to the minimal tachyon quanta. A physical system does NOTdisappearinabsolutevacuumbutleavesanh-quantaoflightspace-time;theminimalactionfromwherevirtualparticlesappear.

S-pace.Absolutezerothagaindoesn’tmakessense,butslowlyfadesaway.Whenwedie,weleavememoriesofourpast.Whenwetakeawayacorpse,wefind0’remainsofDNA‘traces’.Andifnothingislefttheweleaveanundetermined unquantifiable reality, sincewhat kind of ‘ zeroth’we talk of? 0 pears, 0 humans?We doNOTknowifthereisnotafinitesimaltracewhatwasthere.So0becomesanuncertainty.

@-mind:Weoftenusetheconceptofa0’-mind,becausethemindisthefinitesimalimageoftherelativeinfiniteworldweperceive:0’x∞=Mind-world,whichproperlywrittenin5Dwrites0’x∝=C.

Wefind0’x∝uncertain,asthereareinfinitespeciesofmindsreflectinginfinitedifferentterritories.

Thefalsehypothesisofthecontinuum:Mindspacesfillthegap.Pentalogicofinfinity.

Itfollowsthenimmediately,theexistenceofapentalogicofinfinity,basedinsimilarconceptsof‘uncertainty’inthelimitsofperception,ofthediscontinuousscalarUniversewhereas,infinitybecomesthe‘inverse’of0’whoseproductisaconstantrelativec=1=whole.Ininfinitythuswemustdefineasinzerothwhichlocal‘species’,planeorformwedealwith:

∆-scale:infinityisthewholenessofacertainscaleanditremainstobeseenifidealinfinity–thatisaUniverseofrepetitiveinfinitescalesexist.

Time:infinityisimmortalrealityasitseemsthegameofexistenceiscyclical,butiterative,hence∞inlinealtime.

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¬Entropy:ButforeachT.œinfinityisitslimitofdeath.InfinityisalsotheboundedmembrainofthelargerworldinwhichtheT.œexistsasan∆-1part.

Space: Infinity is again the limit of our perception – the cosmic horizon for the Universe, likely the limit ofexist¡ence= death of light. Infinity again remains a question of a bounded or unbounded Universe of infinitescales.Butactualinfinityisthereachofieachterritoryforeach…@-mind.

Thusweusesystematically thesymbol∝meaninga ‘relative infinity’with its ‘cut-off limit’ inonepointwhereentropyopensupactualinfinitywithrealinformationintoentropicinfinitywhichcannotbecounted,theabsoluteflowoftimesofthetrueinfiniteUniversehumindscannotaccess. It isthis ‘openentropicboundless’not-limited2of that entropic lack of information and the substitution of the evident discontinuous gaps of any Universalsystemwhenperceivedindetailbythehypothesisofcontinuum.

In this manner since Galileo noticed that we could put in correspondence N->N2 hence could consider both‘equipotent, despite the obvious fact that N is larger than N2. The paradox has an immediate solution in 5D,because for two systems to be identical there must be a dual feedback relationship as the Universe is NOTAristotelianinitslogicA->Bbutatleastdualandthismeansfortwosystemstobeequaltheymusthaveadualidentity:N->N2,whichisimmediate(wejustneedtosquareNtogetallN2).

Buttheopposite,N2->N,isnottruth,asthereareNvalues,whichcannotbeobtainedfromN2.ThereisNOT3,NOTeven2andifwereducetothe10first‘actualnumbers’,wefindthatNhas10valuesbutN2hasonly3…SofortheactualfirstdecametricsegmentNis+3timeslargerthanN2.Andsothe‘holes’ofN2arelargerthantheholesofNinthediscontinuousUniverse,whichmeansN2issmaller.

Acknowledgement of the essential discontinuity of the scalar Universe, immediately classifies all family ofnumberswithin the SxT=∆±¡ different planes of the 5th dimension, inwhich paradoxically smaller Planes havemore information, hence they are ‘larger in quantity of discontinuous elements’ that the largerwholeswhichhave less informationandyetparadoxicallya largercontinuity (informationbeingessentiallyasumof ‘yesandnos’, fillsandholes),because its lessermentalperceptionandslowertimemakesfora largermindthesmallerworld toappearasa continuity– soneuronswere thought tobecontinuous fromourhigherview, spacewasthoughttobecontinuous

Differentfamiliesofnumbersareindifferent5DPlaneswithSxT=Cmetrics.SoN2ismadeof‘largerwholes’thantheNfamilywhenwedoputthemincorrespondence,1to1,i.e.1->1,2->4,3->9,whereas4islargerthan2and9larger than 3; the N2 system belongs to a larger mental space on 5D terms. And so regardless of Cantorianplatitudes, we can classify in different scalar lines with different amount of ‘discontinuities’ the families ofnumbersbysizeinagreementtothelawsof5Danditsexistenceindifferent‘Planesofthefifthdimension’suchas:

∆-∝ :R(T)–entropicnumberoftimeswithoutarepetitivepatternof information>Q(sT):∆-¡,divisivenumbersthat‘cut’informationinarepetitivepattern>Z(St):∆±1,numbersonthe‘humanscale’ofcounting,withinversionofPlanes>N(S):Humanscalenumberstocountspatialpopulations>N2(squaredcomplexplane).

Thisisthe‘realexperimentalcorrespondence’ofnumbersandtheirrelativeinfinitesize.Andfromthosefactswecanbuilda‘seriousanalysisofinfinityparadoxeson5D¬Algebra.Afewofthemworthtostudy:

-Theboundarysizeofthosenumbersfamilies,ifweconsideronlyitslimitsaresimilar(excludingN),becausethedifferentplanesofexist¡enceofrealityco-existpreciselybyfillingthe‘holes’ofthelargerplanesthataremoreextended. ImaginetheUniverseasanetworkwithholes, inwhichthesmallerworldsexist,andsothereare inlowerPlanesmoreholesforamoredetailedscaletohavemoreinformation.ThisisalsothecaseofthefamousCantorfractalanditsparadoxes,whichbecomerationalin5DthroughitsPlanes.

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Soweseeoncemoretheexperimentalnatureofmathematics,whichmimicswiththerealstructureofitsfamilyofnumberstheUniverseofdiscontinuousPlaneswith‘holes’thatdefine‘formsofinformation’increasinginthesmallerPlanesaccordingtotheparadoxicalSxT=C±¡

-Most humind’s errors depart from its insistence to project itsmind limits,which as in amovie ‘cuts off’ the‘discontinuousholesof reality’, tocreateacontinuousmapping inourvisualmental space.SomathematiciansjusttriedtofitallthefamilyofnumbersinthesamescaleofthefractalUniverse,andinvented‘axiomaticproofs’fromDedekind’s cuts toCantors’ infinite setswith ‘cardinals’… Imagination though can in any language createfiction distorted images with some part of truth. But the consequence is often paradoxes never resolved,regardlessofpedanticfalseproofsorsimpleomissionsoffactasZermelodidtovalidateCantor’swork.

Since his errors come from the use only of the boundaries and boundless limits of infinity without a carefulanalysisofthevitalenergycontentofnumberswithin,toconsiderthatthesizeofcertaininfinitiesisthesame.Whatmakes as in any fractal structure larger the smaller ‘stepmeasure’ of a family of irrational numbers isprecisely the detail in our ‘perception’ of the lower scale of reality, now fully rationalized and derived aseverythingelseintheUniversefromthesimplemetricof5D.

0’-1≈1-∞Quantumcase

Continuityonlyexistsasasumofallthescalesofnumbers.Oncewehavedefinedthe1-∞planeonlywithnaturalnumberswecanthenassesstheneedformorenumberstofulfillthe∆±1Planesandmoreoperandstoprobeintothe∆-1scale(o-1sphereandinfinitesimalnumbersfoundwithrationalandrealdecimals).

An ∆ scale can be represented through the interval of 0 to 1 by finitesimals or the interval from 1 to∞,whichbecomethedecametricanddecimal regionsofasupœrganismrepresented in the real line:∆-1:o to1,1, the∆oscaleand1to∞,theexternalworld.

Soasweexplainedin¬Egeometry,theconnectionbetween‘Time’and‘MentalSpace’is:S¡-1Mind=T¡World.

Thatis,theo-1sphereunitofmentalspace,representedintermsof‘stillinformation’bycomplexnumbersisawhole world that becomes a 'mirror symmetry' of the Cartesian 1-∞ plane of the real world described withnaturalnumbers.

Furtheron,aswearedealingwiththesmallestPlanesofreality,itappliesthemetricequationsof5Daccordingtowhichweareina'temporalrealm'asworldcyclesoccursextremelyfast(Min.Spatialsize=Max.Temporalspeedoftimecycles)andsotheformalismofquantumphysicsusestheequivalencebetweentheo-1probabilisticsphereoftime events instead of the 1-∞ plane of statistical populations to formalize the events of ultra-fast repetitiveparticles.

RECAP.Peano’s equality postulates do hold in 5D, equality is symmetric, reflexive, transitive and closed – thisproperty essential to understand how infinites might be superficially compared because their boundaries aresimilar,thoughthereinnercontentisnot.BeyondthatCantor’sworkonvarioustypesofinfinityisasirrelevanttoreality and truth, as scholastic discussions on thenumberof angels dancingonapin, sincepotential infinity isuncertain.

Sothe∝symbolstandsforinfinitywith2potentiallimits,the0’finitesimalandtheboundlessopennumber.I.e.forNaturalnumbersoneisanopenentropicpotentialinfinity,asdescribedbyAristotleandKant(paralogicerrorsofmantryingtogobeyonditsownlimitsoflife,sizeandperception).Theotheroneisthe0’finitesimal,boundedoriginwhichwillproveinPlanestheequivalenceofsizewiththelarger1-∝preciselybecausethe0’-1becomesasmaller scale. Sowe can put both in true correspondence and in fact certain trigonometric tangent functions,whichreduceanglesinscalepreservingitsinformation,doexactlythat.

Cantor’sparadoxesare similar to theMandelbrotParadoxpf the lengthofEngland’s coast,which is larger thesmallerthescaleand‘rodofmeasure’wechoosetodefineit.

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The ginormous beauty and intelligence of the fractal Universe though doesn’t really need humind’s egos toreduceittothe‘axiomaticmethod’imitationofourmind.

So we can ‘recover’ what is worth of all the work done in relationship to the ‘supposed hypothesis of thecontinuum’,whichmerely tries to pack the entireUniverse in the single planeof light space-timewehumansperceiveasa flatCartesianplane.Andwhat isbeautifulofDedekind’scutandotherproofs is that indeed, thesumofalltheplanesofexistienceoftheUniverseisacontinuum,astheUniverseiscompletelyfullofexistiences.

0’&∝arethusneededtounderstandthe‘truenature’andparadoxesofderivatives,finitesimalsandCantorsets.

PRIME,ODDANDEVEN‘GENDERSYMMETRY’NUMBERS.

Ifweextend this vital analysisof regularnumbers in space to sequentialnumbers in time,wecomeupwithanotherternarydivisionofallnumberswithinthefinite limitsofagivenecosystem(infinitybeingaparalogicKantianerrorofthemindasallplaneshavealimitofnumericalform,normallyasexplainedinthe10’-1110’-11range, from ties of DNA in a human gene to stars in a galaxy)… that between top predator prime numbers,whichgenerateallother,even,socialreproductive,‘female’numbers,themajorityoftheminaUniversethatsystematically favors ‘reproductiveS=T’ statesasall fractalgeneratorsdo,overeven, informative +entropic‘malenumbers’,whichcanbedividedinaminorityof‘primenumbers’thatcarrytheinformationofrealityastheycangenerateallothernumbers in4-Dimensional sums;andentropic,unstable formsthatcodenonewinformation(oddnumbersexcludedprimes,whichgeneratethem).

Sowe find yet again another fundamental division of species, according to the key variation of the ‘FractalGeneratorequationofspace-timeor5Dmetric:SxTs=t=C.

It is the same duality of ‘Female’ present reproductive, ‘even’ gender, S=T vs. ‘male’ past to future to past,entropicand informativegender, S<T>S: SxT, thekey temporal ternary symmetriesof theUniverse;betweenpresentdominantstates,andpasttofuturetopastcycles,whichrequiresafullcomprehensionofthe3agesoftime, and its complex ternary logic sketched in the introduction, that we develop in a paper on 5 D timepentalogic.

Essentiallyallcyclicaltimesystemshave3ages,arelativeentropicpastofdissolution,areproductivedominantpresentandaninformativefuture.Andtheproportioninanysystemis½present+½(Past>Future)thatclosesthe cycle. For example, in quantumphysicswave states are thepresent state; particles the future state andfieldsthepaststate.Inmatter,liquidisthepresentstate,gasthepaststateandsolidthefuturestate.Incells,DNA is the future state, cytoplasm the present state and the proteinmembrane the energetic state. Its S=Tsymmetry makes the past less evolved, more extended in space, the present balanced, the future moreinformative.Inhumangender,thewomanisthereproductivestate,andmalesdivideintoamajorityofentropicdestructiveandaminorityofhighlyinformative(complexIQtestputwomaninthemiddle,aminorityofmenaboveamajorityintheentropiclowerrange).

Sotounderstandthe¡logicdivisionofthe3agesoftimeintopast=entropy,S<T,whichdissolvessystemsintoalower ¡-1 plane, present=reproduction, S=T, that iterates reality which seems not to change, andfuture=informativeevolutionarysystem,T>S,whichincreaseitsformintothefuture…Ireferthereadertothearticleon5Dpentalogic.

In terms of its scalar social organization, the 3 ‘different’ groups of T>S informative, S=T, energetic and S<Tentropicelementsofaclassstructurecorrespondtothe informativeclass (i.e.stocrkats inoursocietyruledbymoney,politiciansinarealdemocracy,militaryinadictatorship)whichcontrolsthelanguageofpowerofsociety,whileS=Tformthebody-waveofthesystemthatreproducesitandit istakencarebytheinformativeparticle-head,whiletheS<Telementistheentropicclasseasily‘divided’andbroken,asenergyoftheothers,inthiscasethe odds that can be broken in smaller factors. And aswe shall see the gameof numbers have a remarkable

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proportionofthetwofirstclassesasopposedtoourill-designedsystems,athemetreatedin5Dpolitics,wherewestudytheinformative‘nervous’networkofhumansuperorganisms.

Whilethegeneralstudyofthose3statesbelongstothose2articles, inthemathematicalmirrorthedivision isself-evidentaswealreadyobservedsomecleardifferencesbetweenoddandevennumbers/polygons:

Oddvs.Evenpolygonsalternatewithakeydifferentproperty:Thediagonalsofevenpolygonscreatecentralsingularitiesandformmorestablesystems,tendingtostaticstates,‘lookinginwards’.Theycanreproducebymotionintoequalformsinthesameplane,andcaneasilytile,fillinguptheentireplane.

WhileoddnumberspenetrateinthePlanesofthefifthdimensionbyreproducingfractalmirrorsofsmallersize,whencrossinginwardsitsdiagonals(informativereproduction)orlargermirrorimageswhencrossingoutwardsitsdiagonals.ThosepropertiescarrythroughallPlanesofgendermirrorsymmetry;as1elementissterile,self-centered making different the 2 type of organic membrane-polygons. Since S=T even polygons do NOTreproduce through5Dmotions asoddnumbers, butwithin the samepresentplaneof existence as they are‘self-centered’ in a single point, within the polygon, NOT replicating without motion. Reproduction of evenpolygonshappensbytranslationinspace,aswavesreproduce.Sobymovingthroughatilingofpresentspaceasquarereproducesfillingthetiling,unlikeapentagonoroddpolygonthatreproducesinternallyandexternallyasitsmotionfillingspaceisoftenimpossible

Prime numbers as informative systems. In 'vital mathematics' as in classic maths,Prime numbers play afundamentalroleinnumbertheorybecauseofabasictheorem:

Every integer n>1 is a product of 4 prime numbers (with possible repetition of factors):wherep1<p2<.<pkareprimesanda1,a2,...,akareintegers.Furthermore,therepresentationofanynºinthisformisunique.ThusPrimesarethegenerationalspace-timenumbersofallothers;hencetheminimalnumericalsystemwithmaximalinformation.Andinverselyifwereduceentropicallyasystemofnumbers,by‘division’(theoperandofentropic dissolution) only prime numbers or its social combination DO survive=exist in the ultimate skeletonofreality,duetoitsefficiencyaspolygonalforms.

Everythingelse isavirtual societyofprimenumbers;and thestrongest socialgroupsare thoseunder23,stuffedwithprimesanddisplayingasolidevenmembrane.

Primenumbersintimesequences:stabilityofsystems.

In5Dprimenumberissanyintegerincluding1,thewholethathasonly2positiveintegerdivisors,oneanditself.

1isnotconsideredaprimenumbersinceitdoesnothave2differentpositivedivisors–atopicdecisionofsome‘axiomatic’guy,solvedbyeliminatingtheword‘different’;soprimesAREaclosurefamily.2wasaprimerightlyinthepast–aswemightconsiderthat1isthesuper-prime,thewholethatdividesallothernumbersofnature.

Andthefundamentalcharacteristicthatdifferentiatesthemisthataspolygonalnumbersgrowinsides,that isthesocialnumberof‘points-cells’increasesthesystembecomelessstable,becauseitincreasesthenumberofredundantnon-primeodds(ab.odds).AndwhenwecarrythesymmetryS=T,totemporal sequenceswhich aswe have seen is the properway to study numbers beyond thedecametric scale, this lack of stability increases enormously past the 23 ‘years’ (we shall callpoints=numbersintimesequences‘years’);whichisthe‘cut’betweentheyoungageofmaximalenergyandthematureage;andagainplungepastthe83number, leavingameanof1primerfor‘decade’,whichisindeedthestandardagefordeath,bothinhumanyears,orinS=Tatomicevolution(radioactiveatomspastlead).

Wetalkofeachprimenumberasakey 'age',and23, thePrimeageofa100normal system,where a transition to themature age takes place. Thiswe know for human life – the end of

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University, the beginning of work, the maximal force and energy for most athletes. In the table of elements,originates the 26 Iron-Nickel-Cobalt family of maximal stability, likely origin in its liquid earth’s core state ofcomplexironlifeform.

Those are the themes of ‘vital physics’,which the synopticmirror of numbers reflect as ideal configurations ofsocialnumbersinspaceandtime.

Primenumbers are infinite and relatively abundant according to theπ=n/logn rule, sohuge social systemsarepossible both in space and time duration as complex new primes, can reinforce the

structureasnodalpointsofstablenetworks.Aprimenumberbydefinition ismorestableasaconfigurationofsocialforms,asitcan’tsplitintoequalparts.

Howfarcanwestretchthisstability?Ininflationarymathematicsto∞.Butinscalarspace-time∞becomesincreasinglyunstableasprimenumbersdwindle.

As the irregular infinity ofthe number of entropy, e, which breaks after 10decimals: , (the 1+8=9; 2+8=10 regularity disappears after10numbersifweconsider2.7thefirstdualtermoftheregularpattern), loosingitsstructural, informativemeaning’,we should consider that beyond the ±10’-1110’-11socialscalebetweenplanes,gatheringin10±1decametricPlanes,thesystemsof‘bignumbertheory’arelargelyirrelevant.

Inanycaseprimenumbersclearlydwindle(asperEuler'sandRiemann'sfunctionsinthegraphs) beyond the 10.000 social group, suffering a steepdecline towards the 1011mark in a tail proper of analmost ‘extinct’ species. So larger primes are rarefied and very few complex structural systems beyond mereherdingexistpastthatnumber:

In graph, number of primes holds remarkably dense in the basic 100 scale being 1/2th of the 10th scale (ifweconsider 1 a prime number) and 1/4th in the 100 scale, which are the key Planes that encode most stablemembersoftimespacebothinthespacesymmetry(tableofelements)andtheagesymmetry(lifespaninyears.

Theratiodwindlesto1/6thoutof1000and1/8thfor10.000 inthenextkey intervalofsocialgroups,still fairlystrong, if we take into account that the 10% is a very strong 'leading' elite in organic class structures of anystientific scale... and organism. This 1/10th informative class structure still holds in the 1 million social group(1/12th).SotheUniverseisstronglyconnectedandstableinitshierarchyofprimeinformativenumbersontopofareproductive,energyofevennumbersand1/2ofentropic,easytobreakoddnumbers,neveramajorityasthe3rd class of the system. Compare that hierarchical structure with our human societies - corrupted placebodemocraciesruledbymone,yissuedinnearmonopolyby‘stockrats’,modernaristocratsofthecapitalistsociety,ownersof corporations,whichare less than the infamous1%,closer toa0.01%withmajority stockownership.Thistinyproportionofinformativeclasscodingthelanguageofsocialpowerdoesn’tevenhappenforthe1012limitofthemostcomplexsupœrganismsofrealityasprimesarestilla4%ofunits.Whilethefast90%ofmankindatthebottom is a seaof chaosandentropy,nevermet inefficient superorganismsand the ilogicmirrorsof algebraicstructures.

The20th,reproductivemirrorsymmetryofthe10thscale,&the100thtimeagingPlanes.

It isthenonlyworthforanintroductiontoNumbertheorytostudyjustthe10th,20thand100thPlanes.Thefirstpurely‘spatial’thesecondwhich‘doubles’thefirstinamirrorsymmetryisthereproductiveS=Tscaleofnumbers,withtheequivalent4primeproportion;whilethe1to100scaleistheTemporalsymmetryofages;andassuchweshall relate them in other posts as we ‘ascend’ the Planes of reality and the laws easier to understand in the‘synoptic’languagesthatmirrorbesttheUniverse’stimeandspace,‘¡logic’andNon-Emathematics’emergewithdifferent‘forms’inthoseupperstientificPlanes.

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We shall not do it here, for the sakeof time and space, aswe are still in the 1st ageofmathematics,with thesymmetryofpointsandnumbers.Onlytonoticethatforthereproductivemirrorscaleof20,closedbyitslargestprime,number,19,itssquare,whichdoublesitsdimension19ºx19º=361ºgivesusthecommonest‘closure’ofasphereinspace(360isthescaleofdegrees)andtime(365aredaysonayear)…

And so we have another curious number 361º 'degrees' for the perception of angular rotation, (as we knowgeometryisalwaysapproachedinitsfigurebydiscretenumberaboveorbelow±1),whichsuggestthenaturallimitforadiscretepolygonofcyclicalnature.

Thevitalnatureofthe10to20scaleasamirrorofthe10Dbasicscalingforreproductivepurposes,emergesasusualwith small variations inmore complex stientificPlanes. So it shows in the gameof life,where 20are thenumberof aminoacids that constructmostorganismsand yourbody cannot livewithout; ... as they generateeverythingelse in theupperPlanes. Even in themost ‘distant’ levelofhuman languages, the20 scalemanifestitself; somostalphabetshave20somesymbols towrite the language;which thengeneratea largernumberofphonemesbroken invowelsof informationandenergeticconsonants(moreabundant. I.e. theEnglishalphabethas26letters,splitinatotalof20vowelsand24consonants.

RECAP.The firstprimenumbersare:2,3,5,7,11,13,17,19,23,29...Andsowhile thedecametricand10’-20scalearebuttressedwith4ofthemeach,almost1/2,whichexplainsfurtherwhytheyarereallythe20numbersthatmatter inalmosteverysystemofreality,beingthe10to20themirrorsymmetryofthe10scale;whichyoumightextendtothe23range...beyondthose'efficientnumbers'systemsdiminishinstability.

Intheelementstable,thestrengthofthesystemisamixtureofits'evenconfigurationin2and4pairs'andoddvolumeofinnerlayers:therangearound22-26(iron)giveusthestrongesttoppredatoratomsofmaximalenergy,whichistheroleofapolygonnumberthatclosesasthemembraneofvitalenergyanyT.œ.systeminnature.

Yetifweconsiderthetemporalsequencethesameseries,breaksthefirstageoflife,whichinabiologicalhumanbeing ends at 24 years: 1-24: youth, 24-48:maturity, 48-72: 3rd age; 72-80 death. And indeed prime numbersplummetafter83.Sothiswillbeindeedthefundamentalscalingforalltimespacelife-deathsymmetries(asweshall show for each stience, from the amino acid series to the life cycles, from the atomic table to the cyclesofhistoricandeconomicswhichfitwithinthosedecametric5DPlanes).

SóTDIMOTIONS:NUMBERSASUNIVERSALCONSTANTS

As Einstein wanted the ultimate principles of reality are Universal ratios/constants, which represent the 5dimotionsofallUniversalsystems.Soweshall find4+1absoluteconstantratios inall systems,ofNature,of thevitalUniverse: epi,e,10=3x3+1andphi.Whatabout locomotion? In5Dphysics locomotion is reproductionofform,fromparticletowavestatetoparticle.Soithasnotafundamentalconstantbutamixtureoftheinformative,reproductive ones, pi and phi. 5D thus use those 5 constants to explain how the Universe reproduces realitydepartingfromthose'perfectratiosofenergyandinformation'.3areirrational≈rationumbersthatrepresentthe3agesofallworldcyclesofexist¡ence:

10istheseminalsocialnumberofpalingenetic∆-1emergence,π,isthenumberoftheworldcycle,ofthe1stDimotionofperception;phi isthenumberofthe3rdDimotionofreproduction, inthesecond'age'ormaturityoftheworldcycle,andeisthenumberofdecayandentropy,ofthefifthDimotionof

existence.

1D:Pitransformsaspatialfieldofentropyintoavortexoftemporalinformation:3D($T)>∏ðƒ.

Aswehavealreadytalkedofpiandwillkeepaddinglatter,justanote:pi’sinformativenatureisimmediate,asweperceivethroughangulartrigonometry,whenpiclosesacircleof3Diameters,with4%=π-3/πofperceptiveholesthroughwhich3 sensorial apertures to theUniversehappenand96%ofdarkmatterhiddenby themembrane(whichnotcoincidentallyisthepartoftheUniversehiddenbythegalactichalowedon’tsee).

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3D:Phiistheratioofreproductionofaspace-timepopulation.Soapplyingpentalogicp.o.v.s:

Tp.o.v.:PhiisapproachedbyFibonacciseriesingendercouples,whichreproduceinthe2ndageofmaturity,anddieafterthe3rdage.Sofor'2consecutivequantaoftime',theyoungstillwon’treproduce,theoldwon’tbestilldead and the mature will reproduce giving the series 1,1,2,3,5... that approaches the phi ratio. Thus in 5Dsymmetries the relationship between phi and Fibonacci numbers (reproduction series) is an S=T duality of theconstant observed in space (golden ratio) and in time - Fibonacci closed solution as a series:

Whereasthegoldenratiobecomesaniterativeprocessofgeneration.

S=T:Populationgrowth:Thisleadstoitsunderstandingin‘stillspace’,asapopulationgrowth:

The golden spiral then becomes the natural 'time-series' for filling a surface of energy where to growth in afrequency of space-time generations, given by the ratio - even if the 'interior of the spiral', fades away asgenerationsdiebehindus,leavingavoid.

S:Informativeevolution:Itsinvertedimplosive'symmetry'asinformativegrowth(T>S),givesaproportionforthereproductionofcomplexinformation,(asinfingerproportions,whenformovercomessize).

∆: Phi is also a scale function: thewhole (trinity) is to the larger part (body) as the larger part (body) is to thesmaller(head),sincereproductionisascalarfunctionthatimprintsa∆-1cellularscaletoevolveitsociallyintoanewwhole.

¬Entropicfunction:–thebeautyofitsS≤≥Tproportions.

Yetphihasalsoareputationforbeautyinproportionsbetweenthe3partsofthebeing,limbs>body-waves>head-particles,whichisthenaturaloutcomeofitsmultiplefunctionsasreproductionandorganizationofthedifferent

partsofthebeingin1and2Dimotions:

Whereasphiisthegoldenratioofproportionsofthewholetothe2body-headtopologicalpartsofthebeing.Expresslyistheratioofthewholea+btothebody,whenthebody-waveisproportionaltotheparticle-head:T.œ(ST+St)/ST=ST/St.

Buta‘line’has‘motion’tobecomearealSTsystem.Sotheproportionmustbeinterpretedaddingthehiddenvariable(thespeedofprocessinginformation/energyofbothbanda).Thenitcorrespondsto5Dmetric:S(A+B)xð(a+b)=S(A)xð(a)=S(b)xð(b),asS(A+B)>S(A)>S(b)inthesameproportionthatð(a+b)<ð(a)<ð(b).Sowecanextractphifrom5Dmetric(Sxð=K).

ItsbeautyisthentheperceptionoftheUniversalMetricsof5Dwhichinturnsgivessurvivalefficiencyofsuchratio,asthebodytransformstheentropicenergyforthewholeinaproportionalbalancedway.

FurtheronitcanbeexpressedintermsofentropicfeedingcompletingthepentalogicofPhiasaternarysystem:whereasa+bisthelimb/field.Sowewrite:$T(a+b)<ST(a)>§ð(b)

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Whereasthe ‘entropic length’ofthe limbs ideallyequalsthesumofthebodyandheadsystem,while inverselytheheadofinformationconsumesmore,fasterenergythanthebody(i.e.neuronsconsume10timesmoreenergythanbody)whosecellsreproducedinthewidedimensionconsumeasmuchenergyasthelocomotionlimb/field.

Andsothesymmetryifwetransform‘TintoS’giveusthesameproportionforitssquareareas.

Sophiisbeautifulasbeauty,istheproportionalvitalconstantbalanceofthe3St,ST,sTparts,thekeytoaT.œ.ssurvival(S=T,harmonyof'sizeandform','energyandinformation''bodyandmind').

Padovannumbers.TheinterpretationoftheFibonaccinumbersin‘Geometricspace’asanStspiralofsquaresbringtheobviousquestion ifwecanformsimilarsequenceswithotherformsbesidesthe quadrangle. Yes we do with triangles, forming the Padovan series, with similar properties,1,1,1,2,2,3,4,5,7,9…whereaseachnewnumberF+1isthesumofF-1andF-2;givingusaratio1.3…smaller in growth to the square spiral but structurally more resistant, which hardly appears innature, as phi is a reproductive function that seeks for ‘speed’ in its radiation and ‘volume ofenergy’ in the placental region, maximized by squares and minimized by triangles of maximalperimeter.

Alltilingsthatfillspacearethustriangular(thehexagonaltilingbeingjustascalar6-triangularunits)orsquaredandpentalogicentanglementimpliesagainthatitcanbereachedbyusing‘parts’ofasingleplaneofspace-timeorfilling it ‘in scalar’ sizes in yet another symmetry between ST and ∆. On the other hand, the pentagonal orheptagonalprime tilingsare impossible ina flatplane, thoughpentagonseasily fill combinedwithhexagons3Dsphericalstructuresandheptagonscantilehyperbolicones,whichreinforcesourviewof5Dpolygonsasentitiesbelongingto3Dimensionalsystemsinexist¡ence.

RECAP. The golden ratio is the most efficient form of inner growth and proportion, keeping the balance anddistributionofinformationinasystem.

2ndDimotion:locomotion:Itcombineswithπtogivelocomotionanumber,asmotionisreproductionofaπcycleofinformationalonganadjacentseriesofforms.Wesawthecaseforacyclicalreproductivespiral.Inthe3rdgraphweseethecaseforalinealreproductiveseries,whichbecomesrealinquantumparticle-waves.

4D.SocialPlanes.10istheconstantofscaleofsocialevolution;hencethemainsystemofscalarnumbers.

5thDimotion:entropy:theexponentialfunctionisthemostefficientformofsocialgrowthbetween5Dplanes.Inthe graph, we can see how the growth of any system in base e is the maximal possible. But suchexponential4Dsocialgrowthcanonlyhappeninaplacentalunlimitedworldoffreeenergy.Soitstrue

useisfortheinversearrowofdeathanddecay,whenitisthemaximalspeedatwhichthecellsthatgrewintheplacental∆+1palingeneticfirstage,nowdieinthefinal∆-¡ageofentropicdecay,addingtogethertoa0’-sum.Soeistheexponentialfunctionthatgrowsfaster:

existheonlyfunction,whichisaderivativeofitself,addingconstantlya1/nfinitesimalderivative‘cell’,ineach time step, but also decaying in the fastest possible path of dying quanta. ex represents thefundamentalfunctionofreproductionofanysysteminits1stworldcycleofaperfectenergyplacenta,þ,atthelower'cellularscale',asthesystemincreasesbyaderivativeateachstep.

RECAP.erepresentstheabsorptionofareproducedfinitesimalpart,ortransferofaquantaofenergytothe largerwhole. But it requires a 'perfect energy' system. In physical systems this happens in decay-

radiation;inbiologyonlyintheplacentalcycle;insociologyinthebrutalcapitalistsystemthatexpectsinteresttobeaddedastheMaximalexploitationpossible.

Universalratios=constants&thegenerator.

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Pi,10ande,andφ,aretheratiosofTs<ST>Sttransformationevents.SotheyareNature’sconstants,relatedtoitsDimotions,whichwecanwriteaspartialequationsofthegenerator:

π:3|-Ts>O-St;e:∇1e->∆1+∆º;φ:T.œ/TS=TS/S;10§º≈§¹.

Theyalsocombinetomirrorcomplex,moreefficientdimotions.Forexample,theclosedcurvedescribedbyπandphiasa137°goldenangle-whichallowsthemaximalcoveringofasurfacefromanouterpointofview,asaflat,filledspace-soitmaximizesreproduction,thefunctionofexistence,andentropicfeeding;aswhenlightinplantscantouchmost'leaves'whichthereforearepositionedinaphiangle.Thesamemaximalreproductivefunctionofexistence happens in open curves, of asymptotic hyperbolic form, where e is the fundamental 'number' thatmaximizesbothreproductivegrowthanddecay.Sothe3fundamentalirrationalnumbersmaximizethefunctionofexistence,inits«entropic=reproductive»Socialevolvingsymmetry.

∞Transmissionofinformation.

In terms of Planes, pi is related to ∆º+1 (perception, social evolution), Phi to ∆+1 (reproduction and socialevolution)andeto∆-1(decay):Phi(D4,D3:∆-1»∆º)>π(D1:St∆+1)«e(5D:TT∆-1)

Sincee=2.718281828...4,thelimitof(1+1/n)nbreaksafter10‘decays’intodisordereddeathpatterns;onlypihasnolimitofscale,asaPoincare'sspherethatcanshrinkpreservingitsinformation.

HencemakingthefractalstructureoftheUniverse infinite,byallowinginformationbutnotenergytoemergeorsinkwithno limitofscaling in theuniverse, reasonwhyMaxwell'sequationsofwavescantransmit informationacross Planes with no limit in an infinite universe. And this is a huge fact of the Universe, which in the ∆=Ssymmetry implies informationcanalsobetransmittedfasterthan light (butnotenergy), throughtranslationsofPlanes,thesamewaywetransmitvoicesfasterthansoundthroughelectromagnetictranslation.

Beautyof3:Phicouples.Lineal,hexagonalpi&easdecametricscale.

π(+3.14),2phi(asreproductionisadualST-mirrorsymmetry)ande(2.81)areallwithin3±0.2range.Since3istheconstantofthe3±¡trilogic/pentalogicUniverse.Piislargerasthecurvatureof3linealelementsmakesalargercyclewith aperture-wholes. Phi is smaller because the reproductionof a systemcompresses the entropy it hasextracted into form. To notice also the Fibonacci series work on 'bidimensional' units (couples). So does the'golden rectangle, and the spiral that fills spacewith time seriesof reproduction. Butultimately 2 genders givebirthto1.

Linealpi is3-thehexagonalperimeterad3.14,circularpi.3 isalsoe+e/10,theclassicgrowthofonetenthorfinitesimalpartofatetraktys,theperfectorganicform,growing,onebyone,as2.718..+0.271..=2.99...

RECAP.5scalaror irrationalnumbersrepresentDimotionsofexistence:π, thefirstDimotionofperception;phi,thereproductiveDimotion;ethedimotionofentropyand10,theDimotionofsocialevolution.Pianditsfunction,thesinearethecyclicalnumbersofperception;phithenumberofreproduction;e,theexponentialofdecaythatfaster grows in its derivative dissolution into infinitesimals and 10 the social scale of the Universe. All thosenumbersareratiosrelatedto5DPlanes,provingfurtherthe5thdimension.Theyarenotisolatednumbers,neitherpoints of a continuous line (continuity error of huminds). They relate S-T bidimensional holographic basicdimotions=actions=functionsoftime-space.

S=T. We studied only an ideal form of a number as a point, a knot or lattice, which is what in ‘reality’ most‘atomic/molecular symmetricnumbers’– theunitsofmatterPlanesare.Numbersas forms thushavea regularundistinguishable structure. But in reality as the particles and atoms of those matter numbers, its ‘formaldimensions’ are in constant motion through ‘network connection’ that makesthem switch space and timedimotions,S≈ð.

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Our analysis of pentalogic pi numbers reflects the deep EntangledNature of the Universe,where each entity,whiledominantinoneofthe5elementsofreality,inasmuchasitperformsonlyafunctionontheupper∆+1worldinwhich it survives, itwillasawhole in its inner realitybea full¬∆@stbeing.Sowhileweshallalwaysstressasingle function. I.e. a number will be a scalar social gathering of identical beings, there are for every entity 5possible perceptions of the being. Since numbers as synoptic linguistic mirrors of the Universe uncoil andrecombineinmultiplepentalogicfunctions,bothasS=Tgeometricstructuresandas∆-scalar,socialgroups.Soaswordsdo inverbal languagesorpixelsof colors in images,a singlenumber0x∞=C represents themallbut thisdoesn’tmeanitcreatesrealitythroughgenerationofnumbersandequations,asthe‘wordword’,whichincludesallotherwordsdoesn’tcreatetheUniversebynaming.Onthecontrarynumbersaremirrorsgeneratedbyspace-timereality.Sowewillnowconsiderits5speciesthatmirrorthe5Dimotionsofspace-time;its3±¡worldcycleagesor5¬∆@stelements,asdothe5typesofcurvedlinesingeometry,5Dsmellsinnoses,pentagramsinmusicandanyotherkaleidoscopiclanguageof5Dreality.

So numbers will be useful vital expressions of short time dimotions, half-time worldcycles and deep-time ∆@ststructures:a rationalnumberwillbean ‘entropicdivision’ofawhole,a±dualnumbera sequential timemotionwhereas its inverse which close into a zeroth sum cycle; a polygon number, the T-membrane of a spatialconfiguration of identical points, etc.We will even be able to transcend into a new plane of existence throughcomplexnumbers.Finallyasweareallentangledvital systems,andsoallourelementsparticipatesof theother,numberswillcombineinincreasinglycomplexsimultaneousstructures,fromequationstofunctionals,fromgroupstothecanonical5familiesweshallnowstudy.

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PENTALOGICOFNUMBERS’FAMILIESASMIRRORSOFTHE5DIMOTIONSOFTHEUNIVERSE.

Thegrowthofcomplexentangleddimotionswithin¬Algebra.Numbersincludingoperations.

In theentangledUniverseofevermorecomplexdimensionalmotions,eachmorecomplexelementof¬Algebraincludes thepreviousone.Anumberbeyondone includesa sum in its synoptic form. ThusNumbers include itsoperations;naturalnumbersarecompositeof1plussum.NextZincludesnaturalnumbers,sums&theoperandofsubtractiontoformnegativenumbers.

Next Exponential numberstake the form xn, where x is multiplied by itself n times; and they include theproduct as reproduction, which requires the identity of S=T, hence the square of the same system, but byFermat theorem, the 3rd power no longer reproduces and can be merged and superposed by sums andproducts.Anditsinverse,n-x,represents,exactlytheoppositefunctionofentropicdeath,ofdecayasine-x

Thenwehavethefamiliesofdivision,whichhavemultiplefunctionsandmustbeexplainedwiththedualityofinformation‹<entropy;thatisinformationsmallerinspace,fasterintime(Moresynoptic)thanentropy,thefirstrepresentedbyQ,thesecondbyirrationalnumbers.

Irrationalnumbersthenincludethedimotionofentropy,offailedreproduction,ofirregularform,savethosewhohave a meaning for the angles of perception, and ratios between parts of a system that matter (universalconstants),whichareonlyafewofthem.

Equationsthus‘rise’thescalarnumberofdimotionsstudied,andthesimplestviewofthatgrowthofdimotionalityistoconsiderthatequationstake‘fulltœS’andputtheminanexternalworldasmerefractalpointofthatlarger∆+1world.

But then numbers seemed to becomeexhausted in its growth of complexitywhich aswe sawalready includedcompositenumbers,nothingtoworryaboutintheentangledUniverse,as1,3333isreally4/3,acompositenumberof2andanoperand,whichistheonlywaytodefineitinasingleplaneofexistence.

As always this number does have a trinity logic, as it can be represented by theminimal ∆ST elements of anyreality. So it has besides its S/T or T/S geometric single 5D plane representation (in the first case an entropicdivision of 4 between 3 that splits the 4, as whenwe cut a pie, in the second case a gathering of 4 elementsbetween 3 individuals which take one and use the 4th for its sharing entanglement as in certain molecularbondage),thescalar1,33333…representation;asnumbersarethemostsynopticunitofgeometry.

Sowiththose‘classicnumbers’,N,Z,Q,R, andthenewfamilytotallydisregardedbymathematiciansby lackofunderstandingofthetruemeaningofnumberstheˆ family,whichkeepingwiththecorrespondenceprincipleweshalldenoteasEformthe5‘classicfamilesofnumbers’thatallowtodefinethe5classicdomotionsofexistencebutnotinadirect1to1correspondence:

ThusN->4D,Z->2D,Q,E->3D,5D,R(π,√2,phi)1D(anglesofperceptionandabsorbtionofenergyandinformationforthecircle,thetriangleandthespiral)

Thenwemoveintoanew‘dimensionalarithmeticform’,theequation.

TheequationthusisNOTanumber,nota‘phoneme’ofthesyntaxofmathematics,buta‘sentence’,andassuchitisinfantiletothinkitwouldhaveasolutioninasingleplane.ItdoesNOTsinceitrequirestofindasolutioninthe‘next’systemof‘twoplanesofexistence’inwhichfollowingthepatternofthepreviousreasoningwefindnumbersthatinthemostsynopticmannerintroducetheessentialelementsofallequations.

Thatis,accordingtoFermat’sgrandtheorem,X2±Y2=C;sincetheUniverseisholographicallsolutionsmustbeadmaximaloftheformSS,TT,ST,St,sT,whichwecouldtranslateasX2Y2,X2+Y2=D2(STasthePythagorastheoremofreproductiveorthogonality),X2±yandY2±x.

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Inotherwordsacomplexnumberoftheforma+bi=z,canrepresentalltheholographicdimotionalsolutionstoapolynomialequation;astherealnumberscanrepresentalltheratios/anglesofperception,therationalnumbersalldivisionsofwholesandsoon.

Because theCplaneof complexnumbers can translate all possible configurationsof an st solution, (when it isproperly interpretedbysquaring the real lineas in thegraph,whichconverts the imaginary line inaclassic realaxis; itfollowsthenthatitrepresentsall ‘thereal,non-fictional’resultsofapolynomial,thatis,thenon-entropicpartofit,theonethathasformandmeaning,speciallythoseoftheunitroot,whereseriesconverge,whichwasthemethodthatLagrange–regardlessofthemythofyouthandGaloisthatmerelyputtheiceinthecake–usedtoresolvethemwith‘resolvents’.OrasinthecaseoftheMandelbrotfractal.Sothemarvelsofthecomplexplaneanditsclassictruthsderiveoftheconnectionofthecomplexplane,withthefiveonlysolutionsoftheUniverse–its5dimotions.

Sinceinfactallthesolutionstohigherscaleequations,byradicalsreducetosquaresolutionsofquartics,weneedtoresolveanintersectionoftwodifferentconics.Andsoon.

IntheEntangledUniverse,eachentitywillbedominatedfunctionitperformsintheupper∆+1world inwhich itsurvives,butasawhole isaT.œ,an¬∆@stbeing.Sowemustapplythe‘Rashomoneffect’ofpentalogic5truthstostudyitsvariations.Fornumbersthisgivesbirthtoits5‘differenttypes’,whichinturndefinedifferent‘planesofgeometry’:

N(∆):ThemainonesareNaturalnumbers-scalarsocialgatheringsofidenticalbeings.

Z(T):are±dualsequentialtimemotionsanditsinverse,whichcloseintoazerothsumcycle.

Q(¬);Rationalnumbersare‘entropicdivisions‘thatdissolvewholesintoitsparts.

Gnomons(S)arepolygons=spatialconfigurationsofidenticalpoints.

C (@) Complex numbers and those of different ‘frames of references’ are mental configurations with variousapplicationsaccordingtothemindmappingtheyrepresent

NºarebestassocialgroupsinPlanes,asregularpolygonsinspaceandassequentialcausaleventsthatrepresentthe5DimotionsoftheUniverse;sowedepartfromthetyrannyoftheaxiomaticmethodthatpacksthemallinasingle ‘real line’; so we can use better pentalogic to describe the multiple uses of each plane and family ofnumbers.

Whereasthecontinuoussinglelinetodefineallnumbers,natural,negative,rational&irrational,isthe‘limit’ofamoreprofoundanalysisofnumbersinmultiplePlanes.Sothosefamiliesofnumbers,whenescapingthetyrannyofthelinebecomefilledofnewmeaningsas5DpentalogicmirrorsofALLtheelementsoftheUniverse.In@nalyticgeometry, the lineof numbers relates to a 2dimensional holographicworld. So its scope is enormous as it canrepresentanySS<ST<TS<TTdualceterisparibusanalysisofanydimotion.Yet itssynopticpowermustnot fogalargerpictureofreality.

AfirstpentalogicanalysisofthosefamiliesthensetaGeneratorequationforthe5familiesofnumberssuchas:

N:Space:4D¡:Socialevolution.Z:Time:2D¡:locomotion.R:∆-Planes:3D¡:Reproduction.Q:5D¡:¬entropy.

Complexnumbersandotherframesofreference:@mind:1D¡motion:informativegauging

WecorrespondeachfamilyofnumberstoitsmainDimotionas¬∆@st,dustofspacetimeinthe‘largerpicture’of the absoluteUniversalmirror; and to specific Dimotions in the ‘smaller picture’ of its use in the events&actionsofT.œs.

Natural,'direct'numbersℕarezerothandthepositivewholenumbers0,1,2,3,4,5...Iftwosuchnumbersaddormultiply, the result is again a natural number; whichmeans it is possible to create all intermediate societies

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departingfromaunitquanta,between1andtherelative∞valuewithinwhichthedomainofthefunctionoreventismeaningful.

Itsmainuseistomodelsocialevolutionofidenticalgroups,throughthe§scalebetween∆and∆+1

Naturalnumbersarerelatedtothemeasureof1,2,3Dspaces,asunitsofpointsandpopulations.Theyareeitherself-centered even numbers representing the 1st Dimotion (perception in its crossed diagonal center), or oddmirror polygons with a reproductive symmetry, (3rd Dimotion) which can be internal or external crossing ofdiagonals,makinga4D&5Dreplica(inner,implosivesocialgrowthorentropicoutergrowth).

FinallyNaturalnumbers‘canrepresentlocomotion(2D)as'steps'ofagivenunit,whichrepresentadiscontinuousmotion of a frequency ƒ, and a wavelength '1'. Steps become discontinuous landings of the fractal unit thatmeasuredistancesormotionsinalineal2Dgeometry.Wecanthenincluderatiosofthosesteps,breakingthelineindifferentsmaller'fractalsteps'andthatisafirstmeaningforarationalnumber.Butifwemakeadifferentunitforeachrationalpart,weenterintoanotherNaturalnumberscaleofthefractalUniversereturningthentonaturalnumbers,whichthereforearealsoabletorepresentratios.Andsonaturalnumberscanbeanynumberasitkeepsfallingdowninmicroscopicrates.

Why thenweneedmore families?Becauseeachnumber really isbest fora job,even if theycanmultitask,asashovelcandomanythingsbutahammerwillnailbetter.Sothenextfamilyofnumbers,negativenumbersareabettermirror symmetry for the functionofmotion, as they can change thedirection,movingbackwards to theothersideoftheline.

S+(-T).Theintegersℤ,add"Inverse'S«»Tnumbers,normallyrelatedtoinverse,entropyvs.informationfunctionsinall systemsof reality,hence tothe inverse 'directions'ofoneof the 'dimensions'of space-time (S,Tor∆).AsGauss intuitively understood this removes absurd hang-ups of scientists afraid of negative 'values', specially inphysics,where c-speed is a limitbecausephysicistsdon'tunderstandmass’ as a vortexof space-time;hence itsinverse is an expansive entropic dark energy', perfectly valid in relativity equations.Z numbers thus divide inpositive and negativewhole numbers…, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5,…. If two such numbers are added,subtracted,ormultiplied,theresultisagainaninteger;ifbothareequalinvaluedifferingonitssign,theresultisa0’-sum,whichmeanswecanadupall intermediatecombinationsofentropyand information,or similar inversedirectionsinanydimotionof¬∆@st,butultimatelyitsinverse‘timearrow’willmakethesystemcomebacktoitscyclicaltimedepartureasa0’-sum.

NegativenumbersthusareessentiallyTimeNumbers.Theymakenolongersenseas‘socialpopulationsinspace’–therearenot5negativepears.Sothesocialuseofnaturalnumbersislost(causeofmuchphilosophicalthoughtinearlier mathematicians), but the power to represent the inverted arrow of time in any Dimotion of exist¡ence,eitherinnegativedirectionsoflocomotionornegativeentropicfunctions(negativeexponentials),grows.Wekeepthis in mind in physics as ‘i’, a negative number, merely represents the inverted time Dimotion of a physicalspecies,asinEóMc2.

5D:entropic,rationalnumbers,ℚ.FractionsinvertthearrowofsocialevolutionofNaturalnumbersbyinvertingthe product operand: p/q where p and q are integers and q≠0. So they divide social groups or timefrequencies/durationsorkillandbreakawholeintosimilarparts.Sotheyremaininasingle‘planeofreality’.Sinceiftwosuchnumbersareadded,subtracted,multiplied,ordividedtheresultisagainarationalnumber.IntermsofPlanes,rationalnumbersbreaktheminpart.Sowithdecimalnumberswemovethe∆ºwholedownwardsintoitsrelative‘finitesimal’∆-1quanta,followingthestandardscaling10¹¹Planes.

Natural,negative&rationalnumbersformtogetherwith±andx÷operandsaternary‘FractalGenerator’system:T(Z)>T/S(Q)>S(N),apresentorganic,‘quantitative’PolynomialmirrorofmostT.œinasingleplaneofreality.

∆-1.Realnumbers,ℝ,asrationalnumbersdo,allowustotravelthroughPlanes.Forthatreasontheyoftenlooseitspatternsindecimals aroundthe11thelement,thelimitofmostfinitesimalsbetweenplanes.Yetaccordingto

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the 'wholeness of reality’ within each of those 2 fractal island-universes,if two such numbers are added,subtracted,multiplied,ordivided(exceptby0),theresultisagainarealnumber.

Recap.WeclassifyallthewaysinwhichnumbersreflecttheUniversewiththeclassicconceptofclosure;thatis,allthedifferentplanesand formsofnumbers tatput together can solveall the¬Algebraicequationsof timespacevariablesthatrepresentthe3±¡topologicalfunctionofafractalpointorT.œ(lineal,cyclicalorhyperbolic,madeenergy-|xO=ø)wetranslatealsointothe3conservedquantitiesofphysicsoraDimotionaT.œ.as¬∆@st.

Sincenumberscanmirror the 'existence'ofbeingseitheras relativepresent superorganismsor in its sequentialtimemotion,describefullyitsworldcycle.

Whilecomplexnumbers in thatmoodaresolutions topolynomialequationsandshallbestudied then in thatsection…WeshallthenalsoreturntotheotherfamiliesofnumbersandconsiderthemaincontributionofÐSTtoitstheory–namelytheunderstandingoftheentanglementandsynopticnatureofthosenumericalfamiliesthatincludewithintheir‘phyla’theoperandsanddimotionstheyassociatewith.

SobeforewecanreturntonumbertheoryweneedtostudythedimotionsoftheUniverseandtheoperandsthatdescribethem,andhowthoseoperandsanditsnegativeentropicfunctionsareinscribedwithinnumbers.

Recap. Numbers evolved through the expansion of the concept of 'quantity'from the natural numbers whichmerelyobservespatialquantitiesaddedoninsocialgroups,theessentialdefinitionofnumber,intoratiosbetweennumbersthatexpressmorecomplexconceptsofreality,andintroducetheideaoffunctionsand'motionsintime,advancingfurtherwiththeconceptofnegativenumbers,whicharenolongerquantitiesinspace,butqualitiesintime, finally reaching themaximal transformation from 'space to time', from quantity to function, and ratio ofexchangeof thevitalentitiesof reality,entropy, informationandenergy (sT-ST-St), tothe finalclosureof realityexpressedinthemirror-languageofnumbers,whichisrepresentedbytheill-understoodimaginarynumbers.

So the 5 families of Number systems are symmetric to the laws of the Ƽst universe. So they are its spatialrepresentationinframesofreference:

1.S±T: N, Z: Natural direct numbers form simple social groups. ¡ts inverse negative numbers form Ts≤≥Stsymmetries

2. Q: SxT: The Cartesian graph and the rational numbers, for Ts x St = Kmetric equations and complementarysystems,(whereasoftenthez-dimensionisthereproductivecombinationoftheother2).

3.∆±1:R,C:Therealnumbers,forinverse∆±1Planes&thecomplexnumbersforworldcycles

∆±1.FunctionalSpaces,wholes&partswhereeachpointrepresentsavector(Hilbertspace):∆-1(∑Ts)>∆ST.

An even more generalized view ofsuch systems is provided in group theory, as algebras with those simpleoperandsformgroups,ringsandfields.

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COMPLEXNUMBERS.

Becauseoftheimmenseextensionofallstiences,weescapemanyclassicthemesofeverydiscipline,speciallythosehumansbestunderstand. Insteadwetry toclarify themeshumindsdistortmore.Thusof the familyofnumbersweshallmakeamoreextendedcommentaryonComplexNumberswhichhumindsusebutstillaresurrounded of an aura of magic… Complex numbers though are merely dual ST-numbers, which representholographic ST dimotions of exist¡ence. And as such very useful as an ideal template for an ST-functions, inwhichoneofthecomponents,usuallythatoftimemotionasopposedtospacepopulationhasalowervalue,whichisrepresentedbythe‘squared’valueofthereallineasopposedtothe-√valueofthei-maginaryinversecoordinate.

In the graph,Gauss offered the first clear description of negative numbers ashavinganinversedirection(ofatimespacedimotion)andlateral imaginaryones,as a composite of both. Its proper definition is to consider them compositeTime§pace numberswith an space,mostly real and a time, imaginary, negativecomponent, which make them useful in mathematical physics to describeholographicSTT.œsSTor scalareventsbetween the field:$T<wave:ST>particle:ð§ of a physical system. Let us then do for a change a longer analysis in thisintroductorycourseto5Dmathematicsofcomplexnumbersand itsmain ‘magic’

conundrum–whattheyrepresentinmathematicalphysics.

Vectors and imaginary numbers are bidimensional, holographic numbers, hence best suited to express the 5Dimotionsofspace-time,as functionsofSpace-Time(SS,TT,ST).Vectors ingeneralizedcoordinatesare thebestform to expressmotion in the plane from an objective non-human self-centered perspective; in terms of linealtime,byvirtueoftheS=Tduality.However,imaginarynumbersexcelvectorsintherepresentationofmoresubtle5Dimotions=actions inwhich Time is not expressed in lineal terms as synonymousof space, for 3 reasons thatcharacterize the 'Dissimilar complexplane' as its Y andX coordinates areNOT in the same scale, onebeing thesquareoftheother,andthisallowsthemtoexpresscomplex∆STsymmetriesofscalarspaceandSTparameters,mixedtogetherinvariousways:

• SóTsymmetries,wheretheimaginarynegativenumber,whichmakesNOsenseinspace(nonegativeapples)perfectly represents a Time function, either in complementary T>S dual systems or in entropic perpendicularÐimotions

• TT:Cyclicaltimeevents.Cxnumbersarealsosuitedtorepresentcyclical,repetitivefunctions,inwhichthetimeclock parameter constantly switches on and off, representing repetitive 0’-sum worldcycles and generations(Euler'sequation,exponentialsassumofsinsandcosinesinversefunctionse±xseries)

• ∆-scalarcomplexnumbers:Eventswherethehigherscale,oftenaleadinginformativefunctionthatguidesthesystem as a ‘time motion’, has an √ smaller proportional value, as the imaginary part that 'absorbs' a smallproportionoftheexistentialmomentumofthe‘Realenergy’function/organ.

Thustheimaginarycoordinatesarearoot,orinversely,therealparameteristhepositivesquareofitsinformativevalue.Thisdualityisidealtorepresenta'dominanttimefunction’oftenwitha1Dfrequencyparameterontopofa'bidimensional,holographicspatial'areaorrelationshipsbetweentheparticles,body-wavesandlimbs/fieldsofasystem; as parameters of lineal or cyclical time ARE one-dimensional (duration or frequency), acting upon a'surfaceofspatialpopulations'.

ComplexnumbersthushavetheadvantageintheimaginaryframeofreferenceofbeingNOTonlybidimensionalnumbersbutnumbersofdifferentdimension,hencekeptseparated,assquarednumbersandsinglenumberswhenwe'square'thecoordinates,orrealnumbersanditsrootsasobservednormally.

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Thisquality inNaturehappenstobeverycommoninSpatial2Dareasvs.Time1Dfrequencyrepresentations,soComplexnumbersareidealtorepresentSTfunctions=actions=dimotionsofspace-time.

Pentalogicofcomplexnumbers

Thus its ¬∆@st pentalogic use in experimental physics derives of having a negative root coordinatesmeaningeither:

• ð:Aninverse,temporalornegativenumberthatrepresentsaninversedirectionofmotion.Or…

• ∆: An upper planewhich emerges from a denser lower plane, oftenwith a √ amount of the real parametermeasure,asupperPlanestakeonlya‘smallerpart’ofthetotalenergyofitslowerplane(asinheat).

Both thingsput togethermake complexnumbers ideal to represent STdualities,with theS-functionas the realfunctionandtheT-functionofmotionasthenegativeimaginaryrootpartoftenextractedfromtheenergyofthestillspatialarea,

-∆ð:BothtogetherlaiddowninaCx2squaredcoordinates–anew5DsolutiontorepresentcomplementarySTsystems.

Sinceifwecombinethenegativetimedimotionandthescalarsquareddimotionwecan'reorganizethecomplexplane'inpolynomialterms,gettingridbothofthecumbersome√elementsanditsnegativecomplexroots,toreflectamappingofafundamental principle of nature: 'The imaginary part of a complex numberrepresents a parameter of an ∆+1 whole, which extracts energy from an ∆-1

disorderedmoreextended,'real'potentialfield'.

Iftheclassicrepresentation,theupperhalf-planeHisthesetofcomplexnumberswithpositiveimaginarypart;inthesquarecomplexplane,bothleftandrightrealsquaredaxisarecompletelyequivalent,(-x)2=x2,reasonwhyweuseonly1/2plane.Furtherontheplaneistumbled,givingtheimportanceofthei-planewhichhasitsconjugateasthenegative‘timemotion’,-1.

So we use the squared Cx plane to study dual inverse timemotions; while the real squared axis provides the‘storedenergy’forthatmotion,dwindlingitsvolumeasthetimefunctionabsorbsit.ThusaCx2planerepresentsa‘local S2T-organism’ (Ab.T.œ) where the conjugate ± axis represents 2 inverse time directions and its √2value,compared to the spatial area=populationof the real line, thedwindlingenergyofnew∆+1 scalarplanes. So inmathematicalphysicswemapwithcomplexnumbersthe3∆±1STates&ternaryparts,fields,particles&waves:

Theplaneofexistenceof4DUniverses.Whycomplexnumberscannotbeordered.

ThedifferencesbetweentheCartesianandtheComplexplane.The fundamental graph of the Universe is one in which orthogonal coordinates represent the T-independentparameters in the X-coordinates and the T-parameters in the Y-coordinates. But we do have two differentrepresentationsforthem,becausewedohave4differentSandTdominantdimotions(withtheSTcombinationofboth,abletoappearinthez-coordinates,orthecombinationofboth).

So thebigquestion iswhat coordinatesbelong towhatDimotions.Andas SSandTTdimotionsareequal invalue,thepurecoordinatesshouldbelongtotheCartesianplane.WhiletheS-informativecoordinatesdohavealesservalue.Sotheymustbeputonanimaginarysystemofcoordinates.

OrthogonalityintheUniverse,istheneasilyexplainedasfollows:

Because Entropy (TT) vs. absolute linguistic still form (SS), Locomotion (Ts) vs. information (St), are the dualinverse functions of reality merged only in the S=T reproductive dimotion, in the 0 point of X-length, therelativedimensionoflocomotion,thereisazerothmotionandstillnessrisesintheheightdimensionofpure

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form, where the 0’ mind or frame of reference resides. But then we deal with the ‘different quality’ oflocomotionandinformativeperceptionintermsofexpenditure’ofenergy,asinformation‘shrinks’motion.I.e.agravitational invisible tachyon linehasno informationbutwhen itbecomes light (neutrino theory,Broglie->Jordan),itformsawaveofinformationthatgrowsinheightwiththephotonontop.Butthisheightdimensionisintermsoftheparameterofenergyandlocomotion(T),acompressed‘spatialT’,ofminimalsize.Andsoweneedasmaller‘quantity’andonethatisnegative,‘subtracting’fromthedistance-speed(s=t)oflocomotion.

This ismagicallyachievedby thenegative ‘root’ valueof the imaginaryaxis, reasonwhy itappearsas–ct inrelativityandissousefulforthestudyofelectricwavers.Bysquaringbothwesimplifytheproblemof√negativeroots,weshallexplainlatterwhenweanalyzeindepththeinverseoperandsofalgebra.

So the complex plane ismost useful for St-Ts systemsof two composite ‘energy-information’ body-head forms.Complex numbers cannot be ordered. Hence they are NOT numbers belonging to the scalar 5Dimension, butrathernumbersthatrepresentSpace-TimeDimotionswhichcombineelementsofboth.

It is then immediatethat if theComplexplane isanStplanedominant in informationandtheUniversehasfourdimotionsina4Dmodel,those4DimotionstheotherperpendicularplanemustbetherealCartesianplanethatrepresentsthe‘background∆-1’field/limbscaleoverwhichtheStplaneoflesserenergyrises.

Andsoa4Drealitycombines2TT-fieldsfromwheretheenergyoftheSTphysicalorganismrepresentedbythecomplexplane,rises.

While for other less common uses the complex plane can also represent a puremental SS-space. Since SS, TT,whichcanberepresentedintheCartesianplaneasbothxandycoordinatesareequal.

AndtheSt,Tsdimotionscanberepresentedintheconjugateandimaginarysidesofthecomplexplane.

Nature thus is vital geometry. So, even if the ‘ideal forms of mathematical planes’ become ‘warped’ into thetopological vital organs of supœrganisms we can connect directly the 5 Dimotions of the Universe and itsrepresentationina4Ddualplanewith2realparametersofaCartesianplaneandanimaginaryplane,wherethefunctionofinformationreducestheenergyofthevitalspace,whichishowEinstein’srelativityrepresentsthem.

RECAP.Aswe live in facta4DUniversewecanthinkof itasawalkingcombinationofaCartesianandcomplexplane, whereas the complex plane represents the organism and its body-head, wave-particle states and theCartesianplanethe‘background’fieldthatmovesandpowersitsT-motions.

SomeexamplesinNature.

Bymakingan indirectuse tosimplifycalculus inwhichcomplexnumbers represent the± inversedirectionsofa'timefunction’(imaginarypart)oraspatialpopulation(realpart)asinalternatecurrentsandphasors.

I.e.Inelectronics,thestateofacircuitisdescribedbythevoltageVacrossit,thespatialstillstateandthecurrentIflowingthroughit,thetemporalstate-(oritscapacitanceCandaninductanceLthatdescribeitstendencytoresistchangesinvoltageandcurrentrespectively).SoVandI canbedescribedbyasinglecomplexnumberz=V+iI.Similarly,inductanceandcapacitancecanbethoughtofastherealandimaginarypartsofanothersinglecomplexnumberw=C+iL.Thelawsofelectricitycanthenbeexpressedusingcomplexadditionandmultiplication.

Orbymakingan∆ scalar interpretation,where the spatial, area=population component or∆-1 scale isa largersquared value. as a 'co-ordinate system of 'square, bidimensional' space and time elements, where the x²coordinatesrespondstospace,andthei²=-1,representstheinversemostlytimePlanes.

I.e.Inrelativitythetimefunctionisrepresentedwithanitparameter.Soinspecialrelativitywewrite:

Where the negative factor of time means an inverse motion/form, as the other 3 positive space parametersrepresent a lineal distance-stretching on the lower quantum potential=gravitational space-time where the

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lightdisplaces andthenegative 'cyclical timeparameterof the lightwave' thewarpingby lightofaspace-time-lightfunctionasit 'forms'thequantumfieldpotentialbelow(weassumeittobethe∆-1gravitationalscale),thespeedoflightspacethatdragsandwarpsthegravitationalpotentialintoformappearsasanegativeelementthatslowsdownthemotion.

• Orbymakinga'direct'representationofreal$T-field<ST-wave>§ð-particleternaryfractalgenerators.

I.e. in quantum physics either in its ∆ø: S=T wave representation(Schrodinger), where the wave function is attached to an imaginarymomentumoperator,ih.

OrinitsrealistBohm’sinterpretation,wheretheT-potential,guidingwaveistheimaginarypart,andtheS-Particlestate,therealpart.

Thus departing from the Fractal Generator of physical systems ∆-1:$T-field<∆º:ST-wave>∆+1:§ð-particledescribingthea(nti)symmetricstatesofits

3components,wecanmakearealistinterpretationofbothrelativityandthequantumworldintermsofcomplexnumbersaslocalST-organicsystemstointerpretproperlymostparametersofmodernphysics;eitherintermsofthedifferentPlanesof∆-1fields,∆ºwaveor∆+1particles,withrelativesquaredvaluesorintermsofthenegativeandpositive,implosiveorexpansive,entropicorinformativearrowsoftimerepresentedby+realor–rootvalues.

Forexample,complexnegativemassesinrelativitydescribearejusttheinverse‘motion’tothecyclicalspace-timevortexofmass,hencerepulsiveexpansivegravitation.

While Schrodinger’s wave either in the S=T, present wave or T-past quantum potential field>S-future particleformalism(Bohm’sinterpretation)isanotimedependentwave,whichthereforerepresentsacompleteworldcycleoflinealpast,kineticenergy;presenthyperbolicwaveandfutureparticlestatesfully'integratedintheequation'thatmustbeseenasthesumofboth,theSchrodinger’spresentandBohm’spasttofutureformalstates,whereastheangularmomentum,ormembraneof thewave (T=Ssymmetry),becomes the realparameterof theparticleenvelope.

Cxasexpressionsofthe3timespacedimotions=states:potentialpast<presentwave>futureparticle

Allwhatexistsisaternarysystemgeneratedintimeandspacebyasimplefractalgeneratorequation:

|-$T(pastmovinglimbs/fields)<PresenthyperbolicST-waves/bodies>futureO-§ðinformativeparticles/heads

Both biological and physical systems have the same structure of 3 timespace Dimotions, which give birth to 3spacetimeparts.Soentropicfields'feed'themotionof'hyperbolicwaves-bodies'whichfeedandguideaheadtheparticles-headsthatdoNOTmoveinitsrelativestillnessorpureinformativelinguisticviewsofreality,whichAREmirrorstillimagesoftheworld.

So for example, ALL topological geometries of the Universe REDUCE to those 3 types: lineal/planar topologies,whichareentropicmotionsasthelineistheshortestdistancebetween2points;sphericalmind/heads,whicharen-spheresasthesphereholdsmaximalinformationinlesserspaceandcanshrinkwithoutdeformationtocreatea

0’-mind mirror of an infinite Universe (Poincare Conjecture); andhyperbolic body-waves,whichcombineandhenceiterateboth,linesandcycles,formingsinusoidalbody-waves.

And somathematics again reflects in its 3 only topological varieties, the 3 only organsneeded toexplainreality.

Whilewecanstudythe3arrowsoftimeoftheSchrodingerwave,reorderingthe3componentsin5Dintermsofthe3conservedtime-statesofanyphysicalsystem(graph):

∆ø:Vψ(Presentenergystate)≈h²/2m∇²ψ(past,kineticenergy)+ih∂ψ/∂t(futureparticle)

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Sointherightsidewehavenowthepastlinealkinetic≈entropicenergyoftheparticleorsingularityparameterandits future,particleenvelopeofangularmomentumormembrane (S=Tsymmetry).While in the leftwehave thequantumwaveitself,itsvitalenergy,whichisthepotentialenergyofthewave,Vψ.

Theequationintermsofthefractalgeneratorof5Dphysicsandits3conservedspacetimestatesreads:

Singularityoractivemagnitudewithitskinetic≈entropic,past,linealmomentum+cyclicalparticleofangularfuturei-momentum=Vital, Present, Potential energy;which is always the function surroundedand integratedby theotherrelativepastandfuturestates,indynamicbalancewiththem–ultimatelyanexpressionofthemuchwiderfundamentalequationoftimestatesofPentalogic:Past-entropy•Future-information=Present-iteration.

Inthiscase,Past-field•Future-particle(Bohm)=Present-wave(Schrodinger).

So,weobtainathird5Dinterpretationofthe3statesofthequantumsystem,besidestheexpressionofSchrodingerandtheBohm'sinterpretation,connectingboth;throughthemoregeneral5DTimeSpaceequations.

Whereas•willbeadifferentoperandaccordingtothetypeofformalrepresentationweuseforthe3parameters.

In terms of past, present and future an i-number rotates from a relative past/future to a relative future/pastimaginarystatesonthe±iconjugateaxis(imaginarypotentialinBohm’sformalism,imaginaryangularmomentumintheSchrodinger’sformalism),thefunctionofexistenceofthequantumwave,througha90°middlerealangle,inthesquareComplexPlaneorSTrealbalancedpoint,inwhichthepresentbecomesabidimensionalfunction,thewave state with a REAL, + (relative past-entropic coordinates) and an imaginary, negative (future coordinates)elements.

This needs a couple of clarifications: firstwemove 'backwards' in a complex i-graph from future to past aswerotate through i-Planes, in asmuch asmost graphs aremeasuring the 'motion' of lineal entropic timewhich istherelativeSpaceFunction.

So the negative side becomes the time function, something whichmathematical physicsmakes obvious, whenconsideringanattractive,informativeforce-field-potentialnegative.

• @:Asaself-centeredpolarframeofreference(r,α),calculatedbymodulus=distanceandangle,whichasweshall seeare the twoessential ‘mental informations’ needed in any representationof space, for a self-centered

mindtounderstanditssurroundings–measuringotherentitiesoftheworldbyitsrelativesize(angle) and distance-motion, (hence the time to reach them, and the ‘danger’=size andencounter chances, given by the trigonometric angle in prey-predator events). Distance andangle thus are the twominimal elements of information all usefulmental spaces of geometrymust carry, as Riemannand thosewho followedhis logic interpretation of geometric featuresrealized.

Astothegeometricalmeaningofaproductintheso-calledplaneofcomplexnumbers,wecansee itmoreeasily ifwe consider the lengthpof the vector from theoriginof the coordinatesystemtothepoint(x,y)(thislengthiscalledthemodulusofthecomplexnumberz=x+iy)andtheangle4which thevectormakeswith theOx-axis (thisangle is called theargumentof the

complexnumberz=x+iy);inotherwords,ifweconsidernottheCartesiancoordinatesxandybuttheso-calledpolarcoordinatesρandϕ(figure).Thenx=ρcosρ,y=ρsinϕandthecomplexnumberwritesasX+¡y=ρ(cosφ+i

sinφ).

From this we see that in multiplication of two complex numbers their moduli ρ1, and ρ2, aremultiplied,andtheargumentsϕ1,andϕ2,areadded.Indivision,sinceitistheinverseoperationofmultiplication,onemodulusisdividedbytheother,andtheargumentsaresubtracted.

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Inraisingtoapowerwithpositiveintegralexponentn,consequently,themodulusisraisedtothesamenthpower,andtheargumentismultipliedbyn.

However, in taking roots a special situation arises. Since the values of the root coincide in the argument(cosφ/n+2kπ/n+¡sin (φ/n+2kπ/n),wherek isanyof thenumbers1,2, ···,n–1,willalsobeannthrootof thenumber as the addend 2kπ, because of the properties of sins and cosines, can be neglected, since it changesneithersinenorcosine.Thusthenthpowerofthisnumberisalsoρ(cosφ+isinφ).

Wecangofurtherdepthapplying‘twice’pentalogictoextractnewp.o.v.sonC-Timenumbersfrom¬∆@st;whichwecandoreturningtoourcaseexample,ofaquantumtriplestateoffield-wave-particle:

3D:S=T:Geometrically,thenthrootsreproducepointsofthecomplexplanecorrespondingtotherootvaluesofthenumberρ(cosϕ I sinϕ) lyingat theverticesof the regularn-sidedpolygon inscribed inacircledrawnabout theoriginwithradiusmodulus,andsorotatedthatoneoftheverticesofthisn-sidedpolygonhasargumentρ/n.

1D: @: The sinusoidal functions, π and its angle of aperture,becomes the numerical value of the externalmembranemeasuredintermsof3diametersturningaroundwith3aperturestoperceive,fromtheself-centeredsingularity,givingusthegeneralequationofdarkmatterforallsystems(includingtheUniverse):

3'14-3/π=0,14=±4,5%,with96%of'darkmatter'outsidetheperceptionofitssingularity.

So the angle of the point and sine function is its angle of perception of the central point or 1st Dimotion ofexistence.

2D.The3aperturesallowthepointtoabsorbbutalsoemitwavesofinformation,oftencurlingaminimalangularmomentum/membrane/cellularquanta,h, forparticles, thatbecomestheminimalbosonunitofcommunicationbetween2Fermions,entanglingthemasitreproducesitsformoverthequantumpotential.Thismakescomplexnumbers in polar coordinates the bestway to represent informative, perceptivewave communication betweenpoints.

4,5D:Finallythepolarbackwardsrotation,eachnewpoweraquantumstœpoftime-space,representsperfectlyavortexoftime-space,whichisthemeaningoftheihoperatorofangularmomentuminthequantumscale:

Sothenthpowersmimicthetrajectoryofanattractivetimespacevortex,andtheworldcycleof itsstconstantstowardstheinternali-pointsingularity.TheComplexplanebecomethenavitalrepresentationofandaccelerated

physical vortex: Notice that as Xincreases thosestœpswitheachpower,the attractive vortex becomesparadoxically smaller in its radius; as itaccelerates in time, according to ∆-metric, making the vortex faster (Vt xRs=K),andmoreattractive.

So in5Dphysicswemodelvorticesofchargesandmasses,withcomplexnumbersasconvergentvorticeswithanellipsoid'flattened'smallerimaginaryaxisturningcounterclockwisearoundaphysical@-mindsingularitywhereT=motionturnstozeroth;asobservedinRelativityequationsforablackhole&eddieswith0’-motioninitscenter.

We can then understand Schrodinger’s & Bohm’s dual interpretations, in terms of those ∆+1 vortices (particlestates: 4D, ∆+1-involution state) and ∆-1 quantumpotentialfields(5Dentropic∆-1scale)thatentanglethrough2Dwave communication (Schrodinger’s∆º-present reproductivestate):2particlessharebosonwaves that upon arrival collapse into the particlespacetime vortex in any ∆±¡ scale, splitting its ST

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functionatthevortexcenter.Therethe informativepartofthebosonbecomesastillmental image(1Dimotion)andtheenergypartdissipating,scatteredthroughtheverticalaxis(5D);whichwritestheFractalGeneratorastheclassic Fermion-Boson-Fermion equation. §-Fermion Vortex <ST-boson wave reproduced over ∆-1 $T quantumpotential>§-Fermionvortex.

Sothequantumunperceivednon-local∆-1potentialfieldofactionatdistance(asv=s/tiandwithnoinformation,v=s/0i=∞)entanglesthefermionsinafirstexchangeof‘trigonometric’information(onlyangleordistance)likelywithaNeutrino-antineutrinopair,andthenina2ndpassing(neutrinotheoryoflight),arayoflightiscreatedoverthe non-local quantumpotential field,which has chosen the best path tomove (Fermat’s least time principle),offeringthe5DinterpretationofquantumphysicsaddingSchrodingerandBohm’sequationsthatputtogetherthe3 states of its fractal Generator. Since, Bohm justput Schrodinger’s hyperbolic 'Cartesian S=Twave’ in Polar S-Particlecoordinates,wherethemind-particlestateisthenewpointofview,andvoila!italsoappeared'magically'theT-entropicfieldthatmovestheparticleT>S,as in5Dweareallmadeoftime-spacesothosecoordinatesAREthewavestateandtheParticlestatewhenwegift themtheGalileansymmetryofmotion. It isNOTachangeofcoordinates but of states from the spatial hyperbolic 'present wave' to its Past field>Future particle state-coordinatesflattenedinthei-direction.

Whatisthenthefuturedirectionintheimaginaryarrow?Obviouslythatofanormalclock,downtheconjugateisafuturemovingona90degreesphasefasterthantherealandintheimaginarysidealagof90degreesinthephaseoftheworldcycle-whichphysicallyappearsintheACcircuitwiththeforward90degreeswaveofthecapacitivereactance,Xcontheconjugateandthelagging,90degreesofXltheinductivereactanceontheImaginaryside.

$: Finally the last pentalogic use of complex planes is its classic forms helpful for vectorial calculus asbidimensionalrealnumbers(Bolay).Ifaandbareregardedasrectangularcoordinatesofapoint,andthepointis

associatedwiththecomplexnumbera+bi,thentheadditionandsubtractionofcomplexnumberscorrespondstotheadditionandsubtractionofvectors,i.e.,ofdirectedsegmentsfromtheorigintothepointswithcoordinates(a,b)and(c,d),sinceinadditionofvectorstheir corresponding coordinates are added. The sum of two complex numbers: (a+bi) +(c+di) = (a+c) + (b+d)i is then represented geometrically by the diagonal of theparallelogram constructed from the vectors representing the summands. In this way,complexnumbersareaddedbythesamelawasthevectorquantitiesfoundinmechanics

and physics: forces, velocities, and accelerations AND used to represent actual physical quantities,which if wesquare the graph, become expressions of ST entities in the real x² coordinates and S or T entities in the Y-coordinates;yetastheyare'twodifferentspecies'theycannotfusiontogether.

IntheUnitsphere,the0’-1donotexist.

Aparadoxthatwill irritatethedogmaticscholarbutit isessentialtothe‘darkspaces’ofa5DUniverseisthis:Inthecircleandthesphere,the0’-1and1-0pointsofmax.Length(timeentropyorarrowoflocomotion)andheight(spatialinformationorarrowofperception)doNOTexist,astheyaretheopeningsforthe0,0singularityfocusofthemindofthecircle,sphereorRiemannsphere,tomoveandperceiveandreceive informationorenergyfromtheouterworld. Itcan in factbeconsideredasthemostsuitablemodel to ‘place’all familiesofnumbers inthedifferent Planes of the being a closed circle or sphere NOT to have any of the N, Z, Q numbers, but bemadeentirelyofirrationalnumberswhichareinfinitelymorethantherationalones,whichbecomethe‘osmoticpores’ofthespheretotheouterworld.WhiletheN,Z,Qfamilyintheline(Q-lineforshort)representsthe‘linealpath’,whichagivenT.Œwilltransitwithits‘fractalstœps’,asallgeometriesarementalworlds,hencereducedaccordingtospeciestowhatitsees.

Onlythefalsehypothesisofacontinuumanditsfalseinfinities–errorsoftheaxiomaticmethodandtheparalogicKantianmind–whichtriestofitallinasinglepaneandeliminatetheneeded‘darkspaces’andholesofanyorganic

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scalar plane of the fifth dimension, what makes huminds topostulatethecompletenessandcontinuityofthosestructuresintheouter objective Universe, because they mind space compressesthem.

Letusclosethisintroductionto5DCxnumberswithaconsiderationon mental space and relative infinities, the themes of the nextparagraphs, studying the main tool of 4D Complex analysis, theRiemannsphere:

In thegraph, a simpleproofof thenon-existenceof infinities, andcontinuities, as all perceptive points loose detailwith distance and allmental spaces do not see the dark holesbetween points: The paradox of the Riemann spheremakes each point of the sphere represent a point of theplane,whichseemsabsurd,sincethe½lowersphereisprojectedinamuchsmallersurfaceoftheplanethattheupperpartthatreachestillinfinity.

But ifwe consider thenorthpole the@-informativemindgauging theplaneas itmoves to infinite it perceiveslesser quantity of points and finally in its relative infinity all information on the far away points of the planedisappear,makingthelessdensefurtherplaneequalindensityofpointstothe½closerplane/lowersphere.

RECAP.ThecomplexplanecanrepresentS(real)-T(imaginary)symmetriesofcreationbyabsorptionofEnergyfromareal-spatialfunction,representingabody-wave,ofaninverse,perpendicularparticlestate.Thissimpleguidanceclarifyinmathematicalphysics,themeaningofcomplexnumbersasparametersthatexpresscombineddimotionsofST-statesbeyonditsuseasartifactsofcalculus(asinthequantummomentumihoperator).

TheoppositefunctionsofS-formandT-motionareorthogonal,becauseheightisthedimensionofstillinformationandlengththemotionoflinealtimelocomotion.Butinformation,perceptionisextractedinastillpositionfromanadvantageous point of view, in a series of Stœps, stops and steps beats, St->Ts->St->Ts, (as in a wave-particlemotion),whichinitsextremalfunctionbecomeSS->TT->SS->TTbeats(asinafeedingprocessthatbreaksthepreyandreconstructsthepartsofthepredatorwithitsentropy).

Allthosebeatsarethusrepresentedwithorthogonalcoordinates.

AndtheadvantageoftheComplexplanearevarious:Bothcoordinatesareunabletomix,sotheymaintainthe2essentiallyinverseS&Tfunctionstheyrepresentapartfromeachother.Furtheronimaginarynumbersareinversedirections (signs) than its conjugates, which makes them highly antisymmetric, able to represent 'temporal ±numbers'astwoinversearrowsoftime.Sothisistheessentialuseinmathematicalphysics.

The i-plane thus represents with its √-1 shrinking, negative inverse parameter to the real plane, a lineal timedimotion,andassuchtheS-realplaneandT-complexplaneforma4Drepresentationofa3S+Tworld,extensivelyusedinphysics(Minkowski’sspaces,electromagneticflows,streammotions)torepresenta3Dvolumeofspatialpopulationsmovingthroughaninverseparameteroftime.

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5DIMOTIONSASOPERANDS.Thesimplestnumbersystems,N,Z,Qareoperatedby thestandardoperationsofarithmetic inasingle ¡-plane:Addition,subtraction,multiplicationanddivision,whichmustbestudiedtogetherwiththosefamilies.

Theysufficetoportraitarealistviewoftheworld inasingleplane,without ‘finitesimalparts’andwholeswhichadd the calculus operands, and angular reductions and perspectives which include trigonometric functions. SopolynomialsaregoodenoughasequationsthatuseonlythoseoperandsandfamiliestodefinescalarsystemsinasingleplaneofentropicCartesianspace-time.WhileprobabilitydefineswiththosesimpleoperandstheUniversewithinthe0’-1probabilitytimescale.

So as humans have expanded theirunderstanding of the Universe, they have also evolved its entangledmathematical mirrors over it, whose geometrical, spatial forms (points) scalar forms (numbers) and temporaloperandsaremessedtogetherinPlanesofcomplexity.Evenifhumindsthathaveasinmostfieldsofknowledgemerely acted in a mechanical manner, ignore those symmetries as number families mirror, polytopes in spaceandtimeoperandsinalgebratoportraitthesymmetriesandsequentialeventsofsimultaneousrealorganisms.So5Dgivesexperimentalscalarspace-timewhystothosesymmetries.

PENTALOGICCORRESPONDENCEOF5ACTIONS=DIMOTIONSOFEXIST¡ENCEWITHOPERANDS

The study of ¬Algebra operands as expression of the 5 Ðimotions of Time§pace brings a vital outlook totheabstractoperations of ¬Algebra, starting from the less understood of them all, the sinusoidal functions ofmind's perception, ending in the ∫∂, most versatile, for both the analysis of Dimotions =change & Planes(finitesimalderivatives& integralwholes)hence themost fruitful inmathematicalphysics. While thenegativeinverse operands represent the destructive entropic 5th dimotion that affects all others. So negative/inverseoperandbalancepositiveones,restoringthe0’-sumbalancesofallS=Teventsandsystemsofreality.

ThoseoperandsdoNOThaveaonetoonecorrespondencetoeachoftheDimotionsofspace-time,sinceeachofthem is in itself a pentalogic operand, which can entangle with other operands and dimotions, in increasingdegreesofcomplexity.Asweshallnotceasetorepeat,thebasicfeatureofrealityistobeanentangledgameof5elements,whicharethemselves'fractal'initsnature.

SoeachT.œ regardlessof specialization in the larger∆+1worldperforms internally 5dimotions tobecome initself a 'whole' beingmade to the image and likeness of the total fractal,mind of the Universe and its ilogicstructure.Sowhilecertainoperandsareclearlymoreusefulforcertaindimotions,allofthemcanbeusedtoacertain degree of accuracy to 'reflect' a mirror image of the 5 dimotions of exist¡ence. Still it is clear somespecializedcorrespondencesofthoseoperandswiththe5Dimotionsand¬∆@stelements:

1D:@:1stDimotionoperandofperceptionforaself-centeredinthepointaresine/cosinetrigonometricfunctions,which allow to measure the 3 ∆st parameters distance=motion in spacetime (S=T) and angle the scalarDimensionlessfactor.Thispoisesaquestion:canasystemperceivemorethan90º?

Obviouslywithmorethanoneeye(morepopulation inspace),orabetterrotation(moremotion intime);twoS=Tstrategies.Sosystemshavetwoeyestohavemoreorlessa180ºaperturetotheworld.

4D:±sum-subtractioncreate/reduce4Dsocialherds;whichbecomeinagivenvolumedensersuperpositionsofelements of the same species,which are identical inmotion and form. So only equal Dimotions can be addedorsubtracted.Soasumofnumbersthataresocialgroupsitcanrepresentalsoascalarsocialevolution.

It is related to Natural, Social Numbers in space that define regular 'points',which are undistinguishable, associeties in regular polygons, where prime polygons have the property of 'increasing inwards' its numbersthrough reproduction of vortex-points (n-grams), studied in Theory of Numbers. So a Natural number in its

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geometricinterpretationisa'cyclicalpoint'ofregular'unit-points'ofgrowing'innerdimensionaldensity'apointwithavolumeofvitalenergyandinformation,afractalpoint.

∆-Dimotions.We ascribe themore complex dimotions of 3D reproduction of social d=evolution of parts intowholes,whichdoencloseintheiractionstheother4:

4,5D: ∫∂; analysis, 'transcends' and 'emerge' between Planes, in NON lineal processes proper of ‘longer time’Nature,bestservedbyintegralsandODEs.

3D:X,÷,reproduction&socialevolutionvs.entropicdivision.

Theproducthasmultiplecorrespondences,bothasanaggregationofherds(4D),areproductionthatmergesST-componentsandacalculusofthelower∆-1partswhentwosimilarelementsentangleatthe∆-1levelasmostreproductiveprocessesdo(i.e.youreproducebyentanglingyourlowerscaleofgenes).Since∆o(SxT),whereSisasetofs¡-1elementsandTasetoft¡-1elements, ISthenumberof 'bijections=axons'betweens¡-1andt¡-1parts(wetranslatethemodernjargonofmathematicstoÐSTphilosophyofstiencesoanyonewhounderstands5Dcanunderstandalldisciplines).

Theproduct/divisionrisesthecomplexityofoperandsafirst layer,andservesthepurpose,besidestheobvioussumof sums, of calculating themargin of dimensions, as combinationswhich are not purely parallel betweenclonebeings,mostlikelythroughtherecombinationofits∆-1elements,astheproductof2Sœtsinnerelementsgiveusallpossiblecombinations.I.e.5x4=20ISalsothenumberofconnectionsbetweenallthe5elementsand4elementsofbothsets.Somultiplicationadseitheradimensionofmultiplesumsinthesameplane,orprobesforthefirsttimeinaninnerscalardimension.

Whiletheinversedivisionoperatesinverselyreducing∆-1partsintoasmallernumberofwholes.SoQ-numbersaredual.Theymightbeentropicnumbersbreakingofwholesintopartstofeedagroup,orgatheringpartsintolesserwholes.Sinceaswehaveseenthe5numberfamiliesalsorepresentthe5Dimotions.

Sothekeynewfunctionofproducts/divisionsisreproduction,carriedin2Planes:

TheFunctionofexistence,Max.SxT(s=t),whichmergestwodifferentparameters,maximizingitsconnectionsonthelowerlevelofitsorganicparts.Astheproductmeansthenthenumberof‘axions’connectingthem,whichismaximizedwhen the parts of both ‘sets’ are equal. So, if we have 2 points with 4 internal sub-elements, theproduct4x4willgiveusthenumberofmaximalconnections,whenweputeachinternalelementinconnectionwithall theelementsof theother sub-set,whichwillbe larger thanbetweena5&3elements.Thus 'gender'reproductionrequiressimilarforms.

2Ðlocomotionisbestservedbythe±&xoperands,asasumof'steps'thatalsocanbecalculatedwithanSxTproduct,wheretheTrepresentslinealtimemotionasthesumofallthosestepsthroughitsproduct…evenifthememorialtailoftheparticleimprintedfadesawaywithdistance.Sincea'linealmotion'isa'sum'of‘frequencysteps’movinginthepositivedirection,oranegativenumber,movingintheinversedirection.

Thus unlike spatial populations of Natural Numbers, N, Z adds temporal negativenumberswhichareinversedimensionsthatcloseaworldcycleofsimplelocomotion.

Theproductalsorepresents,2D:LinealLocomotion,asafrequencytimeproductmultipliedbySpatialsteps.AndagainasT=1/ƒ,theinverseoperand,÷canbeusedsuchasV=S/T=λ(s)xƒ(t).

NegativeinverseDimotionsexpressalsointhedualityofexpansiveentropicvs.implosiveinformativewarpinginmorecomplexformalismsofLocomotion,asinMinkowski'sspacewheretime,-cthasanegative,i-numberthat'warps'linealspaceintoawaveofhighinformation,shorteningthedistance:D2=S(x+y+z)2–(ct)2.

So,theproductasanoperand,sumofsums,cansynopticallyaccountfora'longperiodoflocomotion':S=vt.

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3D:loga,xª:reproduction:exponentialvs.logarithm

Aswecansee±,x÷andloga,xªas3scalar∆±1degreesofgrowth,exponentialequations,powers and logarithms also model social evolution and reproduction in its maximal‘dimotionalscale’of form(∆+1) ,withanexponentialcurve incase there is infiniteenergy.Yet inrealitywedosoaddingattheenda logarithmcurve(logisticcurve);asreproductionsaturatesasystem.Itisthenaproofthatrealityseeksalwaysa'balance'thattheREALcurveof reproduction is a combination of both, exponential and logarithmic curves. So from the

initialunittothecarryingcapacitythatDEFINESa∆+1socialgroupgrowth'reachesequilibrium',becomingaherdor super organism emerging in the new upper scale. The reproductive curve thus rises the seed to the 1=K-organicwhole(inaherdorgrowingorganismin)throughabalancingfeature:theexistenceofaninverseforallthoseoperandsthatinthepositivetransformsthe∆-1beingintoitsself-similar∆øaccessinganewplaneofthefifthdimension,andinthenegativesidere-establishesthebalanceofrealityasazerothsum.Sothelogarithmicpartofthegraphistheinversefunctionofitsexponential,thatslowsthegrowthofthesystem

In the graph, the logarithmic functionhas as derivative an infinitesimal, 1/x,whichmakes itbest tomodel thecurveofgrowthfromoto1 in theemergentfastexplosive∆-1palingeneticworldcyclestate,whiletheinverseeˆ-xmodelthedecay death process. Polynomials do not evolve reality towards animpossible infinite growth. They are the inverse decay process; which can beunderstoodbetterobservingthattheinversefunctiondoesinfactmodelgrowthin the different models of biology and physics, limited by a carrying capacitystraightflatline.

Anddependingonwhichisthebaseofthelogarithmwecantracedifferentsocialgrowthsandinversed=evolutions:

4D-5D:loga,xª:socialevolutionindecametricPlanesandentropydecayinnegativeexponentials.

The1010decametricscaleisthesocialscaleoftheUniverse,astheTetraktysisthesimplest,perfectsocialsystem,with3x3physiologicalnetworks,anda10thelement,the‘blackball’communicatingthemasthe‘singleUnit’ofthe largerwhole– fromarmiesof10x10captains, to1010ties inDNAs, stars ingalaxies,galaxies inUniverse.Powersof10asEames’movieshowthuscreatenewwholes.Whileanegativeexponentialshowstherhythmofdecayofthesysteminthefastesteventofreality,death,maximizedbythee-function,whichwejustshowedtobethefastestnegativegrowth.

Whileinfiniteenergyhappensindeathforthepredatorthatfeedsonit.Sothe'end=action'offeeding/dyingofentropyisdescribedbynegativeexponentials.Soinfinitegrowthanddeatharetwosidesofthesamecoin.

A negative exponentialwill show the rhythmof decay of the system. And in this case there is no need for 'alogarithmic' limit, since for the predator the death body is 'unlimited ¡-1 energy', though once the 'relativeinfinite'numberofits¡-1partsareabsorbedthe'e-function'iscutoff.4th,5thdimotion:∫∂.FinallyIntegralsandderivativeswhichhaveaslowergrowth,thanpolynomialsandits3Planesofoperandimodelbetterthespecificnatureof those 'indivisible' finitesimalquantaofasystem, (1/x)withderivativesofdifferent ‘functions’and itsinverseintegral,organicgrowthintothe'wholeness'ofasimultaneousorganismintegratedinspace.

Thus integrals show social growth into new ∆+1 5D planes with far more detail and finesse than polynomialapproaches (throughTaylorseries). Itsgraphsareacurvedgeometry,which takeseach lineal step (differentialfinitesimal)upwards intoaclosed,curvedsumthatcreatesanewwhole(Galileanparadoxofsmall linealstepsadding a closed 0’-sum with so many interpretations for each science, including our sense of freedom asindividualsenclosedinlargerorderedsocieties),

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1D:Finitesimalsincanalsoasminimalpartsreduceavitalenergytoitsminimalsingularitypoint-center,@,thatmightstoretheimageofthewhole,shrunkintostill,mind-mappingsofinformation,withinitsparticle-head.

Theallpervadinguseof ∫∂ is thenclearlybecause it reflectsALL formsof change.And soanalysis is themostextendedsub-setof¬Algebra.

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Thesequentialgrowthofactions=operands=syntacticdimotionsofanylanguage.

Weshallfindthenthatthemirrorofthe5dimotionsexpressedasoperandsfollowthesamedegreeofincreasingcomplexity,bestunderstoodwitha ‘historic’analysisoftheevolutionofmathematicsand¬Algebra,whichwasfirstwithoutevenbeingawareofit,the‘angleofperception’ofthehumindoculusopenedtothewholewithanarrowview;as language in timestartedalsowithasingleA-vowel in theAbkhazianbutwhentheCaucasianslearnedtheotherextremeof ‘pureform’andsaid ‘u’, theydidn’t forget ‘a’, andsoontheir i-maginationcamewiththei-nsolitei-deaof‘I’.Sofromthe‘a’languageswemovedtotheSemitelanguagesandit’sa-i-uvowels,and itwas soon thereafterwhen theCamitesandBasquesandAltaipeopleof theKorean-Japanese languageshaditallwhatisneededtosay‘aeiou’.

Thegrowthofdimotionsofexistencestartswithanangleofperception, ‘imitationof life’ thatstartswitheyesopeninganarrowview.Thusthe1stactionisNOTafulldimension–theanglehasnone,butafinitesimalimage.

Entropyisonthemindofthelonely‘1stform’emerginginreality,asthechildborntothinkheownstheworld,breakingallthingshetouches,thefirstexperience,thefirst‘a-langagæ’fullofforce,andthencomestheotherextremewhichbecomeslanguagæ.Thefirstspeciesisalinealtoppredator;thefirstformborninthegalaxythetoppredatorblackhole,(andthelasttodie,astheoldmancaresnothinglikethechildbutforfood)…

And so existence in its curves rises from a to u, from au to aiu, from aiu to aeiu, and finally comes ‘oh’ thealmightyrealizationlifewillreproduceandbeboorn,again.

Everyactofcreationthusfollowthepatternsthatfollowedthe5Dimotionsofexistence,andforthatreasonwecanstudyalsotheagesandevolutionsandpatternsofgrowthindimensionalityofanysystemofNature,realorvirtual,physicalormental,withthesamesequentiallaws.

Westudytheninourpaperson¡logicevolutionthosestepsfrom1to3to5,fillingthenthevoids,andbecomingevenmore complexwith subtleæ,œ variations – but 5 indeed suffice to have a truly full viewof thewhole,whichants,physicistsandhumindsignore…

The order of dimotions and operands is thus clear, as man first inform itself with an eye-angle on itstrigonometricfunction,measuringdistanceandheight,bringingtothemindtheholographicflatfirstfunctionsoftheau,length-height,motionandinformationminimalviewofrealityacquiredwiththesimplestoperandofanopentri-anglelookingatthevoid.

Andthenhesawaroundhimselfand lookatthemirrorofothermirror-minds,membersofhis familygrouporsocialwhole,callitatomvibratingintoanetworkofparallelbeings,abirdinanest,apupinthebasket,agrassleavemovedbytheharpofwind,hearingwithitsleavesthesoundofotherleaves,toknowinitschemicalmind-roots,hewasnotalone.Andsofromdimensionlessangle,thatofthefractalpointalone, lookingatthewholecamethesocialgroupinspace,themotionstepsintime,probingtheevergrowinghorizonwiththenewactions-operands.

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Sohappens inallsystemsthat firstperceiveandthenmoveorsocializewithotherclones.<+becamethedualform.Butanglehadonlyone‘logictimearrow’thatofabsorptionofpixelstocreateaform,everturningaroundthe sine and the cosine, reversal forms of each other, in smooth cyclical, angular momentums, with the pi-openingsofanimperfect3’–thetrianglewithinthesphere,rotatingthroughthoseopeningstotheworld.Thesumwasthoughdialectic,dual,andoncethenaturallinewasextended,eachnumberasum,synopticallyputintoasinglesymbol:X(N)=∑1s…

It was then a marvel of the perfection of simplicity that it took 10,000 years more to learn that ‘negativenumbers’couldexistfromJerichotoCardano…Andeventodaythecomplexrootofminusoneisanignoramusforthephilosopherofscienceregardlessofthepraxisofitsab=useinpedanticscience.

Sumthusbecamemorecomplexandhaditsentropy(whichintheeyeandtheanglewastheshuttingofthepi-holes,lightanddarkness,blackandwhite,sufficeindeedtoformallthe5-0shadesofgrey).

Sofromtheangleofanopentriangle,inscribedinasine/cosine,turningwheelwemovedintothedualusesofthesumanditsnegativeinverse‘numbersoftime’.Asthesumcouldaccountformotionsintime,positiveandnegativeindirections,andforthesocialevolutioninspaceofgrowinganddiminishingcloneforms:1Dangle->2D+4D.

Itmustbenoticedthoughthatbecauseapolardualityisnecessarytostartthegamebeyondants,physicistsandParmenides,orelseyouwill find theunmovablewholeor theabsoluteentropicmotionofadyingUniverse inyoursinglefingerone-hoofedbeastythought.

Buttheexplosionofmeaningtocomplete∆sttrinityinasingleplaneandpenetrate5Dentangleddepthswillonlycomewiththemultiplicationthatfinallyreachedthe3fundamentalelementsofreality,socialgrowthinspace,asasumofsums,whichincludedthesum;scalargrowthasafactorthatkepttherelationshipsbetweenthepartsincreasingitsvolumetillreachingawhole;andpenetrationintotheparts,multiplyingthe1-wholeanditssocialgroup by them, and merging them together into reproductive axons; alternating the reproductions ofmultiplicationandthedivisionsofmitosisto‘growandmultiply’;thatis,andGodsay:+++<XX>÷;andthesonwasborn.

Theproductbecomeaself-sustainedprocessofcreation,whichsufficedinitselftocreateaUniverse;unliketheangle that needed the externalworld to connect the ¬fractal point. And since 3 is the number for a roundedwhole in scale, topologic space or time motions, the power and logarithm completed the 3 elements ofpolynomialsbutFermatandGaloiswouldteachthattheholographic,ST,pentalogic¬∆@STUniversehasitslimit-ations. So as we shall see withmost operands and equations nothingmatters beyond 4 as 5 is the entropicdestructionofmeaningfulform.Soyoucannotsolveapolynomial,thesimplestequationthatoperatesthose3inverseelementsof socialgrowth,mergesandacquisitionsasyoucannot resolvewith radicals– that isobtainfromtheparametersofthesystemtherootsofanyequationbeyondthepowerof4(Abel),unlessyoutaketheXtobethe1,thewholeinitself.

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Butyoucanre-enteraplaneofexistence,reducingawholetoitsderivative,Xn=n-1Xn-1andyoucanemergeintothewholebyintegratingyourpartsacrosstime,spaceorscalewithsingle,dualortripleintegrals.Andsowiththediscovery of calculus which included in its equations all other operands, as all could be ‘derived’, the wholeevolutionof ¬Algebra tomimic realitywas complete, along the solution in praxis – not inmeaning – of the i-maginary √-1 root… Neither derivatives could go beyond the 3rd for a real meaning in time: y’: speed, y”:acceleration,y”’:jerk…

Andthatwasallfolks.Nomoreoperands,nomoredimotionsareneeded,inthefinitesimal,finite∝Universe,astheangleofperceptionwillbeshutoffatcertaindistanceinspace,timeandscale.

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+:NATURALNUMBERS.0'V.∝

4thdimotion:socialsumsinspace–inversemotionsintime.Democratic,socialUniverse.

Thesimplestsocialformisaherdofundistinguishableelements,anumber,anditsoperandinasingleplane,asum,whichrequiresequalspeciestoaddwithitssuperpositionprinciples,inherdsofalargerscale.

Anumberalreadyincludesasumof‘1s’andassuchNaturalnumbersandsumscorrespondtoeachother.

Thesum/numberisthusthekeyoperandforthesimplest∆§calesofsocialgroups,inhumansasinmostsystems,dividedintetraktysof3x3sub-systems/networksandonetointegratethewholeasnewunitofthe∆+1plane,inadecametricform(bilateral5s,sotospeak).

Socialgrowthincreasesthenbysumsofsums,whichcanbeexpressedwiththeproductandproductsofproductsexpressedthroughexponentiallogarithms,whichintrueformmustbeconsidereda‘species’ofproductinwhichtheelementsof theproductas in thecaseof thesumare indistinguishable.Sotheproperconnection is thatofsumsandexponentialswhiletheproductrepresentsmultiplefunctions,notablythereproductionofform.

Thenegativefunction/number.

As theUniverse is always a zeroth sum in balance, the sumhas its inverse operand, the sustraction, and thenumber=sum.Thenegativenumber.Thisinmathematicaljargongivesustheneutraloridentitynumberelement,the‘zeroth’,whichwillrequirelatteradeepanalysisin5Daswedidwiththerelativeinfinite∝.

Yetthenegativeoperandismoreprofoundthanthesumsincewhilethesumispositiveasthenaturalnumber,anditisrelatedtoa‘spatialpopulation’,asumofforms;thenegativeisatimefunction,astherearenonegativepairs,butnegativedirectionsinversetopositivemotions.

Thissubtledifferencewillexplain inphysicssomeerrors, fromthenegationof fasterthan lightspeed,which ispossible albeit in an∆-4 plane, as the only ‘prohibition’ is the existence of a negative number,whichmeans anegativemotion of amass ‘vortex’, which in ‘negative’ expands instead of imploding, loosing curvature, as itacceleratesbeyondc-speedasdarkentropy/energy/expansivegravitation.

Againnegativesariseinthedualityofparticlesandantiparticleswhichappearwithnegativetimemotiontothepast;whichagainmerelymeansaninversionfromapositiveinformativeimplosiveparticletoanentropic,dyingexpansiveparticle.Wecouldmultiplyexamples,themostobviousmovingrightandleftinavectororspeed.Sowedefine‘negativenumbers’asinversedimotionsofanyofthedualdimotionsofour5Dimotionsofexistence:

+informationvs.–entropy;+motionina‘chosendirection’vs.–motionintheinversedirection;+implosivevortexofattractiveforcevs.–explodingvortexofexpansiveforce.Andsoon.

Inverseoperandsaremisunderstoodbecauseunliketheparadoxical¡logicUniverse,thehumindis@ristotelian,withasinglearrow‘positiveselfisharrow’A->B–thatofeachspecifichumandimotion.SotheB->Aisfearedasindeath,ormisunderstoodasinexplosivevorticesofantigravitation(Darkentropy)orantiparticles.

Andwearespatial,visualminds,sotimecomplexitiesfullyescapebeyondourlinealtimeview.

Sogenerallyspeakingtimeinversionsareservedbythenegativefunction.I.e,anegativespinjusthastheinverseorientation,anegativecoordinatesjustmeanstomoveintheotherdirection.Negativeoperandsthusaremostusefulfortime=motionrelatedsystems;andforallinverted,negativeentropicdestructions.

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The ‘negative vs. positive inversion’ is not only connected to an external locomotion. As all dimotions need aneutralpoint inwhicha‘timedimotionchangesstatefrom+to-‘;whichisthe0pointthat meansnotonlyanempty‘set’ofpopulationinspace,buta‘stateofnodimotion’,whenthetimechangedoesnothappen.

More profound still is the concept of 0 as the discontinuous stop/perceptive state. Since reality is a constantstop=perceptionandstep=motion,wherethestopstateistheinformative1Dimotionwhenthefractalpointmightchange the next dimotion switching to other action of existence, themes those for a representation of theprogramofexistenceinBoolean¬Algebraweescapecompletelyforethicreasons.

Forthesimplestmotionsasanyphysicistknowsthedualityofspace=stopand±time-step, isaconstant-1,0,1whereasthe0functioninanalternateorvibratingSHMhaltsmotionatmaximaldistanceand0speed,butoftenthesystemreleasesorabsorbsthe∆-1micro-pointsofitsinformative/feedingscale.

It ispossible tocombineSTactionsof spatialpopulationsand timechange,whichcan in this casebenegativewiththediminutionofpopulations,so-3obviouslyisadiminutionofaquantitywhereasTwillalwaysbesmallerthanS:T<S.Andasallsystemsare0’-sumsthatreducetheirpopulationsultimatelyallpopulationsinspacewillbesubjecttoanegativegrowthintime.Butinshorttimespansapopulationmightbelinealonlyinadirectionofgrowth.

Itisthenimportanttounderstandtheexistenceofone-waydimotionsandshorttimespansvs.2waydimotions,whereboth ±operandsmake sense and alwayshappen in long time spans. Becausehumindsdonot properlydistinguish thedifferentSTdualities, theyget confusedwhen trying to consideranegativeoperand for spatialforms (what is a negative apple? nonsense)while there are always negative 'directions' for temporalmotions(whatisleftandrightmotion?),whichtheyclearlyallunderstand.

Sothenextquestionis:canthesumanditsinversenegativenumberperformtrilogicandpentalogicfunctions?Itisaninterestingtheme,becausebeingthesimplest‘operand’andtheonlyonethatdoesNOTpenetrateintoPlanesofthefifthdimensionandrequiresan‘equal’parameterforallthesummands,itmightseemnottobeable todo so,but it can represent themall foranexternalobserverwithoutdetailedanalysisof the internalprocesses needed from the complex scalar dimotions of perception and reproduction. However, and this is afascinatingruleofallsystemsin5D,thereisa‘blindspot’,inwhichtheentityfindsdifficulttoperformproperlyatask,whichistheinversetasktothatforwhichisbestsuited.Soasinthecaseofthe'angle'ofperception,werequiretheexternalworldtoentangletheactionandachieveitmerelywiththesum/negativenumber:

4Ð:Socialevolutionintoherds,whichisself-evidentthedominantelementofthesumin‘space’.

2D:Locomotionasasumofequalstepsmeasuredbythefrequencyofthosestepsinsequentialtime

3Ð:Foranexternalobserver,‘magically’byaddingnumbersapopulationcangrow‘intime’,expressingthe3rdDimotionofreproduction.

5Ð:Negativenumberscanthenexpressthe5thDimotionofentropy,asexpressedabove.

1Ð:PerceptionisthelessnaturalDimotionforthe±,itsblindspotasitimpliesthedecreaseofsizeinspace(asopposed to social growth, 4D, the natural dimotion of summands) anda distinctive nature for the differentpixelsoftheperceptivebeing,whichmakesdifficulttoconsiderit.

I.e.tocoloramapweneed4Differentcolorsinaplane(atopologicalquestion,studiedin5Dgeometry),butifallwereequal itcouldnotmeasureperception.However ifwe introducea ‘higherconcept’–apixel,wecanincreaseperceptionby increasingthepixelsof thepicturebut itwon’tbesomuchperception–thedynamiceventoftheobserver,butoftheobservable.

This constant 4 vs. 5 tetra vs. pentalogic viewof reality implies that often a system ignores the 5th entropicdimotionsforwhichitdoesnotcodeatall,butwillbeexternallyprovidedcourtesyoftheUniverse.i.e.wehave

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4geneticcodingletters(buta5thnucleotideformoreenzymatic,destructivecoding),4quantumnumbers(buttheprincipaljumpstoincreaseenergybyentropicfeedingtoalargerorbital).

Death=entropy is thus the fundamental blind spot of all dimotions/operands, but for physicists fond of itsentropicbig-bangtheoriesofreality.

Socialadditionbyholographicsuperpositionofequalformsvs.Informativemolding.

‘Natural’N-Sumskeeptheindividualityof its ‘ones’ inthesystemaspartsofwholes,whichareherds,eveniftheyimplyafusionofthevolumeofspace-timethesystemoccupiesbysuperposition.

Thewholenessofwhichisoftenontheeyeofthebeholder,introducingthethemeofthesubjectiveobservervs.theobjectiveobservablethatholdsahighertruth.I.e.wavesaresuperpositionsasherdsofsmallerforms;evenifthehumindconsidersthem‘singlewholes’.

RECAP.Thesimplestentangledrepresentationofrealitythrough'basicoperand'comesthroughthedualityofadditionandsubtraction,anditsattachedphysicalmeaningsofsuperpositionandfusionof'parts' into'wholenumbers', or its entropic inverse operandi of negative subtraction. While as all operands it has an inversefunctiontomaintainthebalanceofalltheactionsoftheUniverse;andablindspot,notcoded–theinverseofitsdominantsocialdimotion–informativeperception.

Moreover addition can happen in sequential time or adjacent space, forming growing probabilities orpopulations.Soasthesimplestmodeofoperandsextendsitsdiversificationthroughspaceortimeitwillmeandifferentthings.

-:THENEGATIVEFUNCTION:ZNUMBERS

It follows that negative numbers are the first inverted operand required to balance the Universe. Itsinterpretationisthenimmediateastheentropicinversionofaprocessofsocialgrowth;butandthishasbeenasourceofmuchconfusion,theycanbeplacedinthesameparallelorientation(Znumbers)orintheorthogonalperpendicularorientation(complexnumbers),whichareverydifferentpropositions. And inbothcasesthereare limitstowhatdimotionstheycanrepresent.Znumbers inprincipleareNOTrequiredforthe inversionofthesimplestsumofspatialpopulations,forwhichthenegativesymbolsuffices,butsubtractionendsat0’,asnegativepopulationsdonotexist inpurepresentspatial terms–onlyasaprojectionof futurenegativetime(Debt,etc.)SoZnumbersareonlypertinentfortemporalprocesses.Furtheron,theyarewritteninlinealtime,whichisnotsuitedforcyclicaltimeprocesses,whicharemostlongtermDeeptimeeventsandworldcycles.Thisleaves the range of Z-numbers use constrained to lineal timemotions, 2D locomotions, which are not thatinteresting despite ‘enzyman’s’ magnification of it, due to its obsession with the construction of entropicweaponsandtransportmachines.

Weshallthusnotdedicatemuchworktotheirstudy,inthisbyforcebriefintroductionto5D¬Algebra.

Thebeatsbetween±informationandentropystates.

ThefunctionofexistencehasinExistential¬Ælgebraaseriesofpotentialbeats…

Wedistinguish∑ as the sum symbol forherdsof equalbeings,while∏will entanglenetworks, andonewilltransittotheother,sinceadditionbysuperposition inevertighterspacesofsimilarclonespecies,thesimple¬Algebraic expression of social dimotions, finally collapse into a tighter network oncenomore units can beadded, and vital space must be ‘cut-off’ creating form, causing an essential ‘Stœp’ rhythm of Socialreproduction and growthof cells, informativemolding andpartial loss to increase information,which iswhybotharesooftenconnected.

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I.e. the sculptor forms by eliminating waste of material; the hand is formed by killing the cells that werebetween fingers; evolution of form happens always in a diminishing form; so the Homo Floresiensis was ahobbitbuthadthefirstcomplexformofthehead,andallnewspeciesdiminishinform.

Which introduces the opposite functions and entangled beats of the Universe as the aforementioned FastRadiationofspecies/cellularpopulationsinspacevs.slowdiminishing/evolvingformintime.

Bothinitspositive5Ðv.negativeentropic4Ðwhoseadditionofdecaying¡-1T.œsissofastthatitisexpressedbyanegativeexponentialgrowth,whichinthismannerwouldcompletethe3'Planes'ofaddition:+,x,xn

Ifweconsideranevent’sworldcycle,1,lesserprobabilitiesrepresentpartsofthewholeevent.Ifweprojectitinto space it will be a population of similar event, entering the region of maximal frequency. Both will bemathematicallyprojectedasabellcurve.S=T.Samefunctionfortheadditionofeventsandpopulations,intimeorspace.

TheUniverseissimplein itsoriginalprinciples,madecomplexbythedifferentiationacrossthesymmetriesofscale, topology or time. Indeed, something so simple as the sum and inverse subtraction IS still the mostimportantoperandioftheUniverse,whichgivesusnewnumbers,socialgatheringsof identicalbeings,whichherdtogether intoparallel flowsadoptingmost likelyabidimensionalSTsuperpositionon laminarstatesthatkeepadding the3rddimensionof thebeing. Like the simplest firstmasterpiecesofBach, thearchitectonicalUniverseisasimpleprinciplebeforeorganicismtwistsitsform,inwhichbeings,whichareequalcometogether.

Superpositionofbidimensionalholographicfieldsissoimportantthatthewholeofquantumphysicsisbasedinthis superposition principle. The sum thus is still the master of operandi. But for sums to happen, thebeingsmust be externally identical, to be perceived as parts of a quantified mass, each of them the samevalue.AdditionthusistheultimateproofofthesocialnatureoftheUniverse:Thesumisasocialaction:Sumsformherds.

Oncewedefinedanoperationand itsproperties,westudyhowtheymediatetheactionsofbeings.And it isclear that the first operation sum, acts on the first form, social numbers, to form growing §ŒTs of socialnumbers...

Soparadoxicallythemorecomplexaction,startswiththesimpleoperation,thesumofindividualsintoherdsofformalnumbers.

A locomotion is representedasa 'linealmotion'bya simple 'sum'whenmoving in thepositivedirection,oranegativenumber,whenmovingintheinversedirection;whereasthepositivedirectionisthedirectiontowards'energy',andthenegativedirectionmustthenbetheinversedirectionofthe'arrow'ofinformation.Wecanseethisalso,asanexpansivedirection(energyfeeding)andanimplosivedirection(informationwarping),conceptsofimportance, as information 'warps', and since height is the dimension of perceptive information, a negative,shrinking number, i, will define often in mathematical physics the warping direction of information (as inMinkowski'sspace,it).

However in the entangled, fractal Universe each element can be entangled with all the other elements(fundamentalprincipleofpentalogic).Sowecanalsaddfrequenciesintime(+,T),socialelements(5D,+),pointsinasingleplane,andgroupsofsocialelementstogiveusnewdecametricPlanes.Andwecanadreproductionsassumsofidenticalbeings(3D,+).Andwecancountthedecayofapointintoitsinternalpartsasasum(4D).Thuswecanbuildwithonlysumsamirrorimageof4dimotionsofreality.

Theproblemofthei-number:negativesandroots.Properandimproperinversions.

As a general rule,negative numbers exist when they are a direction ofmotion, not a quantity or volume ofspace.

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So happens with roots, which only should exist for symmetric holographic systems of space-time S=T, forbidimensionalentitiesandregularnumbersasforms.

And alternately they Do happen in many cases in which mathematical physicists discharge them asinversearrows of time because they ignore the inverse 4D vs. 5D arrows, as in Einstein's equations, where theycustomarydischargenegativehyperluminalsolutionsthatdohappeninthelarger∆±4darkworldofquantumpotentials(entanglement,pilot-wavetheory,etc)andintergalacticspace(fasterthanlightneutrinobackground,gravitationalwaves,actionatdistance,redshiftoflight,etc.)

Westudythisthemesinnumbertheorywherewedefinethedifferenttypesofnumbers,sononeedforfurtherinfohere.Instead,wewillmakesomecommentson...thelimitsofclassicAristotelianlogicbasedpreciselyinasinglearrowoftime,whichhassomuch influencedasEuclidpretentiousaxiomaticmethod(fullofholesandnewaxioms,postulates,notionsandvariouserrorsof the§@-humind),our 'underlying'aprioricategoriesofthemind.

At the basic level of arithmetic’s the problems that still linger inmathematics and by extension all sciences,whichuse itasamirror,aretheconceptsofnegativeandsquarenumbers,the inversionsofthepositiveandquartic equations, on the real number,related to the two fundamental unknowns of humind science - thefractalscalar'infinitesimalnatureofspace',andthe'properandimproperinversion'ofthefractalgenerator.

NegativenumbersandrootsdoNOTexistforcertaintypeofspace-timeevents,p.o.v.sand∆s≈tstructuresofrealitybutarean inflationaryexcessof 'information'properofall languages that in the isolatedmind space,without entropic, simpler restrictions can ‘escape’ into imagination: informative software systems, withoutdirectcontactwiththehardwarethatlimitsthepossiblepathsofinformationmultiplykaleidoscopicinefficientcombinatoryforms,'free'oftheconstrainsthevitalenergytheymust'shape'accountingforitssimplifiedrulesintheexternalworld–inadrawingof2stilldimensionswecanmakeanyfigure–ina5DUniverseonlycertainones‘stand’.

And this is the fundamental need of mathematics that force it in search of meaning as per Lobachevski'spangeometry andGödel's ¬Algebraic theorems, to recur to experimental science, to discernonce and for allwhataremathematicalfictionsandwhataremathematicalreality.

'Iknowwhenmathematicsislogicbutnotwhenitisreal'thenappliesinthiscasetotheunnecessaryattemptstodrawcomplexfunctions in4thdimensionalspace,whentheyaremostly in its imaginarypartsfunctionsofmotionsintime,bestoperatedinmere¬Algebraicmethods.

Odd=negativeandeven=positivefunctions.

Whenwemovefromsimpleroperationstomorecomplexfunctions,theconceptsofpositiveandnegativenumberscarrytopositivefunctions,whicharesymmetricbothsidesofthe0’lineandnegativefunctions,whichareinvertedpassingtothenegativeside,andsotheconceptofaninversionofDimotiontakeseffectwithclear implicationsinthereal,physicalworld.

Letusconsideranexamplewithwideimplicationsinquantumphysics,thefact thedualitybetweenantisymmetryandsymmetryhappensalsointhe(antisymmetric)vs.even(symmetricfunctions),whichisinmathematicalphysicsthe basis between social evolution of particles (bosons) vs. antisymmetric annihilation (fermions), so ithas animmediateneverquiteclarifiedapplicationtophysics.

Oneofthebasicquestionsinthetheoryofnumbersconcernedthedivisibilityofonenumberbyanother:

iftheresultofdividingtheintegerabytheintegerb(notequaltozeroth)isaninteger,i.e.,ifa=b•c(a,b,careintegers)thenwesaythataisdivisiblebyborthatbdividesa.Iftheresultofdividingtheintegerabytheintegerbisafraction,thenwesaythataisnotdivisiblebyb.

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Questions of divisibility of numbers are encountered constantly inpractice and also play an important role in some questions ofmathematicalanalysis. Forexample, if theexpansionofa function inintegerpowersofx:

is such that all odd coefficients (with indices not divisible by 2) areequaltozeroth,i.e.,if

thenthefunctionsatisfiesthecondition:ƒ(-x)=ƒ(x)-suchafunctioniscalledanevenfunction,and itsgraph issymmetricwithrespect to

theaxisofordinates.But if in theexpansion (2) all theevencoefficients (with indicesdivisibleby2)areequal tozeroth,inotherwords,if:

thenƒ(-x)=-ƒ(x)

Inthiscasethefunctioniscalledodd,anditsgraphissymmetricwithrespecttotheorigin.

Notice y(s)≈x(t) is anodd function, adeepmetaphysical fact,which implies S andT are indifferent agesof time.Ifwe consider the 2 functions of the 1st Dimotion of angular perception, the sin x function related to the heightdimensionofformisodd,andthecosxfunctionrelatedtothelength2D¡oflocomotioniseven:

Thisdualityhasadeep implication in thephysicalUniverse (Pauliexclusionprinciple),whichcanbe resumed inasimple thought: functions which switch between ± temporal motions, something quite common in all type ofalternate currents, but do NOT change for that reason their wave form (even functions) obviously add bysuperpositionitspopulations.

Functionswhichalternatethedirectionofitstimemotion(frompositivetonegativesign)andintheprocessinvertitswaveformfrom+to-energyorintermsofSvs.Tinvertitsdimotionfromentropy,locomotion,toinformationstate,annihilateeachother.Thisisthecaseofasinxfunctionwherethestateschangefrommotiontoinformation.Whileinthecosxfunction,thewavemerelyswitchesmotionbutNOTstate.

Soinevenfunctions,changingsigndoesn’tdimotion;inoddfunctionsoppositeDimotionsannihilateeachother.

As this is not wanted – no T.œ wants to die, somehow the particle-mind state will by all means prevent theirbody/wavesfromcontactingeachother.Inthenextgraphweseethecaseofevenfunctions.Anychemicalstudentcanfindthecaseofoddfunctionswheretheelectronicwaveafraidofannihilation,actingasthebodystateoftheparticleisthefurthestawayfromtheother.

The reader should easily in any case interpret the results in terms of dimensions and theholographicprinciple.And it reads like this: symmetric functionsareholographiceven functions,whichmeans for example they canbe 'superimposed' as theyarebidimensional 'sheets',wavesetc.Whileoddfunctionsareevenindimensions,andassuchtheycannotbesuperimposed.

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Thismeans essentially bosons are bidimensional and fermions are tridimensional, and this is a huge advance forquantumphysics-recentlyprovedbyanexperimentthatconvertedaphoton,initiallyabosonintoatridimensionalformandendeditsbosonconditions.Accumulationofbosonsintoasinglepointofspace-timethenmeansmerelyitissuperposingthebidimensionalthinlayersintoathirddimensionofheightwhichisindeedwhatweseeinpicturesofbosonstates.

Its representation in both ± sides of the planeworks as follows: the axis of ordinates plays here the role of the'asymmetric' statewhich has split into the ± inverse directions but can fusion again both.Or in terms of relativeequality, those functionsDOhave theirposition in space (establishedby the lineal coordinatesused to representlengthsinspace)differentbuttheirform(representedbytheY-coordinatesusedtorepresentformandinformationheightdimensionsformostfunctions)identical.Sotheycanaccordingtotherulesofthefourthpostulateofnon-e,communicateastheycanshareidenticalinformationandmatcheachotherinsymmetricpeg.

Whiletheantisymmetricfunctionis+--+;thatis,inversebothintheY-nformationandX-pacelocationisdissimilarbothinformandmotion,momentumandposition,younameit...whatever2parametersweusetocomparebothsystems.SotheyenterintoaDarwinianannihilatingprocessastheycannotmatcheachothersform.

The5Propertiesof±operations.Itstrilogic

It is then immediatethatforthefundamentaldimotionofthesumasanoperand itspropertiesdefineasocialdemocraticherdofundistinguishableelements,andthosepropertiescarrytothepropertiesof‘numbers’,astheyexistwiththeinclusionof‘thesum’withinthem.

Sothestandardpropertiesofthesumare:

1.Identityelement:Thereexistsanumberzerothwiththepropertya+0=aforeverya.2.Foreverynumbera,thereexiststheoppositenumberxsatisfyingtheequationa+x=0.3.Foranytwonumbers,theirsumisuniquelydetermined.4.Additionhastheassociativeproperty:(a+b)+c=a+(b+c)5.Additioniscommutative:a+b=b+a

5propertiesimmediatelysmelltopentalogic.Weshallnotfollowasweneverdoafullrigorousconsiderationofthemall,butitisobviousthatinlevelsofcomplexitywefirsttalkofthe±wholenessmonologicexpressedby2(x-x=0),wherebothoperationsareincluded.

1.2 The two first properties thus are related to ‘time sums’ of ±directions; the last 2 to ‘space sums’ ofindistinguishablepopulations;andthe3rdseeminglytrivial, isofimportancetophilosophyofscience.ButinthescalarUniverse,everythingis‘deeper’thanitseems.

2.Considerthe±inversion,whichintheidealworldof0’-vacuumisevident,asthe2invertedarrowsofthe5thdimension, entropy=dissolution of wholes into parts and social evolution of parts intowholes, should be alsoideally perfectly symmetric; but neither of both perfect cases is real. There is NOT a 0 but always a residualfinitesimal.Thereisno0temperature,butaresidualmotion;0vacuumbutaresidualh-planckton.Soa-a=+0’isthereal‘thing’intheparadoxicalasymmetryofreality,whereoneofthe‘twosymmetricsides’oftheUniverseisslightly‘weaker’ initsS∨Tparameter,andtheothersideintheoppositeone.Soingeneraltermsthe‘negativeside’haslesserenergyvaluethanthepositivesidethathaslesserinformationvalue.Thusa-a=+0’.

Butwhat isthevitalreason inaperfectlyefficient‘realUniverse’oforganicspace-timeforthelackofaperfectinversion?Theanswerisobvious,aswedoNOThaveapentalogicresolutionforthedimotionsofreproductioninaworldofonly ‘sums’, ifa-a=0.ThusTrinity, reproductionofan infinitesimal formbygender isgivenbythe0’neutralelement.Inotherwordsfemale+male=finitesimalseminal0’.

3.Neitherthefactthatthereisonlyasumistrivial,asotheroperations(√)giveustwo‘possiblefutures’.Socialevolutionthusishighlydeterministicasitalwaysbringsasinglesolution,moreover,thenexttwopropertiesshow

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that social evolution ‘reduces’ the futures, because itmakes commutativeandassociative variations irrelevant.Thisisakeyconcepttounderstandwhysocialevolutionendsupbeingdeterministicandreducingthefreedomofthepartsthatbecomethenanemergentwholethat‘makesallthosepartsundistinguishable’:

4.5:Thelast2propertiesarealsoquiteprofound;astheyaredifferentfor+anditsinverseoperation-.

Subtractiondifferentiatesthroughthose2finalproperties,preciselybecausetheyarepropertiesofidentity,whichsubtraction does not perform. As the 4th Dimotion of social evolution requires a democratic socialundistinguishable number – which we already saw when defining numbers as polytopes, and ‘≈’ the relativeinfinitywiththeunitcircle,ofarelativeinfinitenumbersofirrational‘holes’withinthemaximalpolytope.

Subtractionthoughisbydefinitionthebreakingofparts,anentropicaction,andsoitisnotcommutativeanditisnotassociative,asthetwoelementsofthesubtractionbecomedifferentiatedbytheoperand.

Recap.Thesimplestsumanditsinversesubtractioncanformacompletesetofdimotions,reproductionthroughits identity number; 0’whichmust be taken as the representation of a finitesimal part. Social evolution as itsfundamental,naturalproperty;entropicsubtraction,whichlackstheassociativeandcommutativeproperties.

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x÷:2ND&3RDDIMOTIONS:RE=PRODUCT=ION:

The Ðimotion of reproduction is the product. Since translation = motion in space, or locomotion is a form ofreproductionof information in lowerplanes, theproductalsoplaysa fundamental role inmathematicalphysicsand the studyof sT-locomotions -motionsof a form througha space inwhich theparticle-wave reproduces itsmemorialinformationasawavethatevolvesandcollapsesasaparticle.

Itsmaindifferencewiththesumisthatitallowstocombinedifferent‘substances’,timeandspaceparameters.Itdoesnotrequirethesameform.

Todefinetheproductthenwemuststudyitthroughthedualities,trinitiesandpentalogicentanglementsitallowsasthemostimportantoperandbetweensingleelementsoftheUniverse;parallelintheanalysisofsocialpartsandwholes,tothecalculusoperands.

As such the product will be able to perform different actions for the 5 structural elements of ¬∆@st; sincereproductionisthemorecomplexofallÐimotionsencompassingpartofthemall.

Andsowecouldrightlysaythatthe‘king’ofalloperandsistheproduct,everpervadinginallsystemstothepointweoftenignoreitswriting,asitisagiventhatanymergingofS&Tholographicspaceandtimeparametersmustbedealtwithaproduct.

Unlikethesumwhichrequiresequalspeciestoaddwithitssuperpositionprinciples,inherds,theProductisNOTonly the sum of equal forms, which is its usual definition, but the merging of different species, with is itsfundamental purpose; hence a reproductive act that brings something else, starting processes of creation in asingleplaneofspace-asopposedtothesumthatmerelywillcreateasimplerherdofalargerscale.

Theproductisalsothefirstoperand,pastthe‘shallowness’ofsummitryherding,whichprobesinlowerandupperplanesofthefifthdimension,asthetotalcalculationofasystem’spartsin∆-1.

SowiththeproductandtheillunderstooddivisionanditsQ-numbersasratios,weenterintoamuchdeeperland,andmultipleissuesariseatonce.

Firstweneedtounderstandwhatwecan'multiply'andwecannot.Wecannotmultiply3pearsand5apples,butwecanaddthemwhenweupgradeitsconcepttofruits.ButeventhenwecannotMULTIPLY3pearsand5apples.Stillwecanmultiply3,asocialnumber,orpropertyofscale,and‘apples’,asynchronousorganismofspace.

Soasanintuitivefirstanswerthatgivessuchenormouspowertotheproductwecanstate:

‘The product can multiply=merge, the 3 main parameters of the entangled Universe, ∆-scales, Time-motion &spatialorganisms;entanglingthemintoan∆ST,scalarDimotionofspace-time.

AsithappenallDimotionsintheUniversearebidimensional-theproductofaDimensionofspaceandaMotionintime;andwecanalwaysaddtothatstructurea‘number’,whichisaunitofsocialscaling.SowetalkofDimotionsas the product=combination of a dimension of spatial form and a motion of time and a scalar number. Gist’sprofound realization on themultiplication operand is that it is the essential operand to combine dimensions ofspaceandfrequenciesoftimemotion,andNºscalestocreateabidimensionalspacetimedimotion.

Forthatreasonasinmanyotherthemes(non-existenceofzerothsandlimitsbut1/xfinitesimals,non-existenceofabsoluteinfinitiesbutlocal∝,scalarnatureofthe‘numericalline’,etc.)itisimperativeforaproperunderstandingof the Universe, an expansion of our philosophy of science andmathematics tomatch closer the reality of thefractalUniverse.

Letuselaborateonthiskeythemconsideringtheholographicbidimensionalbasicformsofspacetime.

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Holographicinteractions.ThecalculusofSTDimotionswithproductoperands

Inthegraphweshowinamoregeneralwaythe5bidimensional,holographicdimotionsofexistenceforwhichtheproductNOTthesumareneeded.

Sowedefine in¬Algebra theproductas thekingofalloperations, since it 'merges' into 'anewentityof space-time', two parameters disjoined. The product proves the very same existence of a holographicUniversewith 3elements,scaleprovidedbynumbers,andspacextime.Sopolynomialscanmergethe '¡-1'elementsor 'cellularparts' of the being, and calculate its space-time actions as a ‘whole’ emergent new reality. The holographicprinciple,bestservedbytheoperandofmultiplicationcombinesspaceandtimeparametersintoasingle'entity',thebestknownofwhicharephysicalmomentum,whichcombinesaT-function(speed)andanS-function(mass).Sooftenwetalkofthatproductas‘existentialmomentum’inExistentialalgebra–basiccaseofSxT,3rd‘Dimotion’onphysicalsystems.InthatsenseanST‘dimotion’isanew‘emergent’entity,somethingphysicistsoftenwrestlewith. Momentum is NOT mass neither SPEED but ‘something else’, reason why it acts as a unit in physicalequations.I.e.theuncertaintybetweenmomentumandposition.‘Anelectromagneticwave’,isneitheramagneticoranelectricfieldbutsomethingelsedefinedbyaproductintermsofbidimensionalspeed:c2=1/µe.

How multiplication then merges two elements, SxT at the point of maximal communication, S=T? Easily. Byentanglingthetwoelements,creating∆-1'axons'ofcommunicationwithalltheotherpartsofthebeing,suchas:

X(5¡-1)xY(4¡-1)=(5¡-1)x(4¡-1)=XY(20¡-2)

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Thatis,amultiplicationthatmergestwowholes,XandY,whichdohaveinthe∆-1scale,5and4elements,ifwejoininthe∆-1scaleeachelementofXandeachelementofY,willgivesusanumberofi-1communicationaxonsconnectingatadeeperlevelthetwowholes,equaltotheproductofits∆-2elements,5x4=20.

AndthisproductismaximalwhenS=T,sofor10totalelements,5x5>6x4>7x3...

Wetalkthenofthe‘perfectproduct’,whenS=T,whichisanessentialconceptofÐST,foraproperreproductiontotake place by ‘gendermirror symmetry’, carried on inmathematics to the ‘Polynomial square’ and combined intermsofSvs.TbytheorthogonalityofthePythagoreantriangleof‘maximal’diagonallength,whicharesurvival,efficientÐSTfunctions,allpervadingin‘REALITY’,beyondtheinflationary‘non-survival’entropicmutationsoftheinflationary language (asall languagesare: there ismoremoney thanproducts,morewords thanactions,moreequationsthanreality).

ThisnewbeingthenSxTwillhavecreatedan∆-2scaleofaxonsthatentangletogetherbothpartsasawhole,inthewayabrainiswiredbynetworks.Andsoweusethesymbolofrepeatedmultiplication∏,forentanglednetworksasopposedtothe∑symbolforsumherds.

Soweinterpretamomentumasamassstopstateandawavestepstatemergedtogetherinthepotential¡-2 levelthatreproducesthewave-particlealongitspaththroughaproduct,whichismaximizedwhenbothmassandspeedare in relativebalance toeachother.So largermasseshave lesser speed and lessermomentum thanbalanced S=T states – a common featureof allsystems of reality deduced of the Max. SxT equation of ‘Momentum of existence’ that allsystemstrytomaximizeinits‘classicbalancedage’.

Themostabundantofalloperand,themergingproductrequiresthereforeamorecomplexrulethanadirectsum,whichactsby'superposition'ofequalbeings.EvenifintheentangledUniverseofgrowthofcomplexitytheproductcanalsobeusedtocalculate‘exact’multiplesofsums.

Forthesamereason,aswegrowintheentanglementofnumberfamiliesandoperandsthosemergedexistenceswillalsotobeoperatedbycalculusandderivatives,speciallywhenwegofrom∆-1to∆ºbymeansofintegrationofderivativeparts.Sincemultiplicationtendstohappeninthelowerscaleofthebeingatdifferentstatesoftimeandspace.Sowenolongeroperateasinadditions,withthesametypeofT.œsinthesameplane.

Thebestwaytodescribethemultiplicationsymbolisthroughtheholographicprinciple.Inrealityone-dimensionalpointswithnobreathdonotexist.SystemsarealwaysSTholographicmergesandsoweneedanoperation,whichis the 'queen of them all' as it allows us to 'merge' S and T states of different entities together into 'newBidimensionaloperators',whichinphysicsforexamplemergesM(ð)x$(v)intomomenta,whichistheconservedbidimensionalentityofthespacetimeUniverse.

Multiplication,beyonditsnaivedefinitionasa'sumofsums',isthere=productivecreativeprocessof¬Algebra.

ThesimplestmodesofrealityareSTcombinationsofbidimensionalplaneswhichadbysuperposition;whentheparametersaredifferenthoweverwemultiplythem.Thesecondoperationof¬Algebraisproductanditsinversedivision,andthisprobesfurtherintothePlanesofreality;sothenewoperationsadsdimensionalityandrequiresnewnumbers,namelytheRationalnumbers.Letusthenconsiderthemfirstbystudyingitspropertiesascomparedto those of addition. So THE product IS THE FIRST operand of the fifth dimension, which merges, in3D:reproductionactsthatrequiresbetweentwo'genders'suchinterconnectionatthelowerleveltoformawhole.

5pointsconnectedwithallothersandwith itself, giveus25connections.This is theultimatemeaningof theproductwhenperceivedfroman∆-1perspectivewhichisneededasinmostformulaetheproductisnotreflexivebutcommunicativebetweenspeciesofS-Tdifferentquality.I.e.momentum,mv,isnotmerelythemultiplicationofm,buttheproductofastaticspace-statemandawave-movingstate:v-anotherwholething.

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MULTIPLEMULTIPLICATIONS:ITS∆STTRILOGIC

The fundamentaleffectofnewoperands is toadanewdimension to the system, in thegrowthofentanglementfrom1to5Dimotionsthatallowsthepersistenceofanentangledreality,wherethe5Dimotionsoftime,canworkinsynchronicity in space. And that is first achieved beyond the ‘wave-herd’ of the first operand and its ‘wavesuperposition’rules,‘smeared’onitsbackgroundrealitywiththetighteningoftwoopposite‘genderforms’throughmirrorsymmetryandtheties/axonsthatcommunicateitspartsinternally,creatinga‘newdimotion’ofform.So5x5neuronsareNOT25neuronsbut10neuronswith25entangled∆-1axons.

Thisistotheproductitsfundamentalfunction,re-product-ion,asthesocialherdgrowthisthemainfunctionofthesum. In the parallelmirrors of all stiences and languages, the sumdoes correspond to the herd, and the entropicstate,andtheproducttotheliquidstates,whereasthe‘perfectproduct’or‘powerlaws’,willbecomethesolidstate,asalltheelementsareentangledtoalltheelementsinamaximalnetworkknot.

This said as the sum could reflect other ‘tetralogic’ elementswith the ‘blind spot’ of angular perception,we canapply to multiplication the Rashomon method of multiple perspectives (Pentalogic), to consider which type ofdimension, and in this manner an enormous range of phenomena can be expressed with the same operation,increasingtheiterativecomplexityoftheUniverse.

Indeed,thefirstquestionofexistentialalgebraishowmanycombinationsof5Dimotionsand10STcomponentscanwemakeinasingleplane?Andhowmanyentanglementsmighthappenin2and3∆±1planes?

Theginormousnumberofcombinationsof the10space-timeholographicelementsmakestheproductaversatileoperand,whichaccountsbesidestheaforementioned3Dreproductionand4Dsocialevolutionasthesumofsum,forsTlocomotionastheproductoftwosuchstates:λ(S)xƒ(ð)).Whileitsinversefunction,thedivision,breaksthewholeintoparts,andsoitrepresentsthe5Dinversefunctionofentropy.Yetasinthesumperceptiondoesn'tfitintheproduct.Letusstudytheminmoredetail.

4D:§:SocialSum.Inasingleplane,classicmultiplicationisthesumofsumsofagroupofsocial identicalbeingsbut italreadygroupsthemintwoscalesassocial ‘numbers',whichbecome a sub-sœt,multiplied x times, that is we add equal groups that should share acommonpropertiestoequalgroup,‘addinganewscalardimensionofinformation’.Soif

we insteadofcounting1,2,3,4,5,6,7,8,9,10;wemultiply5x2=10,afteraskinghowmany ‘fingers’youhaveonyourhands,we did so becausewe did two ‘hand groups’with new information,makinga distinction in the otherwise‘democraticundistinguishableherdofunits’.But ifwecount sheeps tosleepas theyareundistinguishableweaddthemonebyone.Somultiplicationevolvesocialgroups=numbersinasingleplane.

TIME: 2D: sT:Multiplicationcombinesa timemotionanda spacedimension togeta ‘distance’.Butherealreadycomplexscalarelementsenter inview.Aswecandothatmultiplyinga ‘shortscale’spacedimension,a ‘step’andtheequivalentshorttimeperiod,afrequencyλ(S)xƒ(ð)=Distance,butwecandoitusinglongdeeptimescale(linealtime) in v=s/t-> Vt=S and for that reason we need to diminish the scale of our space scale of measure to‘instantaneousvelocity’;originofcalculusoffinitesimalspeed.SoV¡-1xT¡+1givesusthesameS-distance.

All thosesubtledistinctionwebringconstantlybetweenscalesdomatterwhenwetry tounderstand indeeptherealityofmathematicalphysics,evenifhumindscompressallscalesintoacontinuouslineandasingleplane.InanycasetheSTproductofstepsandfrequenciesisnowadifferent‘property’unifiedasawhole,adistancewhichinthe first case increases the ‘scale’– it is a scaled step, and in the secondhasapeculiareffect,as it converts twoformsofmotion,speedandtimedurationintoafixedform,spatialdistance,whichmeansobviouslythat‘distance’isa ‘mental mirror’, an imaginary image that ‘stiffens’ the combination of a short and a long time that must bethereforealsomental‘parameters’.Indeed,the‘realthing’isλ(S)xƒ(ð);thatis,amandoingstepsofametertoitsvital frequency of a second, a car’s wheel turning at rpm; the other concepts are humind’s abstractions of timeduration(linealtime),instantaneousspeed(derivative)andadistancewithnopersistence.

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Yet another proof that the real thing is cyclical timefrequencyandfractalspace ‘stœps’.Sincewe justprovedbysuchsimplereasoningthat lineal timedoesnotexist!addsadimensionoflinealdistance.Itisthesimplestandcommutativeform.

SPACES:Sumbringsanumberinasinglelineoradisorderedherdbutmultiplicationadsadimensionofspatialwidthtoformanarea,whosedimensionalgrowthcreatesaholographicsystem, in which the height dimension will be a dimension of information added to thenumerical length, which now becomes a width of energy-volume. So you are reading abidimensionalpageofinformation.

Again,newsubtleenter atplayhere.Because thenewdimensionof form that creates thestatic square, or in a more complex view a circle, might be transformed from a temporalmotion, to become space. I.e. when a mass which is an imploding accelerated vortex ofgravitational motion (equivalence principle of Einstein’s gravitation), uncoils intodeceleratingentropicmotion,(E=Mc2),itdoessobyaCCratio,whichisin5Dphysics,asin

C2=1/µe,aregionofvacuumspacepopulatedbymagneticandelectricconstants,whichspreadsanddeceleratestillmassbecomesanextensionofspace.Soheretheimaginarytermofthehumindis‘Energy’,andtherealconversionistheinverseproductthatbreaksmassinto2∆-1dimensionsofspace.Thisisageneralruleofallphysicalsystemsthat ‘transform’ constantly static space into timemotion. I.e. the same fluxofwater athigher speedand smaller

section goes through a thin tube than a large tube with larger space section and slower speed,themselvesconsequenceof5DSxTmetrics.

Furtheron,at this stagewe findalreadyanon-commutativeproperty formultiplication.As there isadifference in orientation: 4 x 5 ≠ 5 x 4. Hence the non-existence of commutative properties in'matrices',theclosest¬Algebraicrepresentationintime-numbersofspatialpoint-areas.

Knowing thosedifferences according towhichdimensionswe 'operate' canbe a hint for experimental science aswhenHeisenbergdevelopedquantumMatricesduetothenon-commutabilityofitsoperators.

$Tv.$ð:Intopologicalterms,multiplicationworksforboth,linealproductsorcyclicalpoint-like,clocksoftime,buttherearealsosomeinsightdifferenceswhichbelongtogeometryamongthemthefactthat

only lineal forms are ‘deterministic’ and ‘tile’ the entire space, as opposed to cyclical points, which leave emptyspaces thathorror vacuum fillswith smaller scales.Or inotherwords, a ‘rectangularworld’doesnotneed scalar

space,neither‘fractalnon-Euclideanminds’tomirrorit.Itisasbored,deterministicandtruthastheaxiomaticmethodofEuclid.

∆+1<∆-1. A little explored pentalogic form ofmultiplication is essential to themeaning of a‘number’asasocialscale,whereas3is‘3timeslarger’thanone,and10thecommonestscaleoftheUniverseaunitofa largerwhole. In thegraphwecanseethisconceptas1becomes3,alargerscale,2becomes6.We’llreturnlattertoitsimplicationswhenwestudythepropertiesofmultiplication,notablythoserelatedto0’andthe‘paradoxofinformation’.

Scalingisthustheessentialmeaningoftheproduct,whenwedonotmultiplyaparameterwithaphysicalmeaningbutmakeaproductofa‘purescalarnumber’.

3rdÐimotionproduction=multiplication=reproduction.Dimensionalproducts.

Sowereturninthecyclicaluniversetothemainuseofaproduct,entanglementandreproduction,whichisthemorecomplex dimotion that requires a travel between planes as a seed or finitesimal part of awhole reproduces andentanglesinthe¡-1scale.Soaswesaidthatwhenwemultiply5x5,ifweconsider5thenumberof∆-1elementsof2groups,then5x5=25turnsouttobethenumberof'axons'connectingeachofthe2wholesatthelevelofits5∆-1elements.So25givesusthenumberofaxons=ST flowsofcommunicationbetweenthe5sub-elementsof the2

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wholes.AndwhenS=T,5=5,weobtaintheperfectproductor‘power’law:Product’svitalroleistocreateanetworkonthelowerscaleofsystemstocreateanewwholeSTbeing,differentfromSorT,momentumthatisneitherspeednormagnitude,lightthatisnotmagnetismorelectricity.

Firstaswehaveshownmultiplicationcanaddadimensionoflinealmotiontoaparticle-vortex(quantummotion),yetaswehaveprovedinotherpapers,locomotionisthereproductionofaforminalowerplaneofspacetime(theentropic/energy field that moves) from stop-particle to wave-motion states. So the very essence of motion asreproductionofinformationisencodedInthesimpleequation:λ(S)xƒ(ð).

So pentalogic on the main function of the product, which is also the fundamental Dimotion of the Universe,reproduction, connects T-Locomotion (2D) as reproduction of information along the path of a particle; ∆-scalarreproductionastheentanglementofmirrorsymmetriesthroughreproductionofaxons(3D),andthesocialsumofparts,thatreproducesinternal‘numerical’groupsbeforeaddingthem(4Dreproduction).

ProductsofST-holographicwholes.

But the product becomes even more interesting, when it goes into the next scale ofcomplexity,reproducingnotindividualSandTelementsofadimotionbutcompletedimotionsmadeofanSandTcomponent,calledvectors:

Vectorial productmergeswhole space and time dimotions (amass or charge, the active S-magnitude,with a fieldofmotion, the T-element). In the graph, a productof 2different STdimotions, one ‘b’ dominant in red-long entropy and other ‘a’ dominant in blue, relativeheight, the code of information gives birth to a 3rd (that should be green in ¬E geometriccodingofcolorsJ,arelativereproductivewidthperpendiculartotheothertwo.

AvectorialproductisthusoneofthecommonestformstocombineStxTsdimotions,actinginitslimitof90ºorthogonalityasapureReproductionofanSandTrelativeform.

Whereasthenon-commutabilityofitsproductisrelatedasinthesquareproductin2Euclidean(ab.E)Dimensionsto the orientation now in 3 E-Dimensions, establishing a fundamental physical direction for the electromagneticspacetimeofourgalaxy‘clockwiseforward’,asitwasfoundinthepracticaluseofvectorsbyHeavisidetodescribeelectromagnetism,whereastheelectricinformativefield(forourelectroniceyes),andtheenergeticmagneticfieldcombinetocreatethereproductivec-speedwaveofinformationofourlastperceived∆-3scale.

Theconclusion isobviousonlyspatialsumsandproductsofpopulations,havenodynamicorientation,astheyarestatic points, but both timemotions and scaleswhichmust be understood as the SLOWEST, absolute direction oftime,wherepartscomebeforewholesDO.Andsince theproductapplies tospace, timeandscale,mostproducts,which havea T-element are found to be non-commutative.All T(S∧T) products are not: bxa=- axb.Thus, only theproductasasumof‘stillpopulations’,SS,whichultimatelycanbeconsideredasumNOTaproduct,iscommutative.WecomethustotheparadoxicalconclusionthattheproductisNOTasumofsumsandisNOTcommutativeinitstrueform.

Matrixproduct.Andmoreoveritsbidimensionalorientation(Matrixcalculus)and3Dorientation(Vectorcalculus)arecloselyrelated(asarotationofavectorisalinealtransformation,calculatedthroughmatrices)–themesthoseofadvanced5D¬Algebraweshalljustintroducehere:

WecanthinkofamatrixasanorthogonalsystemofT-rowsandS-columns,whichrepresentthe‘∆-1’partsofitsS-form and T-motion organs. Then the re-product-ive operand of two such ST(x,y) Matrices happens through themergingoftheSx-componentsofaMatrixwiththeTyComponentsoftheotherandtheTxcomponentswiththeSy∆-1components,whicharethenaddedtoformanew‘offspring’combinationofboth∆-1parts.

InÐSTpoint terms theproduct isnot commutablebecause for reproduction tooccurbetween twodifferentT.œsrequiresgendermirrorcomplementarity,henceoppositeorientation.

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PRODUCT’SPROPERTIES:ASSOCIATIVE,SCALAR,DISTRIBUTIVE,INVERTED&NEUTRALELEMENTS:0’V.1.

Weunderstandbetternowthepropertiesofoperandsin5Dterms.Thetwomainoneshavebeendiscussed:

Theproductisnotcommutative.

Only the product as a sumof sums signified by a scalar number is commutative... Reproductive dimotions, usingMatrices and vectors,which depart from its simple use as a sumof sums, are subject to the rules of orientationbetweenSandTelements.Sincewecanalsowriteforbetterunderstandingofthoseproperties,the5DimotionsofexistenceintermsofdominanceoftheSpaceorTimeElement,insteadofusingCaps,usingtheorderfrompasttofuturethatdeterminesthedominanceofoneortheotherelement.Sos<Tlocomotionisdominantinmotionoverformv.t>Sinformationdominantinformovermotion.

Inaninteractionbetweenaspatialformandanentropicmotiononeelementwillbecomethe‘relativepast’andtheothertherelativefuture,dependingonwhichoneprey intheother,onwhichonetransformsintotheother.Andthismakesaproductnon-commutative.Whileinsymbioticeventsofcomplementaryreproduction,theorientationinspaceofeachelementalsoplaysafundamentalrolethatwilldetermineaccordingtotheangleofcongruenceifanevent iscreativeordestructiveorskew.Onlywhentheoutcomeisasocialsumintheformofaherdorgroupofgroupsitdoesn’tmattertheorder.

DUALITYANALYSIS:MERGINGTHEPRODUCTANDTHESUM

Multiplicationisassociative:(axb)xc=ax(bxc).

As in thecaseof thesum,multiplication ‘continues’ the flowof timeaftera firsteventbutagainthispropertyholdsforproducts,whicharesumsofsums.

Multiplicationisdistributiveoverthesum:ax(b+c)=axb+axc=e

Theconceptofdistributionoverthesumisinterpretedin5Dintermsof2inverseDimensionaloperandsactinguponawhole,considering(b+c),awholewhichisfirstdividedintwoparts,bandc;hencenotsomuchdistributionoverasumbutpartitionandreproductionofbothparts.Sinceinthatcaseitreducestoaself-similarity:

Ax(Db+c)=Axb+Axc.

WhereAisthedominantpropertyappliedtothewholeD,by'slicing'it,,D=b+c,intotwoseparatepartsenlargingthembothbythesameratio.

SointheleftsidetheproductoperatesinthewholeD,andintherightside,initsrelativeparts;whichdefinesDasaspatialpopulationorherdorwavesusceptibletobebrokeninundistinguishableparts.

For all other more complex Ds, which cannot be subject to an undistinguishable partition, multiplication is notdistributiveoverD.Butastherule‘already’hasbrokenDapriori,thelawappearsasUniversal.

Linealvs.cyclicaladditionandmultiplication.

Thesumandtheproductasasumofsumsarelinealsystems,astheequation,ax+b,whichisalinewithbastheoriginandaasthetangent,shows.Butsuchlinealproductsmustbeseenas∑,asummandofxelements,sincetheproductistakenplacewithaconstantnumber.

Itisadifferentcasefortheproductoftwovariablesoranyotherparameters,whicharenot'constantsums'.

Solinearityandsuperpositionagainhappeninthesumbutonlytheproductassumofsums(4D¡product).

Fromthepointofviewof the3 'elements'ofagenerator,T-potentials<ST-waves>S-particle/heads,potentialsareherds,hencetheirsumscanbetreatedasconstantproducts.

Wavesarebothsumsandre=productivesystems(∑,∏).Andsotheyarebothlinealandcurved,hyperbolic,|xO=ø

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As sumswaves superpose (linealquantumwaves,whichmustbe consideredbidimensional, anddotproducts).Asproducts, they create new dimensions, as in lightwaves are: c²=k/µ; and cross products. Such products are nonlineal,astheyreproduceasinusoidalwave.Andoftenthekeytoclassifywavesassumorproductsistheircapacitytocommute(sums)ornot(products).

If we extend the concept of ¬Ælgebra to other generators and mirror languages, we find 3 categories in theUniversalsyntaxofmostlanguagescodedasgenerators;whereacategorywillbesusceptiabletoadditionandlinealproducts but not the others. I.e. in verbal subject<verb(ST)>Object, the verb is a complex bidimotional non-linealaction,buttheobjectandsubjectsarespatialformssusceptibletolinealsumandproductinquantitativetermsinthecaseofthe‘it’=object,andqualitativetermsthroughitspluralformsinthecaseofthesubject.

Products that do not commute happen when they combine different 'categories' (potential x wave x particle;subjectxverb≠verbxsubjectandsoon).

And so in thenext scaleof equationsonlypolynomials thatuse sumsandproducts ‘over constantnumbers’ arelineal–theotheroperandsthatactoncyclicalforms–sins/cosinesorcurves(integrals,derivatives)arenotlineal,whichmeanstheyareNOTdeterministicandhaveoftenmultiplesolutions.(nπ,differentialsolutions,etc.)–themesthosetreatedindifferentsectionsofthosepaperswhendiscussingpotentialfutures,anddeterministicones,linealstepsandcurvedvariations,sinceonlythelinehasasinglepotentialfuture,orelsestopsbeingaline,whileacurvehas3potentialfutures,dependingonitschangeofcurvature.

0’and∝ intheproduct.

We refer to a previous analysis on pentalogic of 0’, the ∆-finitesimal fractal • point, the T-minimal motion ofaddition, theS-past ‘memories’ofaextinct form,andthe@-mind inrelationshipto the∝perceivedUniversetounderstand0’.

0’inmultiplicationbringsnewparadoxes,because0’=0’xa<0’+a.

Thatis,anoperandofahigherDimotionalrankmultipliedbya‘fractal•0point’islessthanaddingit.

Sincetheproductofa formbyan infinitesimalcausesareductionofdimensionality,andso itreturn less thantheoperandofaddition.Thereasonisthata+0’mustbeinterpretedasanon-operation,‘nothingisadded’totheherd.Soitisatrivialnon-existentevent,onlypresentintheinflationarynatureofthelanguage.

Butsince0’isafinitesimalwhichisbydefinitionthe‘∆-1’seedofare=production,0’xaISarealoperation,whichmakesainteractwith0’;thatis‘produce’aseminalseedat∆-1whichisundistinguishable.Butastimegoesthrough,0’mightgrowtobecomea,sothelimitof0’whent->∝becomesa.

For the same reason0’ x∝ =1=Cmeans that the limited infinitenumberof reproductionsof0’will concludeapalingeneticcyclegivingbirthtothewhole.Thusaftertheparentalformsinksitselfintothe∆-1scale,palingenesisbringsthe'emergence'ofthewhole,1,thefullT.œ.

Inthatregard,whattheneutralelementmeansisanelementnotsusceptibleofbeingoperatedbytheoperand–itsblindspot;whichiszeroth,noadditiontotheherdforthesum,and1,a‘masturbatory’(:actofself-production…withoutinputofenergyorcodefortheproduct,whichrevealsmoreaboutthenatureoftheoperand.

Multiplying±.Fromouteroperandstoinnerpropertyinahigherdimension.

Whichleadustothequestionof'why'-ax-b=ab;-axb=-ab=ax-b.

Themain‘abstract’mathematicalreasonistomaintaintheconsistencyofcalculuswithpolynomialsandequationsthatmixbothoperands.Butwearenothereconcernedwiththeidealsimplificationofmathematics,butwiththehiddenparadoxes,whichwillberequiredforresolvingmathematicalparadoxesinphysicsforlackofunderstandingofthemathematicalones.Letusthenapplypentalogictothe–product.

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Space. We limit such operations to the ‘domains’ in which they are meaningful, which excludes as we alreadyexplainedtheproductof‘naturalspatialpopulation’whicharenotnegative,reducingthereforetotheothertrilogicelementswithaTcomponent,‘negativetimesT-locomotions’and‘invertedscalarTT-entropicdirections’:

@-mindlogic.Asitonlyappliestotimedirections,anotherwaytoresolveparticularcasesistoslidetheminusin–b(ifbisaclassasspatialbeingsthatcannotbenegativeinto-a,thetimeparameter,sowewrite(--a)xb.

Onageneralbasisthismeansthenegativesignisalogicsymbol,aswhenwesay,Idon’tthinkIwon’tgo=Iwillgo;whichfortimeprocesseswhichhavebothdirectionsispossible,asabackandforthvibration,.Andsoweget–ax–b=axb.Wecanalsoapplythisconcepttoanspatialformintermsof‘orientation’,whichistheonlycaseinwhichthenegative ‘concept applies’; that is, amirror symmetry. And so the 'inverse of an inversion' of amirror symmetrybecomesini-logicthesamemirrorbeing,whichjustifiesthenegativeproductincomplementarygendersymmetries.

Scalarentropicmotions:Negativeproducts.ThiscaseusestheconceptofilogicintermsofaTTentropicprocessoftwonegativeflowsoftime,whichistheessentialdefinitionofentropy,death,diminution,inversionofgrowth(4Dvs.5D):Letusputanexample;imagineyouarehavingadebtof1200$andyouremove(negative)everymonth-100$,sofor-12months,thenegativedebtwillbeshrunkbutayearpast,-12x-100=1200$,thedebtwillbepaidtozerothandyouwillnotbesosillyastokeeppayinginthepositiveterritory.

Inthepositivesidethis limit isneverreachedasthe-∝,0 interval isnowthe0,∝ interval;but ineffect,as∝ isrelativetherewillbealsoalimitforapositiveproduct.Thisnegativeproductthoughisa‘time-space’,STproduct,whicharetheonlyonethatmakesense,togiveusapositive(-changex-population,12monthsx100$).

Whatinversionthenmeansindifferentcontextsasamirrorsymmetry,oranyotherkindofsymmetryISTHEREALproperinterpretationwemustdo.Aswhatwearereallymultiplyingareinversionsofinversions.Andsometimesitwillmakesenseandsomenot.

Whatabouttheother2commutativeproducts(-a)xb=(-b)xa=-(axb)?

WepostulatewiththesamereasoningthatbwillbeanspatialS-parameterand–aantemporalchangeofdirectioninthedimotionthataffectsb.SoS(-T)willbecome–ST. While the inflationary language that does not identify numbers by the‘properties’theydescribeallowstowritethose3similarequations,ofwhichonlyonesolutionwillbereal,andtheother2 inflationary,an important findingof5Drealistmathematics,whenwefindtheternarysolutionsofcubicequations,whereoftenwedischargetwoofthem.SothenegativeproductisanSTproduct.

Finallywecanapplythoseconceptstotheperfectproduct;thatis,thenegativexnegativeproductofthetype(-yx–y)=(-y)2anditsrootsandimaginarynumbers.

Which means in many cases we can get discharge the negative root of an X2solutionasscientistsusedtodo;butnotalways.AsasquarenegativecanbeaTT,doubleentropicmotion.Considerthecaseofamass,aninformative,SS,implosiveaccelerated vortex of gravitational forces (Einstein’s principle of equivalence).When it dies entropically it releases a quantity of entropic energy equal to itsproductbyc2:E=mc2.InrealitythisC2istheinversearrowoftheimplodingmass,an exploding, expanding entropic TT explosionof radiation, ultimately a –c x – cfactor,which ishidden intheresultas itgivesalsoc2,butappearsasnegative inrelativityinthefactor–(ct)2nowexplainedalgebraically.

Sothe-symbol isthesymbolof INVERSION,whichcanbeas inthecaseofc2aninner property of the being, 'internalised' by the system, hidden when a larger

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wholeenclosesinitsmembrain(asamass)butreleasedasinverseTTentropy,whenthemassenvelopebreaks.Andviceversa,thenegativeentropicmotioncanbeenclosedasthe∑∆-1pointswhenwemovefrom∆-1to∆º.

Thus if ± are external trans-formative operands that relate 'social numbers' in the ± simpler operands of socialgroups;inaproducttheyareINTERNALpropertiesthatdefinethetypeofSTdimotionwemerge;oftenrelatedtoanscalar entropic transformation; reason why one of the elementary proofs of negative x negative products is thescalarproofofextendingtheproductasascalargrowthtotheinverse-,-quadrantasshowninthegraph.

Butsince therearemanytypesofproducts fordifferentS(œ)Tspeciesofmathematicalobjects, incertaincases±multiplications will make no sense, in others will cover up inner properties, and we have to do in its practicalapplicationstodifferentsciencesacarefulexaminationtofullyunderstandthemeaningofthoseminussymbols,aswehavealreadynoticedintheourintroductiontotheRashomonPentalogiceffectsofcomplexnumbers.

Becausereproductionisthegame,theproductisthecommonestofalloperands,andthroughitgenerationofmostother forms happen. I.e.we saw the fundamental law of prime numbers is to generate all others,which can bereducedtotheprodutof4Primes,whichbecomesits‘tetralogicdimensions’:N=p1xp2xp3xp4

RECAP. The product’s main function is reproduction. But pentalogic expands its use to all other dimotions. Theproductisonlyasumofsums,whencommutative.Butinitsreproductivefunctionsismotcommutativein2D(non-commutativeMatrix)or3Dvectorialcalculus,closelyrelatedbylinealtransformations,whichare∆-scalarproducts.Since orientation in space and angle of congruence are ‘vital elements’ of 5D reproduction. So for the complexproduct of 'ST' Dimotions, we define non-commutableMatrix products that can give origin to different particles(quantumphysics’DiracandHeisenberg’sformalism).

Athemewhichisnotsettledbyanymeans,andhascreatedmuchconfusionistheproductoftwonegatives,-x-=+;asithastodealwiththeconceptsof0’,negativityandrelativeinfinity;whereas0’representseitheraninfinitesimaloranasymptotic ‘barrier’thatcannotbecrossedas it isthelimitofthedomain,representinganstop/informativetimedimotion,asinPv=nkTbetweentwosidesofatime-spacedimotion;henceanegativeproductofanegativehasalimitwhenitremovesallthenegativityout.

ThusasusualmathematicallawsillustratethephilosophicalaspectsofÐSTelements,itsSTsymmetries,Dimotions,scalartransformationsandmirrorreflections,andviceversa,thepentalogic,dualandternaryrelationshipsofthose∆STelementsofrealityexactthefullmeaningofeach¬Algebraicoperand.

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ITSINVERSEOPERATION:÷DIVISION.

DualityonDivision.

The first and simple view on Division is a duality between its positive informative action, when division is self-division; hence reproduction, in conjunction with the product, vs. entropic division, when division is caused byexternaltearingandpredationfeeding:

ST-reproduction: The Pentalogic method applied to entropic Ðimotions represented by divisions and the inversereproductiveÐimotionsrepresentedbyproductsbringsnewinsightstoaninversion,whichinnatureissymbioticinmostprocessesof reproductiontoobtainanewwholesuchas X->X2->∂X2->2x/2=x+x forms inalgebraic termstheseriesofoperandsthatreproduceasystembydoublingitselements,derivingitschangeandsplittingthem.

Inasimplerstœpsequenceofdimotions,withoutderivative/integraloperands,wewrite:x->2x->2x/2->x+x.

TT-entropy: Division has obvious negative entropic consequences such as the sharing of food, the breaking of awholeintoitspartsandsoitclearlyplaystheinverse∆-14D-entropicdimensionofasystem.

The negative, predatory action of bisecting a whole into parts, often digested as the whole 'breaks=dies' (piesharing).Herethenominator isameaningfulentity,butthedenominator isonlya 'bisectingnumber'orherdthatwillnot'create'athirdentity,justdestroythenominator.

Divisioncanbeexpressedasthenegativeexponential,whichisalsoa1/inversedivision,isthenaturaloperationofthearrowofentropy,as itmeans to 'divideawhole' inparts forapredator 'group' to feedon them.Aswhenwedivideapietoeatit.

-Thepositive, collaborationof twoentitiesperceivesasSand/orTDimensions,whichcreatea 'stable'newentitydefinedbytheratio,whichshouldnotbe'operated=dissected'.

Andthisisquiteobviouswhenwedealwithrealsolutionstoproblemsusingpolynomials.

-AsinglerationalQ-numbermightrepresentan∞numberofdivisions.Thus,4isalso8/2,12/3,16/4….Butnotaninfinitenumberofproducts.Only4x1and2x2.Asentropictimeisthe‘background’ofrealitywithfarmoreeventsthanthehard-builtarrowsofpositivesocialevolutionofinformation.

-Itistheinverseofaproductasasocialcommunicationbetween¡+1elementsthatgiveusitstotal'axons',orlinesofcommunicationofthe2wholes(4x4=16axonsconnectingallthe4elementswiththeother4).

PropertiesofDivision

Thedivisionhasnotthesamepropertiesthantheproductasnoinversefunctiondoes;essentiallybecauseasalwaysinrealitythepositivesideofithasafarstrongerappealtoNature,andsoitdoesoftenfollowassociativepropertiesthatcontinuetheflowof times;which iswhattheassociativepropertydoes.Whileentropic inversefunctionsareshort livedduality affairs thatdivideanddestroy theelements involved. Indeed life is a continuous flowof timesfrombirthtoextinction.Butentropicdeathtakesasingletimequantafromontooff.

Soas(a-b)-c≠a–(b-c)->(5-3)-2=0≠5-(3-2)=5-1=4;(a/b)/c≠a/(b/c):(12/4)/3=1≠12/(4/3)=12/1.33=9’

So its difference with the product is its non-associability, as division ends the chain of events in an entropicdestructionofinformation,henceitcannotcarrybeyonddualityinmanycases

Whichsetislarger,theproductofNaturalnumbersoritsdivision?IfweconsideronlysolutionswithinthesetofN,obviously istheproduct.ButItseemsevenifweacceptsolutionsinR,theproduct is larger,asmanydivisionsareequal:6/3=8/4=10/5;whichreaffirmsthedominanceofsocialpositiveformoverdestructivenegativeones.

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TRILOGICONDIVISION.DEFININGTHE3VITALCONSTANTSOFTHEBEINGANDITSRATIOS.

Divisioncanalsoactonfunctionsnormallyexpressedbytheproduct,usingalawfundamentalto5Dmetrics–thelaw of inversion of scales, whichmakes an ∆-1 scale inverse in properties to the ∆º larger state. Let us see thesimplest,alreadyexaminedexampleof locomotion,andextendthecasetothedefinitionofthe3fundamentalSTvitalconstantsoftheUniverse.

sT-Locomotion.Asaratioitsmeaningismorecomplex;sinceoftenspeciallyinmathematicalphysicswheretheunitschosenbyhumansaremisleading(i.e.timedurationisanabstraction1/t=frequencyistheunitofcyclicaltime,soans/tisbetterexpressedasSxT(wavelengthxfrequency),andsoon.

But, it is better to treat whenever possible equations as products instead of ratios, to extract easily the 3fundamentalvitalconstantsoftheUniverse.

St-Information.InverselyDivisioncreatesinformationbybreakingthecontinuumofawholeintoitsparts,thistimeincombinationwiththesum,suchas10/10=1->∑¡=10=1+1+1+1…

Thecombinationofdivisionratios&itssum-as2inversearrowsof4Dand5Dactions(breaking&socialevolution),thus extract the inner information of the system, or sum of a group of social identical beings breaking down its'number'.

But between planes it counts the 'maximal number of connections' established between the ∆-1 elements of 2groups giving us the maximalaxons= ST flows of communication of 2 wholes that reach through X its maximalcommunication. So its role is to create a network on the lower scale of systems - in brains, neural networks orreproductivesystems,reasonwhysooftenareproductiveactionthatcreatesathirdelementasinvectorialproductsissymbolisedbyareproduction.

∆±¡:1D-4D:ScalarSocialevolution.ItisthenfromthecomplexentanglementinparallelspaceofthecombinationofSt-formal division of a length into ratios, and its Ts-locomotion into a vital time motion, and SS, simultaneousperceptionofthosescalarratios-thefirstexpressionofafinitesimal-fromwherethefirstsyntheticworldcyclesanditsmirrorlanguage–musicarose.

ThusMusicanditsratioswerefamouslystudiedbyGreekmathematicians:Pythagorasfoundcertainidealratios,inwhichdivisiondoesn’tbreakentropicallythewholebutcreatestheS-informativediscontinuitiesneededto‘perceiveinformation’whilethewholeremainsasacontinuum,bytheexistenceofaT-motiononanST-body-themotionofthestringweplugtogetasoundbymakingitvibratewithacertainnumber=frequencyofnods.

Hencea full S-ratios<ST-ring>T-motion ‘fractalgeneratorof trilogic is formed,ensuring a constant SóT feed-back equation that allows amusical event totakeplacewithunlimitedvariationsinspaceandrepetitionsintime:

In thegraph, thestring ispluggedbutnotbroken incertainratios,whicharethereforeexpressionsof∑∆=∆+1-aconstructive∆-1division.

Wewillstudyinmusictheory,onthepaperofhumindlanguageswhytheyarethose ratios. Notice that as usual they are simple penta-(do)decalogicvariations,whichsufficetodescribemosteventsoftheUniverse.

This argument leadsus tounderstand thedifferencebetweenmental lossofinformationinarationalnumber,anditspreservationinrealrationevents.

Lossofinformationinmentalspace.

Astheproducthasrisingthecomplexityofoperands,includingnewscalarones,sodoestheinverseoperationofdivision,which brings as the negative symbol did to Natural numbers, a new family of numbers; the rational

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numbers,whichincorporatesasZdid,theoperandwithinthem.ButunlikeZ,complexityhasrisena‘dimension’asarationalnumberisactuallyanoperandon2numbers;andfurtheron,itbringsthefirstofmanysimilarprocessesof convergence or divergence frompresent to future. That is,a relative infinite,∝ number of present divisions,4//2,6/3,8/4etc.endupinthesamefuturesolution:2,whichimpliesthe2ndtimeweobservethephenomenaoflossofinformation(aswesawalreadyintheproductofNx0’).

Thesolutioninthepreviouscase,Nx0’,istoconsiderthattheoutcomeisNx0’finitesimals,soinformationleavesatrace inthe∆-1planeas it isnotabsolute. Inthiscasethesolution istodistinguishthe lossof informationthathappeninanysynopticlanguage–thesolution,2…Fromthefactthatintherealeventinformationispreserved.I.e.if4lionskilla200kiloszebra,eachonewillget50kilos,butwewillsee4lionseatingzebra;if8lionskilla400kilos buffalo, each one gets 50 kilos; and so themental solution is the same as it has ‘erased’ the value of thenumberoflions(denominator)butifweseetheventwewillseethe8lions.

Thesameprocess isobserved in theproduct,whichhoweverretains for theproductasasumofsums, thesameinformation,sincethefinalherdisindistinguishableinitsunits,butnotforproductsthatcombineSandTelements(i.e. ifwemovein longstepswithaslowfrequency,asabigslowanimalwemightbetracingthesamedistanceproductthatasmallanimalwithbigsteps,butwelooseinformationaboutthetypeofspeciesmoving).

Inbothcasesweobservethatthelossofinformationhappeninprocesseswheretheproductordivisionentertherealmof5D.Aswejustdescribedacaseof5Dmetrics: largeranimalshaveslowermetabolismandtimerhythmsthansmalleranimals,buttheproductisthesame,SxT=C

Inbothcasesthoughthelossofinformationismentalandcanbesolvedbyleavingthenumbernotasasingledigitoperationalnumber,whichloosesitsinformationbutasanequation,eitheranSxToranS/Tratiokeptwhenitisleftasafractionalnumberswithoutoperatingit.

RECAP.Aratioofinformationmaintainsthewholewithoutloosinginformationwhenthedivisionsolutionco-existasawholerationumber,expressedinthenumerator,sumoftheparts,expressedinthedenominator.

Lawofinversionin∆stœps.

The duality of division and product, as we observed studying locomotion, based in the fact that ∆TS, scales,dimotionsandvital topologieshavetheir inverse function,allowstocreatebalancedC=1solutionstoequations,usinginverseoperands,includingproductvs.divisiononinversescales,topologiesordimotions.

Thecombinationofsomeof those inversions isoneof thekeyelementsof®lgebraable toexplain thewhyofmanysimilarequationsofscience.

I.e.linealtimeistheinverseofcyclicaltimefrequency,ƒ(ð)=1/Tsowecancalculateinlonglinealtimev=S/Tthedistanceinlonglinealspacewithdivision,orwecandothesamewithproductinshorttimesteps:wavelengthxfrequencyonthelower¡-1scale.

Suchsimilarity isa fundamental lawof ‘scalarplanes’,whichchange functionand formaswemoveupwardsordownwardsinplanes;i.e.atomsofironformrings,butinthemacro-scalebecomelinealsword-likeforms.Proteinsare lineal but in the cell, becomewarped and circular elements of the cellmembrane. It is a case of themuchbroader ‘Galileanparadoxes’betweensmall lineal stepsand largercyclical forms; smalldiscontinuousherdsandcontinuous undistinguishable forms from a larger point of view. And a general rule that allow distinction ofinformation and applies also to $T>ð§ lineal limbs/fields vs. cyclical particle/heads ensemble by intermediatebody/waves (Spatial inversions) and young Ts, lineal youth of max. locomotion vs. old, §ð-age of maximalinformation.

AnyStœpintime,ageorspatialtopologythuskeepsinvertingthroughatransitionalregionaswemovefromonescale, topology or age, into another, ultimately developing 3 relative past->present->future <<pastentropy>present>futurecyclicalpatterns.

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Fromthis facts,wecanconsideronlywithmultiplicationanddivisionsomeofthebasicUniversalconstantsthatdefinethespaceandtimeparametersofT.œs,ofwhich3are3paramount:

s/T:SpeedofLocomotionvs.t/S:densityofinformationwhichareinverseso:T/S=ST/TS=1.

AndSxT=ExistentialMomentumofthebeing,deliveredin‘frequencyactions’ofexistentialforce.

Thuswestarttoseethecloseconnectionbetween¬Algebra

¬AlgebrareflectstheSTinverseDsymmetries,astheyAREtheinversekeyelementsforitsoperations,giventhesocialnatureofthenumberandthepolynomial/integralnatureofitssocialevolutionintovariablesasopposedtotheinverseoperationsoflogarithmsandderivatives.

Considerthesimplest,firsthistoricexample(asprinciplesbecomeeasiertoseeinitsbeginnings).

Anexampleofgeometric¬Algebrawouldbesolvingthelinearequationax=bc.

TheancientGreekswould solve this equationby lookingat it as anequalityof areas rather thanas anequalitybetweentheratiosa:bandc:x.TheGreekswouldconstructarectanglewithsidesoflengthbandc,thenextendasideoftherectangletolengtha,andfinallytheywouldcompletetheextendedrectanglesoastofindthesideoftherectanglethatisthesolution.

Thesolutionseemsthesame,butitisnot.Theratioisadivision;thesquareisamultiplication,andbothareinversefunctions,whichgivesustheidentityelement.

Yetthemeaningofitsgeneralcaseisdeeper,asitallowstoidentifytheconstantsofthebeing:

IndeedasaratioofSTdimensionsmultiplicationdefinesthe3fundamentalvitalconstantsofanybeing.

Itsspeedofreproductionofinformation,S/T,itsdensityofinformation,T/Sanditsexistentialforce,SxT(spatialsimultaneousview)≈TxSorexistentialmomentum(activeview),whichalltogetherdefinestheidentity'element',thebeingin'iTSelf'.

Let us put an example with the most important Universal constants, those of the Galatom, based in its S(h)constant, the Planckton and its T© constant, the speed of light, and those of the human being, with its timeconstant,thesecondanditsspaceconstantthemeter,tounderstandhowClassicalgebraandexistentialalgebrarespondtothesameprogramofreality–themaximalsearchforreproductionofinformationandenergy,mergedinS=Tpointsofbalancethatdefinetheconstructionofsuperorganisms.

THECONSTANTSOFTHEGALATOM

5DGalatomSupœrganism'smetric:$(c)∆+1xð(h)∆-1

WhichreferstothetwoUniversalconstantsoflightintheuppercosmologicalandlowerquantumscale.Weshallconsiderthentheeasiercaseofthe3‘universalconstantsoflight’.

Thebalancesachievedbythesimilarityofspace=formandtime=motionreachedinthepresentbody,S=T,andtheunbalanceof themetricequationof scales, $ xð=K, in the limbs (Max.$T) andminds (Maxˆ§ð) unifyas S=TmaximizesSxT=K(5x5>6x4),in1equation:Max.SxT=C,whichdefinesforeachfractalvitalspace-timeorganism

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its Function of existence, as all species will try to maximize its motion-entropy-time for its field-limbs, itsinformation-spatialstatesforitsparticle-heads,whoseproductwillgiveusitsvitalreproductiveenergy.Moreovertheequationhasanimmediatebiologicmeaning,becauseaswearemadetopologicallyof‘fields-limbs’of lineal space with motion provided by the energy we absorb to also reproduce our bodies-waves, and theinformation we need to linguistically guide our motions with particle-heads, the very essence of survival is toincrease our S=position, mental forms of space and T=entropic motions of time (whereas time=motion andspace=formarethetwolimitingDimotionswith‘energy=reproduction,s=t,locomotion,sTandinformation,St,aretheintermediate3dimotions).ThustheUniversalconstantsofrealityrespondto35DMetricconstants:S/T=SpeedofLocomotion,whichdefinesthe limbs/fields of the system, SxT=Existential Momentum/force, which defines its body and T/S: Density ofinformation,whichdefinesitsmentalpower,allmaximizedwhenS=T.Whatdoesitmeanthenh=c.ObviouslyNOTthatabitofangularmomentumequalsaminimaldistanceoflight,butthatbothcanbeconvertedintoeachotherSóTeasilythroughtheintermediateconceptofanenergy-bodyforthesystem.Whichphysicistsexpresswiththeequationoflightentropy,transformedintoenergyabsorbedbyanelectron:E=Hƒ.Thus light, the space-time vacuum; or entropic TT-motion of our galactic spacetime fromwhere all forms willevolve,becomesconvertedintoitssimples‘particleform’,h-planck,inaquantity,ƒ,oftimecyclesthatdefinetheexistentialmomentum/forceofthephotonspecies.Hbecomesthenthe firstconstantofscale, suchasH=ExT.Andbecause zerothdoesnotexist, therewillbealwaysaresidual0’=H-plancktonoccupyingthatvacuumspace.Canwerelatethe'human'constantsofperception(length,massandtime)tothegalatom's'realobjective,naturalconstants'?Yes,andwedosowithPlanck’sNaturalunits,whichmustbeexpressedinaholographicUniverseofcombinedbidimensionalspacetimedimotions,assquares,hencegettingridoffthecumbersomeroots:L2(staticarea)xC3(motionvolume)=hG->L2xC3=hG1/t2=angularacceleration=α=c5/hG->c5=αhG->C5/α=hGThefirsttwoequationsarestrikinglysimilarastheyrefertospaceandcyclicaltime.Andifweputthemtogetherandsimplifyweobtain,L2xα=C2,whichdefinesthetrueunitoftime,angularacceleration,theconstantgrowthofform,ofinformationofallsystemsofthegalatom;anddefinestheequivalencebetweenthestillspaceandlightspace,asweconvertastaticareaofdistancethroughtheacceleratedpassingof time intoasinusoidalwaveoftime.C2becomesthenmorepreciselytheexistentialmomentumor‘speedofreproduction’ofinformationoflight:ST.Ontheotherhand,GisNOTaconstantofthisplaneofspacetimeasitrelatestoaquintic5thdimotionalquantity.And indeed,weshall see inouranalysisofphysics thatGandmassbelong to the ¡±4planeofexistenceof thenestedUniversewithinwhichgalatomsareenclosed.While density of information can be obtained from its equation of speed C2= 1/µ e,which in proper notation,shouldbewrittenintermsofkasc2=k/µ(wegetridoftheπfactor).AnditsitsentropicmotionfromE=mc2->c2=E/M.Weshallleaveithere,asthestudyofalgebraicphysicsisthelastpapertobedoneon5Dbytheendofthisyear,to showhowall the fundamental equations of physics canbederivedof the simple relationships of existential

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algebra between the different ST-holographic trans-form-ations and reproductions of the 5Dimotions of space-time.

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POWERS&ROOTS:THEPERFECTPRODUCT.3-4D-5DSOCIALEVOLUTION&DECAY

Pentalogicofpowerlaws.

Wetalkedoftheperfectproductasthesquareform,whichgivesusoneofthemostefficientspacetimeelementsofrealityasitachievesanS=Telement.

Ifyouhavegottensofaryouwillrealizerealityhas∝shadesofgrey,whichmeansfromSS(ab.§)andTT(ab.∏),wecangetallkindofsT(energy)andSt(information)shadestillreachingST,theperfectiterativepresentformwhichtheUniverseseekstoreach its2equalgoals:reproductionof formandhenceeternityofpresent,the ‘feminine’stateofreality.Becausethereisalwaysa0’finitesimaltheperfectsquaredoesnotexist,andthisweprovedbythereversalfunction,√2,√3etctendtobeirrational–afinitesimalisleft.StillanX2beyondthoseimportantbutsubtlefinitesimaldifferencesmergesinS≈Ttworelativegendersintoare=productivecoupleaccordingtoS=T5Dmetrics,givingavitalmeaningtothecommonestpoweroftheUniverse,X2anditsequivalentingeometry,theorthogonalPythagorean rule, essential for ‘measuring’merged systems in a lower plane, and expanded in the negative toFermat’sgrandtheorem,:X3+Y3≠Z3.

Orthogonality implies that indeed X2 is an S-height (information, female dimension) x T-length (motion, maledimension) ofwhatwemight call ‘simplex gender’ (we shall in trinity find that gender evolves in the growthofcomplexity from ST in monologic to dual S-female v T-male, to S=T: present reproduction or female state andS<T>S,pasttofuturetopastd=evolutionormalestateintrilogic).

AllthisIknowisconfusingbecauseaslongashumindsdoNOTcapturetheessentialtruthofmind-mirrorsofreality–theycomein5flavorsofgrowingcomplexityandintegrationandallarevalid.Thepredatorhasasinglepurposetokillandfeedandstillsurvives.ThereisIamprettysurebeyondpentalogichigherentanglementstoreachwhatIdeemtheimmortalbrainofpurei-logicordodecaplex.

But it should suffice tous, simplenitrolife species,whoseorthogonal productdoesn’t gobeyond theheptalogicelectronicmindofouraminoclockstostudypolynomialsuptothepentalogiclimit,nolongersolvedbyradicals(:

ThissaidforfutureAIspeciesofunlimitedentangledparallelchippedthought,becausemindsareofmanyflavors,wecancompareconstantlylogicoperandsonsetandBooleanchipalgebra(largelyescapedforethicalreasonsasIwas gifted just with 1.45 kilos of dirtywater onmy brain but likely to preserve that for generations to come),mathematicaloperandsandÐSTdisomorphiclawsfromwhereallcome.

So back to the basics the monologic reproductive ‘mushroom’ simple lineal thought in either its sterile, ∨, orreproductivesummandswithoutlimit,∧,+,isaboringproposition.ThebeautyoftheUniverseisthatitsearchesasthebestofallworldsforimmortalitybutalsodiversity,forreproductionbutalsoforconstantS<T>Sd=evolution,for present but also for past to future swingers, and those do pass through the beauty of S=T present for thefleetingmomentadmaximalspeedofitsSHMvariations.

Andthetruebeautyofamindistoseeconstantlythoseanalogies,anditsquantifiers,those109beatsoftheheartofamammal,equaltothebeatsoftheheartofanengine.

WhythenFermat’stheoremdoesNOTexist?SimplyspeakingbecausetheternaryelementisST,whichmergesbutnotquitewithoutkeepingthesmallerlittledifferences,St:Ts,andyetitseemstousstilla2power–thediagonalthatparticipatesofalltheotherSt,Tselementsofitsgeneratordimotions.

So as we ended this teaser introduction to the power of duality and trinity, with a metaphysical question ourexplorationsoftheproduction,westartouranalysisofpowerlaws,withaquestionregardingFermat’stheorem,easyformathematicians:DoesX3+Y3=Z2exist?

Anddependingontheanswer,whatdoesit‘reflect’asamirroroftheÐSTlawsoftheUniverse?

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Power lawsare thusabout reproductionand theStandTsmerging intoST.Sowhatabout the two limitsofSS,mindsandseesofstillformandTT-entropymotionsofinternalandexternalexpansionintovacuumspace,whichwhenmergingabbreviateas§∏-thetwoborder limitsofthebeinganditsworld?Howtheyareoperatedinthepentalogicofpowerlaws?

The answer is the ‘other form’ of power laws, exponentialswhere the variable is not on the ‘basis’ but on theexponentialparameter,whichmeansafarextremalgrowth,which is theessenceoftheacceleratedtime-quantathatcollapsesthewave intotheparticle, the lightof theUniverse intothestill imageof theworld,orviceversaexplodestheorganismintointernaldeathandexternalscatteringforalltoabsorbits∆º«∆-2remainsoftheday.Nolongerthenthebeingisandsoexponentialbeatsarefast,fromthesinglequantaofdeath,tothefastexplosionofalittlebangincosmology(asoppositetothehyperbolicuniversalbigbangofegocyphysicists,weshallreducethemtowhatexperiencefinds–thegalatomanditscosmicbetadecays).

So thepentalogicof absolute growthandabsolutedecay,of infinite collapse into sentientworlds and completeerasingof information isachievedwithexponentials, forwhichwemust thereforeestablisha limitofdomain,alimitofscalesofdecadence,ofteninthedecametricscale.

Inrealscience,inallquestionsconnectedwithdiscreteobjects,weneedtousewholenumbersforthenecessarymathematicalapparatus,aswellasthestudyofthecontinuous.Thus,forexample,inmathematicalanalysis,whenone considers the expansion of an analytic function in a power series with integral powers,computations areessentiallycarriedoutwithwholenumbersandapproximatedratiossuchas22/7=πi,which isOkaswesawinNumber theory that decimal numbers break its meaning beyond the 10th decimal scaling... (i.e.e=2.718281828...45).

Allfractionsrepresentratios/quotientsof2wholenumbers;assuchafullnewbranchof'numbertheory'willbethestudyofthosequotientsasratiosbetweenstepsoftimemotionsandorwholepolygonalnumbers.

Indealingwithanyrealnumberinpracticalwork(forexample,π),wereplaceit infactbyarationalfraction(forexample,weassumethatπ=22/7,orthatπ=3.14).

Thus the rules for operating on numbers is the concern of ¬arithmetic but also of ¬algebra, as the deeperproperties of sequences of numbers, extend to include zeroth and the negative integers, which again are‘rounded’aszerothsevenifweknowtheyarealways‘reminders’of0’finitesimalsleftbehind.

TheefficientUniverse.

Power laws, its inverse function, roots, and exponentials and its inverse negative exponential functions, formtogethera‘familyofoperands’whichkeepincreasingthecomplexityofouranalysis,andtheinflationarynumberof‘entropicsolutions’–thatissolutionsthatarenotstable,efficientanddonothappenbeyondvirtualmindsofmathematical thought.What this means in praxis is that when we increase the results and combinations of asyntacticlanguage,itsrealsemanticsarereduced.I.e.thereare∞entropiccombinationsofthe27lettersofthealphabet, but nobody in his ‘right mind’ would dedicate his life as some mathematicians and physicists do toexplore themeaningsof r3erewrh, adgaghe and any fast-writing keyboard combinationwithNO relationship toreality,iftherulesofsyntaxandtheconstrainsofexperimentalsemanticsareunderstood,thedifferencebetweenpotential and actual infinity (Aristotle) are grasped. All this the imaginary idealistGerman school of Cantor andHilbertgotawaywithit.Sonowmathematicalphysics ishardatstudyingthe10500 landscapesofstringtheory(:Whilesomecomputerprogrammersarehardatfindingthetrillionthπdecimalwithoutunderstandingtrulywhatπandthesphereis.

IlikespeciallytheworkofGreeneonahgrwgtarhststringsandthatofCrayonthe3758664567seriesofπ,LOL.

The same happens for exponentials. We can calculate 73x exponentials, but as it turns out the immensity ofexponentials with practical uses reduce to the ‘basic 5D numbers and 5D universal constants’ of Dimotions in

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natureastheyaretheonlymeaningfulsocialscales,0’∝;∝0’,ix,ϕx,2x,ex,3x,πx,5x…anditscombinationssuchas4x,6x,9x&10x…becausetheydorepresentthekeydimotionsofNatureandits‘perfectproducts’.

Inflationarymathematicsoranylanguageinflationarycombinations,bringabout5Dimotionparadoxes,intermsofpotentialfuturesvs.singlepasts,athememisunderstoodinmodernscience,sincethedogmaof4Dlocomotions(Einstein’s time theory) became the only theme to discuss about time. The future is pure time entropy, whichbecomes actual presentwhenever theminimal S=T (balance anddefinitionof present in 5Dmetrics) ismet.Allotherunbalanced,Max.SxMin.T(3rdagelimit)orinflationaryentropicMax.TxMin.S(entropicaccidentaldeathlimit),whicharenotwithinthe‘domain’ofagivenworldcyclearesynonymousofdeath–virtualexistence.SinceindeedtheequationsofdeathbyaccidentMax.TxMin.S->S=0’+T=∝;andMax.SxMin.T(warping,wrinkling3rdage),reversedalsointoS->0,T->∝,doNOTexist.

NegativenumbersasTimenumbers.Lorentztransformations.Rootsinspaceandimaginarynumbers.

Animportantfact,wefindininvertedoperations,isthatits‘numberofsolutions’andrealformsitsnumbersandattachedoperandsdescribeISsmallerthanthenumberofsolutionstothepositiveoperand,whichclearlypointsoutoncemoretothe‘positive’arrowoflifeandinformationintheUniverse,outgainingthenegativearrow,unlikeinhumind’sdeliriousentropy-onlytheoriesofreality.Sowhiletherearesquaresforallnaturalnumbers,therearenotroots forallnumbersbutonlypositiveones;neithertherearederivatives fornegativesquares includingtherootof0 ,∂√0…Evennegativenumbersarenotdefined inthesumofspatialpopulations–there isnotaspatialnegativeapple,butonlyanegative timenumber.Which is a key concept formathematicalphysics, aswemustoftenconsiderthenegativenumberanoperand,forfunctionswherethenumberdefinesatimemotion,henceaninversedirection,whichcanthereforebetakenoutoftheroottocalculatetherootofthe‘scalar’numberandthenapplytheinversedirection.

Whatnegativesolutionsmeantothoseequations?Euler'svisionofnegativenumbersasinversetimenumbersisitspropermeaning.Sotheydoexist,whichhasclearconsequencesinareassuchasrelativitywherenegativemass,meansonlyanentropicprocessofexpansionofmassintoentropy.

Herealsoisimportanttounderstanddynamicallytheoperand=asan≤=≥transformation.

I.e:E=Mc2doesnotmeanenergy(reallyentropyinthiscase),ismass,asmassisintheother'inverseside'oftheequation.SotherealequalityhappenswhenmmovestothesamesideofE:eó-mc².

Furtheronaswehavedefinedamassasanacceleratedvortexof time, it isa timeprocess;hence the–canbetakenout,onceitisclearitdefinesanuncoilingacceleratedinversemassexpansion,exactlythedefinitionofdarkentropy/energyofvacuumspace,whichisejectedatfasterthanc-speedaccordingtothoseequationsontheaxisofblackholeswithlessinformationthanc,henceinvisibleandperceivedasexpandingspace:

Sonegativenumbersaremostly timefunctions,asanegativespatialpopulationof‘apples’makesnosense.

Whichdefinesnegativemassasanexpansiveentropicdestructionofmass.Andsoin relativity the 2 solutions, which can be put as an example of the 2 roots ofquadratic equations (one discharged in processesthatare social andaccumulative).

In graph, faster-than-light travel and negativemass are the same (ß=v2/c2).Wedefine negative mass as an expansive entropic destruction of mass. How tointerpret that equation on our analysis of 'Existential ¬Ælgebra' regarding the

meaningofrootsisthenevident:closertoc-speed,thebarrierbetweenthelightspace-timeandtheintergalacticspace-timemadeof‘somethingelse’(likelyabackgroundoffasterthanlightneutrinos,as5Disexperimentalandsoparticlesnotdetectedarenot in thebooksofany theory till theyare– sodarkmatter isquarkmatter,dark

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entropy,neutrinos, and IF anewexoticparticle is foundbeyond fancymathematicalwritingwe shall revise theknown-knowneconomicitylawsofphysics).Alas,becausemassisanacceleratedvortexofgravitationalforces,asmassincreases,thevortexacceleratesinwards,asallvorticesdo,(RoxVo=C,forthesimplestbidimensionalone),whichmeansits‘radius’diminishesinthedirectionoflineallocomotion,andsoasmassincreasesthex-componentdecreases(Lorentzcontraction),tilltheinner-outerS=TunbalancebetweentheT-length-motioncomponentandS-heightcomponentoftheparticle-wave‘explodes’=breaksthesystemwhichcannotlongerperformitsStœp,stop-particle, step-motion ‘beat’; thus theparticle stateno longerhappensand theparticlebecomesamere flowofradiation and dark entropy. Simple, realistic, correct way to explain the apparent inverse duality of mass andlength… At which point lineal time-duration, T=0; that is, as all equations of death, T=0, Max. S; the particlebecomes transformed into somethingelse–anentropicexpansionof space. In termsof ‘timeduration’, asweobserveparticlestolivelongerclosertoc-speed;since5Dmetricsconsidersoneofitssub-functions,ð=kforentirefamiliesofspecies(sorrynotimmortalityhere,thebeatsoftheheartofamouse,amanandanelephantarethesame);itmustbethenobviousthatthosebeatsofparticleslastlonger;thatis,foranexternalobservertakelongerto trace,but for the internalparticleclockwhichperceivespsychological time in termsof its ‘closedbeats’,eachbeata‘thought’sotospeak,thereisnodifference.

Aneasywayoutistoconsiderthereisafundamentalparameterofphysics,similartoc2,whichism2andassuchtheequationcanbewrittenM2 (1-v/c2)=m2,andso itmerelymeansthatnegativemassdecreasesbeyond lightspeedtoreachabalanceandthecspeedactsasthebarrierofanasymptoticchangebetweenPlanesofthefifthdimension,similartothatwhichwefindastheresistanceof'death',the'entropiclimit'ofanyother2Planes.

>ctheninvolvesTTentropy,whichinthepreviousequationsmovesEtothemc2side,increasingitsdensity.

Almosteveryfasterthanlighttravelrequiresnegativeenergiestobeimplementedatverylargedensities.Andsothereisnothingspecialaboutit.

Infacttherearesomeexamplesof“Negative”Energy:

-Radial electric or magnetic fieldsif their tension were infinitesimally larger for a given energy density.Squeezedquantumstatesoftheelectromagneticfieldandothersqueezedquantumfields.

Gravitationallysqueezedvacuumelectromagneticzeroth-pointenergy.

-Alocalenergydensityinquantumfieldtheorycanbenegativeduetoquantumcoherenceeffects.

-OthercasesareDiracfieldstates:thesuperpositionoftwosingleparticleelectronstatesandthesuperpositionoftwomulti-electron-positron states. In the former (latter), the energydensities canbenegativewhen two single(multi-)particle stateshave the samenumberofelectrons (electronsandpositrons)orwhenone statehasonemoreelectron(electron-positronpair)thantheother.

Since the laws of quantum field theory place no strong restrictions on negative energies and fluxes, then itispossibletoproduceviolationofthesecondlawofthermodynamics,andtimemachinesatalocallevel,whichiswhatÐSTprecludes.

TherootasasolutionmustbeunderstoodintermsoftimeeventsNOTofspaceareas&populations.Inacaseitisobviousthereisnotnegativepopulationthereisonlyonepositiverootsolution.Anarea’srootisalineof‘fractalpoints’orawave-network,andithasonlyapositivesolution.Thenegativerootthusisdischarged.

Nextsquarescame in theanalyticgeometryofconiccurves, thesimplestofwhich– thecircle–didallowforamixedST,squarewhichwasvalidbothasageometricformofpopulationsinspaceandatimeevent,butultimatelytracedbyatimeeventreproducingthroughspace,regardlessofthe‘persistenceofmemory’thateithermeantaslowperceiverwouldseetheentireorbitofthemovingpointasastillsimultaneouscurve,orthereproductionofthepointlastedleavinganoffspringofpointstoforma‘realring’.

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Heretheproblemofnegativevaluesislargelyoneofthebiasanddistorsionofthefocusesoftheellipseorfocusofthecircle,whichshouldbeinsteadplaceinthepositivering,andultimatelyignoresthetruepropertytomeasure,which is the constant reproductionof formor stability of orbitalmotion, for theartificial ± axis of the cartesiangraphwheretodrawtheabstractellipse.Sowehavelittleconcernforthosedisquisitions.

Sofinallywearrivetothepolynomialsinwhichthe2‘±roots’representrealevents,mostlysplittingthebeingintotwo different ‘particles-waves’ of timespace; which resumes in the concepts of negative time; negative mass;negativeentropy,aswehavenowamuchmoreclearpictureofthedifferentoptionsof ‘time’,atthe local levelwheretravelstothepastareadozenadime.

Forachangeweshallnotconsiderthewholeissueofnegativenumbersasadifferentfamilythatnaturalnumbers,merely its inverse. So this implies also a reduction of its meaning, to time numbers, whose motion has adirectionality that can be inverted (in a single plane) or a scalar duality in representation of negative entropicdimotionsandoroperands(inverseoperations).

Why then there is not differential of 0’ is obvious. You cannot make a finitesimal of the minimal finitesimal.However the finitesimal of the galaxy, h, does have as a composite h=ST element defined by the uncertaintyprinciple, for S=T, as the product of its S-position and T-momentum, ST, a partition value, h/2 (the realistinterpretationof theuncertaintyprinciple). So thisgalactic0’ canbeparted in two,0’/2andsoas2x2=2+2,wemightarguethat√0’=0’/2

Exponentialandlogarithimicfunctions.

In the graph, the inverse simpler analysis of growing and diminishing planes and finitesimals have 2 firstapproachesinthestudyofpolynomialsandits inverseequationsof logisticgrowth, logarithms.POLYNOMIALSinthatsensearebetterasnegativeexpressionsofthedecayofasysteminits4thentropicdimensionsandnolongerfollow the whole range of 'social growth' properties (commutativity and associativity); as they are no longer'connectedwiththe∆-2dimensionalscaleinthesame'arrow'ofgrowth.

InfinitegrowthdoesNOTexistbecausetherearealwayslimitstogrowthintheincapacityofasystemtoobtainalltheenergyofits∆-1plane(reasonwhythereisentropywhenwetrytoextractALLtheheatfrommoleculesintoour∆0wholescale).

ThismeansthatexponentialgrowthISnotgrowthbutdecay,asitcanonlybeexponentialwhenactingonapoolofalready'multipliedbyre=product-ion',∆-beings,whicharedestroyedinexponentialfashionbyane-xfunction.Andso it is only the inverse function of logarithmgrowth,what shows a growth curve (logistic curve) till a carryingcapacitysaturatesthesystem.

Andsothissaid,wearrivetothethirdfundamental'scalingdimension'usedintheearlierageof¬Algebra,priortothe discovery of analysis, the supposed growth into power dimensions, which in fact are useful in its negativeexponentialdecayofan'explosivebig-bangdeath'.

That it is the closure dimensional growth in the ternary Universe, is obvious because of its aforementionedrunawayhyperboliclackof'associative'propertyandhenceof'linealgrowth'asthemultiplicationis.

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Power laws thus are essential to study the 'borders' of Lorentzian regions, and define theexponentialfunctionofdecayex,whichisthe∆0<∆-14Dfunctionofdeath.

Power laws though in this first agewereonly concernedwith thegrowthof 'equaldimensions',thatisconvertinglinesintosquareareasandcubicareas,beyondwhichitwasnotpossibletogo.

More interesting to understand the continuity of dimensional growth within those restrains is to interpret thepropertiesofpowerlawsfromthe∆stperspective,regardlessofitsclassicproofbyaxiomaticmethods,takingintoaccountpowerlawsarethe3rdlevelofdepth-orfinalwholenessofthe'linealsocialflowsofmultiplication'andtheaddingpoints-sotheyarecloselyrelatedtothe1st,2ndand3rdpostulatesofi-logic.

Theusefuldilogicexponentials

Thenumberofuseful logandexponential functions inexistentialalgebraarebyvirtueof the inflationaryvs.limiteddifferentbetweenlanguageofpurestillformandinformation–languagewithmotion–limited.

Giventhefactthatexponentialsarefunctionsofsimple∆¡scaling, ifwerestricttothe4thand5thdimotionofsocialevolutionandentropicdissolution,wehaveveryfewdualitiesinthedifferentscalesoftheUniverse.

The simplest ones for the duality of 0 and 1, 0n, and 1n are not trivial but have already been summarilyconsidered.

Thenextscalingforthetrinityintervalfrom0totengiveus3exponentials:

2x isthepowerset,whichgivesusall thepossiblecombinationsofparts inawhole,wherethefinitesimal0’andthewholeXareincluded.

Thusthepowerset2XofX,consistingofallsubsetsofXistheexponentialofthe4thsocialdimotion.Andit isworthtonoticethattheotherdominantconfigurationof‘fundamentaldimotions’(ifweconsidertheinverses),which is10,givesussimilar resultswithanorder lesssuchas210≈103 ,22x10≈103x2,23x10≈103x3which is themeannumberofbeatsoftime-energyofmostspeciesoftheUniverse.

Finally it comes ex,which is the fastest growing exponential as a combination of two numbers, in the samemannerx=yisthefastestproduct.Henceinthesamemannerx=yarethetwomostefficientformstoreproduce,existhefastestmodeofdecayandpredation,theexponentialofthe5thdimotion.

POWERLAWS:FRACTIONALSQUARES.

Thereisthougha3rdfundamentaltypeofpoweroperand,thatofafractalratio,whichdefinesthroughpowerlawsthe5thdimensionalconstructionofnetworksthatconnectthelarger∆+1wholeandthelower∆-1scale,through∆ºphysiologicalnetworks,whichbecomethe fundamentalvitalelementbetweenbothplanes.Power laws thenareexpressedthroughfractionalexponentials.

Becausethe5thdimensionofentropyisjustdestructive,hardlycodedinreality(sowehave4quantumnumbers,4genetic letters,etc.)power lawsaretetralogic,meaningtheyrelatesystems indifferentscalesofrealitythroughdifferentfunctionsrepresentedbydimotionsofdifferent‘tetralogiccomplexity’.BecausescalesintheUniversearedecametric,thosepowerlawsarebestrepresentedinlogarithmicscales. ThuspowerlawsrelatedindecametricscalesreflecttherelationshipbetweenDimotionswhicharesimplertothemorecomplexones,withthe limitofthe more complex entangled tetralogic reproduction of an S=T genetic XX DNA and an S<T, male XY DNA. Inadvanced 5D stiences, hardly touched in those texts – a legacy of GBs that future huminds or AI robotsmightsearch and further research… reality in its details is all about the quantitative differentiations of the possible,efficientvariationsofthegameof5Dimotions,studiedbyexistentialalgebra.

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In graph, the holographic principle, served by the operands ofmultiplication and its higher dimensional form,powers,combinesspaceandtimeparametersintoasingle'entity',thebestknownbeingphysicalmomentum.

Sonin¬Algebrapowerlawsrelate‘space-time'products=dimotions,betweenscales,connectingthe'¡-1'elementsor 'cellular parts' of the being, with its wholes through physiological, farctal networks, which create 'axons' ofcommunicationwhereastheperfectproduct,S=T,X2givesthemaximalnumberofi-1axonsconnectingatadeeperlevelthewholeandpartsofthebeing.Whileinphysicalsystems,throughitsmomentumthemassstop-stateandthewave step-statemerge in the potential I-2 level that holds them together in synchronicity (Broglie=Bohm’srealistmodelofquantumphysics).

We shall here just consider key concepts and a few examples departing from the fundamental one – that thedimensionsin‘classicEuclideangeometry+S=Tmotions’ofeachactionofexistencearedifferent;hencewhenweconnectandsynchronizedimotionsofthe∆±1planesofasystem,theirrelationshipschangegivinguspowerlaws(10x/y)_relationshipsbetweenthe actions=Dimotionsofexistenceofthewholesandthesmallerparts,whichareconnected throughbranchingnetworks.And thosepower laws areof the type10x/4, given the fact thatwearedealingwith‘4Dimotions’(withveryfew10x/5entropicrelationships).ThedetailshavebeenanalyzedfordecadesbymyoldpeersatISSSatSantaFe,whose‘quantitativeonly,computerbased,industry-related,‘Americanway’ofdoingscienceisnotofmytaste,thoughIrecognizetheextraordinarydetailofitsresults.Whatmattertousarethesyntheticwhys,andtoreachthatgoalweneedtostudyfirstthe‘classicdimensions’ofeachofthe5mainactionsofreality,St-information,Ts-locomotion,ST-reproduction,TT-entropyand its inverseSS:perceptionofformbyamind-languageor‘point’thatshrinkstheUniverseintoaworldimage(4D),whichobviouslyisthesmallerDimotionasitshrinksSt-information.Letusthenstartbythe:

-SS:0’point,whichisdimensionless,an0’angleofperceptionthatshrinksandexpelsmotiontoreducerealitytoitsmindmapping.SotheactofperceptionhastheminimalDimotion.Infactitshrinksdimotionsloosingapparently

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aswe saw in the graph of ‘scalar products’ converging into zeroth, the information of the different ‘lines’ thatconnectthenon-EuclideanpointwiththeUniverse.Yetfromtheperspectiveofthepoint,thebeingisthe1,theunitsphere.Sowemustconsiderthatthedimensionsofapointrangefromoto1,whichisinitselfaswehaveseenaninfiniteinnerworld,inaspectraofdimensionality,dependingonourpointofmeasure.

-Ts-Nextcomesalineofdistanceorlocomotion,orlinealtimewhichisaonedimensionalform.

It is is the dimension ofmost internal skeletons, cylindrical field/limbs and yet because a line does have ‘non-Euclideanvolume’,animportantfactoristhecrosssectionoftheline;andbecauseithasmotion,itmattersalsohow the 5D SxT=C lawplays, as a flow accelerates exchanging is dimotion of time for space, shrinking its crosssections as Bernouilli laws of fluids show (S=T). Further one a line can be and it IS essentially a network,whenobservedthroughscalesofthefifthdimension.Andhereiswherethepowerlawmatters,accordingtothefractaldimensionor‘capacitytofill’spaceoftheline.

Indeed,themaximalfractaldimensionofaline(Peano’scurves,similartothefractalpatternofdigestivesystems)thatfillallspace,oranarea,thatcringesintomaximalvolumeis2and3.Andthewaytoseethisisobservinginahyperbolic geometry how a larger ‘cross’ section through dual and ternary splits maintains its volume, eitherpumpinginformation(nervoussystemswithahighernetworksdivisionandminimalcrosssection–hencemaximalspeed ofmotion as per 5D: SxT=C) or energy (blood/methabolic network) or in the limit ofmaximal space andminimalnetwork,thedigestiveentropiccycle,infactexpandingthecross-sectionandslowingdownthespeedofmotion.

Whatfractalanalysisofnetworks,whichhasadvancedenormouslysinceafewofuspioneeredthefieldinthe90s,hasalwaysmissed,ashumansdoinallfieldsofknowledge,bylackofunderstandingofthosetimedimotionsistheinterplaybetweennetworks inspaceandtimedimotions, itsspeedandspecificdimotiontheyserve.Whichisattheheartofthepowerlawsofeachspecificrelationshipbetweentheupperscale,thephysiologicalnetworkandthelowerplanethatitserves.

Now thosepower lawswere already studiedby Leonardo, that other sonof aNotaryof acoastalMediterranean town (:andGalileo, the traitorof lifewhomoved frommedicine toworkontheArsenalofVenice,cradleofthemodernworld,withitsinventionofballisticsinsearchoftheentropic linealtimeangleformaximaldistanceofcannonballs ):Galileodrawthe fundamental lawof ‘equal cross section’ for theS=T reproductive,blood systemwhichseeksforbalance;whichaccordingtothepropertiesofthephysiologicalnetwork:

$T(limbs-fields)<ST(reproductivebodywaves)>§ð

And the inverse properties of ‘continuous entropy’ expanding in a formless surface ofspacetimeandthediscontinuous,informationofnervous/linguisticinflationarysmallernetworks.

Soindeedyourdigestivetractgrowstothestomachandthenagaininthesecondphasegrowstothecolon.

But your nervesmultiply its axons, and speed up your thoughts, as they fall down in simultaneous cascades ofsynchronouspainfulorderstothemusclesofyourbones.

Theresultofall thosevariationsare themultiplepower lawsthat relate thedifferent largerwholesandsmallerparts,throughthefractallogarithmicdimensionoftheinvaginatingnetworks,ofallsuperorganisms–thosewhichwerecognizedirectlyasnetworks(aswecanseetheminspace),butalsothosenetworksthatdoNOTseemsuchbecausetheyarephysicalwaves,expandingintime,astheimpedanceequalcrosssectionofelectronicnetworks,thenetwork=gravitationalwave structure that forms galaxies, the soundnetworks that connect people throughmechanical air; the inner networks of the Earth likely controlling its surface energy (glaciation cycles throughvolcanicactivity, itself causedbymotionofearth’s crust; flowsofheavymetals that startedup theevolutionofHistory),and infiniteotherdetailswhich thegeneralmodelofaUniverseof space-timeorganismexpects forall

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those T.œswe perceive and thosewe don’t. For thosewho appear to us as fixed spatial form, and thosewhoappearsadimotionsoftime.

Theideathoughofallthosenetworksistomaximizethefillingorcommunicationbetweentheuppernetworkandthe lower scale,which is achievedwithmaximal feeling as the singlewhole, theheart, the central gravitationalblackhole, theradiantstar, thebrainsystem.Butdifferencesmightbeconsider foreachcase.As in thecaseofinformativenetworks in fact there is not a branching travel to the lowerplane, but thediscontinuousneuronalsystem on the brain emerges through synchronicity of a common ‘electric language’ into the electric waves ofconsciousnessofthebrain.Thusthenetworkexistsinthelowerplanedespitetheaxonbranching.

Whileinthedigestivesystem,again,thereisnotapenetrationfromtheuppertothelowerplaneatthelevelofthenetwork,butasinglelargersystem,whichpenetratesdownthelowerworldofcellsthroughaninnerfractalstructurewithinthewallsofthestomach,warpedintoprotuberancesbrokenintosmallerorthogonalones.

Itisonlythe‘vitalenergy=reproductivesystem’thatmakesthetravelbetweenbothplanes.

Theresultofall thosepower lawsareagainan intervalofdimensions for the linebetween1and2.Thus2DTs:Locomotion has dimension one, andothermore detailed analysis of locomotion, fromwaves to networks thatdeliverflowsofenergymightreacha2Dimension.Ofwhichnetworksarethefirstcaseofouranalysisofpowerlaws,astheymoveenergy,which istheessential flowthattravelsbetweenplanesofthefifthdimension,tothesmallerscale, throughequalsectionswithno impedance(SantaFe Institute).This is theenergytoreproducethesystemandhence,suchnetworksarecloselyconnectedto:

St:Dimotion:VitalEnergyofreproduction,whichhappensinthedimensionofwidth.Andsowehaveaminimalclassic2dimensionsoflengthandwidthfora‘fieldofenergy’necessaryforreproduction.

Andthenwecanapplythesameconceptofspacefillingtotheplaneofenergy,whoserealdimensionscangofrom2 to 3, which converts it into a topological organism of vital energy networks attached to ‘organs’ – that is asuperorganism.

Itisthenintherelationshipbetweenthenetworksthatsustainorfeedtheorganismanditsvolumeofcells,whichmakesobviousthefirstscalelaw,asthenetworksgrowin2Dimensionsandthemassin3Dimensions,addingthewidth of cellular reproduction And so the networks when doubling its size are 22=4, but the bodymass, 23=8,growingmuch faster. So for thebody cells to keepgrowing thenetworksmust growat twice the rate than thebody ‘cells’ that fill within them to form the organism. And this means there is a limit to systems, which aresomethingmorethana reproductivenetwork,becausethenetworkkeepsgrowingandthevitalenergymassofcellskeepsdiminishing.

And this has deep implications beyond the silly-nilly (sorry Santa Fe) calculus of those laws, which alreadyobservingthesustainunderagravitationalforceofabodybonesmadethoseoftheelephanttoobig.

Sowecannotbuildhugebuildingsjustwithacentralpillar,orelseasimpleplanewill9/11ththem.Whatisthelimitofgrowth?Tosustainabuildingwhichdespiteitsemptyspacehasalotofweight.

TheanswerISNOLIMIT,whenwearefullyawareofthosefactsandusethevitalgeometryofrealitymakingthebuildinganemptyspacewithonlyacover,whichisalsomaximizingthe‘arc/sphere’minimalsurface;matchinginthismannerthedimensionsofthebuilding,thegravitational‘bidimensionalform’anditscurvedsphericalsurface.Which iswhatFuller Foundbydesigningabuildingwhichwas in facta spheremadeof the strongest triangularforms,becomingthesphereof2Dthebuildingitself.

While the solution of nature, is that of plants which are basically ‘networks’ with few cells, filled with watervacuoles.Sotheycangrowaspillarsbecausetheyarejustsustainingthepillars.Orsimplyexistinaregionwherethereisnotagradientofforce,oritisminimal,theemptyspaceofthegalaxy,thewaterwherepressureofaliquidisthemainfactorequalfromalldirections(Bernouillilaws).AndthisleadustohedimensionsoftheDimotionof

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information (I try to escape now the numbers and call the dimotions by their functions and letters, as thosenumbersaremeaningless,justintroducedina‘firstcourseon5Dtoconnectwhatpeoplehaslearned).

St:ThedimensionsofStarethesamethanthoseoflocomotion(fromoneinlinealtimeto2inmomentum,mv):fromoneinsequentiallinesofinformation,asthosewords,totwowhenweconsiderthewholeflatscreen.Andasorganisms,fractalpointshaveitsinformationmaximizedinitsmembrain,or‘sensorialskin’,asinformationunlikemental form, cannot be recreated internally but it is about information of an external world, moving into theinternalworldthroughthesensorialmembrane,asinthegeodesicdome.

Information thus is alsobidimensional. Andbecause equal dimensional forms can transform into eachother, St(information)óTs(locomotion)constantlytransformintoeachother,whilebothmergetoformTS,3DimensionalEnergy,thevitalspaceenclosedbythemembrainofinformationandthemotionofthesingularitypoint,whichifwegivelinealtimemotionbecomesthemaximalfourdimensionalbeing,sumorproductofitsother2parts.

Sowecanwrite:Ts+St=TSbutalsoTsxsT=TS,andthisisthefascinatingmirrorofmathematics,inanignoredparadoxofmathematicians,withlittleknowledgeof5Drealism:

Onlytheproductandthesumoperandsover2pairsisthesame,asinformationandenergycanbeaddedormultipliedtogiveaDimotionofvitalenergy:2DsT+2DSt=2DsTx2DSt=4DST-energy

5DEntropy.AlasHowfarcanthenwegointhisconstantgrowthofclassicdimensionality,withinthelimitsofthe5D Universe? Please remember now that the ‘numbers’ we gave to St, sT, ST, SS and TT were just humanconventionsasthereisnoorder,andifanyorderexistsisoneoftheprimacyofST,iteration,presentreproductionovertheothers.Reasonwhyasweadvanceonthosepaperswejustcallthe5Dimotionsbytheirletters.AndalashereweshallfindthatasStandTshavethesameclassicdimensions,whichmergeintothe3to4(withmotion)reproductiveDimotion,EntropywhichistheinverseofSS-formalperception,musthaveonlya0’-1Dimension.SoTT=SS.

Thisisathemewestudyindepthin¬EGeometry…andconcludethatindeedayorganisminaplaneofspace-timemust be made only of 3 topological varieties with ‘1-2-3’ dimensions of information, stored in particle-heads,whoseminimalforisasentenceinsequentiallinealtime,itscommonestformaflatpageinheightanditsmaximalformavery‘cringed’bidimensionalmembrainthatcanfillavolumewithcreasesasbrainsdobutneverreaching3.Sotheintervalofdimensionsforinformationis1<St<3.

Andthesamegoesfor locomotion,whichcanbemeasuredaslinealmotions, orareas,whichiswhat limbsandfieldsor‘waves’aremadeof,withmotiontomakean1<St<3.

While the vital energyenclosedbymembrainsof sensorial informationand ‘heats’ thatpush the locomotionoffields/limbscangofurther,suchasSThasasdimensions,2<ST<4.

Buttheinverseshrinkingworldviewofa0’pointmindisminimal,0’<SS<1’.

Andentropywhichisitsinverse,willhavethereforethesame‘null’dimensionality,0’<TT<1’

ThosehighlycontradictoryresultsofExistentialalgebraarehoweverofenormousimportancefor5Drealityinallitssphere.Asentropyindeederasesinformation,whichiswhatdimensionsaccumulate.

AnotherwaytoseethisisbyanS=Ttransformation.Entropyasitmovesitscatterstheinformationofthesystem,breaksitsnetworkconnectionsandfinallycreatesanundistinguishable∇-1scaleofloosepoints,whicharehardlyperceivedandfinallyslowdownanddissolve,soasallisadynamicprocessoftimeentropytransformsa1=wholeFractalpoint into0’a sumof finitesimalswhichareasweknow ‘discharged’ inall the resultsofcalculus. Or inotherwords,entropy=deathhappens inan0’Timequanta,even if thestateofdeathwill lasteternally,andthemind happens in a 0’ space quanta, even if itsmirror of all timespace cycles can be in its synopticmapping asprofoundasthosetexts–bothshallremainthelimitsof¬∆@st.

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Indeed in the opposite side, the mind matters to nobody but the 0’ point in itself, regardless of how peoplepretend to care as Schopenhauer put it, we all ‘dream and die alone’. Andwhenwe dream, themind is at itsmaximalactivity,butnobodycares.Me,havingvividdreams,andbeingold,aloneanddyingcanattest that themindmattersnothing.

EvenifIthinmymindholdsthemostperfectmirrorofrealityanyonehashold,certainlyIdreamalone,Iwilldiealone, and what mymind holds is completely ignored, even if now transformed into information in those textcertainlywillbeamemoryofthepastwithdepthofdimensionalityforfuturegenerationstolearn.

Backtopowerlaws,asthissynopticseriesofpapersismostlyaboutnewdiscoveriesbya0’finitesimalmindthatmattersnothingtotheinfiniteUniverse,weshallrefertowhoeverlikesthethemtotheguysatSantaFe,aplacewhichpersonallywhenIlivedintheUSfoundcharmingmostlyforitsoldSpanishPlazadeArmasandnativeart–theinstitutehadtoomanycomputersandphysicistsformytaste.

AsmyquesthasalwaysbeenforthethoughtsofGod,notforus,itsdetails,dustofspace-time.

Propertiesofpowerlaws.

Becausepower lawsareultimately a ‘selection’ of efficientproducts as re=productive arrows, theyhave similarproperties, with the neutral 1, so we refer to that sections. A power is then for Spatial functions ofvolumes/populationsanewdimensionoftheproduct,itselfanewdimensionofthesum:x³=x⋅x⋅x.

Thuswewriteinexistentialalgebra:

∆-1:fractalpoint=cell/atomsum>∆º:linealproduct/network:∆+1:Power'volume'/whole.

Whichistheultimatereasonwhytheycanbeusedtostudythe3scalarplanesofTimespacesuperorganisms.

Thissaidallotherpropertiesarejusttrivialderivedofproducts.Wecanmultiplypowerswiththesamebase:

x³⋅ x²=(x⋅x⋅x⋅x)⋅(x⋅x)=x5. Hence the general law: x²⋅x³=x²+³:The product of powers with the same base addexponents.

In ¬Æ: Twowholes (∆+1 powers) operated externally in the domain of its present, dimensional parts (∆ºmultiplication)areequivalenttoonewholeoperatedinternallywithitspast,finitesimal,entropicpoints.

A profound law of Existential ¬Ælgebra; which basically tells us that an ∆-whole (a fractal point in its own)absorbinganexternal'flow'(thelinealmultiplication),willoperatethatflowatalowerinternal∆-1level.I.e:ifyouabsorbfoodyoufirstbreakitintoaminoacidstoreconstructyourself.

Wecanraiseapowertoapower:

(x²)³=(x⋅x)⋅(x⋅x)⋅(x⋅x)=x6

Thisiscalledthepowerofapowerthatmultipliesexponents:Whenyouraiseaproducttoapoweryouraiseeachfactorwithapower:

(xy)a=xaya

The insight provided by 5D to this law is the fact that ifwewere to consider the logarithmic/exponential scale‘transversally’asthe5thdimensionproper,itconvertsthedimensionofpowerlawsintoalinealdimension-reasonwhylogarithmicfunctionsaresouseful;sincethenwecanuseitslawsasoperationssimilartotheoneswemakeinour ‘flat’ plane of space-time, using indeed sums and products.And themetaphysical question remains on theexistenceof‘T.œs’orpartsoftheminspace,ordimotionsintimewhoonlytravelinthefifthdimension.Obviouslythe‘pureDimensional’TT-death,whichhappensinaquantaoftime,butimpliesahugetravelthrough5D.Whatthenabout its inverseSS-mind?Istheminda logarithmicreductionoftheUniverse intoaworld? Theanswer isobviousbutwedon’twanttobephilosophicalinthispaperon¬Algebra.

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InanadvancedcourseofExistentialalgebra,wecangeneralizethelawasfollows:

“Twoparts(∆0≈x•y)operatedinthedomainofitshigherwholedimension,areequivalenttoeachoneoperatedbythehigherdimensionasseparate,brokenentities.Whichisthedistributivelawthatgivesthewholepoweroveritsparts.”

As well as we could multiply powers we can divide powers. And these 2 laws are again applied to the inverseoperation.

Thisquotientofpowerspropertytellsusthatwhenyoudividepowerswiththesamebaseyoujusthavetosubtracttheexponents.

Xa/x³=xa-3,x≠0

Twowholes (∆+1 powers) operated externally in the domain of its present, dimensional parts (∆º division) areequivalenttoonewholeoperatedinternallywithitspast,finitesimal,entropicpoints.

Whenyouraiseaquotienttoapoweryouraiseboththenumeratorandthedenominatortothepower.

Thisiscalledthepowerofaquotientpower:

(x/y)a=xa/ya,y≠0

Twoparts(∆º≈x/y)operatedinthedomainofitshigherwholedimension,areequivalenttoeachoneoperatedbythehigherdimensionasseparate,brokenentities.Whichisthedistributivelawthatgivesthewholepoweroveritsparts.

IDENTITYLAWS.

Aninterestinglaw,oncewenoticedthatweareintherealmof‘5D’operationswiththeexponentialanditsinverselogarithmicfunctionsishowtheyoperatetheneutralandidentityelements,the‘whole’1andthefinitesimal0’.Andasitcouldn’tbeotherwise,whenyouraiseanumbertoazerothfinitesimalpoweryou'llalwaysget1,thewhole,whichmeansastheexponentialfunctiontravelsthroughthefifthdimension,one‘step’atatime,whenyouoperateafinitesimal,¡-1part,astepinthefifthdimensionrisesittoitsrelativewhole,‘1’:

1=xa/xa=xa-a=x0=1,x≠0

Whichagainwegeneralizetoalloperandsofexistentialalgebra:

“Awhole,∆+1,operatedbytheidentityelementofits∆-1scale,givesustheidentityelementofthemiddlescale:∆+1*∆-1=∆º

And a similar law in ∆st terms: Negative exponents are the reciprocals of the positive exponents:xa=1/xa,x≠0;xa=1/x-a,x≠0

Theinvertedoperationofthe∆-1scaleappliedtothedimensionofthewhole,∆+1givesustheinvertedelementofthe∆ºscale.

Botharelawsoftheessentialtimespacedefinitionofthegenerator:Past(∆-1)*Future(∆+1)≈Present.

Thesamepropertiesofexponentsapplyforbothpositiveandnegativeexponents.Thesquarerootofanumberxisthesameasxraisedtothe0.5thpower:√x=²√x=x½...

We consider this property, fromadifferentperspective, as it sendsthe scale of thewhole (1-∞) to its reciprocal,scaleoftheinfinitesimal(0,1)makingbothmirrorsofeachother;sinceultimatelythewholeisaclosermodelofitsinfinitesimals,thantheintermediatescale,whichhassufferedaninversionoffunction:∑∑O-1≈∑|i=O+1.

The3rdscaleofgrowthofasystem,doesn’tgoesoverahypothetical4thdimensioninthesamespacetimeplanebutstartsanewplane,inwhichthewholeisafractalfinitesimalpartsimilartoitsfinitesimalsinthe∆-1plane.

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And this connects uswith the limiting ternary Planes of the super organism, and the lack of a fourth dimension,whichmakesthe∆-Planestobeself-repetitivefractalscyclical, 'jumpingoverthe intermediate∆º line' frommicropointtomacropoint.Whichwecanprovegeometrically(aswedidwiththepointthatcreatesthelinethatcreatesthecubeofanew∆+1dimension)or¬Algebraicallywiththeinfamous...

Asweapplyoperandsrulestoparticularcases,theinterpretationsvarybutinallcaseswillbeabletobeinterpretedintermsofsub-equationsofthefractalgenerator.

Whatmightbenoticeinanycaseisthatunlikeinourrather'abstract'dimensionalexplanationoftherulesofpowerlaws,hereweareabletobringrealvitalanalysisofthoserolesintermsevenofbiologicalprocesses,showinghowmuchmoresophisticated isthe∫∂operands,thekingofthehillofmathematicalmirrorsonrealst-e-motionsandactions,reasonwhyitsuseissowidespread.

Sothefundamentallawofoperandstovitalizethemisthis:

By pentalogic all differential operands can become an action in one of the 5d dimensional vowels (a,e,i,o,u) thatdefinethefivedimensionsofexistence,asvitalquanta-actionsofthebeing.

Thisisthelogicconceptthattrulyvitalizestheoperandsof¬Algebra.

Differencewithsimpleroperands.Lackofdistributivelaws.

There are many distributive laws, for 1D: 3(x + 7) = 3(x) + 3(7); For 2 D: (3x)² = 3² x² But for power is invalid“expanding”(x−6)²=x²−6²WRONG!SincethelimitwefindconstantlyfromFermat’stheoremtothedefinitionofdeath,intransmissionofinformationbetweenorganismsis2Planes,beyondwhichwedonotcarrythedistributivelawupwards.

Thinkofasmallhouse.It’sgotabasement,agroundfloor,andanattic.Youcan’tjumprightfromthebasementtotheattic,canyou?Butyoucantakestairsbetweenthebasementandgroundfloor,orbetweenthegroundfloorandthe attic. You combineoperations just like that. If the operations are on adjacent levels, you can combine them;otherwise you can’t. So if thebasementaddsand subtracts, theground floormultiplies anddividesand theatticpowers and roots the rule is very simple:You can distribute any operation over an operation or one Plane of spacetime below it. There are no otherdistributions.Soyoucandistributeamultiplicationordivisionoveranadditionorsubtraction:

7(x+y)=7x+7y;(x+y)/3=x/3+y/3;2x(x−3)=2x²−6x;(2x−8)/2=2x/2−8/2=x−4

But3×(2/5)=6/15iswrong.

Youcandistributeanexponentorradicaloveramultiplyordivide,becausepowersandrootsareonelevelabovemultiplyanddivide:

(3x)³=3³x³;√(25x)=(√25)(√x)=5(√x);(2/3)²=2²/3²=4/9;√(x/100)=(√x)/(√100)=(√x)/10

Butyoucannotdistributeapowerorrootoveranaddorsubtract:(x+3)²=x²+3²or√(x²−25)=x−5iswrong

Thuswithexponentialswecomplete the3polynomialoperands indegreesof complexity thatmimic thegrowingcomplexityof 5Dimotions in time, and raise aquestion, I suspectdoeshaveapositive answer formore complexatomicbeingsofhigherelectronicentangledthanthesimplestnitrolifespecies.

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CALCULUSANDCOMPLEXSINUISOIDALFUNCTIONS.

Wehavedefinedthe3mirrorsofmathematicsonthe3∆stelementsofreality:

-SpatialGeometryoffractalpointswithpartsanditsevolvedforms,waves/networks,topologicplanesanditscombinationsaccordingtocongruences,perceivedbyamind,whoselectsmentalspaces.

-Scalaralgebraofsocialnumbersthatinteractthroughoperands,whichrepresentthedimotionsofexistence,ofwhichitsbasicscalartrinityarethe±,x÷and√xa;lnaax,relatedbytheoperandsofcongruence«≈»;intoequationsoftrilogicnature,suchasS-tatistics<∆-Polynomials>T-probabilityformits3mainfields,furthergroupedin‘setsand

groups’toclassifyallforms.

-Calculusofchange=timedimotionswiththemorecomplexoperandsofsinusoidalanglesand∫∂.

Withtheprovisothatintheentangleduniverseallthosefieldsintermarryinpentalogicstructures,andeachofthemisabletomirrorallthedimotionsandelementsof¬∆@st

Thissaid,intheevolutionofmathematicalhumind’s,theageofcalculusandthestudyofdimotionscameafterthefirstageofspatiallinealgeometryandarithmeticscalarnumbers,astheanalysisofyoung‘deterministic=axiomaticlinealforms’properofalllanguagesandrealitiestheymirrormovedontotheclassicageofbalancedcurvilineal

forms.

Thisisobviousifyouhaveunderstoodthe‘3agesofanyspacetimeprocessthatgoesfromlinealdeterministicyouthtospiraling3rdageofcurvedinformationthroughanintermediatebalancedcurvilnealage.

Sotopologicallyweareintheageofcalculusandwaveforms;algebraicallyweareintheanalysisofthe∆±1planes,NOTyetprobingintothe«limitsofthewholeofwholesandtheminimalistparts(SœTinflationaryanalysisof

transinfinitesandemptysets).

Soifweweretodivideinamoreextensive3agesdivisionparalleltothehistoryofmankindcalculus,itwouldbelongtotheclassicageofmathematics,itsmatureexperimentaltime,whenitbecomeaclearrealistmirrorofphysicalsystems.Whilethethirdagewouldcorrespondtotheeclecticexcessiveageofinflationaryredundant

formationwithsettheory,thetopofthemind-viewofrealityinmathematicalterms,becomingitsfoundations,inanactofhumind’segocy–thatisconstructingrealityfromthemirrorandthemirrorfromtheaxioms‘imaginedbythehumind’.SoHilbertwilldefinepointsandlines,withanastoundinglyactofself-egocy:‘Iimaginepoints,linesand

planes’,justbecausehecouldn’tunderstandfractalpointsasmirrorsofreality.

AllthisleadustothebeginningofcalculusinwhichweshallseethesamefightbetweenLeibniz,therealistandNewtontheidealist.Buttheproperwaytoclassifytheworkofboth,isasthe‘scalar,viewofcalculus(Newton,whicharrivedtoitthroughseries)andtheST,geometrylook(LeibnizwhoarrivedtoitthroughthetangentS/T,betweentheorthogonaldimensionofheightinformationinitsratiowiththedimensionoftime-linealmotion).

Those2approacheshadalreadybeenforeseeninthefirstageofgeometrybyArchimedesandotherpioneersofgeometry,butlackinganalytictoolstomergeS=TintheCartesianplane,couldn’tgomuchfurtherthanthe

Newtonianapproachofexhaustionoflimits.SoNewtoncouldbesaidtoconcludetheworkofArchimedesandLeibniztostarttheageofmoderncalculusandtrigonometry–sinusoidalfunctions,thementalviewonchange.

Insimpletermsweconsider2twootheroperands,relatedtothepurestformsofspaceandtime:anglesandsinsandcosinesthusrelatetotheSS-trigonometricperceptivefunctionoforganisms,andcalculustoTT-puremotionsintime.Butwhile intheUniverseeachentityhasadominantspecialization,allofthemcandomultitaskingandworkfordifferent5Dimotionalpurposes,albeitdistortedandusuallywitha‘blind’spotfortheinverse5dimotionofexistence,whichmathematicssolveswiththeuseofinversefunctions.Soanglesofperceptionandderivativesactasfinitesimalsactionsof1D:SS-perceptionand5D:TT-entropy.

Let’s consider those2operands,briefly, before studying its combination in∆+1algebraic sentences=equations.

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1D:@NGLESTRIGONOMETRICPENTALOGIC

Becausesinusoidalfunctionsrequireforafullunderstandingastheydealwiththeshrinkingoftheworldintoamapofperception, integralcalculusandderivatives,weshallstartwiththesum, insteadofthefirstdimensionalangle,andresolveits3Planesandpolynomials;evenifasweexplainingeometry,theangleandtrianglecamefirstlikelybeforeevenmanlearnedtocountjustbythefactofopeninghiseyestotheworld.

Though trigonometric functions belongmore properly to vital topologies as they embodied the first principles ofcreationofmentalspacesasmappingsoftheinheritingqualitiesoftimeandspacethatmatterforthesurvivalofanyfractalpoint,suchas length=distance/motionandheight=informativepower,ofthespecieson‘view’measuredby‘angles’whichare ‘scalar’parameters,hencedimensionless,as theycan ‘travelwithoutdeformation’ through thePlanesofreality,wedoneedabitofan introduction,sincesinesandcosinesareessential functionsof¬Algebraicequations,afterall,theanalysisofherdsoffractalpointsmovingthroughitsvitaldimotionsofexist¡ence.

Let us start with the trigonometric science of perception, which allows a pi cycle to observe through its natural'membranes'-its3diametersthatbecomethecyclicalmembrainthatisolatesitfromreality,the'angles'thatgiveusinformationbyparallaxonthe3rddimensionofrelativesizeanddistanceofsystemsoutsidethebeing.

The simplest 'monologic view' of trigonometric functions is therefore, the fixed perception of an angle that'measures' the height-information and length-distance between the being and an outer T.œ. The sine thereforeprovidesthedimensionofform,andthecosinethelargerdimensionofdistance,energyandmotion...

Thesimplest levelofperceptive trigonometry isaPinumber:pidoesexistasaperfect formofspace, from3 (thehexagon) to pi (the circle) but it does not as a number (irrational) since time is dynamic, not static, so constantlymovesbetween±π,allowingtheopen-entropy/closed-informationdualityofmembranesthatenclosesystems:

Itsdominantuseandfirstreasonitbecamethefirstdevelopedfieldofmathematicsisitscapacitytomeasurefromapointof viewdistancesaccording to ratiosandparallax,which is theoriginof tridimensionalperception (bilateraleyes),andFertileCrescentmathematics.

That ̦πisanscalar∆±∝number,whoseangularfunctionisdimensionless,isprovedbythemerefactthatasLeibnizfindout,itcanbewrittenasasumofseries,theveryessenceof5Dmathematics,ase,is.Andhencealsoitdilutesitsmeaning beyond a certain decimal. We shall thus return to this theme when considering the two differentcomplementary approaches to calculus and analysis, the Newtonian, ∆-discover of calculus through series andLeibniziandiscoverythroughSpatialderivatives.WillisandLeibniz’sdiscoverythatπ/2andπ/4couldbewrittenasaseries–henceasaseriesofeverreduced,finedetailedpixelsofasingularityview–beingthose90ºand180ºanglesthe mean and maximal perception of eyes – reinforces the essence of the game: entanglement andmultifunctionalityofallin∆-scale,Spaceandtime.

Howthiswork in itssimplest form,needstounderstandhowa 'spherical, idealmind-membrainof3πdiameters,and0.14Dapertures,allowsamindtoperceivethroughthem, 'rays'todistantobjects.Themindthuscanalwaysmeasuretheanglecoveredbyadistantobject,andwithaminimaldisplacement,anewangle.

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3rd-centuryastronomers firstnoted that the lengthsof the sidesof aright-angle triangle and the angles between those sides have fixedrelationships:thatis,ifatleastthelengthofonesideandthevalueofone angle is known, then all other angles and lengths can bedeterminedalgorithmically.Thesecalculationssooncametobedefinedasthetrigonometricfunctions.

Sotrigonometricfunctions,thefirsttoappear,as1Dperceptionisalsothefirst'action'areoperandsforthefirstDimotionofperception.

1D:Ofthoseoperandsthenweconsiderthesimplest 'operands'asthosewhoactdirectlyonapoint-number.WeneedthentofindtheoperandofthefirstDimotion,self-centeredinthepoint,perception.

Since all points are also numbers as all have internal structure, we can operate both in S-pace with points andtopologiesandinscalewithnumberstoshowthoseinternalpartswithnumericalvalues:

Thegraphshowstheelementsofthepoint-numberinwhichanoperandcanact.Eventhoughmostoperandswillactthrougha similarity ≤≥ symbol in twodifferentelements. Yet the simplestoperandsare thosewhich canact in asinglepoint-number.

Anentirefieldof¬Algebraorrathergeometryisthenthestudyofthelawsofperceptionofthecircle,basedonthelawsoftrigonometry.

Ithastobenoticedonafirstsightthatagain'inversions'intheeternalUniversedoNOTfullycanceleachother,butbringsomethingelse.I.e.thereisapointat45ºofbalanceinwhichsineandcosineareequal;andsoitisthepointofminimalcancellationofinvertedfunctions,reasonwhywithoutanyfurtherargumentitwillbethepointinwhichthereismaximalreachofanstœpprocess(Galileo'sfirstdiscoverywithcannonballs).Thiswecanexpressitsayingthatthesineisthe'stop'perceptivestatefunctionandthecosinethestep-motion,linealone.

However in the entangled Universe, we can consider that 'all systems' are alive, and somehow evolve 'addingdimensions' as a world cycle does, developing new perspectives, and indeed the sine cosine didn't stop in thestoppinggeometryoftheGreeksthatobservestheouterworldstatically.Thecirclecanbegivendifferentmotions,upanddownandthenitbecomesanSMHsystem,oritcanemit'energy'imprintedbyitsformalmotionandproduceawave.Inthismanner,thesineandcosinecanperformalsothe

2ndÐ-3rdÐ:motionandcommunication:Thesinusoidalfunctions.π.

Inthegraph,anSMHcanbeviewedasthesimplestmentalrepresentationofthewholeUniverse,athemeweshalltreat inmetaphysics. It is the simplestmotion added to an 'angle of perception', able to communicatewaves ofinformationoveran∆-1'undistinguishable'fielditwillformwithits'perpendicular4thnon-Æmotion.

Yetastheseconddimotionofallsystems,onceitperceivesismotionandcommunication,trigonometricfunctionscanmoveafractalpointcreatingtheformofthewave,orcommunicateafinitesimalbitofitsinformationthroughawave,whichcarriestheinformation(amplitude,frequency)oftheoriginalcircle.

Pithenbecomesthenumericalvalueoftheexternalmembranemeasuredintermsof3diametersturningaroundwith3aperturesto'see',fromtheself-centeredsingularity:

The point-number has a central point of perception, which therefore can be defined externally by an angle ofaperturetotheworld.Themembranenormallyisapi,3diameternumber,whichleavesanangleofaperture,3'14-3/π=0,14=±4,5%,with 96%of 'darkmatter' outside theperceptionof its singularity. But as themembraneof thepointturns,theangle 'sweeps'with its3aperturesbetweendiameters inarelativediscontinuousUniverseall theworldoutside.

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So the angle of the point will finally allow a full 100% view in 3 different 'ages' of the full world cycle of themembrane. So in dynamic 5D analysis we can consider not only a sine and cosine function; but the sine 'main'function that calculates the aperture, also as a function of time, such as for a full turn of the 3 diameters thatcompletethemembrane(2πradians),theanglehasseentheentireexternalworld3times.

Whatthismeansthenisthatthesineofasingularitypoint'grows'witheachturnoftheworldcycleofthebeingandintimecanbehigherthanone.

Thissaidweshallnotgointosomuchcomplexityandconsiderthe'static'conceptanddefinethesinemerelyastheangleofperceptionofthecentralpointandhencethefunctionthatreflectsthe1stDimotionofexistence.

SMHFUNCTIONS:SóTFRACTALGENERATOR

Itmustthoughbeclearenoughthatthefundamentalfunctionoftheperceptivesingularityisthesinusoidalfunction,soitisinitsactiveSHMmotions,whichtrytomaintainthebeingthroughitsworldcyclesinthe'centralpoint'ofthesystem,its'mind-singularity-zerothpoint':

Inthegraph,theternaryfractalgeneratorofanySHMDimotionsysteminits3States:

Γ(SHM):$T(linealvibration)<ST(sinusoidalfunction)>§ð(angularmotion)

ThefractalgeneratorofSingleHarmonicMotionsisperhapsthemostimportantgeneratorofmathematicalphysics,inasingleSpace-timeplane,similarinimportancetotheHamiltonian/Lagrangian,whichisthemostimportant

‘scalar’generatorofphysicallocomotions.

Bothtogetherformtheessential∆STstructureoflocomotion,reasonwhywecanreducealmostallmathematicalphysicstothose2typeofequationsandthosecloselyrelatedtothem(Laplacian,Fourier,etc).

TheSHMfractalgeneratorthrowslightintotheworkingsofphysicalSHMsystems,asthe3elementsformtogetheraco-existingorganism:Itsparticleheadwithlinealvibration,itsbodywaveanditscyclicalmembrain.

Sinusoidalfunctionsareconcernedwiththecentralpointofviewofthe@-mindinitsworldcyclesof'angularperception',(sinefunction),whichistheinformativefunction,initsheightdimension,§ð.Whilethecosinefunctionconcernswiththelengthaxisofspatialsize,$T.Thusingeneralsinesareinformationfunctionsandcosinesare

energy-entropyfunctions.Bothrepresentfromthecentralp.o.v.,itsexternalperceptionoftheworld,andinternalperceptionifitsinnerwhole.

The inverseoperationsof sineand cosine findasusual apointofbalance,which is the45º,where therefore themaximal'momentum'canbereached(maximaldistanceofanythrowing).45ºisthereforethe'eye'positionofthesphere;theplacewherethereisalwaysarelativeview.

Thesinemustthenbeconsideredtheinformativefunctionandthecosinetheenergeticfunctionor'stopandstep',formandmotionstatesofthebeing.Generalrulestoenlightenthemeaningofmathematicalphysics.

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It also follows immediately that angular momentum is the best external parameter to measure the 'perceptivespeed' of a system, measured in terms of frequency. So in quantum physics,it will be the measure of the firstDImotion,andasitappearsasaconstantofallsystems,wededucethatinquantumphysicsthereisasimplemetricequationforallspeciesofthegalatom,h(ð§)c($T)=K.

5D:∆:SOCIALEVOLUTION:FOURIERSERIES.

If the world of SMH functions isthe fundamental mathematicalcycle and fractal generator ofinformation through space andtime; its fundamental expressionin social scales is the emergencethrough synchronicity of largerwaves sum of its components,which can be then merged oranalyzed in its componets,throughthefourierseries.

The Fourier series form themorecomplex dimotion of the sine cosine functions, as they allow by social

evolutiontodrawany'function'ofinformationasthesumofthemainharmoniccomponents,showinghowdespitethecomplexityoftransmissionofinformation,italwaysreducedtocycles,linesandscales.AndhowUniversalaretheparadoxes of 5D scales, notably the inversion of form,motion and function between them. So the accumulatio ofcyclicalsinesbringsalineallargerwhole,andviceversa–thesmalllineal,freestepsarealwaysbentbythelargercircularworld..

THEKINGOFOPERANDS:∫∂

Whenweconsidertogetherthe3simpleoperandsof±,x÷,√xa,anditsS=Tpolynomialfunctions,finitesimalsintime(probabilityevents)andintegralinspace(Distributionofpopulations),theyformtogetheradiscretewholerealitymirrorthatsufficesinitselftoexpressallrealities.Soaworldcanbeconstructedwiththemwithouttheneedof

further‘analysis’.Whereasprobabilityisequivalenttoderivativesandspatialpopulationstointegrals.

Whatthenbringsthenewoperandsofcalculustothetable?Theansweristrinityof∆STadvantages:

-SPACE:∆S:Ononesideitincreasedenormouslythefinesseofdetailwhenpopulationsgrewextremelyinnumbersandwerenoteasilyhandledbystatisticalmethods.Thisistheoriginoftheconceptoflimittozeroth,whichalso

expressesthatthenumberofeventsgrowstowardsarelativeinfinity,andtheminimalquantabecomesa‘finitesimal’almostazeroth,0’.Andparadoxicallybecausethequantabecome‘¡ndifferent’,undistinguishableastheminimalunitofaplaneofexistence,itallowedtotreatpopulationswith‘functions’thatgatheredthemallfromthepointofviewofthe‘property’extractedbythewhole.Soamassnolongerdifferentiatesitsatomicquantabut

measures‘weight’;amotiondoesn’tcareanylongerforstatisticalcountingbutmeasuresheatwithaFouriertransform.Andsoon.

ThisalsoimpliedthattherearealwaystwoalternativemirrorstoconsideracertainanalysisofandS=Tparallelism,thatofStatistics=probabilitiesinthequantumrealmofdiscretepopulations(henceusedinquantumphysics)andthatofseeminglycontinuous,becausethemindmakesitsquanta¡ndifferentuseofcalculus,wherederivativesaretheequivalenttoasingleeventandanintegraltoawholestatisticalpopulationorfullsumofalleventsintoa1-

wholeprobability.

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Italsomeanttomovethe‘Cartesianregionofanalysis’fromthe0’-1sphereinwhichprobabilitiesbyvirtueoftherenormalizationto1intothe1-∝sphere.

-TIME:Butthetrueadvanceofcalculusoperandswastobe‘ameasureofchangeoverafunctionwhichalreadywasmeasuringaspecificdimotionoftime-space’,thusbothincreasingthespecificationoftheanalysisofchange,and

performingbyvirtueoftheanalysisofaspecificdimotioninitsintegralform,awholeanalysisofanorganicsequenceofthatdimotion.ItonlyremainedtobeabletostudyfunctionalsofmultiplefunctionsthatwouldrepresentthedifferentdimotionstobeabletoportraitentireT.œsinsimultaneousspaceorsequentialtime.

Becausethefinalpentalogicoperandthattransformsallothersandfindsthefinitesimalpartsofscales,theratesofchangeintime,anditsequivalentS=Tcurvatureintopologicalspace,issoimportanttoallsystemsofrealityasthedimotionofscalarsocialevolution,reproductivestates,usedtocalculatethepointofexistenceofaworldcycleandintegratepartsintowholes,weshalldedicateamuchwideranalysistoitstwooperands,evenifweshallonlytouchthesurfaceofitsdepthof‘analysis’,andthenapplyittotheunderstandingintimeoftheworldcycleofexistence.

¬Limits:Stillasthederivativeandintegralswillbethelimitofsophisticationintheanalysisdonewithmathematicaltools,andactsessentiallyinthe∆-1scaleofdimotions=action;mathematicsatitsbeststillwillfailtodescribe

properlythelargerplanesofexist¡enceofwholesfromthepointofviewofitsco-existingorganicelements.AnditisfoolmathematicalcreationismtoEITHERdenytheexistenceofthoseproperties,becausetheycannotbereachedwithfinitesimalactionsofminimaltimedepth,ortrytoinventnewmathematicaltoolstodescribethem,when

verballogiclanguages,builtpreciselytodescribethelargerrangeoftimedepth,cansynthesizeitscausalprocessinalanguagenaturaltoman,whichwepreferwhenwedealwithsentientandorganicqualitativepropertiesthat

mathematicscannotproperlydescribe.

-@:Thisobviouslyisdeniedbylinguisticcreationismsoscientificzealotsexpecttoexplainitallwith∫∂.Thatisthemethodalsoofcreationistphysicists:‘whatcannotbemeasureddoesnotexist’(Reductionismthatpreventsto

understandthefutureofsynthetic,organicsystemssuchasthoseofhistory.

Recap.Amongalltheoperandsofmathematics,derivativesandintegralsarebyfarthemostimportant;astheyrevealthe inside workings of the 5 dimotions=changes of space-time and its scalar structure,with far moreprecision,stœpbystœpofS<T>Schange,thanallotheroperands,whicharealsoincludedwithinthem.

Calculus thus is the summit of the scalar growth of complexity in ¬Algebraic operands as mirrors of theworldcyclesofreality,paralleltothesame‘growthofcomplexity’fromfractalpointstowavesandnetworkstotopological organisms (inner view) or outer minds (logic, congruent view) in vital geometry and the stairs ofgrowth in logic time, from themonologic of points, to the duality and trinity of lines to the pentalogic of 5Dnetworks…andevenprobinginthedodecalogicofworldcyclesofexist¡ence.

Because of the use of a ‘finitesimal, ∆-1 part’, unlike the rough use of polynomials to measure change,‘wholesale’, analysis is able to measure different types of change – of scale of space populations and time‘motions/speeds’–indetail.Asimpleexamplewillsuffice.

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∆+1:2ndAGE:EQUATIONS:SENTENCESOFMATHEMATICS.

≤≥≈ó:EQUIVALENCE;SIMILARITYANORDEROFOPERANDS

Asasystempossess¬∆@stelements,equivalencemustimplyidentityinthe5elements,suchas:∆: Similarity in scale. S: Simultaneity in space. T: Similar phase/state/age in time.@: a common language ofinformationandpointofview.¬:Equalentropiclimitsinform,timeandscalardepth.

Obviouslythenthesystemis‘identical’and‘equal’.Butsuchcaseonlycanhappenforthebeinginitself.Becauseeven if two beingswere identical, theywill not be able to occupy the same space (fermions),unless they areperceived fromthe∆+1pointof viewofawhole,madeof internal identicalpoints thatappear fromtheupperpoint of view so small as occupying the same space (bosons). Those subtle differences of pure metaphysicaldelightcorrespondhowevertoexistentialalgebra.Soweshallnottreatthemhere.WhatmathematicsinclassicEuclideancalledthenequalityorcongruencewasonly¬equalentropiclimits;thatiscongruenceinthemembrainoftheobjectasabletobesuperposedandcoincideinexternalform.

Thiswillbeamantraofallourpaperswhenjudgingtheextraordinarysimplicityofænthropicmenwhichreducealltoentropicconcepts.Socongruenceisontheentropiclimitsandtheinternalformmattersnothing.Equalityisalsodefinedasaone-dimensionalequivalence, ina singleproperty.Suchconceptsaremereequivalences thatallowtwosystemstoentangleonthe‘dimotionofequivalence’.Andyetasmoreelementsbecomeequivalentthebeing’sbondage increasestillwhenall theelementsareequivalent,a formofabsolute lovecalledboson-fusionhappens.

Operandsfacilitatingequationsandfunctions

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Themostimportantoperandsareinfactthose,whichconnectthe2sidesofanequation.AndsohappensinExistentialalgebra.AsweshallseewhenwestudyitastheGroupofallgroups.Therelationshipsofsimilarityandequivalence,congruenceandidentity,andthoseofdissimilarity,ruledbythefourthpostulateofnon-Euclidean

geometry,thepostulateofcongruenceanditstranslationintoalgebraicoperands.

Soletusfirstrecallfromourpaperongeometrythedifferencebetweencongruenceinidealizedmentalmathematicsandcongruenceinthe∆STworld.

Identitydoesnotexist.Itdoesnotevenexistinthebeingwithinitself,astheflowoftimeconstantlychangesthebeing.Butitdoesexistadegreeofcongruenceinthe‘information’,inthe¡logicformofthebeingasavariationofthegameofexist¡ence;whichalsomakespossibletherepetitionoftheself.Identitywouldrequiresomeshort

oftransmigrationofsouls–anentitywhichcouldbeentangledbetween‘ilogons’ofexist¡ence.

Equalityisproperlyexpressedequivalence.Andthisisthefundamentaluseinequations.Twothingsareequivalentinsomeproperty.Whenwesaytwothingsweightthesame,havethesamenumberofelements,etc.AndthisisthemagicofMathematics,whichallowsits‘simplfiying’conceptstopermitequalitybyeliminatingallthepropertiesthatmakethingsdifferent.Sowe‘equalize’mathematically2entitiesbyasingleproperty.Equality

or‘equivalence’isthereforemonologic.Athemewhichisofenormousimportancewhenstudyingtheparametersandmagnitudesofphysicalstiences,andtrytofindatheoreticalmeaningtokeyconceptssuchasenergy=time,information=space,wave=particle,etc.Weusebothsymbols,=,≈,preferringthesecondone.

ClassicMathematicsofcoursehasitsaxiomaticmethodweshallasusualcriticizeorrather‘punctualize’fromthepointofviewofpointwithinnerpartswedonotknowas‘onlythepointhasalltheinformationaboutitself’.

Itisofinterestthentonoticethatthe3postulatesthatdefineequalityinmathematicscanbereducedto2includingacircularproperty–keyelementtounderstandthecyclicalityofspaceandtime.ThusallXisequivalenttoX,ifXisequivalenttoY,YisequivalenttoX(symmetry)andifXisequivalenttoYandYtoZ,Xisequivalentto

Z.

Yetasthefirstis‘trivial’,thesecondandthethirdcanbewrittenasacircularequivalence:IfX->YandY->Z,Z->X.

Whichdemandsatrinityofterms,thatisa‘3=πcycle’,hencetheminimalunitforacircularidentityinwhichXreflectsuponitself,justifyingtheπformofallcircularmotions.

Thepropertiesofequivalencearenecessarytoexplainhowasystemwhichisidenticaltoitselfreproducesasitmoves,inthe3modesofphysicalmotion;asiteithertranslates,rotatesorreflectsinamirrorsymmetrywith

itself,sinceitispreciselythose3relationshipswhatallowthemotionsinspaceasareproductionofform,whichneverceasestochangeandyetappearsasadynamicpresent:

Thedynamicsymbolstoexpressequivalencesymbolsforapresent,reproductiveS=T,dynamicrelationshipare:

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ST; Se≤≥Ti (information=energy), TieS, Exi=St; X, ≤≥,ó ; ≈;Ø:ConstantPresent,balanced,Space-time; iterativebody-waves;femalegender,tiesofspace-time,pointofbalance,work.

Motionisreproductionofforminalowerscale.Bohm’srealism:quantumpotentials.Howa systemmoves in a crowdeduniverse,wherewe are vital space-time? The only answer that resolves alsoZeno’sandquantumcomplementarityparadoxes, is ifwedonotmovebutreproduceour information,translatedintoalowerfasterwavescaleofthefifthdimension;aswereproduceoursoundinfasterelectronstotelephoneornerveimpulsesintochemicaldopaminetojumpdiscontinuousneurons.Somotionbecomesscalarreproductionofform,andsinceallisaformofmotion,allisreproduction,whichisthedefinitionofamathematicalfractal,afeed-backreproductiveequation;5Dmetrics,whichbecomethenthe‘functionofexistence’whosegoalistoreproducetheformofallsystems–thesimpleroneswithmaximalmotion-translation inspace,thecomplexoneswithmin.

motion as a reproduction that emerges between scales. And this givesbirthtotheworldcycle.Considerthecaseofquantumphysics:

In thegraphweseeaparticle reproduced inadjacent regions that fadeaway, and the result is the perception of awave ofmotion. In Bohm’srealist model this reproduction happens in a lower plane of quantum

potentials,wherealsoentanglementhappens,whichisthe∆-4scalewhichisv>cin5Dmetrics(Min.SxMax.T=C),hencereal.

Motionthenisreproductionofformoversuchpotential:thewaveerasesformintomotion,theparticleisastillstatethatgaugesinformationentangledtootherparticle,fermionandboson,stilltoeachother–despitetheperceptionof relative motion in our scale – hence the information electrons share has always a c-constant speed. Thus theLorentztransformationareobjectivelyrealformankindwhoeliminatesthestopstateofparticlesaswedoinamovieeliminating the stop frame but if we were observing reality from the perspective of an atom, we would ‘stop’,entangle in the quantum potential, neutrino scale & so eliminate the spooky effects of ‘time dilation’ & ‘length’contraction,fromourperspective(butnotofmassincreaseasitisascalareffect).Thisisthe'rational'5Dexplanationofboththec-constantoflightandentanglement;aselectronicbeingsperceiveinformationin'stoppositiontoeachother’andmovein'wavestate'.Motionsareperceivedbyparticlesthatstopmotionintoform,intoinformation,asdistances.

Again those relationships have traditionally better understood inmathematics and theoretical physics through theconceptof ‘Symmetry’,whichnowwerenewandcleanofallglobalpresumption. Inessencewhatall thismeans isthatwedonotexisteverasidentities inmaterialform,butinpurein-form-ation,whichistranslatedbackandforthscalesofthefifthdimension,whichofcourseestablishesasalwaystheultimatequestionofall5Dmodelofrelational,generationalscalarspace-time–the‘identityofallminds’,astheultimateilogicgameofexistence.

Inmathematicsthisprocessisakintotheequipollenceofvectors,whichisultimatelyasimplifyingrepresentationofatranslation in space as a reproduction, themes those proper of physics.Whatmatters here though is that only 3‘translations’,motionsareallowed,relatedtothe3propertiesofequivalence:reflectioninamirrorwhichallowsthebeingtoreproduceitsformbypeggingthroughmirrorsymmetry‘facingeachother’–asit isnotpossibletoassesstheidentityofreflectionwithonebacktoback.Linealtranslation,whichhappensthroughthetransitiveproperty,andangularmomentum,whichhappensthroughthecircularproperty.

Somotionsareindeedreproductionsofformbasedinthe‘assessment’ofaformofthecapacitytorepeatitsidentityinanotherregionofspace-time.Ifnotitwouldnotrequiremirrorsymmetry,circularandsymmetricproperties.

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AsecondessentialelementthencomesbeyondMONOLOGIC,whentwoelementsalreadyreproducedenterinrelationshipofequivalence,forthepurposeofevolvingsociallyintolargerscaleorenteringindifferent

relationshipsofpredationwitheachother.

Duality:4thaxiom=postulateofrelativecongruenceofexist¡ences:similarity,¡ndifference,trans-form-ations.4th Postulate: Equality is no longeronly external, shown in the spatialperimeter of any geometrical form(Euclidean congruence) but alsointernal and further on it is neverabsolute but relative, since wecannot perceive the entire innerform of a point – hence thestrategies of behavior such ascamouflage. Forms are self-similarto each other, which definesdifferent relationships between

organic points, according to their degree of self-similarity. The 4th postulate is thus the key to explain thebehaviorofparticles as thedegreeof self-similarity increases thedegreeof communicationbetweenbeings.Someofthemostcommonbehaviorsand‘eventsderivedfromthispostulateare:

1)Reproductive functions in caseofmaximal self-similarityor complementarity inenergyand form;ei->SeiorMaxExmin.I(male)=Min.exMaxI(female).

2)Socialevolution,whenpointsshareacommonlanguageofinformation,i=i->2i.

3)Darwiniandevolutionwhenformsaresodifferenttheycan’tunderstandeachother’s informationsoinsteadthey feed into each other: i ¹ i. In such cases if those 2 entitiesmeet theywill start a process of ‘struggle forexistence’,tryingtoabsorbeachother’senergy(whenE=E)orsimplywillnotcommunicate(whenE¹E,sincethenthereisneitheracommoninformationtoevolvesociallynoracommonenergytofeedon).Yetbecauseanypointabsorbsonlya relativequantityof information fromreality, similarity is relative, faked forpurposesofhuntingwith biologic games such as camouflage or sociological memes that invent racial differences, allowing theexploitationofagroupbyanother.

The4thpostulatedefinessystemsasidenticalwhenits3ternarypartsareequal:theouterangularmomentumor‘membrane’,itscentralActivemagnitudeorsingularity,focusoftheforcesandthevitalenergy,enclosedwithinthem, and all others as similar with different ‘angle of congruence’. We distinguish 2 different interactionsaccordingtothedegreeofequalityofitsternaryparts,assystemscanbesymbiotic,iftheirindividual,cellularoratomic‘fractalpoints=parts’aresimilarenough,interactingthrough its3physiological‘lines=networks’evolvinginparallelcreatinganorganicplane,asthosedescribedinthenextgraphforeachscale=science,ortheycanbeentropic,destructive,predatory,whentheyaredissimilaranddon’tspeakacommonlanguageofinformationtocoordinateitsactions,whereasthestrongersystemwillperpendicularlybreakandfeedontheweakerone.

The geometric complexity of the 4th Postulate is caused by the topological forms created by any event thatentanglesMultipleSpaces-Times.Sinceitdescribesthepathsandformsofdualsystems,whichconnectpoints:Self-similarity implies parallelmotions in herds; since equal entitieswillmaintain a parallel distance to allowinformative communicationwithout interferingwith the reproductivebodyof eachpoint.Darwinianbehaviorimpliesperpendicularconfrontations,topenetrateandabsorbtheenergyoftheotherpoint.Finally,absolute,innerandouterself-similaritybringsbosonstates,whichhappenmoreoftentosimplerspecieslikequarksand

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particles that can form aboson condensate as theydoinblackholes,wheretheproximity of the points ismaximized.And indeed, thesamephenomenonbetweencells with the same inner

information/DNAoriginatesthe‘collapse’ofwavesintotighterorganisms.Finally if there isno similarityneither inbodyormind, itsexistenceas 'catalleys', thatnever cross (relativeinvisibility).WetalkthenofSkewT.œ.s.:

4thNon-EpostulateisimplicitintheworkofLobachevskiandRiemannwhodefinedspaceswiththepropertiesofself-similarity(Riemann’shomogeneity),whichdeterminesitscloseness(Lobachevski’sadjacency).

4thpostulateofrelativecongruence&angleofparallelismasamirrorofits5pentalogicdimotionsandvariationsofangledefineDarwinianorsocial,reproductiveoutcomestocommunicativeeventsbetweenfractalpoints.

Thus in praxis we assess similarity by an ‘angle of parallelism’ that increases social evolution into herds andsupœrganisms,orperpendicularitythat‘scatters’systemsintoentropicdestruction–elementsthoseofanentirefascinating new field of 5D topo-biologic studies that analyze in geometric terms, the vital topology andrelationshipsbetweenformandfunctioninallsystemsofNaturefromparticlestoorganisms.

Thissimplegeometricaltruthhoweverisessentialtoallsystemsofnature,whoseanglesofconnectiondeterminethefunctionsandsymbiosisbetweenparts.

The Universe always starts with an asymmetric being, which can go both ways: towards a social evolutionarysymmetrythatlastsintimeandimpliesamirrorparallelism,oranantisymmetricdestructive,perpendiculareventinwhichonepartpuncturesandabsorbstheenergyoftheother.Itisthetopo-biologicalternaryprincipleofnon-Euclidean,Non-AristotelianI-logicgeometrythatputstogetherboththebiologicalandmathematicalpropertiesofreality. The concepts of symmetry=parallelism, antisymmetry=perpendicularity and asymmetry aremirrored bythe4thNon-EPostulateofsimilarity.Butwecanextendtheconceptofasymmetriesalsotoasymmetriesoftime,betweentheyoungageoflocomotionandtheoldageofinformation,ofactions=Dimotionsbetweenthestepandstop similar actions, and the entropy and social evolution actions,which bring us the final asymmetry of scalesbetween theupperarrowofwholewithmore spatial sizeand the lowerarrowofpartswithmore information.Whenthosedualities: step-motion/stop-perceptionandscaleup (5D: socialevolution), scaledown (4D:entropicdissolution)areputtogetherweobtainthemostcomplexbalancingdimotion,reproduction,andwhentheyarealladdedupintheexistenceofabeing,wegetitsworldcycle.

Inthegraphwecanassessthedifferent5mirrorsinwhichmathematicalSpaceandlogicTimereflectsthegameof5 Dimotions=actions of existence, which then expressed by territorial monads GENERATES its logic REALITY. InGeometryfractalpoints=monadswillotherthroughwavesofcommunicationofenergyandinformationthatgrowintoreproductivenetworksaterritorialplane,creatingasuperorganism,whichwillrelatedtotheexternalworldaccordingtoitsrelativesimilarity=congruence,assessedbyitsangleofparallelismorperpendicularity.

Inlogicterms,asuperorganism,bybreakingitsformlessasymmetryintodifferentspatialconfigurationsaccordingtocongruence(socialparallelsystems,complementarygender-mirrorsystems,Darwinianperpendicularsystems,or systems thataredissymmetricanddonot shareany reality)buildsa casualpyramidofgrowth froma fractalpointthroughwavesofcommunicationintosocialnetworksthatbecomereadytoact-move,feed,perceiveandevolvesocially.

Since wemust add to the mathematical and logic languages-properties of reality the 5 actions, or organicpropertiesofthescalarUniverseasessentialtothegameastheyareitslogicandmathematicalmoreabstractlaws

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- a fact the egocy of æntropic men of course reject, as it must remain in its monad-subjective monologic theonlyclaimanttolifeproperties.

Thuspentalogicofscalarspace-timeisestablishedbyits¬Efractalpoints,its¡logiccongruencewithrealityinwhichit will order a territory to perform its 5 vital actions=Dimotions of existence, and the mathematical, logic andorganic laws of those 3 languageswill be therefore the bottom line of the 'Creative process' of the Universe -nothingchaoticexcepttheentropicDimotion,whichconformsthemonologicofhuminds.

TrinitythroughCircularity

Thelastcaseofequalitytoconsiderinthissimplifiedintroductiontoathemeiscircularitybetween3elementsthatareidenticalinitstranslation,closingapicycle,whichofcourseisneveraperfectclosedcircleasthereproductionof each of the 3 elements are discontinuous to the next reproduction, leaving 3 π holes, π-3/3=4%, which inpentalogic represent for thecentral singularity the ‘canonical’percentageofperceptionapointhasof theouterworld, often just the openings for its network connection to other self-similar points, to construct a network,walledfromthedarkspaceitdoesnotseesoitbelievesitdoesnotexist.Suchblindnesstoallothernot-self-similarpointsofparallelcatalleysconcludesourbriefanalysisofthelawsofcongruence.

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RELATIONSHIPSOFORDERANDITSOPERANDSINEXISTENTIALALGEBRA.

Letusnowconsiderrelationshipsoforderanditsinteractionsanddifferenceswhichobviouslywillfallwithinthe4thPostulateofCongruenceineithersymbiotic,≤or≥vs.Darwinian<or>relationships.

TheyareessentialtoexistentialalgebraandanyofitsderivedGroups,SœTorBooleanAlgebra.

The5Сmotionaloperandare: <:∆of locomotion.«∆ofentropy. >∆ information. »∆of form.≤≥=balancedpresentstœps.

Anyofthose5Dimotions(Dimensionalmotion)oftime-space,allowtheinteractionbetweenlocalT.Œs:

Ts,Tes;‹,<;|,$T:P$T:Pa$T,temporalenergy,lineallocomotion,gaseousstate,limbs/fields.

St; Sit, ›, >;O, §ð; ƒð§: ƒuðure§,spatial informationcyclical, informative,space-time,solid,crystalparticle-headstates

TT;Te2,TeTe,«,¬;∆-1:EntropicTime-energy,death,dissolutionofformintoitspartsofalowerplane.

SS;Sisi;»;§,@;∆+1:Linguistic,mentalspace,stillmind-membrain,whichorganizespartsintowholes.

SequentiallytheyshowchainsofdifferentS,Tand∆statesof5dimotionsofexist¡ence.

The differenceswith those of classic algebra, is their duality of both order and temporal dynamic relationships,whichimpliesnotanabstractorspatialpropertybutadynamicfeedbackinteractionbetweentwobeingsthatareputinrelationshipbytheoperandoforderorequivalence.

Thatis,while<meansA<B,AsmallerthanB;aspatialrelationshipassizeinspaceisthemostevident;inexistentialalgebraitalsomeansthatAcan‘explodeitstighterdensity’inanentropicexpansioninspace,tobecomeB,andmoreoveritalsomeans,B>A,potentiallyBcanimplodeintotheevolutionaryhigherdensityofB.

Since Existential algebra is not ‘Aristotelian’, each of its symbols have a trilogicmeaning, which often happenstogetherestablishingfeedbackequations:A<B>A.

Thus we can say the relationships of order of classic algebra are a ‘limit’ in the relationships of symbiosis anddualityofexistentialalgebra;whichisthekeyconceptforthemorecomplexplurilogicsystemsofrealitythatco-existinscales,whoseelementsperformseveralfunctions,andaredefinedbytrilogicfractalgeneration,pentalogic0’-sumcyclesandevenmorecomplexlogicentanglementsbetweenitsparts.

It is thenA<Barelationshipofhierarchy? Itmightbe if it isaccordingtocongruencearelationshipofpredationfromBtoA.Howcanthendistinguishthem?Firstbyreferencetoreality.

Unlikethepurelyabstractrelationshipsoforderofidealmathematics,theconnectorsofpentalogichavemultipleinterpretations,astheyareconnectorswith∆STtriplemeaninginscale,intopologicalspaceandintemporalorder.I.e. > in fact is A bigger than B; A dynamically shrinking informatively as it ages into B; and if we look at it‘orthogonally’inthescalarDimension,A∇B(whichIconfessnottouseforfinger’slaziness);meansthelargerAisinanupperscale,aboveB.

And furtheron those relationshipsmight imply thatA ‘feedsonB’, if theangleof congruence is ‘negative’, inaDarwinianrelationship,orAprotectsBasamembrainthatenclosesit,etc.

ThusthepotentialofthesamesymbolsoforderwheninterpretedinExistentialalgebra,showsthemultiplicityofpentalogicmeaningsandfunctionsofeachpartofreality.Earlierinmyexplorationof¬Æ,Iusedtotranslatehugetractsofscientificequationto¬Æsymbology,tillIrealizethiswastobeasolitaryjourney;soIusedthemtowritelong rolls and then conceptualpaintings, and then I constructedpentalogicAI-gebra (Algorithmsof Information,Artificial Intelligencewith them)whichas Leonardowithhis seaweaponsdidn’tpublish formanwould justusepentalogicAItokillhimself,andsoIremainedmute.Itisthoughanimmenselyintelligentworld.

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The distinction then of those multi-actions for this very introductory course on the Universal Grammar ofexistential algebra, ismadewith the4 ‘degrees’of identity and congruence, from≈ to≤ to< to«,which is thesymbolofentropicannihilation,indecreasingsymbiosis.=impliesarelationshipofequivalenceinthesameplane,≤oneofsymbiosisbetweentwoplanesasA≤BmeansthatAisapartoftheSœtofBitsT.œs(largerwholes),butisalsoA≈Bsymbioticbyarelationshipofmutualdependence,asAismoreinformativethanBandsoitmightcodeB.Forexample,A(gene)≤B(cell),A(electron)≤B(atom),etc.

Soforanimaginaryenlightenedscholarshipthattranslatesallrelationshipsbetweenpartsandwholes,SœTofTœs(whereweuse the inverseSœt for the∆-1partsof∆ºTœs), intoexistentialalgebra, its symbolswouldallowtotranslatealleventsandformsofreality,asa ‘musical language’oftheflowoftime-spaceevents,topologiesandscales. Ipublishmyself30yearsagoafewbooksonthat language,whichnobodyeverunderstood;somecopiesstillextantonmylibraryandindeedyoucouldreadonthem,themusicofthespheres.Becausetheadvantageofthislanguage,likethatofBooleanMentalAlgebra,whichasweshallseeisnoincidentallytheclosertoit,canbeused formultiple ‘languages’. As Boolean algebraworks for digital numericalmathematics and first order logic,Existential algebra can mirror all languages and events of reality, and also establish with its set of alloweddimotionsandrules,addingthe|xO=Ø,topologicsymbols,whatoperationsarepossibleandwhatarenot.Beingthemost importantwithmultipleapplicationsaswesawinouranalysisofthepropertiesofexponentialsanditsdistributivelawsthatinformationcannottransitbetweentwoscaleswithouttranslation,anditcannotberegainedifittransitsdirectlybetweentwoscales.

Soafterlayingdownthesymbolsofexistentialalgebra,aswhenwewritedowntheneumesofmusic,wecanthenestablish what are the possible harmonies allowed by the existential game, which are self-contained in itsmeanings. Indeed, the previous reduction of symbols to only 5 implies that A<B is allowed and preservesinformation,A«BisallowedbutasitisadimotionofentropyiterasesinformationandthereisnotA<«B;thatis,asystemcannotjumptwoscalesofthefifthdimensionwithoutdeath.

What about the relationshipwith propositional logic? I am not so interested in advancing that information butessentiallyitisthesameconceptofalargermorecomplexilogicstructureforthemeaningofclassicdogmassuchasAiseitherAorB,A->B,thenBdoesn’timplyA,etc.aswehavediscussedinmanyoccasions.PhilosophicallyitimpliesthattheBooleanAlgebras,andAristotelianlogicisalineal,‘actionoriented’,subjectiveself-centeredlogic,whereA,thesubject isalwaysright,whatmeansAImodulo-2willbe ‘nazi-like’andadmitnopitybutforceas italreadydoesitsAI,Algorithmsofinformation.Doesitmeanapentalogicwouldworkbetterforman?Certainlynot.Asitwouldgiveitmorefreedombutstillremainself-centered,sincesurvivalassensorialawarenessistheprogramthatunderliesallothersunlessapoptosisisstrictlyprogrammed.

Finallytostressagainthedynamic,temporalnatureofexistentialalgebraasopposedtothe‘infantile’,spatialviewofhumind’ssymbols. Indeed,A<B isaspatialpropositiononhumind logic inthesamemanner inverbalthoughtthe child learns first names of spatial forms, and only latter according to intelligence is able to understand theconceptofatemporalverb,andyetthenitwillusefirstthespatialpresent,andinfactlinguistics,asubdisciplineoftrinity logic and the universal grammar of the fractal generator is the tool to understand the complexity andentanglementofaculturewiththelivingUniverse.Somesimplemindedspatial‘visualNeanderthal-like’languageslikeEnglisharesubjective,startwiththeIthatdoesn’terase,lovespresentformsandhasnoimaginationtoformsubjunctive,etc.

Sinceallsystemsstartasvisual,spatialinitsfirstlanguage,dogmaticbelievinginabsolutetruthsofe-videntforminitsmentalattitude,andveryslowly,manynevermoveofsubjectivityoutofspacetoanentangleddynamictimeflowlanguageofperceptionoftheilogic,underlyinglawsofreality.

RECAP. It is then thanks to relational operands that ‘sentences’ of mathematics are possible. Without themmathematicswouldendsimplyinnumbersfamiliesanditsassociatedoperands,asthelanguageofachildendsinnamingthings.

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In most functions the independent ‘submissive’ parameter is a function of time-energy from where the, thedependantdominantparameterthatformsit,amentalortopologicsimultaneousspace,arises.

Sowecanwriteageneralfunctionofallfunctions:F(Sy)=G(Tx)

In most functions the independent ‘submissive’ parameter is a function of time-energy from where the, thedependantdominantparameterthatformsit,amentalortopologicsimultaneousspace,arises.

Thedetailsofthoseequationswillbespecificfunctionsthatcreateaformofspacefromdifferentdimotionsoftime.Thusultimatelyformsoftheexistentialfunction.Afurtherdetailofthosefunctionswillbeanequationforaparticularspecificcaseinwhichthespecies,eventsandentropiclimits(domain)arefurtherspecified.Andfinallythesolutiontotheequationwillbethedeterministicdescriptionoftheformandeventstudied.

ExistentialFunctions

Ifweweretoresumethemaindifferencebetweenexistential®lgebraandclassicAlgebraisthis:®lgebraisinterestedinfunctions,Algebrainequations.Afunctioniscloselyrelatedtothe5Dmetricfamilyoffunctions,asoftenrespondstoarelationshipbetweenthe5Dimotionsofspace-time,whichinthemostgenerictermswrite:

S=TorSóTindynamicterms,andcanbedevelopedin3ages,ormaximalpoints,betweenbirthandextinction:

SS(0$T:sizemax.§ð:Form)»Max.Ts(energy)xMin.St(information):youth<S=T:Maturity>Max.StxMin.Ts(3rd

age)«TT(0§ð:frequencymax.$T:distance)

Whereasweuse both the dual symbols for Space-formand T-motion, and the topologic symbols for lineal $Tspacetime,cyclical§ðspacetimeandbalanced,ø-STandthedynamicsymbols«»‹›.

Functionsofexist¡ence then formmostof the importantequationsofscience,whicharedetailedvariationsofthosefunctionsofexist¡enceforeachspecies.

Themainsetof functionsofexist¡encearethosewhoshowoneofthose3relativeequationsofage/topology,whichwecanwriteas3functions:S=T,thefunctionofpresentreproduction,whichoftenbecomesSt=Tsastheinversepropertiesofenergyandinformationcometogetherintoamiddlepoint;andSS=1/TT,whichshowstheirreconcilable limits of seeds of informationwhich occupy 0’ space but packmaximal information, vs TT, pureentropy that lasts in themoment of death a single finitesimal quanta of time, 0’, but acquiremaximal spaceextension.The limitofwhich isthebig-bangthatexpandedspacead infinitumfroma0’seedofmaximal form(butwereducetothequasarbig-bangofblackholes,akatopquarkstarsin5Dcosmology.

Thosetypeoffunctionsasequationsofmeaningfulsystemsofrealityarethustheinterestof®lgebra.

WeshallbeablethenveryoftentotranslatesuchfunctionsasY(st)=Nº(s),С(x),with3components:Yasthedependentvariable,normallyafunctionofspace-time;Nº,asthenumerical=spatialvaluesascribedtoX,whichdetermines its growth or diminution, the Ðependent variable,which changes in time,within certain ‘entropiclimits’

Equations on the other handmight be as trivial as the polynomial equationswe study in elementary algebra,wherevariablesaresimplenumbersinreferencetoasingleparameterofspacetime(aprice,avolume,etc.)andareoflittleinterestforthiswork.Sowewillmakeonlyafewremarksaboutthem.

The notion of a function as used inmathematics ismore restricted thanwe have thus far indicatedwith thedefinition of the function of existence,which considers how the parameters of ‘holographic space-time’ (its 5Dimensionalmotions,TT,Ts,ST,St&SS,determinetheactionsofbeings).

Soweextendtheconcepttoenclosefunctions,whicharenotmeasuredwithquantitativenumbers.

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Thestatementthataperson’sobligationtosocietyincreaseswithhisageexpressesafunction,thevariablesofwhichare4D:∑œ:socialevolutionandTs>ST>St-age. BotharerelatedbythegrowthofSt-informationastimepasses inourworldcycles. So the function IS a truism,a lawderivedof space-timeagesanddimotionsand itsentangledlaws.

However, it isnotpossible tomeasure theextentof this social increasewithage innumbers.But thatdoesn’tmeanasub-functionofthefunctionofexistenceisexpressedbythatlaw.

Mathematics then is concerned also with partial equations of the function of existence, but only with thosevariablesofspaceandtimeandscaleandentropyandmind(the5Dimotionsexpressedasstructuralelements)whosevaluescanbeexpressednumerically.

This is a limit ofwhatmathematics can describe, and the limit becomesmore stringentwhenwe analyze themore complex functions of existence, which are best described with deep time long organic worldcycles.Mathematics is best for the other extreme of detailed, small functions, where ‘numbers’ increase (steps oftimespacethatmeasuredistance,interestratesonshortperiodsasthequantaofgrowthinmoneyaccumulatinginabankandsimilarvariableswhosevariousvaluescanbemeasuredandexpressedasnumbers.

TheFunctionofExistence,Max.SxT,isthefunctionofallfunctionsthatcanbeexpressedinalllanguages.

Anopen‘function’–agoal…hasasaresultthecreationofaconstant‘C-plane,topologicalorganism,orpartialoutcomethatresultsinanaction,whosepurposeistomaximizethefunctionofexistence.

Allactionthenbecomesanexpressionofasurvivalmandateofthefunctionofexistence.Allfunctionbecomesa‘broaderfamilies’ofpossibleoutcomesandvariationsofanaction.Andstillwecanexpressitwithaformulaofexistentialalgebra.Whatisthenanequation,anevenmorerestrictedterm?ItisafunctionexpressedasanS»=«TorSxTmetricequation.ThecommonestexpressionsoffunctionsthusareF(Sx)=∏∑Ty

Whereasthespatialparameter,ismostlikelythe‘outcomeofanentangledspace’,productofdifferentdimotionsintime,whichactasthevariablesofthespace-function.

InalgebraictermstheparadoxicalessenceofNaturecomeintoplay:Alimitisnever0,butafinitesimalratioofchange,an∆-1element.WhenweapproachthislimitthenweapproachtheS=Tlimitoftheexistentialfunction,meaning,ultimatelywedoreachanstopandstep,S=T,‘perfect45%anglethatmaximizethefunction,asSx=Ty.

Thesimplestobviousexperimentalproveisthefactthattheshotofacannonballismaximizedbya45%angleofmaximal efficiency and reproductionof the locomotionof form. Sowe can consider in general that equationshavealongertimeview,asMax.SxT=Cfunctions,whichhaveitsbestmostefficientform,whenits‘derivative’ofminimalquanta,bringsarelativeS=Tstate,whereasthe‘rateofchange’byunitoftimeisaquantaofspace.Inpraxisthoughatthelevelofthequanta,wecanalwayschoosethistobeso.Forexample,humanshaveasecondas itstimequanta,which isthebeatof itsheart-bloodthatprovides informationandenergyto its∆-1cell,butalsothebitofmentalinformation(thoughtspersecond)andsteps,whichvarybetween½and1second.

In this manner partial equations and specific cases of the function of existence become when measuredquantitativelyfunctionsandequationsofalgebra.

Soanequationisafunctionwithaconstantresolution,anoutcome,apresentresult,arealpraxis,adetailofthefunctionofexistence.Theelementsare likely thesame,Spaceand timeparameters thatmightvary forminga‘familyofactions’.Soanequationtendstoberesolvedwhenthedetailsofthefunction-0itsentropiclimitsareset,reducingtheuncertaintyandmultiplevariationsofapotentialfunction.Theequationisanactualfunction.Anditssolutionrequirethe¬entropiclimitsofitsdomain.

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Similarrulesofexistentialfunctionsandmathematicalfunctions.Dilogic:orthogonality.

The first obvious one is between TT and SS, pure entropy and pure still form. Both are opposite but itsrepresentationisorthogonal,akeylawofexistentialalgebramimickedbyclassicAlgebra.SinceasystemiseitherinthelimitofabsolutemotionanddissolutionasTT-entropy,whoseparameterswillbequantitativelydifferentaccordingtothefunctionsweusetomeasuremotion,buttheotherextreme,§isabsolutestillness.Nomotion,0’-motion,andthatisadefinitiveparametermeasuredfromthepointofviewoftheobserver.SowecandrawasimpleCartesiangraph,withtwoorthogonalcoordinates,whereastheo-pointistheperceivedinitspointofnomotion,stillness.Hencethe0x,Y-ordinatesbecomesthepositionofstillnessandinformation,whilethe0YandvariableCbecomesthecoordinatesofmotion.

Werepresent functionsof transformationof SS intoTTas an ideal SS/TT=Chyperbola,whichwill comeout inmultiple sciences to represent the tug of war between ‘position’ (form) and motion (momentum). So weestablish3orthogonalcoordinatestorepresentthecomplementarities,andoppositionsbetweenactions:

SS vs. TT and St vs. Ts., Entropy-Energy vs. form-information can be represented in a Cartesian graph; andfunctionsofbothparameterssuchastheLawofBoylePv(S)=NkT(T),withanidealperfectasymptoticgraph,ortheLorentzTransformationsbetween‘motionandmass’.

In certain cases form arises from the system’s energy, as a complementary form with not a partialtransformation,aspartoftheenergy-bodyremains.ThenwecanuseacomplexplanewherethevalueofY(S)isamagnitude smaller than that of the body in terms of energy (sT vs. St), and in this case the function is alsonegativeas itdetracts from the sTenergyof the ‘real line’. So for such typeof functionsbetweenenergyandinformation that do not imply a full transformation of one into another the complex planeworks better. Butorthogonalityisstillmaintained.

Finally for the S=T, reproductive merge of two relative functions orthogonal to each other, produces a dotproduct,withanoffspringofnewcombinedSTelements,inthe3rddimensionofrelativewidth.

TheCartesianplane,Complexplane,vectorialplaneanditsorthogonalpropertieswillbeusedin¬Algebra(Non-Aristotelianalgebraofmultiplecausaldimotions)torepresentthemainrelationshipsbetweenSSvs.TT(positionvs.motion),Stvs.Ts(-informationvs.Energy)andthereproductivefunctionS=T.

There is another use of Cartesian coordinates, as an X- sequence of growing social numbers, hence able torepresentthe4thdimotionofsocialinformationandgrowthwiththepassageof‘longlinealtime’,whichareoftenmetwithsimplepolynomials(i.e.therepresentationsofgrowthdistanceinspacewiththepassingoftime).Wecannotbeexhaustivebyanymeanswiththelawsofexistentialalgebra,soalientothepresentviewofachaoticUniverse, properofænthropicmen;but inourpaperon¬EGeometry and¬Algebra (non-Aristotelian algebra)anditscorrespondencewith¬Ælgebrawewilldevelopitslawsinrelationshipwiththelawsofmathematicsandgeometry;sothesubjectdoesn’t looksoesoterictothereader,andinthefutureanyscientistcanconnectthelawsofitsdisciplinewiththelawsofexistentialalgebra.

Theangularfunctionbecomesanessentialoperand,appearingineveryplaceofmathematics,fromthesimplestorthogonal lawsofPythagorastothe∝dimensionalityoforthogonalHilbertspaces,whichshowthat indeedinthequantumscaleofthefirst‘sensoriallivingforms’,h-spinsalreadyallthevitalpropertiesofrealityco-exist.

What matters of the previous Planes of growing complexity is its ‘entanglement’ in successive systems thatincludethepreviousone,anditsrelativeangleofcongruencethatwilldefineitsoutcome.

So thewaytostudyanysystemof reality isan increscendoanalysis from theone, thewhole, thepoint,asobservedexternallywithits2‘polarparts’,themembrainofinformation,thatfocusitssensorialperceptionintothesingularitycenterwhichembodiesthewillofsurvivalofthesystem,astheycreatetogetherthe3rdelement,thevitalenergybetweenthem.

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This is the first stairofgrowth inanalysisof systems. I.e.wecan takeanegg, thewhole,and thenconsider theplacental ‘sac’, itsmembrain to the outerworld that absorbs energy and information, then focus on its centralgenetic material and observe the vital energy within it. Then as developmental evolution takes place we shallimmediately observe two poles, the animal and vegetative pole,which follow 5Dmetric – the animal sensorialinformativepoleissmallerthantheenergeticone.Andasthesystemgrowsagainitwillformtheectodermwiththesensorialelements,theendodermandthemesoderm,andfinallyitwillbecomeeachofthem,theblueprintforoneofthe3physiologicalnetworksofthesystem.

Trinity is thus the easiest way to grasp reality as it is in a single plane of the 5th dimension. But even in ourdescriptionof theeggwe realize the4thdimotionof socialevolutionkicks immediatelymultiplying the layersofcells till the whole thing emerges into an ∆+1 fetal form and all the way already in its morphological stages,entropictimeerasesbyapoptosissomeofthosecellstoformthefingers.Andsoon.

Intimemonologicandinspacethegrowthofnon-Euclideanpointsarethustheeasiesttwosidesofthedescriptionoftheentangled,fractalcomplexUniverse,reasonwhyweservethemfirst.Iftheyaregrasped,thenwecanputthemtogetherestablishingavitalprogramofexist¡enceforallsystemsofnature,whichwillbetheprogramofthe5dimensionalmotionsoftime-spaceofanysupœrganism.

Sobyadding¬EGeometryand¡logic,wegettheactions=dimotionsthatformthesequencesofrealityandwillbeenactedbyall secondary languages inexist¡ence.Wecan indeedapply thevital topologiesof¬EGeometry, thelevels of ilogic entanglement and structures of monism, duality, trinity and pentalogic to resolve the 5actions=functions=dimotions performed by the vowels of a language, the type of cells of an organism, theoperandsofalgebra.

And as each part of a fractal is a fractal in itself, we can apply then pentalogic to each element of reality toilluminatethesystemorlanguagefromdifferentperspectives,sinceindeed,intheentangledgrowthofcomplexityforamonistsingle formtoexist,willstillhavetoperforminan imperfectmanneralong its inverseform,all thefunctions of reality. So even the smaller point is a world in itself (Leibniz), even the simple dimotion can bedecomposedinpentalogicelements,eventhesimplest±inverseoperandscanrepresentthe5¬∆@stelementsofanysystem.

Because we have studied the 5Geometric ¬E Postulates, in depth in our 1st Book on ¬E Geometry and the 5dimotionalprogramofexist¡encefromthep.o.v.ofthe¬Algebraicoperandstheyreflect,weshallnotgofurtherinhere;justbearinginmindthebeautifulvitalsentientgeometryofsomethingsoapparentlyinsignificantasananglewhichhasnotevenadimension.

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THEFUNCTIONOFEXISTENCEINPROBABILITYANDSTATISTICSHowtemporaleventsbecomespacepopulations.Themaindistributioncurves.EntropicNormalvs.lognormal

Sointhenextgraph,weobservehowtrulystochasticsystemslook,bothintimeasa'cumulative'populationofevents, and in space, as a distribution with a mean of maximal probability. The erf (error function, asmathematicianscall it) is reallythetimesymmetryofthedistributionfunction;hence itsSigsmondi-cumulativeform:Distributionoffrequenciesintime,equalpopulationsinspace,bothdefinedbythesame'magicformula',whichweshallstudyindetail.

It isoneofthefundamentalsymmetriesoffractalspace(measureinpopulationsof identicalentities/ 'points')and cyclical time (measure in frequency of time events/numbers); and it is ultimately one of the clearestexpressions of S≈T and the equality of the 2 'real units' of spatial geometry (points) and temporal ¬Algebra(numbers);asthetotalofeventsbecome1,andsodoesthetotalvolumeofthecurve.

ButtheessentialdifferencebetweenspaceandtimeIsthis:spaceissymmetric,bothleftandright,butthecurvethatrepresentstimeisNOT,ithasdirectionality,andithasnotsimultaneity.

Thusstochasticpopulationsinspacetendtoasimultaneousdistributionaroundthemean'0’-line'positionconsideredthecentral'point-line'ofthe'phasespace',whilethetemporalcumulativearrowtendstowardsaternary distribution in 3 clear ages of differentgrowth, with anexponential growth in the 'mature'ageof the system (middle region),andabeginning andanend,which are 'reversed in form' (first youngageoftheevent,and3rdfinalageoftheevent).

Forthematterthatbringsthosetwocurveshere–itsrelationshipwiththe function of existence, wemust perceive the distribution curve asthespatialevent,whichmakestheperfect‘reproduction’(meanform),justa1event–atthesummitoftheS=Tclassicageofexistence,andallother ‘stœps’ of spacetime, slightly erroneous reproductions, whichthereforedetermineas they ‘accumulate’aspopulations in the0 to1

sigsmondicurve,agrowingagingandmalfunctionofthesystemthatdragsits‘tailofmemorialpastrepetitions’alongitsway.

Thebigquestiononhowthe0’-1timeprobabilityformbecomesa1-∞space∆+1scaleismoreimportantthatwhat it seems,as it showsacertaindeterminismofgreatnumbers that isof thegameofexistence,which inmanywaysthedistributionfunctionsignifies.

PopulationsaroundanSTcenterwithavitalenergypopulationandtwocues,whichoneseemstothinkshouldbe the parameterization of the singularity and membrane of equal value, as a hypothesis of work, showshowevertheuncertaintyofwhichpointtomakethecut,andthisisprovidedbytheprobabilitydensity...atthenormal, which gives us a huge difference over themedian, more according with the properties of ∆st: over2/3rdsofthepopulationisvitalspaceenergy,153/4thsgoforthesingularityandthemembrane.Thoughinrealmeasuresthehigherdensityofbothreducesfurtheritspresenceinspace.

Soweconsiderthatthedeterministicfunctionofconstructionofasuperorganism,bothinthe1cyclesandthe1-∞isomorphic∆+1scaledefine'whythelawsofgreatnumbers'bringustheequationsofdistribution.

Howwe get from one to another then can be 'deduced' axiomatically for further 'insights' in the process oforganicstructureinspaceandtimefrequenciesthatbuilda1-beingfromitsfinitesimaloccurrences.

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Butthespatialsymmetrygivesusalsoanotherkindofinformation,astheregroupingofthosefrequenciesintoprobabilitiesofpopulationscompletelychangesitsroleintheStsymmetry.

ThefundamentalthemeofProbabilitiesandstatisticsbesidesthiss=Tsymmetryistheconceptof¡ndifferenceofbeings-events.Indeedwhenwegeta6isequaltocreateanindistinguishableparticle.Thetypeofparticlesthusis equivalent to the set of events in time. And as sun they accumulate into the networks that construct thesystem.Aminimaldistributionof3onlyelements,theidentitypresentelementandtheinvertedones,seemtobe by far the commonest, with the point ofσ as the place where the symmetry reaches a breaking pointbetweenboth'asymptoticareas.'

Networksintime.Determinism.

Reality though is not dominated by entropy but by information and information bias towards a certainproperty all distributions, precisely because it ‘forms’ reality, and so as we get into systems that have anetwork condition probability change biased by multiple elements of information. The results are biased,skewed structures in time and space. Let us consider one fundamental for the growth of supœrganismsaroundacentralpoint,whereitshouldresidethemindofthesystem.

Astimepassesthenetworkgrowsaroundthe1stclonedualitywhichremainsatthecoreofthesystem.Eveninthemorecomplexsuperorganismsshouldbepossibletofollowthepositionoftheseminaldualcell,whichlikelywillremainintheleadingpositionandpossiblyemergeinthecontinuityofthemindexists,asthemindof the supœrganism. It is then possible to consider a continuity of all i-logic minds proportional to its

informativecomplexity.

Determinismgrowsfromherd to organism asinformation organizes asystem around itsnetworks.

CALCULUSON

ENTROPICLIMITS.SPATIALSTATISTICS

The beauty of it all thoughcomes when we compare

thatgraphwithoneofstatisticalpopulations:anditsequivalentinprobabilities,asbotharegaussfunctionswhicharethefunctionofexistence,whichhaveallthosepoints,becauseitisunavoidablethatrepetitionscommiterrors,astheyareinfactnormaldistributions,ofallpossiblemaximalentropiccombinationsofbeings.

Yes,itisallfittoexistwitherrorsanderrorsbecomefrequenciesoftime,thatformpopulationsofspace,andthoseerrorsarecausedbytheneedforthemaximalentropyofallpossiblecombinationsandvariations.TheUniverselikestomutateyourcyclesoftimeandpopulationsofspace,andthentheylingeraroundthosefree

radicals,thoseslightlyodddistributions,oneofthem,excepttheperfectresonanceofaDirac’sfunction,canlastforeverbecausetheyarenotallequalyourrepetitionsofbeings,youractions,youre-motions.Animportant

insightof5DalgebraontheS=TmirrorsymmetrybetweenprobabilityandstatisticsisthefactthattheGaussdistributionthatcorrespondstoamaximalentropicfunctioninprobabilityhoweverrepresentsthenormalstatisticaldistributionofawell-ordered

informativepopulationarounditsmean‘evolutionarycenter’.Thuswhatisentropyintimebecomesinformationinspace,reinforcingallthe5Dconceptsofthegeneral

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modelofgenerationalspace-time,asentropyisthe‘timesubstance’fromwherementalspaceandinformativeseedsextractitsenergyofexistence.

InthegraphtheMaxwell-Boltzmanndistributionforthevelocitiesofgas=entropicstateofmatter=molecules.Forsmallv,thisquantityisclosetozerothbecauseofthefactorv2intheequationfortheweightofastate;afterthezerothpointitreachesamaximumandexponentiallydecreasestozerothagainforlargevelocities.

Wethusseethatagascontainsmoleculeswitheverypossiblevelocityvalue.Thegreatestnumberofmoleculeshaveavelocitycorrespondingtothemaximumofthedistribution.Wecanseetheskewnessofthedistributioncausedbythenon-existenceofareal0’pointofnospeed,andtherelative∝,whichislimitedalsobythec-limitofthelarger‘galacticenclosure’(speedofthebackgroundspacetime,asV∆-¡>V∆¡-nandthespecificmembrainofthecontainerthatreducesit,absorbing‘energy’fromitsenclosedvital

space.Itillustratesalsotheinteractionofeventsintime(speed)definingpopulationsinspace.

Anotherreadingofthesameconceptisthis:deterministicsystemsarethoseinwhichitsentanglementandbondagebetweenits¬∆@stpartsincreasessignificantlywithtime,asthesystemevolvesfromaloosegroupofindividualsthatformaherdunderacommondimotion/property,intoawholeorganictiedupasysteminwhicheachpartisconnectedtotheothersthroughanetwork,whichevolvesintoaternary,S<ST>Tnetworksystemto

formatopological,organicplane,orsupœrganism,wheredeterminismgreatlyincreasesandprobabilityisminimal.

Thusprobabilityandstatisticstendtobeusefulforsystemsinaherd/waveconditionfarmorethansystemsinaparticle/head/superorganismhighlyordereddeterministstate.Andforthesamereason,itisbettersuitedfor

theanalysisofthe∆-1partsofthewhole.

Theconceptofaneventintimeorpropertyinspacewhichisstatisticallymeasuredinacollectivenumbersofparts,whichhaveadegreeofentropicfreedomthatmakethemsusceptibletoenoughvariationtobemeasuredstatisticallycomesthennaturallyasaconsequenceofallthis.Essentiallywearedealinginstatisticalanalysiswithnon-wellorderedsetsofdisconnectedpartswhichshareacommonpropertyordimotionofexistence.SostatisticsisimportantbutitdoesNOTdefineastatisticalentropicuniverse,unlessassomanyphysicistsdowe

excludethesolidstateofbondage,wheretheperfectordermakesstatisticsquiteirrelevant.

Limitsanddistortionsoffunctionsin-betweenplanesofexistences.Thefunctionofexistenceinspace.

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Whatistheformofthefunctionofexistenceinscalarpace?MathematicallywecanseetheequivalenceexpressingtheGauss curve both in statistical and probabilistic form. Butwhenwedealwith the realUniverse, it is farmoretellingtoconsideragraphofthefunctionofexistencefromthepointofviewoftheparametersof‘energetictimeandinformativespace’,oftheworldinwhichitisinscribed.Asweshallfindwithinthatworld,limitsofentropyandformthatkillthesystemoutofbalance–limitsthatexplainwhytheGaussiancurvehasnotinfinitedeviation,asthoseareformsthatwouldnotbestable.Inthegraph,froma30yearsoldbookwithafunny‘nomenclature’weseeapointofbalanceforanyfunctionofexistence,whenthespatialformandtemporalmotionofthebeingisequalized,S=T,onthetwoextremesofthefunctionsofentropyandform,theparametersofthebeingbreakdowninimbalance,whichimpliesthedeathofthesystem(Max.SxMin.T∧Max.TxMin.S),asitchangesfromthebalancedS=T"Newtonian'regiontooneofdistorsionor‘Lorentzianregion’(takenfromgravitationalspace,wherethesystemdiesnearc-speedwhenformandmotionbecomedistorted).

Thoseextremal regionsare theones inwhich the systemdiesbyexcessof formormotion,provoking itsentropicdissolution.Whileintheborderwith∆-1thesystemisbornasaseedthatwillexpanditsvitalspace,asitimprintsitsgeneticinformation,tillfindingabalancebetweenboth,whenthesystemfinallyemergesasan1-wholeandaslongasitstaysinthatregionofbalance, itwillsurvive.Butif itmovestotheregionwithparametersproperofits largernested ∆±1 Universe, it will either collapse by excess of form- density (spatial form) or by excess of energeticacceleration(temporalmotion).

So we talk of the graph also as the region of domain of the function of existence – i.e. if the parameter weretemperature,formanitwillbefrom35to40degreeswherethehumanorganismcansurvive.Foranatombetween0Kandcspeed,andsoon.

Soweconsideradifferent'finiteview'ofcontinuity,astheregioninwhichthefunctionDOactuallysurviveentangledbetween its S and T elements; and the discontinuity barrier where its s=t elements (particle-wave, head-body)disentangle by overdrive of energy or information –where the spatial view of the function of existence tends tohyperbolicform:

Inthose'verges'theS=TbalanceofT.œ,breakbecausethenestedUniversehavelargerS=Tparameters,andtheT.œwants to reach those∆+1 limits,but it canonly ‘attempt’ to reachoneata time,breaking its innerbalance. So itschangein∆,SorTbreakthesystem:

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S: In space itmeans its topological form suffers a topological tearing or transformation. Then themembrane andsingularitysplitandthesystemdies.

∆:Inscaleitmeansitssocialevolutionovercrowdsthecarryingcapacityofitsworldandtheherdcollapses.Theyaretheminimalandmaximalregionsinscale.

T: In time itmeansanexcessofwarping/wrinkling for themembrain thatdetaches from its vitalbody,exhausted.Theycanalsobetheinitialandfinal'conditions',thatisthevalueofthefunctioninitst=0andt=t,beginningandendofitsworldcycle.

@:Inthemirrorofthelanguageofcalculusthoselimitsmeanregionswherethedifferentialequationshavenolongermeaningfulsolutions.

¬:inentropicterms,theyarelimitsforthefunctionofexistencesanditsspace-form,andtimemotion.

Conclusion:AfractalUniverseofdiscontinuousregionsofexistence.

TheUniverseisdiscontinuous.TodifferentiateafunctionwedoNOTneedabsolutecontinuitybuttheexistenceofaninfinitesimal1/n,andnojumpbetween'neighbourhoods',whichshouldbenofurtherthan1/ndistanceeitherintheX or Y coordinates. 'Adjacency' of the function then is defined by discrete 1/n intervals, which sufficeinNature=reality,regardlessofmathematicalmethodstodefinethem.

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IAGE:TRILOGICOfEQUATIONS:∆-POLYNOMIALS.T-PROBABILITIES.S-TATISTICS.

Wearenowequippedwiththesimplestfunctionsthatformtogetheratrinityoforganicelements,fromthesumtotheproducttoexponentialsandpowerlaws,toconstructa‘world’inwhichonlythosedimotionaloperandsdoexist,formingsentencesthatrepresent‘trans-formations’ofspacetimethroughadimotionaloperand.

A function f is amathematical rule that assigns to a number x (in somenumber systemand possiblywith certainlimitationsonitsvalue)anothernumberf(x)throughasetofoperandthatreflectsthemodeinwhichonetransformsintoeachother.Forexample,thefunction“square”assignstoeachnumberxitssquarex2,whichmeansithasbeen‘operated’,XxX,throughare=productivedimotion.

Thuscommonfunctionsaredefinablebyformulas,whicharerelatedtotheoperandsofequivalence∆s≈∆Tandatleastadifferentoperand=dimotionoftransformation,orelseitwouldbeamereidentity,suchas:

∆§:Polynomialsofthetype,f(x)=x2.Thelogarithmicfunctionlog(x);&theexponentialfunctionexp(x)orex(wheree=2.71828…;andthesquarerootfunction√x.

∆T:Trigonometricfunctions,sin(x),cos(x),tan(x),andsoon.

Andsowecanapplythedifferentilogicstructuresofreality,ofwhichthesimplestoneistheS=Tduality,whichinitsmorecomplexformbecomestheT-worldcycles≈S-superorganismdualitythatmergesbothspaceandtimeelements.

Inthesimpleworldofthe3stairsof±operandsthisgiveustwohugefieldsofmathematics.

S=T: Probabilities in time and statistics in space, of which the most interesting element are the study ofsimultaneouscurves(Conics),whichwehaveconsideredonourpaperongeometry(spatialmathematics).

∆-scales:polynomials.Soweshallconsideronlyafewquestionsofpolynomialsandlinealalgebra,inspaceandstudywith deep philosophical insights on the nature of time, the laws of probabilities, before we get into the ‘wider’operands, able to penetrate further beyond a single plane of the 5th dimension, the trigonometric and calculusfunctions.

We can then consider that the age of functions divides clearly in 3 degrees of complexity understanding of the 5Dimotionsofexistencewithdifferentfinesse:

Thelinealyouthofdiscrete‘herds’andsimplexdimotions:∆0

-Theageofpolynomialsandprobabilities,whicharethe ‘derivative’ finitesimals in timeofaneventandstatisticalpopulations,whicharetheequivalenttotheintegralofawhole.

-The finalculminationsof theageofpolynomial simple functions,with the introductionof thecomplexplane, theangularfunctionsandtheEuleridentitythatrelatesbothphasesoftheworldcycleofexistencethroughexponentialequivalenceswithsinusoidalfunction.

Thecurved,entangledageofcalculusandsinusoidalfunctions:∆+1

- The age of calculus, where we add derivatives and integrals that play the roles of probabilities and statistics,perceivedwithmuchmorefinesse.

Themodernageoffunctionals,functionsoffunctions:∆+¡

TofinallyendintheXXcenturywiththeexpansionofthediscipline,alreadyinthe3rdageofmathematicstofunctionsoffunctionsindifferentmentalspaceswithn-levelsofdimensions.

To understand the subtle differences of precisions between one and the other age, we need to incorporate theapproximationsofpolynomialswithderivatives…

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Theexistence oflimits to any linealapproximation of the cyclical reality of the Universe, a deep philosophicalquestionstudiedinouranalysisofthedivergenceandcreationoffuturesthatareultimatelyclosedisalsodeep,aninessenceestablishalimitofvalidityforanylinealapproximation,obviousintheTaylorseriesofdifferentialarounda0point, which onlyapproaches (with the notoriousexception of the exponential decay function of pure entropy, inwhichbothcoincide),thecurvedfunctiontoalimitnormallythatofasinglecyclicalrepetitionofthefunction(asinthesineapproximation).

Linearitythusonlygetstoapointorlimitgivenbya'cyclicalrepetition',eithertothenextplane,ortheendofaworldcycle,ortheendofastœpDimotionofexistence.

The key ¬Algebraic concept of ∆st systems is the existence of a STable region of balance between planes ortopologies where the asymmetry of the system is fairly lineal operated in decametric Planes of growth andsuperposition,andtheregionsofrelativepastandfuture,|orO,∆-1or∆+1,wherethereisasplittowardsthepurityofmotionorform,disconnectedpartsorwholes,acceleratedvorticesorlinealscatteringandmustbeoperatednotwithscalarpowersbut finitesimal integralsandderivatives,moreprecise in theirmeasureof the 'curvature'of thephasespacewestudy.

Itfollowsthatwhenstudyingmorethanaplaneofexistence,itisbettertoapproachthequestionthroughthemoresophisticatedprocedureoftheintegralandderivativeoperands,thatfirstlocalizestheminimal'finitesimal'ofchangethroughaderivativeandtheintegratesitalongavarying'curve'thatbetterreflectsthe3'different'sectionsofaflowof space-time evolving through Planes, with its central lineal region, better suited formultiplications and simplerpowerlaws,whichbecomehyperbolicinthedecayingandemergingfrontiersoftheplane.

Thenthereisthequestionoftransformationsbetweenspaceandtimeand5Da(nti)symmetries,whichisanessentialpartofclassic¬Algebraandweresumeinthoseterms:

Integraltransformsmakepossibletoconvertadifferentialequationof5Dspace-timewithincertainboundaryvalues(timemembrane,which limits theequationasa 'real system',notan infinity, into termsofan¬Algebraicequationthat can be easily solved (a polynomial which is a result in a single space-time plane). And this transformationobviouslyshouldbeoftwocanonicalforms.Andasithappensthereare2canonicaltransforms:

-Aspatial,linealtransformation,andthisistheLaplacetransform:f(p),definedbytheintegral:

F(p)=∫∞0e-ptF(t)dt.

ThelinearLaplaceoperatorLthustransformseachfunctionF(t)ofacertainsetoffunctionsintosomefunctionf(p)anditisusedmostfrequentlybyelectricalengineersinthesolutionofvariouselectroniccircuitproblems.

Atemporaltransformationandthis istheFourieranalysis,whichprovedthatafunctiony=f(x)couldbeexpressedbetweenthelimitsx=0andx=2πbyaninfiniteseriesofwaves:

F(x)=1/2α∑acoskx+bsinkx.

Thatisanequationcouldbecomeacyclicaltimedependentequationdevelopedasasumofharmonicwaves.

Andfinallytheinverse,thefactthatafunctioncouldbeconvertedintoa5DanalyticalequationbetweenPlanesofthe5thdimensionisprovedbythethirdmostusedapproximationoffunctions,theTaylorseries,whichexpressesafunction f—forwhich thederivatives of all orders exist—at a point a in thedomainof f in the formof thepowerseries:

∑∞∆=0f(∆)(a)(z-a)∆/∆!

InwhichΣdenotestheadditionofeachelementintheseriesas∆rangesfromzeroth(0)toinfinity(∞),f(∆)denotesthenthderivativeoff,and∆!isthestandardfactorialfunction.

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So this 3 transformations a means - and its applications enlighten an infinite number of real equations that thedifferent5DPlanesofrealitycantransferenergyorinformation.

That is, a5D flowofenergyand informationcan travel intoa singlemembranewithabsoluteaccuracy (no lossofentropy,noneedoftransformsorgroupstoresolvethem.

Butthereisaminimallossofentropywhenwetransformbetweenplanesbackandforth(ofinformationorenergy)asthetransformisNOTabsolutelyexact-asforittobeexactthenumberoftermsnormallytendtoinfinity,whichisnotpossibleinthefinitedurationofanyflowbetween∆±1Planesofthe5thdimension.

Furtheronitisimportanttounderstandthemeaningoftheoperandsandthelawsofrelativeequalityanddynamictransformations of ¬Æ where equality never fully exists, but we transform, F(t)<=> F(s) as in E<=>Mc², or weapproximatevaluesthroughanevidentproperty≈.

2D:±Wehaveseenhowtrigonometricfunctionsdescribe ingrowing layersofcomplexitybycombiningwithotherDimotionsasvitalbeingsdothemorecomplexdimotionsofcommunication,reproductionandsocialevolution.

Sothenextquestioniscanthesumanditsinversethenegativenumbers,suchaseeminglysimplefunctionsdothesamepentalogic?

Theanswerisyes,butagainasinthecaseofthe'angle'ofperception,thingsmustbecomemorecomplexthroughthecombinationofthedifferentoperands,whilethesum/negativenumberremain'dominant'.

5Ð:Socialevolutionintoherds,the5thDimotionisself-evidentthedominantelementofthesum.

2D: Easy to explain is theDimotionof locomotionas a sumof stepsmeasuredby the frequencyof those steps insequentialtime(thoughbetterexpressedbymultiplication

3Ð:Thenthesumcan'progress'in'accelerated'growth,andthatwillexpressthe3rdDimotionofreproduction.

4Ð:Whilenegativenumberswillexpressthe4thDimotionofentropy

1Ð:Anglesofperceptionaresines,andassuchitsderivativescapturea‘quantaofperceptionintime’.

The5Dimotionsofspace-timethatencode,thesocialevolution,darwinianlocomotionandentropicfeeding,decay,reproductivegrowthandperceptionfunctionsofasuperorganismaremirroredbythe5actions=motion=operatorsofmathematicalspace:s,±,x÷,xªlog,∫∂...Astheyareencodedinspeciesbygenes,inhumanmindsbysentencesofwords,inlightbyitsdimensionsofformandspeed,inatomsbyquantumnumbers.Andmanyotherwaysindifferent

syntacticformsasthe‘languagesofgodareinfinite’.

Suchoperandscanformsentenceswithrelationalequivalentandorder=≤≥«»symbolsofits5dimotionsindynamicform.

Inthosesentencesthequestionofcausality,isexpressedintermsofindependence.

Itseemsthenthatmostspatialfunctionsaredependentontimetheindependentfactor:$=ƒ(t)

Weinterpretindependenceintermsoforderfrommotion,theoriginalsubstanceoftimeintospacethatextractsitsinformation:as functionsare firstmotions in time that stopandbecome 'forms'of space, leavingapast-memorialtrace.whichifNOTerasedbecomesapopulationofspace,whichmovesagainandthenbecomesapopulationandinthismannerreproductionofdimensionstakesplace,buildingabeingofgrowing∆Dst.

Inmostfunctionstheindependent‘submissive’parameterisafunctionoftime-energyfromwherethe,thedependantdominantparameterthatformsit,amentalortopologicsimultaneousspace,arises.

Howthisisexpressedin∫∂termsbecomesthenclear:sincetimeisdiscrete,discontinuous,madeofT.œ.s,moving,stopping(oftenperceiving),movingandstopping,wemustfirst'encounter'theminimalstepofthetimemotion,and

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thisiswhatweshallcalldt,andthenmove,stopandmovestopanumberofsteps,whichweintegrate,buildinginthismanneranewdimensionofspace-time.

So the combination of ∫∂is in fact a process of creation of an ST dual dimension of space-time. And that is theultimatemeaningofit.

Sowhen study those simplest equations of physics, we shall consider those inwhichwemake a 'ceteris paribus'rhythm of considering it first from the point of view of 'time' steps and then from the point of view of 'space'integratingthemasasimultaneousspace,whenwehave'traced'enoughstepstomakethatsimultaneitymeaningful.Andthisisthemeaningofadefiniteintegral.

Ratesofchange.Thestopandgomotion:stœps.

FinitesimalchangesarerelatedtothefundamentalbeatoftheUniverse,thestop-form-space-perception,go-motion-time,beatoftheUniverse,whichweshallcallastœp,thediscretewayofmotionoftœsthroughSPace,whichoftenasinmoviesweperceiveincontinuousmodeeliminatingthestopelement:

∆S(top)->∆t->∆S-≥∆(S)t(ep).∆±¡:Planes,Planes

Inpentalogicweshallconsidersomebasicelementsofanalysisinitscalculusofpartsandplanes.

Galileanpxinanalysis:finitesimalsteps(derivatives)integratedtocalculateacyclicwhole.

Furtheronanalysishasoverallotherbranchesofmathematicsaspecialqualitytostudy'changes'betweenplanesofthefifthdimension,asmultiplederivatives 'jump'(albeitwithdifferentdegreesof 'focus')betterthanmostly 'linealpolynomials'betweenplanes,andthe'curvilineal,Lorentzian'variations,slowdownsandaccelerationsontheSxT=KparametershappenbetweenPlanes:

Theformalstienceofthe1stand5thDisomorphismsinthemathematicalmirrorisanalysis,whichdealsdirectlywiththerelationshipsbetween ∆-1 'finitesimal'parts' and(in)'finities'.Twonewtermswestillaccompany,withthe lostinflationary term 'in'; since infinitesimals and infinities are a Kantian paralogism; as all planes have a limit in itsquanticunits,andallwholesafinitecirclethatenclosesthemintoarelative0’-1'circleunit'.

Besides the duality of the 0’-1probabilistic mind unit whichreflects the external 1-∞universe, a second duality thatweights heavily in analysis isthat of perception of lineal vs.cyclical form: We are minds ofspacethatmeasuretimecycles:∫@≈∆ð.

Hence the equation of mind-measure defines theunderstanding of differentialcalculus:Asalwaysinpraxis,theconcept is based in the duality

betweenhuminds thatmeasurewith fixed rulers, lineal steps, over a cyclical,movingUniverse. SoMindsmeasureAristotelian,shortlines,inalong,curvedUniverse.

So thequestioncomestowhichminimalist linealstepofamind isworthytomakeaccuratecalculusof those longcurvedUniversalpaths.

Thegeneralruletoidentifybothpolynomialsandanalysis,isthis:Y=S=ƒ(x=t)

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Thedifferencebetweenlinealpolynomialsandnon-linealanalysis

In thegraph,weexplain thedifferencebetweenapolynomial 'regular'descriptionofasystemas itchanges in the"Newtonian',socialscaleinwhichchangesarenotofqualityasmuchasofquantityandtheanalyticalregioninwhichthereisachangeofstructureandhenceofqualitywithirregularitiesbettershownbyanalyticoperations.

Itfollowsthatmoreimportantthan'variables'areto¬Algebra'operands',whoseencodedmeaningand'magic'wayof relating systems to get a 'future or present' outcome by merging them according to certain rules of 'creativeengagement',trulygivesthepowerto¬Algebratomirrorthea(nti)symmetriesoftheUniverse.

ThekeyconnectorofT.Œwithclassicscienceisthefullunderstandingofthedual¬Algebraoperands,±,x/,∂∫,√xªaspartofthe¡logic,pentagonalgameofrealityinallitsmirrorsymmetries;thatis,asdimotions≈actionsandstructuralelements,whereasan immediate correspondencebetween thoseoperandsand the ternaryelements∆@st canbeestablishedasfollows:

Thesum-restaretheinversearrowsofthesimplestsuperpositionsofdimensionsbetweenspecieswhichareidenticalinmotionandform.

Theproduct/divisionrisesthecomplexityofoperandsafirstlayer,andservesthepurpose,besidestheobvioussumofsums,ofcalculatingthemarginofdimensions,ascombinationswhicharenotpurelyparallelbetweenclonebeings,mostlikelythroughtherecombinationofits∆-1elements,astheproductof2Sœtsinnerelementsgiveusallpossiblecombinations. Ie.5x4=20 ISalsothenumberofconnectionsbetweenall the5elementsand4elementsofbothsets.Somultiplicationadseitheradimensionofmultiplesums inthesameplane,orprobesforthefirsttime inaninnerscalardimension.

Thenwearrive finally to thepowers-root systemsand integral-derivatives,whichoperate fully on the∆§cales andplanes of the system, which require two slightly different operands. As §¹º'social decametric Planes' are lineal,regular,sowecanoperatethemwithpowers,rootsandlogarithms.

∂∫ But when we change between Planes into new wholes and new planes of existence we are into 'a differentspecies' and soweneed tooperatewith themagicof finitesimalderivativesandanalytical integrals,whichkeepabettertrackoftheinfinitesimal'curved'exponentialchangesthathappenbetweentwoplanes,wherelinearityislost.Theintegral/derivativethuswillberelatedtothecloselyconnected'mindintegration'ofinformation.

Ultimately a derivative of a larger world, measure in a still time point of zero latitude) and the processes ofintegrationofpartsintowholesthatalwaysdischargespartofthebeing,reasonwhyaderivativeisessentiallysmallerthan the power operand, as those processes eliminate part of the whole. This being a key technical element ofanalysis (which often is approached by binomial series - McLaurin, Taylor, etc. - connecting both operands, butreducingthepowerseriestothat'a'constanttimespacepointinwiththementalorwholeintegrationtakesplace.

Philosophyoftime.Pentalogiconprobabilities&Statistics.

Probabilitydespiteorperhapsduetoitssimplicityisanessentialbranchof5Dasitdealsdirectlywithfundamentalthemesofphilosophyoftime,worthtoexploreevenbeforewestudythelawsofprobability.Toframeourdiscussion,firstweshallconsider,thePentalogicofProbabilities,whichisimmediate:

Duality:S=T:ProbabilitiescarryontotimebyvirtueoftheS=Tsymmetry,populationsinspace.

∆-scale:Probabilitiescarryintothe01fastertimesphere,thelawsofthe1-∝,entropicCartesianplane.

SS-TT limits become then for probabilities the0’ finitesimal probability and the1-whole event; the first of certainimportanceintheoreticalquantumphysics,insuchprocessesasquantumtunneling.Asusualweshallonlyconsiderthemesrelatedto¬Ælgebra(ab.Non-Euclidean,Non-AristotelianExistentialAlgebra).

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SPATIALSTATISTICSAnimportantinsightof5DalgebraontheS=TmirrorsymmetrybetweenprobabilityandstatisticsisthefactthattheGaussdistributionthatcorrespondstoamaximalentropicfunctioninprobabilityhoweverrepresentsthenormalstatisticaldistributionofawell-orderedinformativepopulationarounditsmean‘evolutionarycenter’.

Thuswhatisentropyintimebecomesinformationinspace,reinforcingallthe5Dconceptsofthegeneralmodelofgenerationalspace-time,asentropyisthe‘timesubstance’fromwherementalspaceandinformativeseeds

extractitsenergyofexistence.

Theinterchangeabilityofprobabilitiesandpopulations.

Inthegraph,thebeautyofpurestochasticprocesses,anditsbellcurveandlogisticcurve,whichfinallysaturatesthe'field',residesinitsuniversalityandcapacitytodescribethefundamentaleventsofrealitybothinspaceasa

seriesofreproductionsthatbecomespace-populationsandintimeas a curve that accumulates populations and errors or constrains(internalandexternal¬limits)tofurthergrowth,endingintodecay.

Now there are many themes that connect the law of probabilitydistributionwiththoseofÐSTparameters.

Consider a view of the curve in time as the 3 ages law with themiddleageasthemost'abundant'.Aswehavedescribethe3agesofthelogisticcurveofgrowthwecannowconsiderwhatshouldbeobvioustotheÐSTprofessional(me,iandmyself:).

The present ST age between the points of inflection, should beworth the fundamental 'harmonics' of ÐST, bidimensional andternary functions, either 1/2 or 2/3rds,of the total value of thefunction,andwithinthe±3iconicvalue,asithappenswitheˆxweshall find all the values worth (emergence, first age, 50% ofmaturity, second age and age of extinction). And so we do findindeed,thecommonestnormaldistributionsof±2/3rdsatsigma1andover99%atsigma3,whichisthereforethewholeexistencein

timeandspace,ofanysystemoftheUniverse.

Whilethe50%comesatsigma0.6745abeautifulnumber,whosesecretshallremain(withme,Iandmyself:).

I’ll leavethoughsomeinterestingrelationshipsforfuture5Dmathematicianstoconsiderthearcanesecretsof‘magicnumbers’.ThedarkmatteroftheUniverse,whicha‘lineal’circledoesnot‘measure’,becauseitismadeof 3diameters, isπ-3/π=0.045%; the limited reality that a central perceiverobserves if closed in a cyclewithsuch3apertures;whichasithappensistheleftoversofthenormaldistribution,outside2sigmas.Thuswhatweperceivedclosed in thestandardπ-3circle iswhatdeviates fromthemeandistribution– theoutsideworld isequivalenttotheoutsidermeasures.

Thedifferencebetweentheabsolutenormalsigmaandthe2sigmasontheotherhandis1/10thofe,0.2718;thestandardderivative=finitesimalofthetetraktys,decimalscale.Themesthoseofadvanced¬ælgebra…

Sobitsoffrequenciesoftimevs.quantaofpopulationsofspace.Asfrequenciesoftimebecome'populations'ofspaceoncetheyhavebeenbornintime,and'settled'inspace.

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So, we shall talk of the symmetries between frequencies of future time that becomespace populationaccumulated in the past, till they both recede in size as we keep growing in Planes, and becomeundistinguishable, continuous, and quiet. And then we are studying pie space'surfaces',throughtopologicdefinitionsofadjacency,equality,perpendicularityandparallelism.Nowwithavitalsense-meaningweshallevolveinourupgradingofthe'Universalgrammar=syntax'ofthoselanguages,wheninthe3rdlinewestartinearnesttoconstructthedifferentrhythmsoftimespace-change.

Naturethoughalwayshasagoal:iterateabody-waveofenergy,withthoseparameters.

Sinceattheendofthejourneyweshallseethatwhatnaturecaresaboutistoreproduceitsfractalstassembliesofcreativepatternsofspace-time.Asall isreproduction.Theultimatesubstanceofreality ismotionwithform,andmotionisthereproductionofformalongan∆-1disorderedregionofquantaofspacethatawhole∆-beingwillmoldand'rise'.

And again those processes will be described mathematically with some basicoperandi, which describes theunion of a surface of smaller continuous simultaneous quanta of space from the past larger, less informedPlanes,andofquantumfrequenciesoftime.

So the obvious rule as time and space planes are perpendicular is that a function of st is a multiplicative,reproductivewave functionandoneof±superpositionsoneof fieldandparticlesusesapower law,beinganintegral formof expressing thedifferentPlanes inoneparameterwhich is growingdecametric through thosePlanesorexponentially.

TheefunctionISofdeathanddecaytransitions,the10scaleISofreproduction,andthebodyandhead,initsreproductionanddecaymergeallthosemeaningstogether.

SopatternsinNaturearejustfrequenciesofbitsoftimeandpopulationsofquantainspace.

And so the subtle differences betweenboth concepts andwhen to use them, for entitieswhich often scienceconfusesduetolackofperception(asinquantumphysicswhereoftentimeprocessesareconsideredspatial),willbeessentialforthestreamliningmadebystiences.

Ingeneralwetalkof timecycles,whichcreatespatialpopulations,withdifferent 'degrees'ofpersistence intotherelativepast.Somotionsinspace(locomotions)havehardlyanypersistenceintime(wedonotleaveatraceof parallel forms) except in the simplest beings (waves of light and so on); but reproductions in time leave apersistence of populations. And so the time bit becomes a space quanta. And this again is an importantphenomenainthesimplestforms(wavesandparticles)ofthe∆-3,4Planes.

TheimmortalityofDirac’sfunction.

This leadus toconsider if there isa formof immortality.That isone,whoseeventshaveaDirac’s distribution where the deviation from the norm, tends to 0’=zeroth (term for afinitesimalzero),withoutreachingit.Itwasdiscoveredbyoneofthe4‘talented’physicistsofUK(Newton,Maxwell,Heaviside&DiracNOTHawking).The Dirac delta is used to model: - S: a point charge, point mass or electron point. That is the S=T function side in space. But it is more interesting, to consider it in trilogic as:- T: The function of immortality in time, when the reproduction of a form is perfect in all its stœps, which correspond precisely to those particles that are ¡indifferent in all its replications through its quantum potential stœps as waves-particles (hence equal in form).-∆: It models a tall narrow spike function (an impulse), and a perfect resonance, which transfers the information of an ∆-1 element into an ∆º whole, hence emerging as a one. Resonance thus emerges a perfect distribution which leads to its perfect Log scales.

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S≈THOWTIMEEVENTSBECOMESTATISTICALPOPULATIONS.

The1-11¹¹infinity.

The KEY question to formalize probability as population is how to ‘translate Planes’, which tend to bedecametric,inpopulations...good!

OneofthefewthingsthatworkrightonthehumanmindanddonohavetobeadaptedtotheUniversalmind,fromd@stto∆ûst.

Shallwestudythemdownwards,through‘finitesimaldecimalPlanes’orupwards,throughdecametric,growingones? Answer, an essential law of Absolute relativity goes as follows: ‘The study of decametric, §+ Planes(10§≈10•10∆≈∆+1)issymmetrictothestudyoftheinverse,decimal∆>∆-1scale’.

Orinitsmostreduced‘formula’:(∞=(1)=0):(∞-1)≈(1-0)

Whereas∞istheperceptionofthewhole‘upwards’inthedomainof1,theminimalquantatotherelative∞ofthe∆+1scale.While1istherelativeinfiniteofasystemobserveddownwards,suchas∆+1(1)iscomposedofanumberof‘finitesimalparts’whoseminimalquantais0.

Soinabsoluterelativitythe∆-1worldgoesfrom1to0,andthe∆+1equivalentconceptgoesfrom1to∞.Andsonowwecanalsoextractofthe‘infinitorumthoughtreceptacle’,akeydifferencebetweenbothmathematicaltechniques:

A conceptual analysis upwards has a defined lower point-quanta, 1 and an undefined upper∞ limit.While adownwardsanalysishasanupperdefinedwholelimit,1andanundefined‘finitesimalminimum,+0).

Sothesmartreaderwillnoticethisabsoluterelativedualityof∆±1where∆@isthe‘observer’,impliesrelativityofknowledge,alwayswithaself-centeredelementtodefineit,and‘therelativedefinitionoffiniteinfinities,or∆+1limit(ab.∞)andfiniteinfinitesimals(+0).

Thisbringsanessential isomorphismofabsoluterelativity (doNOTconfuse∆-equality,withS-yncrhonicity,Ti-somorphismand@dentity;we‘repeat’asIknowwhen,ifanyhumanevergetstoreadthosetexts,thereisTOOmuchupgradingandnottogetdizzy,weDOrepeatessentialtruths).

I-somorphism is the concept of equality in time-information and at the heart of the possibilities to domathematicalproofsindifferent,seeminglynon-identicalspace-timedomains.

When we apply the identity of ∞|0 (here written in inverse fashion) as in the title of this post on ‘numbertheory’,poisedtocompletewhatmy fellowcountryman,Fermat, started,weunderstandthewhyofnumbersanditstechniques,sofaronlymadeexplicitasmostscienceisonhow-terms:

The real numbers NOW include always an inverse infinity between each 1+1 interval. YET they can providesatisfactorymodelsforavarietyofphenomena,eventhoughnophysicalquantitycanbemeasuredaccuratelytomorethanadozenorsodecimalplaces;as0nowistheundeterminedlowerlimit.

Itisnotthevaluesofinfinitelymanydecimalplacesthatapplytotherealworldbutthedeductivestructuresthattheyembodyandenableduetotheequivalenceof0≈1≈∞.

Analysisanditsinverseintegralandderivativecalculus‘drinks’onallthis.

Thus it came into being because many aspects of the natural world can profitably be modeled by thoseequivalences,asbeingcontinuous—atleast,toanexcellentdegreeofapproximation.Again,thisisaquestionofmodeling,notofreality.Matterisnottrulycontinuous;ifmatterissubdividedintosufficientlysmallpieces,thenindivisiblecomponents,oratoms,willappearandfinallywewillfindthefinitesimal+0quanta.

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But atoms are extremely small, and, for most applications, treating matter as though it were a continuumintroduces negligible error while greatly simplifying the computations;whenwework on the 1-∞ upper ∆+1scale,thatofthecosmologicalrealm;whereastheintermediatescale isthatof∆ohumanthermodynamics.Sowecanthenstateinphysicalsystemstheequivalenceof:

+0(quantumphysics)≈|-thermodynamics≈∞(boundinfinity):Gravitationalscale.

All what is above quantum effects, in physics, can be studied in a continuum modeling, which is standardengineeringpracticewhenstudyingtheflowoffluidssuchasairorwater,thebendingofelasticmaterials,thedistributionorflowofelectriccurrent,andtheflowofheat,andsoon(all∆>∆+1physicalsystems).

The2GreatFieldsOfG-Statistics

GStatisticsstudiesgenerationofspace-timesystemsbymeansofstatisticalmultiplicationofpopulations.

Theclassic fieldofprobability ISconcernedONLYwiththestudyofstochastic, randomprocesses;hencethosebelongingtothearrowofentropy.

SothefundamentalnewfieldintroducedbyGSTisthestudyofcausalprobability,asallpointsofspace-timehave3 possible dimensional motions, and the choice of them establishes aclear connection between probability,statisticsandtopologicalevolution.StochasticprocessesthuslooseinGSTagreatdealofchaos,andbecomefarmoreordered.

Sothenextandsimplestanswertothatternaryprobability,iswhichofthe3arrowsisequallyprobable?Andtheanswerisobviouslypresent,asitisconservedanditsvaluetheproduct,superpositionorPythagoreansumoftheother two (the commonest combinations/operandi of s<st>t metric). What then about the entropic andinformativearrows?Theirvaluemustremainthesame,buttheyareimbalanced:

Max.TtxMin.Ss(entropyarrow)vs.Max.TsxMin.ST.

Sosometimestheyarenoteasilycomparable,andmosttimestheyareinvertedinitsparameters

Further on, as in all fields of reality, we can reorder what we know of stochastic processes of 'entropy' -memoriless'Markowianprocesses',accordingtothe∆º±STfivedimensionalelementofallsystems(10Dforthefullmodel).Sothosearethethemesofthispost.

MainThemesofGSTatistics&Probability.

Oneofthemostfascinatingsub-disciplinesofGSTisthestudyofthemathematicaldualitiesthatrepresentspaceandtimeandcombinebothasmirrorsymmetries.

Ofthemthemostimportantisthatbetweenpopulationinspacewhichdistributesinthesamefashionthatitsoriginal time event probability. As beings are born in time and becomememorial space, so both are similarconceptsruledbysimilarequations.

Sothedefinitionofbothstiencesisimmediate:

"Probabilitydealswithfuturetime-space,statisticswithpastspace-time".

Andentropyandtheoryofinformationinthewayitisconceivedbyclassicsciencewiththeir'presentstate'.

Sodepartingfromsuchdefinition,asusualwecanstudythedisciplineswiththe(∆º±1)S≈T5mainperspectives(∆º,∆±1,S,T,S≈T)ofanysystemthemesareessentialtoprobabilitymodels intimeandpopulations inspace;andattach toeachperspective themain themesof theclassic science, reorderingandenlightened themwithwhys and new insights, born of our 'advanced' structural understanding of the organic, fractal ternary,symmetricUniverse.AsthesubjectsdealdirectlywithSTsymmetries,andarefoundationalofoneofthemain

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mindmirrorsofmankind(beingtheothertwoverbalthoughtandart);thesubjectistrulyrichandwewillonlytouchafewconcepts:

• S≈T: The duality of sequential future events in time (probability) and synchronous past populations ofeventsinspace(statistics).

• ∆º:TheepistemologicalconsequencesofthepresentdeviationofknowledgeNOTasthestudyofinternalcausal processes that lead to the future deterministic creation of events, (classic concept of epistemology ofscience),butass'mechanical’external,non-thinkingprocessofrecollectingdatawithmachinesandputtingitintoaprobabilisticmapping todetermine the 'likely'event, considering this tobeallwhatmatters toknowledge,as'evidenttruth'.Itispartofthetendencytosubstitutetimelogicforspatialevidence,regressingintheevolutionofthescientificmethod.

• T: And hence the proper understanding, in the path of the Russian school of probability (Kolgomorov,Markov)ofthetrueinnercausalityofeventsintime,in3greatareasaccordingtotheternaryprinciple:disjointedpureentropicevents($T-events)withminimalcausality(whosestochasticrepetitionsareindependent),eventsoffull causality, which are strictly determined by a guiding informative pole, 'soul' or singularity(§ð events) andeventsofmixedternarycausalitybothintime(stronglyinfluencedbypastresultswhichconditionsthefuture)andspace(divergingfreelyintothe3possiblepathsofallevents,aneventwithmoreentropy,ormoreinformationormoreiterativespace-timerepetitions).

• ∆±i:Theconfusionofspatialpopulationsandtemporalevents,inthosePlanesofrealityofwhichwehavelittleevidenceandcapacitytodistinguishboth(quantum,∆-max.ilevels),sincethetimeclocksofthosePlanesrunsofastweseetheminitsfullworldcyclesasiftheywereformsofspace.Inthatregardakeyrulefordistinguishingthemisthefactthattimeeventsare3-sequentialelementsandspace-eventsarebidimensionalholographicformsinsimultaneity.Letusconsiderfirstthisaspectofprobability.

• S: The spatial entropy of systems and its choices of future paths, according to the 'partition' of presentpopulation, which divide themselves in branching ternary forms, till filling up all the possible variations of thesystemwithdifferentpopulationsaccordingtoprobability.I.e.forexampleinthegalaxytherewillbeaseriesofdifferent fractal histories of mankind, according to the possible human vs. mechanical vs. Gaia vs. extinctionfuturespossibletohistory(3±deathpathsoffuture).

Somostplanetsareextinctbynukesintoblackholesandstrangelets(max.probabilityoffuture),someevolveintoametal-earthglobalorganismof robots, somewillnevergivebirth to life intelligenceof thehumantypeandfinallythelessofthemwillmakehumanstofullyunderstandandrespectthelawsoftheorganicUniverse-and evolve avatar like into the final stage of Gaia, as collective mind of the earth. So indeed, the theme isextensiveandapplytoallkindofquestions.This5thbeingstudiedelsewhere.Soweshalldealwiththeother4.

Thesamelawsworkfortimeeventsandpopulationdistribution.

Beforewegetintoit,letusconsiderthemainerrorintheunderstandingofprobabilityandstatisticsistoknowinsmallfastPlanesoftheUniversewhatisatimeeventandwhatisaprobabilityone.

The whole confusion happening in quantum physics about the way the Universe works departs from thatconfusion,whichaddedtotheidealistageofbaroquemathematics(whenHilbertsaysthathe'imagines'pointsand lines), and thecomputer age of approximations and other 'epicycles' of calculus and measure,and theincreasingvisualageofthoughtwhencausalitymattersnot,givesuswhatisthemostprominentofthoseerrors,whichplaguethephysicsofthequantumworld:theconfusionofternarytimeeventsandbranchingbetween$T,ST,and§ðpaths,offuturetimechoicesandprobabilities,withspatialpopulationsandfixedspacesymmetries,whichtendtobebidimensional(holographicprinciple).

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Bohmvs.Bohr.

It is remarkableof the timeswe live in that theabsolutelyobvious truthofquantumphysics,Bohm-Broglie's,deterministic model has been obliterated by the absurdity of the probabilistic interpretations, just becauseidealistplatonicphysicistsdonotcaretounderstandthewhysofmathematics.ButunlikevonNeumanndon'tevenacknowledgethisfact.

Indeedscientiststendtoconfusethose2symmetriesofrealityinsmallscaleandviceversa,forverylongtheythoughtandmanystilldothatgalaxiesofslowevolutionaryages,were'differentspatialspecies',NOTspeciesindifferentmomentsoftimeevolution.

Amassoferrorsproperofphysicshappenintheuncertainquantumrealmwithnopracticalevidence,hencewithmodel-theory, whicherrs because of itsmisunderstanding of cyclical time processesthat imprint its formsinfactualspace.Sooftenphysicistsconfuseatimeparameterwithaspacesymmetry(aswhentheyconfusetheelectro-weaktimeflowofinformativechangewithaspatialforce,orthe3agesofquarkswith3forcesinspace,astheyhavenotunderstoodtheexistenceof 'timeforces',whichtrans-formrealitybesides itsspaceforces.Soweshallendupwiththattheme.Butobviouslyaswestudyallstienceshereweshallputotherexamples.

As usual we will try to simplify all maths ad maximal, as the maths are always right, the conceptualmisinterpretationiswhatweseektorepair.

Symmetryof∞:StatisticSpacePopulations=ProbabilisticTimeFrequency

Theculminationofnumbertheorywouldbeprobability intimeandstatistics inspace,the lastofthemanyS=Tsymmetries between space and time; whereas we can observe in time also ‘scalar’ processes not of multiplepopulations in socialgrowth,butofmultipleevents,andnot surprisinglybothbecomeself-similar,which is themaininnovationof5D,suchasprobabilitiesintheo-1sphereoftimeeventsisequivalenttopopulationlawsinthe1-∝entropicplaneofpopulations.

Thislawhasmultipleimplications.Themostobviousisthatitresolvestheconundrumoftheequivalencebetweenthe quantum scale, we measure as a o-1 event probability sphere (because the speed of life and death of itsparticlesaccordingto5D,SxT=C,issuchthatweconfuseitsshort-livedpopulationswithtimeevents),andthe1-∝entropiccycleofmatter(thethermodynamicstatisticallaws).

Becausethequantumworldhas3‘Planes’ofbeings,theparts(particles),theorganicwhole(atom)anditssocialensemble (laws of molecules and matter or worldcycle), obviously those 3 cycles differ. But its treatment asprobabilitiesvs.statisticsstillrespondstothefundamentalequivalenceaforementioned.

Tosavespacewedealmostlywithprobabilityontimeasstatisticsinspaceiswell-knownandunderstood.

Onprobabilitytherearemanythingstobesaid.Theobviousonesarethemostinteresting.Wejustwillconsiderafewthemesforthesakeofbrevity

WeshallfirstmentiontheentanglementofS=Telementsinthemorecomplexfieldofstatisticsandprobabilitieswhichweshalltreatinour2ndbookonMathematics(¬Algebraandtimetheory).

S≈Thowtimeeventsbecomestatisticalpopulations.interchangeabilityofprobabilitiesandpopulations.

Thesymmetryisprofound:bitsoffrequenciesoftime=quantaofpopulationsofspace.

Asfrequencies of time become 'populations' of space once theyare born in time, and 'settle' long-lasting inspace;giventhefactthat‘probabilitybirth’isafasterplacentalcyclethanpopulationlifeinthelongercycle.Sothere is a clear proportionality between both, reason why the bell curve of statistical populations andprobabilitiesisthesame.

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So, we shall talk of the symmetries between frequencies of future time that becomespace populationaccumulated in the past, till they both recede in size as we keep growing in Planes, and becomeundistinguishable,continuous,quiet.

Soagain,wecancastbothwithNature’svitalfunctionofexistencewithitseternalgoal:iterateaT<ST>species.

Reproductionandmotion,twosidesofthesamecoin:±,Xallpervadingoperand.

Sinceattheendofthejourneyallwhatnaturecaresaboutistoreproduceitsfractalst assemblies of creative patterns of space-time. As all is reproduction.Theultimatesubstance of reality is motion with form, and motion isthereproductionofformalongan∆-1disorderedregionofquantaofspacethatawhole∆-beingwillmoldand'rise'.

AndthoseprocesseswillhavealwaysitsS=Tsymmetriesdescribedmathematicallywith re=product=iveoperands, which describe the union of a bidimensionalpopulation surface of smaller simultaneous quanta of spacemoving, reproducingthroughfrequenciesoftime.

Soweuse3‘scalaroperands’forthoseprocessesofsocialevolution,reproductionandemergenceintoanewscale:

Equalbeingsaddsociallyinbidimensionalfieldsby±superpositions,inthesimplestsocialact.Butasdensityandgendermirrorsymmetrieshappen,andSTequivalencesmovetheherd,perpendiculartimeandspacefunctionsrequireanstmultiplicative,reproductivewaveformorperpendicular,field>particlessystem.AndfinallypowerlawsandintegralsandlogarithmicPlanesofdecametricgrowthcompletethe∆º>∆+1creationofanewplane.AndthereisNOmoreoperandsastheUniverseisdiscontinuousandsoonceagameiscompletedanewscalestartsfromnothing,asanewfractalpointemergesinanentirenewworld.

Soprobabilitiescanbeseenasfrequenciesofreproductionintimeofpopulationsofspacequanta.

Thepersistenceofmemory–livesthatlastconvertprobabilitiesintopopulations.

Timecyclescreatespatialpopulationswithdifferent'degrees'ofpersistenceintotherelativepast:Motionsinspace (locomotions)havehardlyanypersistence in time (wedonot leavea traceofparallel forms)except insimplerbeings(wavesof lightandsoon);butreproductions intimeleaveapersistenceofpopulations.Sothetimebitbecomesaspacequanta.Andthisagainisanimportantphenomenainthesimplestforms(wavesandparticles)ofthe∆-3,4Planes.HencetheneedtoknowtheS=Tsymmetriesbetweenbothdisciplines,toknowwhentouse them,as inquantumphysicswheremostprocessesare timeevents,even ifhumanswithamuchfaster time clock consider them spatial.We illustrate both cases with 2 key equations that mix those vitalconstantsdefiningbothT=locomotionandS-populationsasreproductiongames):

Euler'sformula(left)describestheS=Tsymmetryofaworldcyclereproducedinalower∆-1‘plane’.SoitcanbeinterpretedasthewaveformderivedoftheconstantreproductionoftheUnitcyclein∆-1points'definingaworldcycle of space-time in ∆-1, with faster time life hence minimal persistence of memory. Its reproduction, thusappearasmerelocomotionastheintensityofthewavefadesawayinthelowerherd-plane.Intheotherextreme,theprobabilitydistributionofapopulation in space (Gausscurve)persists so long,as itspopulationsare in thesameplaneinwhichtheywerecreatedthatbothSandTstayroughlythesame.AnditsmaximalvalueisinS=T,thecentralpointthatmaximizesthefunctionofexist¡ence,whichisdefinedatthatpoint.Bothcurvesfurtheronaresimilartothebelllikecurveofthefunctionofexistenceintime,sothecentralregionS=Tintimethematureageofthesystem.Thusfieldssoapparentlydisjoinedareequalizedby5Dmetricshowinginarathersimplebutprofoundwayhowall is about super organisms tracingworld cycle of space-time in synchronous or sequentialforms.

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Conclusion:S=TSymmetryofprobability=unitcircleandpopulation=1-∞.

The basic concepts of the theory of probability, namely random events and their probabilities, are completelyanalogousintheirpropertiestoplanefiguresandtheirareas.ItissufficienttounderstandbyABtheintersection(commonpart)oftwofigures,byA∪Btheirunion,byNtheconventional“empty”figure,andbyP(A)theareaofthe figure A, whereupon the analogy is complete.Thesameremarksapplytothevolumesof3Dfigures.

Themostgeneraltheoryofentitiesofsuchatype,whichcontainsasspecialcasesthetheoryofvolumeandarea,isusuallycalledmeasuretheory.

Itremainsonlytonoticethatinthetheoryofprobability,incomparisonwiththegeneraltheoryofmeasureorinparticularwiththetheoryofareaandvolume,thereisacertainspecialfeature:A

probabilityisnevergreaterthanone.ThismaximalprobabilityholdsforanecessaryeventU:P(U)=1

Thesymmetryo≤1≥∞isthenafundamentalgraphofthefifthdimension.LetusseehowwithasimpleexampleofDisomorphismsbetweencertainpointsoftheunitcircleandthe1-∞complexplane:

Mirrorsymmetriesbetweena0’-1universeanda1-∞areinterestingastheysettwodifferent'limits',anupperuncertainboundforthe1-∞universe,inwhichthe1-world,∆ºexists,andaloweruncertainboundforthe0’-1Universe,wherethe1doesnotseethelimitofitslowerbound.Arethoseunboundedlimitstrulyinfinite?

Wecanconsideritintermsofthehomologyofboththemicroscopicandmacroscopicworlds.

Ofcoursetheaxiomaticmethod'believes'ininfinity-wedealwiththeabsurditiesofCantoriantransinfinitiesinarticlesonnumbers.Butasweconsidermaths,afterLobachevski,GödelandEinstein,anexperimentalscience;we are more interested in the homologies of ∆±1. For one thing. While 0 can be approached by infiniteinfinitesimal 'decimals', so it seems it can never be reached, weknow since the 'violet catastrophe' that theinfinitesimalisa'quanta',a'minimum',a'limit'.AndsowereturntoLeibniz'srightfulconceptofan1/nminimalpartof'n',thewhole'1'.

Thisimpliesbysymmetrythatontheupperbound,theworld-universeinwhichthe1isinscribedwillhavealsoalimit, a discontinuitywith∆+2,which sets upallinfinities in theupper boundalsoas finite quanta, 'wholes ofwholes'.

OneofthemostimportantS=tsymmetriesofthemathematicalUniverseistheonebetweentimeprobabilitiesin the 0=1 unit circle and the 1-∞ plane of statistical populations (space-points), as it is both,a symmetrybetween the ∆-1 scale of finitesimals (unit circle, where a finitesimal, 1/n is defined as the inverse of a realnumberofthe1-∞plane);andonebetweentime-cyclesandspace-planes(topologicalsymmetry).

TheconjunctionofEuclideangeometrywitha focus in thec@rtesiangraphworld represents thebestway toconsider ∆-Planes, as themost beautiful of those scalingsis alreadyincorporated on it, giving us the key '∆-symmetry,betweenthe0’-1unitcircleandthe1-∞larger∆+1scale,whichhaveanaturalcorrespondence-asthemathsthatworkinthe0’-1unitcircleworkonthe1-∞scale,with3differences:

1. Theunitcirclehasadefinedmembrane,butitsminimalistinfinitesimal'decimal'isnotdefined,inversely,the1-∞scalehastheminimalinfinitesimal1definedbutthe∞elementisnotsothosearetheelementsforabestchoice of 'mental geometry' to study a problem depending on which element we do know, the 'singularity 0’-undefined,1-defined'andthemembrane,1-closed,∞-open.

2. Aunitcircleisað-cyclical/polargeometry,the1-∞scaleisanopenunconstrainedone,anotherchoiceforsolutionofproblemsdependingonitscharacteristics.

3.Aunitcircleis∆-1.The1-∞scaleis∆+1,withthe1-membrainastheopenorclosedborderamongthem.

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TheUnity circlewhere1 is∞and0, theunreachable 'quanta'of the∆-1 scale is thebestmirror symmetry between a self-centered 'polar' mind and its universe (1-∞ region). Inmaths in fact most processes that matter can be in geometry proved by a bidimensionalinformative (holographic principle) graph, and most key processes of space-time worldcycles, can be observed to happen in the 4 first digit, around the three key number(e+e/10=3,π-aperturesofthecycle=3andsoon).

Whenweextendthedomainoftheunitcircletothecomplexplanewithitsbidimensionalholographicnumbers,theconceptofa'world'representednowbythehigherdimensionalityoftheRiemannsphere(unitcircleofthecomplexplane)isclearer,aseverypointofthespherecommunicateswithonepointoftheComplexplane,whilethe 0 point of the sphere becomes the inverse of the ∞ point of the 'complex plane'. In this manner as inprojective geometry, the o-mind point becomes the mind-mirror of the ∞ real Universe: O-point mind x ∞Universe=constantworld.

Leibniz’smonads,whichdonotcommunicate,arethesimplestfractalpoints,asminds=mirrorsoftheUniverse,where'eachpointisaworldinitself.'.

Inmathematical number theory,we say that: The interval o-1 is the same ∆-1 infinity that the interval 1-∞.Cartesian points in that sense were for the pioneers of science, more than a mathematical artifact, but themathematicalmindinitselflookingatthetime-spaceUniverse.

Inthemathematicalsectionswestudythatreflectionmind-languagemirror.Andthemanyhomologiesbetweennumbersandspace-timeevolution.Torenormalizeabeingfrom1to∞reducingittoafunctioninthe0’-1rangein that sense mimics the fact that the whole in each scale of diminishing size holds the same quantity ofinformation,andindeed,the0’-1and1-∞limitmakesenseasthemindo-1representationofthewholeworldoutsideofhim1-∞

Now,thereare2other'spaces'besidesthe3topologicalspaces(Sp-cylindrical,ST-Cartesian,ðƒ-polar)worthtonotice,toexplainallthefractalspace-timecomplexworld.

Letusrememberafundamentalprincipleof∆sttheory,pentalogic,whichIalsocalltheRashomonEffect–theneedfor5¬∆@stperspectivestounderstandtheUniverse:

'Everysystemthatexistsin5D²space-timehas3±∆Disomorphicfunctions=forms.'

Toexistyoumusthaveafunction-formconnectedtoeachofthe3present'space-timedimensions'explainedinthepresentfractalgenerator:Γ:§t<ST>§ð;andthetwoscalardimensions,∆±ioftheUniverse.

In praxis we can often reduce the 5 roles of the being to theΓº∆±1 dual roles which will give us the bareminimumfunction and formof the system. Sowhen the bestway to start the description of any system is toconsideritsfunctionsinpresentternaryspace-timeandinscalar4D/5Dinverseentropy/socialspace-time.

Andsowemustfindforcomplexnumberstwofundamentalrolesas'numbersable'todescribethe3statesofthegenerator,$T<ST>§ðandthe∆'polynomial'or∫∂functionsbetweenplanesofexistence.

The thirdgreat fieldofnumbers theory,probabilityand statistics; focuses in theanalysisofnature's space-timeeventsfrombothpointsofview,thespatial,statisticalinthe1-∞graphandthetemporal,probabilisticinthe0’-1sphere.Sowe find2parallel subdisciplines,S-tatisticsandT-probabilities,asanS∆º≈T∆-1mirror, similar to theone@nalytic geometry performs between topological and ¬Algebraic solutions, Those dualities are of specialinterest,astheybringbothelementsofS=Trealityandallowtoobservetheirdifferencesbyconsideringinwhattheydiffer,evenifthefundamentaltheoremofmoderntheoryofmeasureisthatforeach'theoremofprobabilityin theo-1 cycle, it correspondsanequal theoremonpopulationmeasure in the1-∞plane', givenweight to theconceptofafractalUniversewhichperformsthesameeventsandformsinallits∆-planes.

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Bridgingthe5Dtheoryofprobability-statisticsandthepresentstateofthesubject.

Ultimatelywhataprobabilitysphereshowsishowpalingenesisofarepetitivesumofpartialevents/cells/beingsintimeaccumulatestowardsthe 'project'ofcompletinganemerging∆+1whole, fromits 'pieces,bitsandbites',which firstmustbe reproduced inenoughnumbers (statisticalaccuracy requiresanN->∞frequencyofevents),andthenmustcollapsealongthelawsofaggregation,whicharedescribedinprobabilityandmeasuretheory.

So the a priori condition for the lawsof probability tohappen is amassivereproductionof events, the a prioriconditionfor a distribution of populations to become exact in the standard bell curve and itsdeviations is amassivereproductionofpopulations.Onlythentheorganicgameofexistencebecomesefficientandexact,whilewhenwedealwithminimalistnumbersofsocialevents/populationsthestructuresareinefficient,morefree,lessdeterministic. Itsmathematicalexpression thenconsidersthe∆-1planethatofprobabilities,whichsurface intothepopulation1-∞planeonlywhentheeventhas'happened'withprobability1andhencebecomesa1-unitofthepopulationplane.

5DPROBABILITYV.STATISTICS:RANDOMV.CONDITIONALEVENTS

Fromtherepresentationofprobabilityasthestandardvalueofthefrequencyƒ=m/n,ofaneventA(m),wheremaretheoccurrencesofA,n,thetrials,0≤m≤n,andthus0≤ƒ≤1,followsthattheprobabilityP(A)ofanyeventAmustbeassumedtoliebetweenzerothandone:

1. 0≤P(A)≤1

Howeverifweconsiderthelawoflargenumbersandclassifyeacheventasanundistinguishable,finitesimal0’,since0doesnotexist,aminimaluncertaintymighthappen,aslongaswearewithinthelimitsofpossibility–thatis,theeventAispossible,orelsewewouldnotbothertocalculateaprobability.Soasn->∝ ;0’becomes.

The first distinction thus between 5D probability and classic probability is that neither 0 nor 1 have absoluteprobabilities.Since0’doesexist.Unlessisanimpossibility;thatisafalsehoodwhichdoesnotbelongtotherealmof mathematics but of logic, a language in which the Algebra of truth and false makes sense. The error ofprobabilitiesthatinclude0aswellassettheoriesistoconsiderthatfalsehoodisequivalenttozeroth.Butwetalkofadifferentlanguage.Falsehoodsdonotenterthefunctionofexist¡encewhichiswhatnumbersstudythroughsocialscales.

However1,thewholemightexist,whichsetsacuriousasymmetry,anessentialpropertyoftheUniversethatwewillfindthenintheinverse1≤x<∞,onthe2ndworldcycleofexist¡encewheretheasymmetryisreversedtothatofthe0’-1.As∞iswhatisuncertainanddoesnotexist.

Sowesetaswedidinnumbertheoryanewlawofprobability.Sowerewritetheclassiclaw,as0’:

1.0<P(A)≤1;orusingthesymbolofafinitesimal0’≤P(A)≤1

In all themes related to0’ practical purposesmake ‘acceptable’ theeliminationof the residual ‘finitesimal’, asclassicmathematicsdoeswiththeidealinfinitesimal.Soinpraxiswemightsayclassicmathematicscorrespondsto¬Æmathematicswiththesimplificationofdischarging0’.Whythentobringitintotheplay?Becauseofboth,philosophical reasons and experimental sciences, such as mathematical physics that enters into paradoxes(singularities,infinities,lackof0temperatures,0motion,h-uncertainities)solvedwiththeunderstandingof0’.

Itdoesalsoimplyadiscreteuniverse,wherecontinuityissatisfiedwhenthechangeonYisproportionaltoh≈0’.Againthetruthisdiscretenature,themindanditslanguagesthoughsimplifydischarging0’andmakingitinto0.

Intuitively,acontinuousrandomvariableistheone,whichcantakeacontinuousrangeofvalues—asopposedtoadiscretedistribution,where thesetofpossiblevalues for the randomvariable isatmostcountable.While foradiscretedistributionaneventwithprobabilityzerothis impossible(e.g.rolling3½onastandarddie is impossible,andhasprobabilityzeroth),thisisnotsointhecaseofacontinuousrandomvariable.Forexample,ifonemeasures

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the width of an oak leaf, the result of 3½cm is possible, however it has probability zeroth because there areuncountable many other potential values even between 3cm and 4cm. Each of these individual outcomes hasprobability zeroth, yet theprobability that theoutcomewill fall into theinterval(3 cm,4 cm)isnon zeroth.ThisapparentparadoxisresolvedbythefactthattheprobabilitythatXattainssomevaluewithinaninfiniteset,suchasan interval,cannot be found by naively addingthe probabilities for individual values. Formally, each value hasaninfinitesimallysmallprobability,whichstatisticallyisequivalenttozeroth.

Formally, ifXis a continuous random variable, then it has aprobability density functionƒ(x), and therefore itsprobability of falling into a given interval, say[a,b]is given by theintegral:

Inparticular, theprobability forXto takeany singlevaluea(that isa≤X≤a) is zeroth,becauseanintegralwithcoincidingupperandlowerlimitsin5Disalwaysequalto0’–afinitesimal.

Howthisworksout is simple.Unlike in classicmathematics5Dmathematicsdohaveentropic limits,where thedomainofthefunctionisdefined,outsideofwhichitdoesnotmatter.I.e.theprobabilitythatIamGodiszeroth,butthatdoesnotmatter.Itisameaninglesspropositionwithinthe‘logic,rational’natureofthecosmos.Evensowe could ascribe a probability of belief if we forget rationality, and then itmight be as large as 1 if I found areligion,whichcertainlywouldhavemoreknowledgeonthe ‘thoughtsofGod’ thanthemusingsofG.Bush, theGodbushcreepinginthedesertthatanassbreederoftheHebrewsawinthebronzeageandinterpretedastheword of god. And since all is ultimately mental space, why not? This kind of scholastic argument shows theabsurdity of breaking beyond reason, beyond reasonable domains, and beyond what matters to us – theprobabilityofcertaineventiscountedonlywhenitismeaningfultotheevent.Andsoeveninameandistributionofrandommeasuresaroundacertaindeterminedvalue–howeverignoredbytheobserver,thetailsofsigmagotoinfiniteinfinitesimals(0’∝).

It follows immediately by themean distribution, which is by the S=T parallelism also themean distribution ofpopulations, that the concept of an ‘error’ in measure and an ‘error’ in the repetition of information (cloneindividuals of an species that will vary in size etc.) that while the futurewants to be determined, exact, in itsreproductionofan infinitepresent, itdoescommitalwayserrors, in thereproductionof its finitesimalparts sothatdeathistheunavoidablefactofanaccumulationoferrors;ifweconsidertheeverlastingequalityofallformsastheNatureof immortality(indistinguishableinfiniteentropy,whichthenormaldistributiondoesmaximize). Italsofollows inverselythatreproductionofpresentwillalwaysbethwartedbyaminimalsigmarelativepastandfutureofthatproperty;thatis,its±variationsinpopulation(statisticalmirrorinspaceofdistributionintime).

The same concept applies to conditional probability, when we consider the simplest sequential monologicAristoteliancausalityofthesimplestlevelofeventsinreality,thechainsofdimotionofspace-time,asaseriesofactionsofperception,locomotion,energychasingandfeedingfollowedbyrepetitionofour∆-1finitesimalpartstoevolvesociallynewtissuesthatwavewornout.

AstheconditionalprobabilityoftheeventhappeningrequiresaseriesofconditionalstepsI(rightperception)->A(right motion) -> E (right feeding) -> O (right organic reproduction) -> U (right universals evolution), and theconditionalprobabilityofU≤O≤E≤A≤I, it also follows thatNatureextinguishespopulationswhichdoNOTevolvesocially; that is errors in IAEO, do not achieve U; do not reproduce, that is errors in IAE not achieve O, andextinguishsystemsthatdonotfeed;thatis,commiterrorsinIA…

Realitytendstofavorentropicerrors,entropicdistributions.SohowtheUniverseachieveitssurvival?Simplybyincreasing enormously O for those who achieve ∆+1 U; social evolution of parts into wholes, which then canreproduce in∆-1a relative∝numberof seeds.And thusultimately thepossiblevariationsof realityhavebeen

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limitedbythelawsofarealistprobabilisticmethodwhereprobabilityis0<0’≤P<1<1’,whereas1’isNOTonly1,thewholebutthemindabovethewholethatmakesitalltogether.Thatis1,isnot9,3x3parts,but10,3x3+1…

Whatabout theother ‘chains’ofmorecomplexentanglement?Here the lawsofprobabilitydochangebecauseunlikeclassicalgebra,in¬Ælgebra(existentialalgebra),conditionalprobabilityreverses,suchastheprobabilityofaneventtohappenwillincreasewiththeconditionaleventshappeninginsimultaneity.Soconditionalprobabilityincreases beyond 1 the happening of an event, to a discontinuous set of repetitive stœps of events that bydifferentmeansthatmultiplereproductionincreasestheeventsofprobabilitybeyondone.

Inthatregard,5Dprobabilitybridgesbeyondthesingleevent’schancestoconsidertheanalysisonhowthe0’-1finitesimalscalecrossesthe1limittoenterintothestatisticalplaneofgrowingpopulations.

Thesinglepast,thevitalpresent,thepotentialfuture,andtheroots…ofequations.

Let’s start those comments on probability with a theme hardly understood: The difference between the pastwhich isalwaysoneand the futurewhich ispotentiallyan infiniteentropy field,whichwill reduce toone,as it‘passesthroughthepresent’,duetothedeterministicnatureoftheactionsofsurvivalofthe‘actors’ofpresent.

Indealingwithrootsmathematicianshavebeenforverylongatlossandhavenotunderstooditskeyelement:thefactthatgoingdownwardsfromanumberthroughitsroot,we‘split’andgrowthenumberof‘potentialfutures’.Soweneedto introducesomeconceptsonthesinglepast, actionpresentandpotentialfuturesthatstructuretheconstantflowoftimesintheUniverse.

Inessencethefutureistimeentropyandassuchhasseveralpotentialpaths.Whilethepast isasingleoneandhence it precludes that the future will be highly deterministic. And this determinism is performed by thesequentialprogramofsurvivalofalltheactorsofaneventthattrytomaximizeitsfunctionofexistence.Sothefutureseemstobequitedeterministicwithaproviso:iftheactionsofsurvivalofthepresentfail,thefuturewillnot be,will be ‘negative’, a lack of future for oneof the contenders. So in thepresent theprobabilistic futurealreadyhintsat;butas the individual ismodularandcanchoosebetweendifferentactionsandnoactions, thefutureprobabilisticpotentialvirtualdualpathsofbeingandnotbeing,alreadymanifest inthepresentofactingandnotacting.

Andsothefuturehastwopotentialoutcomes,onepositiveandonenegative,exactlyasthe2rootsofanumber.Speciallywhentheyaresolutionstoaneventintimethatistakenasthefuturelogicpathoftheequation,whereitwillsettledowntoapointthatwillbecomethepastandnolonger‘move’.

Thisofcoursebringsanecessarydistinctionbetweennegativerootsolutionstospatialproblemsofpopulationswhichsimplydonotexistandmustberejected,and±solutionstoaproblemofaneventintimewhicharevalidastheyshowthe2potentialpaths,butmightnothappenbothtogetherandsoweneedfurtherinformationtosolvetherealoutcomeandfinallytosplitsolutionsinwhichthe±valuesrefertoa‘perpendicularsplit’ofthepartsofthe being represented by X2, which is now broken in the S and T elements one of which due to its inversepropertiesisconsideredthenegativesolution.

Itisnotthatsimpleandabstract,afactthatthosewhousemathematicstoassessrealproblemsshould,becausethey need to discern what are the proper solutions of the 3 possible outcomes of a root: we discharge thenegativesolutionforspatialpopulation;weanalyzetheoutcomeas–entropicor+informativeforeventsandweacceptbothwhenthesolutionisthesplitofanSTsystem.

Thissimpleconceptappliesthentothecubicrootwheneveritisrelevant,aswecanalsotalkofa‘ternarysystem’,sT<ST>St,andsoaternarysplitispossibleandintheveryfewcaseinwhichaquarticisrelevantofadualsplitofSTandST,beyondwhichtherearenosolutionsbyradicalstoquintics(Abel-Ruffini),afactworthtomentionasthere is really nomore positive dimotions (the fifth being entropic dissolutionwith explosion back to the ∆-1,whicherasestheinformationoftheplaneofexistenceweweredealingwith).

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Theindifferenceforentropicfuturesandthosewhodon’tchange.

Animportantconsequenceofthatstructureofsinglepast,vitalpresentandpotentialfutures,isthatthefuturesthatmatterarethosewhosolutionsare‘meaningful’tothesurvivalofasystem.Thismeanstherearemorethan2‘rootsofthefuture’asentropyispotentiallyinfinite,foreventsthatdonotmatter.I.e.wecanthrow2dicesandget36differentresultsbutforthe‘dices’–theactorsoftheevent,theyareindifferenteventsthatdoNOTchangetheirstructure.Sotheatomsofthedicesanditswholegeometrywon’tchange.Andyetifthedicesareconnectedtoaplayerintheentangleduniverse,whohasrisked1000$tomake36thousandtopayadebt,orelsehewillbekilled, it is a life vs. death=entropy event. And as such is a dual eventNOTone about the dices but about theprobabilityhewillliveordie.Andthatmakesita‘conditionalprobability’.

We must then consider memoriless Markowian processes where entropy makes indifferent the result andconditionalevents;butourinterestarenotintheBayesianrulesofthatcalculusortheKolmogorovformalisminthepedanticlanguageofsets,butonthestructureofthe3agesoftimeregardingitsprobabilitiesasfollows:

Becausethepresentis‘conscious’andvitalguidedbyactorswhoseektomaximizeitsfunctionofexistence,theprobabilityof ‘survival’ isalways larger. I.e inthepreviouscase, likelytheplayerwon’triskhis lifebuttryothermeans to enhance the survivalwith theMafiosi. In hunting processes prey escapemore often that onewouldthink,becausethestakesforthemarehigher.Inplanetaryorbits,massfallsfarlessofteninastraightlinetothestarbutturnsaroundfeedingonthegravitationalvortexwiththeminimalrateoffallingtoelongatethelifeoftheplanet.Andsoon.

And so theprevious graph inwhich adistribution tends always to the S=Tpoint of balancemerely notices theobvious fact that S=T maximizes survival and it is always the maximal probability of an event, the middlereproductiveclassicageofabeing.Andsoon.

Determinismthenisalsoontheeyesoftheobservertoseeornottheerror,andconvertthepreviousgraphfromthe‘perspectiveofaslowtimethinkingmind’henceabletoseemultiple‘trials’intoa‘DeltaDiracfunction’sotospeak,withathindeterminisminthecenter.

RECAP.

Numbertheoryachievesacomplexevolutionwith itssymmetricanalysisofnumbersas time-likeprobabilitiesvs.numbersasspace-likestatistics.

Ofthemanywaystoexpressthiswecansaythattimeprobabilitiesinthedomainof0’-1isequivalenttospacestatisticsinthedomainof1-11¹¹thenextscaleoftheUniverse.

Thiscorrespondencecanbeusesasageneralexampletothatofthemindmirror,0’-1observingtheworld1-∞inwhich ishosted,andmanyother symmetriesof scale.Wecan seea superorganismas thememorial scaleprintingofthose11¹¹citizen-cellsofthesuperorganism.

Soforallpurposesagoodapproximationwouldconsiderthetotalnumbersof∆-1elementsofthewholesuperorganismasthe∞limitofthe1-∞sphere,normallyatthe11ˆ11level.

In thismanner the symmetriesbetweenmotionand slow,whole stillness,mindandentropy, virtual and realdevelopthemselvesinthekaleidoscopeofperfectmirrorsofmirrorsofworlds,mirrorsofuniverses...

So we shall close our brief view of number theory with some basic notions on that symmetry, essential tomodern maths, specially with the development of mathematical computers and its capacity to do recurrentsequencesofquantitativemethods.

Letusthenfirstdefinethepentalogicofprobabilitiesandstatisticsinitsminimalduality:

Γst:T:probabilities=S-populationisthe¬Algebraicduality.

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∆±1: Time probabilities happen as they have been formulated in the sphere 0’-1 micro universe to spacestatisticsinthe∆+1,1-∞self-similarUniversetowhichtheo-1mindworldreduces.

Differentialequationsasexpressionofmodernprobability.Deterministicv.RandomMarkowianprocesses

Wesaidthatthemodernversionofprobabilityarederivativesthatextractasingleeventandgatheritalltogetherintoawholewithanintegral,withthevirtueofbeingmorespecificforeachtypeofDimotionofspacetime.Andforthatreasonsgivingusmoreinformationonthecausalreasonsofevents,whichprobabilitylacksbymerely

connectingthepartsandthewholewithoutinvestigationontheconnectiveelementsbetweenboth.

Letusconsidertheclassicexampleofanalysisoflocomotions.

Theprincipleofcausalrelationamongphenomenafindsitssimplestmathematicalexpressioninthestudyofphysicalprocessesbymeansofdifferentialequations,towhichweshalldedicateanentirechapterof‘time-maths’.Soletusjustintroduceacoupleofconceptsneededtoconsidertheconditionalprobabilityofthe

existentialchainsofactionsthatdeterminethesuccessofagiven‘functionofexist¡ence’.Letthestateofthesystemunderstudybedefinedattheinstantoftimetbynparameters:x1,x2....xn

Theratesofchangeoftheseparametersareexpressedbytheirderivativeswithrespecttotime:

Ifitisassumedthattheseratesarefunctionsofthevaluesoftheparameters,thenwegetasystemofdifferential

equations: Thegreaterpartofthelawsofnaturediscoveredatthetimeofthebirthofmathematicalphysics,beginningwithGalileo’slawforfallingbodies,areexpressedinjustsuchamanner.Galileocouldnotexpresshisdiscoveryinthisstandardform,sinceinhistimethecorrespondingmathematicalconceptshadnotyetbeendeveloped,andthis

wasfirstdonebyNewton.Inmechanicsandinanyotherfieldsofphysics,itiscustomarytoexpresstheselawsbydifferentialequationsof

thesecondorder.

Given the values z0 andυ0 at the initial instant t0, the valuesof z andυ for all further instants t are computeduniquely,uptothetimethatthefallingbodyhitsthesurfaceoftheearth.

Theproponentsofmechanisticmaterialismassumedthatsuchaformulationisanexactanddirectexpressionofthedeterministiccharacteroftheactualphenomena,ofthephysicalprincipleofcausation.AccordingtoLaplace,the state of theworld at a given instant is defined by an infinite number of parameters, subject to an infinitenumber of differential equations. If some “universalmind” couldwrite down all these equations and integratethem,itcouldthenpredictwithcompleteexactness,accordingtoLaplace,theentireevolutionof’theworldintheinfinitefuture.Thisistruthbutbecausethenumberofelementsthatareinfluencingeachotherisenormous,thehumindcannotaccessthemallandsoabsolutetruthisnotaccessibleexceptforthebeginningandendpointsofSS(absoluterestrespecttotheperceiverandabsoluteentropicdeath).

Itisnecessarytounderstandthisdistinctionbetweenthemindthatperceivesandtheeventitperceives:

‘Onlytheeventcarriesalltheinformationaboutitselfinthepresenttimeinwhichithappens’.

Thismeansanexternalobservercannotcalculatewithabsolutecertaintywhatishappeningbeyondthose2pointsofrelativeSS-TTorviceversa,TT-impulse-SS-rest,whichare in fact thepointscalculatedas ‘Cauchyconditions’,etcfordifferentialequations.Sinceevenifhewerecalculatingspace=positionswithexactitudeitwouldtakeittoo

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longintime=motionandwhenalltheinformationweregathereredthepositionwouldhavechange.Thisisalsoattheheartoftheuncertaintyprinciple,anditsultimatecause isthatmotionNOTformisthesubstanceofrealityandDISCONTINUITY of infinite fractal pointswithWILL to act not a single ‘God-will’ in aworld of ‘mechanical’mindlesssystemsisatplay,andfinallyfractalpointsmoveinstopandstep,particle-wavestateswhichallowthemto‘zig-zag’.Soknowledgeisalwayssynthetic,limitedtocertainparametersandequivalencerelationshipsontheleveloftheobserver,whichthenegocytransformindeterministictruthwhentheyarereductionistones. I.e.wecannotknowthemovementofalltheparticlesofacanonicalensemblebutextractitstemperature,whichisNOTaparameterofthescaleofthemolecules–speedis–butofour‘simplifying’scale.

Itishowevereasiertocalculatefromsmallerpartswhoarebyvirtueof5Dmetric,SxT,fasterininformationandnumbers,theslowersyntheticbehaviorofthewhole,whichinfluencesthem.Moreover,thewholelikesthesmallerparts toknowthathis ‘big fieldbehavior’– changesoncollectiveenergy=heatand topological formparametersmustbeobeyed,anddoesn’thideinformation.Forthatreasonsyntheticorganictruthsareeasiertoknow.Andwecanguessthefutureintheorganiclevelofnetworksthatequalizeparametersofthesmallerparts.Wecanbemorepreciseinmeasuresof‘largerwholes’thansmallerquanta.

Thus in fact this quantitative mathematical infinity is extremely coarse in comparison with the qualitativelyinexhaustible characterof the realworld.Neither the introductionof an infinitenumberofparametersnor thedescriptionofthestateofcontinuousmediabyfunctionsofapointinspaceisadequatetorepresenttheinfinitecomplexityofactualevents.

Allthismeansthatthestudyofactualeventsdoesnotalwaysproceedinthedirectionofincreasingthenumberofparametersintroducedintotheproblem;ingeneral,itisfarfromexpedienttocomplicatetheω,whichdescribestheseparate“statesof the“system” inourmathematicalscheme.Theartof the investigationconsistsrather infindingaverysimplespaceΩ(i.e.,asetofvaluesofωorinotherwords,ofdifferentpossiblestatesofthesystem),such that ifwe replace theactualprocessbyvarying thepointω inadeterminatewayover this space,wecanincludealltheessentialaspectsoftheactualprocess.

Butiffromanactualprocessweabstractitsessentialaspects,weareleftwithacertainresidue,whichwemustconsider to be random. Theneglected random factors always exercise a certain influenceon the course of theprocess.Very fewof thephenomena thatadmitmathematical investigation fail,when theory is comparedwithobservation,toshowtheinfluenceofignoredrandomfactors.

This is the state of affairs in the theory of planetarymotion under the force of gravity: The distance betweenplanets is so large in comparisonwith their size that the idealized representationof themasmaterial points isalmost perfectly satisfactory; the space inwhich they aremoving is filledwith such dispersedmaterial that itsresistancetotheirmotionisvanishinglysmallthemassesoftheplanetsaresolargethatthepressureoflightplaysalmostnoroleintheirmotions.Theseexceptionalcircumstancesexplainthefactthatthemathematicalsolutionforthemotionofasystemofnmaterialpoints,whose“states”aredescribedby6nparameterswhichtake intoaccountonlytheforceofgravity,agreessoastonishinglywellwithobservationofthemotionoftheplanets.

But as soon as we introduce those parameters or put a ‘third body’ element the outcome while it will bedeterministic in the sense it will not split into many worlds events, not even in those cases it seems to do it(quantum paradox) since the electron merely ‘dissolved’ from electron-particle state, to wave of light bosons-densityduringitsmotionbutwillcollapseandhittargetasasingleelectroninitsendSS-tate.

Sinceaswesaidtheprecissionofinformationismaximalonlyinthestillstateormind-mapping,andinthatcaseisachievedthroughselectionofformanditsreductiontomappingthatcanbemoredenseininformation,orintheendstateofentropy,achievedbydissolutionofthewholeanditsdetailsintoanentropic‘heat’,scatteredstate.

I.e. In the flight of an artillery shell under gravity and resistance of the air the perturbing random factors aresignificantlylargerandthescatteringoftheshells,i.e.,theirdeviationfromthetheoreticaltrajectoryreachestens

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ofmeters, or for long ranges even hundreds ofmeters, as the deterministic networks of the state of particle-organismhavebeenlostandsoexternalnon-measuredelementsactuponeachbrokenpart.Thesedeviationsarecausedpartlybyrandomdeviations inthe initialdirectionandvelocity,partlybyrandomdeviations inthemassandthecoefficientofresistanceoftheshell,andpartlybygustsandotherirregularitiesinthewindandtheotherrandomfactorsgoverningtheextraordinarilycomplicatedandchangingconditionsintheactualatmosphereoftheearth. It is then when the causality and determinism is lost when we can use the methods of the theory ofprobability,whichonlyconsidertheoriginandendwithoutthecausalelementsinbetweenandmeasuretheendbygeneralmoreextendedproperties-i.e.theareaofthescatteringofshellsessentialforthepracticeofgunnery.Sowestudyrandomevents,whentherandom“residue”foragivenformulationofaphenomenonprovestobesolargethatitcannotbeneglected,thenapossiblewaytoproceedistodescribethephenomenonmoreaccuratelyNOTbyintroducingnewparametersandtomakeamoredetailedstudybythesamemethodasbeforewhichwillrunintheextremeamountofdata,onlyabletobereducedbytheuseofafaster,moreefficientmentalsystemofsmallerSxT=Celements (chipparadoxandgrowinguseof computers to calculateonbigdata,our larger slowerminds–withtheobviousfutureconsequenceforhumanfreedom,aswebecome‘managed’bycomputersthewaywe‘manage’plantsofslowerchemicalminds).

Insteadifwedon’tcheatthescaleanduse‘superbrains’atfasterratesofprocessinginformationtheonlymethodavailabletohumindsistodeterminetheinfluenceoftheseresidual“random”factorsforalongintervaloftimeorforalargenumberofrepetitionsoftheprocessunderstudywithprobabilisticandstatisticalS=Tmethods,averagelawscalledstatisticallaws;startedinthekinetictheoryofentropicgases,whichshowshowthejointinfluenceofrandomcollisionsofmoleculesgivesrisetoexactlawsgoverningthepressureofagasonthewall,thediffusionofone gas through another, and so forth – that is,weobtain larger, less detailedparameters onour∆ºplanebymaking¡ndifferentthevaluesandconsequencesofthe∆-1scale,whichisinfactwhatcomputerbigdatasystemsare doingwith the growing ‘herding’ of humans as anonymousbeingswith ‘credit’ ratings for the physiologicalblood-reproductive-economicnetwork,etc.

RandomnessofMarkowianprocessesinspacevs.CausalityofdeeptimePlanes.

Butagain itmustbeunderstoodthosemethodsareasubstitutetothecausalunderstandingofnetworksandorganismswhere information ismorespecificanddeterminismhigher,and there isa languagespecific to thenetwork to be able to understand it inmoredetail. I.e. thebiological hormonal languages to understand thespecificsofacells’actionsandbehavior,theanalysisofthefinancialsystemtounderstandhumanscontrolledbyflows of money in a capitalist system (even if here this control of society by bankers is censored and soeconomists pretend it is all random, better studied with statistics). The same goes for the preference of adifferentialequationoverastatisticalmethodtostudyspecificdimotionstargetedbyspecificfunctions.

Sothekeyfundamentalelementtouseprobabilityandstatistics,ashumansconceiveit,isthestudyofchaotic,entropic randomness,which implies formost cases study to study an ensemble of indifferent entitieswith anentropicfuture,ofdemocraticensembleswithequalopportunitiesforallthe‘parts’inexistenceofthesystem.

Yet even random phenomena becomes ordered. Why? Because even in the most 'memoriless' process, thechancesof futurewill be reduced toaminimal setof 'dual and ternary'processesofbranching, and theonlycharacterofpurerandomnessisthatallofthemwillhaveasimilarrelativechance.I.e.adice,hasthreespatialdimensions(3doublefaces),whichwecanconsidereachtobethe±dualinversedirection(oppositeones).Soweare ina3x2system.Andalldirectionsareofequalvalue,becausetheyare 'spatialorientations'withoutany'topologicalbias'(i.e.not$T,ST,or§ðdifferent'faces',whichcouldtrickthedice,asforexample,aregionwith'heavier§ð-mass',an§ðcentreofgravity,displaced,andsoon).

So in principle,randomphenomenahave a predictable nature shown in the regularity of the 'grandnumberslaw', and this brings us also a reflection on 'time events', since as they repeat, theyalways tend to closeaconservativeenergy-zerothsumworldcycle.Whichincreasetheprobabilityanddeterminismofaprocess.As

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onthelongtermallworldcyclesbecomeclosed,andtheUniversebecomeinitsrelativeinfinitedurationalwaysabalancedworldcycle,withallthedimensionsofscale,topologyandtime,comingtoa0’-sum.

Thus thoseare the insightsofGST in themysteriesof 'probability: zeroth sumworld cycles imply the lawofgrandnumbers (regularityandequalprobability)as longas theprocess is 'spatial' (memoriless, simultaneous,withnopreferreddirection,etc.) Soas ingeometrywe say that from 'long spatialdistances' all linesbecomegeodesics (aswhenyoucomeout inscaleof the flatearth), in timewesaythatallprocesses findequilibriumamongallthepossibleeventsofthesystem.

Classiclawsofprobability

ST:Thusasamathematicalfoundationforstatistics,probabilitytheoryisessentialtomanyhumanactivitiesthatinvolve quantitative analysis of large sets of data. Probability theory is thusconcerned with the analysis ofrandomphenomena:randomvariables,stochasticprocesses,andevents.

Ƽ:FurtheronMethodsofprobability theoryalsoapply todescriptionsofcomplexsystemsgivenonlypartialknowledgeoftheirstate,asinstatisticalmechanics.Hereweobservetheessentialdualitybetweenthehumanobserver,which introduces uncertainty and a possible certain reality;which humans often confuse (quantumparadoxes).Butandthiswillbethedifference,theresultsofhumanuncertaintycanbe'clarified'andconsideredcertain, if theoutcome isnotpurely random,with theusualbell curvesetc. properoftrue stochastic systems.Thenwemustconcludethattheonlyrandomnessisthatofthehumanobserver.

Hence, the great discovery of twentieth century physics, which was the probabilistic nature of physicalphenomena at atomic Planes, described in quantummechanics can be judged to be 'an error of the humanobserver', in asmuch as its results are clearly 'quantized', fairly determined, NOT purely statistical. (And thesolutiontothisconundrumisthewave-pilotdeterministictheory).

Thedifferentcausalitiesthatgivebirthtoprobabilities.

Itisnotpossibletopredictpreciselyresultsofrandomeventsthoughwehaveanticipatedandproved,properlyinterpretedthatthecurveofdistributionfavorsthepresentstateof1sigma,with2/3rdsandmakesparalleltheinversephasesof 1st, third age, emergenceanddeath, and its symmetries in space (dominanceof thebody-mass,overtheheadandlimbs).Thoughheretheanalysiswouldrequireamuchmoredetailedstudy,nottobedoneforthetimebeings.

Instead,wewouldliketoconsidertheessentiallawsofprobabilitiesandstatisticsin'classicscience'.

Whenasequenceofindividualevents,suchasthementionedrollofdice,isinfluencedbyotherfactors,suchasanimbalanceofthegravitycater,itwillexhibitdistortedpatterns,fromasymmetricspatialdistribution,whichcanbealsopredicted.

Tworepresentativemathematicalresultsdescribingsuchpatternsarethelawoflargenumbersandthecentrallimittheorem.

Bothareessentiallythesame,thoughthecentrallimitmightbeconsideredacompositeƤ,resultwhereeach'distribution'isapartialsumofevents,andthelawoflargenumbersisdefinedforsingleevents/cells/forms:

Inprobability theory, the lawof largenumbers (LLN) isa theoremthatdescribes the resultofperforming thesameexperimentalargenumberoftimes.Accordingtothelaw,theaverageoftheresultsobtainedfromalargenumber of trials should be close to the expected value, and will tend to become closer as more trials areperformed.

TheLLNisimportantbecauseit"guarantees"stablelong-termresultsfortheaveragesofsomerandomevents,such as ∆-1>∆, whatever this means. And so goes for the central theorem, with an intermediate §cale ofpartitions…∆-1>...∆§...>∆

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And sowhenwe 'plug in' into the existential game, both theorems theytell us that as § grows, towards thelargest numbers of a finite 'Universe/world',we obtain a 'median distribution',whichmeans in theUniverse,asuperorganism.Indeed,assystemsmultiplyitspopulation,the'partitionsoforgans'(whichcanbemodeledaspartialdistributionsofthecentrallimittheorem,andthescale∆-1,willconvergetoagivenform-forentropicsystems, the gauss distribution; but extrapolated to the whole reality, to the program of creation ofsuperorganisms).

Theyarethusgeneralizedfor 'systems'notofstochasticnaturebutwithcausalpatternsandfuturepredictableresultsadeterminismofcause->effectessentialtoreality.

5D∆>∆+1,expandstheconceptofstatisticalprobabilitiestouseofitsgenerallawstodeterminecausalevents:

∆±1STcausalityanddeterminism.

Whatistheprobabilityofthegameofexistence?InaninfiniteUniverseabsolute.Andforanyeventdeterminedbythegame,eversmootheraswegrowindetailofanalysis(∆ºview),innumbersofeventsintime(§ðview),inpopulationsinspace($T),andsoaswedeemtheUniverseinfinite,allsystemstendass->∞,t->∞and∆->∞and∆º->∞(numberofpixelsofthementalmapping,everfinerindetail)tobecomedeterministic,perfect,enactingtheGSTgameofexistenceanditsworldcycles,ofwhichthereareinfiniteindeedproofsinallsystems:

• Orbitsbecomemoreregularasthemassofthe'comet'->planetgrows.

• Dicethrowsbecomemoreregularwith1/6thprobabilityforeachnumber

• Superorganismsbecomemoreperfectaswemovetowardsthe1trillionmarkofcells

• Crystals lessamorphousandwe imagine theoneson thecentreofplanets tobeperfect 'diamonds',perfectironcrystals(Ours)andsoon.

• Societiesbetterorganized,withlessfriction,asChinashows.

• So happens as populations radiate or time passes by in evolution: the perfect super organism, the perfectplatonicformisreached.

It is the limitof thegame- inall thosecasesweshouldpreviouslydefinethe limitingdomain, thenormalizeddistribution to 1, the proper range; and itwould be interesting just for the sake of theory, to considerwhatwouldbe, ifthat isthecase,thelimitsofthosetheorems.Soaquestionispoisedevenif it looksfarreaching.Whilethesystemkeepsmoothingadinfinitum,orthereisalimitalsointime,towhatseemstobeobviouslythelimitinspaceof'carryingpopulations'?Eventhoughitseemscounter-intuitiveandlikelynotbeexperimented,Ibelieve somewhere in the quadrillion mark the system will break and enter a chaotic, no longer smoothdistributionbutmovetowardsdeath(being4thedimensionsofasysteminasinglesheetofspace-time).

RandomProcessesofMarkovType.Theformalismbasedintheequivalenceofprobabilityandderivatives.

The formalism of probability today for random processes was established precisely when mathemticiansrealized that ultimately there is a direct connection aswe said, between a quanta of time probability and afinitesimalofcalculus,sotheywereabletomerelyintegrateprobabilitydensitiesasiftheywerefinitesimals.

To Markov is due the construction of such probabilistic scheme which is an immediate generalization of the

deterministicschemedescribedbytheequation: MarkovconsideredonlythecasewherethephasespaceofthesystemconsistsofafinitenumberofstatesΩ=(ω1,ω2,···ωn)andstudiedthechangeofstateofthesystemonlyforchangesoftimetindiscretesteps.Butinthisextremelyschematicmodelhesucceededinestablishingaseriesoffundamentallaws.

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Instead of a function F, uniquely defining the statew at time t > t0 corresponding to the stateω0 at time t0,

Markovintroducedtheprobabilities: ofobtainingthestateωiattimetundertheconditionthatattime t0 we had the stateωi. These probabilities are connected for any three instants of time: to<t1<t2 by arelation, introduced by Markov, which may be called the basic equation for a Markov process:

Whenthephasespaceisacontinuousmanifold,themosttypicalcaseisthataprobabilitydensityp(t0,ω0;t,ω)exists for passing from the stateω0 to the stateω in the interval of time (t0, t). In this case theprobability of…passingfromthestateω0toanyofthestatesωbelongingtoadomainGinthephasespaceΩiswrittenintheform:

Equation (35) is usually difficult to solve, but under known restrictions wemay deduce from it certain partialdifferential equations that are easy to investigate. Some of these equations were derived from non rigorousphysical considerations by the physicists Fokker and Planck. In its complete form this theory is the so-calledstochasticdifferential.

Themethodofstochasticdifferentialequationsallowsus,forexample,tosolvewithoutdifficultytheproblemofthemotion in still airof a very smallbody, forwhich themeanvelocity cof its fall is significantly less than thevelocityofthe“Brownianmotion”arisingfromthefact,becauseofthesmallnessoftheparticle,itscollisionswiththemoleculesoftheairarenotinperfectbalanceonitsvarioussides.

RECAP.The fundamentalelementofprobability is curiouslyenough thecertaintyof its sumofopposite results.Thatis,thesumoftheprobabilityofaneventnothappeningandhappeningisalways1.Andsoinintroducesdeepphilosophicalquestionsonthecausalityoftime.

Pentalogiconprobabilitydealsextensivelyonthequestionofcausalitybetweenthesmallerpartsandthewholes,andthedegreeofdeterminismintheuniverse,vs.thenumberofpotentialfuturesthatcanexistandthosewhoareefficientpathsthatwillalwaysoccur;as inquantumphysicalpathsthatalwaystendtocollapse inthemostefficient,leasttimepath,orpointofminimalmomentum,whichmustberegardedastheoppositefunction;thatis, the point ofmaximal information,which is the reason the system tends to it (andalso the point ofmaximalduration,asmax.informationspendslessenergyandlastsmore).

Thosethemesofphilosophyofprobabilityandtheoppositionbetweeneventsandnoeventsandthedualstatesofentropyvs.stillness,momentumvs.informationarethusofmoreinteresttousinthosetexts.

A first insight comes from the realization that from the perspective of the whole 1 probability, theundistinguishablepartsdonotmatter,sotheyhavecertaindegreeoffreedom,butthewholeassuchwillfollowitsworldcycleof existenceas a0 sumandasa1wholewithno freedom.Soprobabilities areultimately theequivalentintimetothe‘freedominspace’ofthesmallstepsperformedbyabeing,whichwillcertainlyendinacyclicalclosedcycleof‘0’-energyexpenditure’–itsequivalentinphysicalterms:freedomtothefinitesimalsisalwayscurtailedbythetheoryofgrandnumbers,bythewholeofwholeevents.Andsotheywillapproachtothe Gaussian center where the whole in populations or events is maximal, which on ‘great numbers’ ofindistinguishablefinitesimalpartswillbecomecertain.

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Asecondquestionaround5Dprobabilityiscausality,whichisentangled5Dimotionalin5D,butforsmallstepsagaincanbeAristotelian,lineal,henceallowingthelawsofprobabilitytobedeterministicforindividualceterisparibusdimotions.

Bothconceptscometogetherinthelawthatforaneventtobetruth,foranorganismtobestabletheGaussiancurvehastohaveasteepcentralpoint,asmalldeviationfromthemean,whichiswhythereisastandardcurveforprobabilitiesthatarerealevents,commontoalmostallofthem.

As in the case of polynomials, with resolutions of multiple equations to find radicals=numbers for a singleparameter,therearealsoinprobabilitiessomeirrelevantquestions,andasinpolynomialstherearelimits.Sofirstweshallconsiderapentalogicofprobabilityfunctions.

Theinversemirrorsofthepastandthefuture,theeventandthenon-existence.

Sowemustnotonlytranslatetimesequencesoffrequenciestosimultaneouspopulations,itsmemorialtailofpastresults,butalso'expand'the0’-1unitcircleintothe1-∞scaletoobserveisomorphiclawsbetweenprobabilityandstatistics.

ThatisbetweenTimeandSpace,betweenthefutureoccurrencesinprobabilitytimeandthepastpersistenceofmemorialpopulations,andhowtheyareidenticalinitsasymmetryasoneofthem,populationsfadeanddieawaywhile the other, possible paths of future, converge and collapse, leaving finally only a trace of past, onlyanoccurrence of future, in the collapse ofmemories thatdisappear, in the collapse of future dreams that areblownaway.

All thisof course requiresacompletelynewoutlookon themeaningofprobabilitiesandpopulations,wemightcarry through the fourth line before my tail of timedisappears, but don't count on it. This will always be anunfinishedjobthatmightbeabortedevenbeforehumindsnoticeitsexistencebythedisappearancenotonlyofhisauthorbutthehuminditself(seethesectiononhistoryandeconomics).

CONDITIONALPROBABILITY

ThedifferencebetweenstatisticsandprobabilityarethosebetweenSpaceandTimestatesandpartsandwholescales. Statistics is a space state,made of amaximal number of indifferent elements. IT does act therefore onsimultaneous parameters and cares not for the causal time process; so it uses quantitative procedures andintegratestheindifferentmassintomeanvalues.

Probabilities are time, sequential causal events; of which we can differentiate independent events, closer tostatisticscalculatedbyaddingfrequencynumbersandconditionalevents,inwhichoneeventconditionstheotherdiminishing the probability to reach certainty 1 (the whole event or population). Let’s consider both types ofprobabilities.

Inthe0’-1sphere,twoeventsaresaidtobemutuallyexclusiveiftheycannotbothoccur(underthecomplexofconditionsS).

Forexample,inthrowingadie,theoccurrenceofanevennumberofspotsandofathreearemutuallyexclusive.AneventAiscalledtheunionofeventsA1andA2ifitconsistsoftheoccurrenceofatleastoneoftheeventsA1,A2.Forexample,inthrowingadie,theeventA,consistingofrolling1,2,or3,istheunionoftheeventsA1andA2,

whereA1consistsofrolling1or2andA2consistsofrolling2or3.Itiseasytoseethatforthenumberofoccurrencesm1,m2,andmoftwomutuallyexclusiveeventsA1andA2andtheirunionA=A1∪A2,wehavethe

equationm=m1+m2,orforthecorrespondingfrequenciesƒ=ƒ1+ƒ2.Thisleadsnaturallytothefollowingaxiomfortheadditionofprobabilities:

2.P(A1UA2)=P(A1)+P(A2)...

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iftheeventsA1andA2aremutuallyexclusiveandA1∪A2denotestheirunion.Further,foraneventUwhichiscertain,wenaturallytake:

3.P(U)=1

Thewholemathematicaltheoryofprobabilityisconstructedonthebasisofsuch3axioms.

One theeventhashappenedhoweverweare in the realmof countingpopulations in the1-∞sphere, and thecorrespondencehappensbetweentheprobabilitydistributiononthe0’-1planeandthepopulationdistributiononthe1-∞.

• The formal structure of theory of probability is simple: events are mutually exclusive if their intersection isempty,i.e.,ifAB=N,whereNisthesymbolforanimpossibleevent.

• TheaxiomofprobabilitytheoryistherequirementthatundertheconditionAB=Nitholdstheequation:

P(AUB)=P(A)+P(B)

Thedistributionofsocialgroupsasprobabilitiesintime,ismoreaccurateasits¡ndifferenceincreases.Modelsofthe1plane,asFourierorprobabilityshowstheformsofreproductionofacomplexplaneinlower@-geometriesofexistence.

Theanalogyisbynomeanssuperficial.Itturnsoutthatthewholemathematicaltheoryofprobabilityfromtheformal point of view may be constructed as a theory of measure, making the special assumption that themeasureof“theentirespace”Uisequaltoone.Whichprovesthattheo-1probabilitytimesphereisequivalenttothe1-∞statisticalpopulation,andsetsincorrespondencequantumphysicsandmolecularstatistics…

Such an approach to thematter has produced complete clarity in the formal construction of themathematicaltheoryofprobabilityandhasalsoledtoconcreteprogressnotonlyinthistheoryitselfbutinothertheoriescloselyrelatedtoit intheirformalstructure.Inthetheoryofprobabilitysuccesshasbeenachievedbyrefinedmethodsdeveloped in themetric theoryof functionsofa real variableandat the same timeprobabilisticmethodshaveprovedtobeapplicabletoquestionsinneighboringdomainsofmathematicsnot“byanalogy,”butbyaformalandstricttransferofthemtothenewdomain.Whereverwecanshowthattheaxiomsofthetheoryofprobabilityaresatisfied, the results of these axioms are applicable, even though the given domain has nothing to do withrandomnessintheactualworld.

Theexistenceofanaxiomatized theoryofprobabilitypreservesus fromthe temptationtodefineprobabilitybymethodsthatclaimtoconstructastrict,purelyformalmathematicaltheoryonthebasisoffeaturesofprobabilitythatareimmediatelysuggestedbythenaturalsciences.Suchdefinitionsroughlycorrespondtothe“definition”ingeometry of a point as the result of trimming down a physical body an infinite number of times, each timedecreasingitsdiameterbyafactorof2.

Withdefinitionsof this sort,probability is taken tobe the limitof the frequencyas thenumberofexperimentsincreasesbeyondallbounds.Theveryassumptionthattheexperimentsareprobabilistic,i.e.,thatthefrequenciestend to cluster around a constant value, will remain valid (and the same is true for the “randomness” of anyparticularevent)onlyifcertainconditionsarekeptfixedforanunlimitedtimeandwithabsoluteexactness.Thustheexactpassagetothelimit:µ/n->p,cannothaveanyobjectivemeaning.Formulationoftheprincipleofstabilityofthefrequenciesinsuchalimitprocessdemandsthatwedefinetheallowablemethodsofsettingupaninfinitesequence of experiments, and this can only be done by amathematical fiction. This whole conglomeration ofconceptsmightdeserve seriousconsiderationif the final resultwerea theoryof suchdistinctivenature thatnoother means existed of putting it on a rigorous basis. But, as was stated earlier, the mathematical theory ofprobabilitymaybebasedonthetheoryofmeasure,initspresentdayform,byaddingtheconditionP(U)=1

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Ingeneral,foranypracticalanalysisoftheconceptofprobability,thereisnoneedtorefertoitsformaldefinition.It isobviousthatconcerningthepurelyformalsideofprobability,wecanonlysaythefollowing:TheprobabilityP(A/S) is a number around which, under conditions S determining the allowable manner of setting up theexperiments,thefrequencieshaveatendencytobegrouped,andthatthistendencywilloccurwithgreaterandgreater exactness as the experiments, always conducted in such a way as to preserve the original conditions,becomemorenumerous,andfinallythatthetendencywillreachasatisfactorydegreeofreliabilityandexactnessduringthecourseofapracticablenumberofexperiments.

In fact, theproblemof importance, inpractice, isnot togivea formallyprecisedefinitionof randomnessbut toclarifyaswidelyaspossibletheconditionsunderwhichrandomnessofthecitedtypewilloccur.Onemustclearlyunderstandthat,inreality,hypothesesconcerningtheprobabilisticcharacterofanyphenomenonareveryrarelybasedonimmediatestatisticalverification.Onlyinthefirststageofthepenetrationofprobabilisticmethodsintoanewdomainofsciencehastheworkconsistedofpurelyempiricalobservationoftheconstancyoffrequencies.

We see that statistical verification of the constancy of frequencies with an exactness of ε requires a series ofexperiments,eachconsistingofn=1/ε²trials.

TheunionofanygivennumberofeventsA1,A2,···,AsisdefinedastheeventAconsistingoftheoccurrenceofatleast one of these events. From the axiom of addition, we easily obtain for any number of pairwise mutuallyexclusiveeventsA1,A2,···,AsandtheirunionA,

P(A)=P(A1)+P(A2)+....+P(An)(theso-calledtheoremoftheadditionofprobabilities).

Iftheunionoftheseeventsisaneventthatiscertain(i.e.,underthecomplexofconditionsSoneoftheeventsAkmustoccur),then:

P(A1)+P(A2)+....+P(An)=1

InthiscasethesystemofeventsA1,···,As,iscalledacompletesystemofevents.

WenowconsidertwoeventsA,andB,which,generallyspeaking,arenotmutuallyexclusive.TheeventC istheintersection of the events A and B,written C = AB, if the event C consists of the occurrence of both A and B.Forexample,iftheeventAconsistsofobtaininganevennumberinthethrowofadieandBconsistsofobtainingamultipleofthree,thentheeventCconsistsofobtainingasix.

This is all very trivial and indeed independent probability is trivial because we completely work on non-causalprocesses,akintothoseofstatistics:Thepropertiesofprobability,expressedbyformulas(1),(2),and(3),serveasasufficientbasisfortheconstructionofwhatiscalledtheelementarytheoryofprobability.

CONDITIONALPROBABILITY.

Wheneventsare connectedhoweverprobabilitybecomes somewhatmore complex, still though suingonly thepropertiesofproductanddivision,butestablishinglesserchancestoachievewholeevents,thus introducingtheconceptofanuncertain‘goal’,asB,isconditionedbyAwhichnotalwaysiscontrolled.Andthisismorehowrealityworks,given the fact thatAandBcanbeconsidered ‘two fractalpoints’wills’whichmightdiffer.Letusbrieflyintroducetheformalism:

Inalargenumbernofrepeatedtrials,lettheeventAoccurmtimesandtheeventBoccurltimes,inkofwhichBoccurstogetherwiththeeventA.Thequotientk/miscalledtheconditionalfrequencyoftheeventBundertheconditionA.Thefrequenciesk/m,m/n,andk/nareconnectedbytheformula:

k/m=k/n:m/n

whichnaturallygivesrisetothefollowingdefinition:

TheconditionalprobabilityP(B/A)oftheeventBundertheconditionAisthequotient

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P(B/A)=P(AB)/P(A)

Hereitisassumed,ofcourse,thatP(A)≠0.

IftheeventsAandBareinnowayessentiallyconnectedwitheachother,thenitisnaturaltoassumethateventBwill not appear more often, or less often, when A has occurred than when A has not occurred, i.e., thatapproximatelyk/m≈l/nor:

k/n=k/m•m/n≈l/n•m/n

In this lastapproximateequationm/n= ƒA is the frequencyof theeventA,and l/n= ƒB is the frequencyof theeventBandfinallyk/n=ƒABisthefrequencyoftheintersectionoftheeventsAandB.

Weseethatthesefrequenciesareconnectedbytherelation:

ƒab≈ƒaxƒb

FortheprobabilitiesoftheeventsA,BandAB,itisthereforenaturaltoacceptthecorrespondingexactequation

4.P(AB)=P(A)•P(B)

Equation(4)servestodefinetheindependenceoftwoeventsAandB.

Similarly, we may define the independence of any number of events. Also, we may give a definition of theindependenceofanynumberofexperiments,whichmeans,roughlyspeaking,thattheoutcomeofanypartoftheexperimentsdonotdependontheoutcomeoftherest.

WenowcomputetheprobabilityPkofpreciselykoccurrencesofacertaineventA inn independent tests, ineachoneofwhichtheprobabilitypoftheoccurrenceofthiseventisthesame.WedenotebyĀtheeventthateventAdoesnotoccur.Itisobviousthat

P(Ā)=1-P(A)=1-p

From the definition of the independence of experiments it is easy to see thatthe probability of any specificsequence consisting of k occurrences of A and n–k non-occurrences of A is equal to: Thus, forexample,forn=5andk=2theprobabilityofgettingthesequenceAĀAĀĀwillbep(l–p)p(l–p)(1–p)=p²(1–P)³Bythetheoremontheadditionofprobabilities,Pkwillbeequaltothesumoftheprobabilitiesofallsequenceswithkoccurrencesandn–knonoccurrencesoftheeventA,i.e.,Pkwillbeequalfrom(5)totheproductofthenumberof suchsequencesbypk(l–p)n–k.Thenumberof suchsequences isobviouslyequal to thenumberofcombinations of n things taken k at a time, since the k positive outcomes may occupy any k places in thesequenceofntrials.

Finallywegetthebinomialdistribution:

Directexaminationof themassofobservationsmakesclearonly theverysimpleststatistical laws; ituncoversonlyafewofthebasicprobabilitiesinvolved.Butthen,bymeansofthelawsofthetheoryofprobability,weusethese simplest probabilities to compute the probabilities of more complicated occurrences and deduce thestatisticallawsthatgovernthem.

Sometimes we succeed in completely avoiding massive statistical material, since the probabilities may bedefinedbysufficientlyconvincingconsiderationsofsymmetry.Forexample,thetraditionalconclusionthatadie,i.e.,acubemadeofahomogeneousmaterialwillfall,whenthrowntoasufficientheight,withequalprobabilityon each of its faces was reached long before there was any systematic accumulation of data to verify it byobservation. Systematic experiments of this kind have been carried out in the last three centuries, chiefly byauthorsoftextbooks inthetheoryofprobability,atatimewhenthetheoryofprobabilitywasalreadyawell-developedscience.Theresultsof theseexperimentsweresatisfactory,but thequestionofextending themto

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analogouscasesscarcelyarouses interest.Forexample,as farasweknow,noonehascarriedoutsufficientlyextensiveexperimentsintossinghomogeneousdicewithtwelvesides.Butthereisnodoubtthatifweweretomake12,000suchtosses,thetwelve-sideddiewouldshoweachofitsfacesapproximatelyathousandtimes.

The basic probabilities derived from arguments of symmetry or homogeneity also play a large role in manyserious scientific problems, for example in all problemsof collisionor near approachofmolecules in randommotioninagas;anothercasewherethesuccesseshavebeenequallygreatisthemotionofstarsinagalaxy.Ofcourse, in these more delicate cases we prefer to check our theoretical assumptions by comparison withobservationorexperiment.

TheLawofLargeNumbersandLimitTheorems

Itiscompletelynaturaltowishforgreaterquantitativeprecisioninthepropositionthatina“long”seriesofteststhe frequencyofanoccurrencecomes“close” to itsprobability.Butherewemust formaclearnotionof thedelicatenatureoftheproblem.Inthemosttypicalcasesinthetheoryofprobability,thesituationissuchthatinan arbitrarily long series of tests it remains theoretically possible that we may obtain either of the twoextremesforthevalueofthefrequency

Thus,whatevermaybethenumberoftestsn,itisimpossibletoassertwithcompletecertaintythatwewillhave,say,theinequality

Forexample,iftheeventAistherollingofasixwithadie,theninntrials,theprobabilitythatwewillturnupasixonallntrialsis(1/6)n>0,inotherwords,withprobability(1/6)n

wewillobtainafrequencyofrollingasixwhichisequaltoone;andwithprobability(1-1/6)n>0asixwillnotcomeupatall,i.e.,thefrequencyofrollingasixwillbeequaltozeroth.

In all similar problems any nontrivial estimate of the closeness of the frequency to the probability cannot bemadewithcompletecertainty,butonlywithsomeprobabilitylessthanone.

For example, itmaybe shown that in independent tests,with constantprobability pof theoccurrenceof aneventineachtesttheinequality

7.|µ/n-p|<0.02

forthefrequencyμ/nwillbesatisfied,forn=10,000(andanyp),withprobability

8.P>0.9999

Herewewishfirstofalltoemphasizethatinthisformulationthequantitativeestimateoftheclosenessofthefrequency μ/n to the probability p involves the introduction of a new probability P.Thepracticalmeaningoftheestimate(8)isthis:IfwecarryoutNsetsofntestseach,andcounttheMsetsinwhichinequality(7)issatisfied,thenforsufficientlylargeNwewillhaveapproximately

9.M/N≈P>0.9999

Butifwewishtodefinetherelation(9)moreprecisely,eitherwithrespecttothedegreeofclosenessofM/NtoP, or with respect to the confidencewithwhichwemay assert that (9) will be verified, thenwemust haverecourse to general considerations of the kind introduced previously in discussing what is meant by thecloseness of μ/n and p. Such considerations may be repeated as often as we like, but it is clear that thisprocedurewillneverallowus tobe freeof thenecessity, at the last stage,of referring toprobabilities in theprimitiveimprecisesenseofthisterm.

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FurtherRemarksontheBasicConceptsoftheTheoryofProbability

Inspeakingofrandomevents,whichhavethepropertythattheirfrequenciestendtobecomestable, i.e., inalong sequence of experiments repeated under fixed conditions, their frequencies are grouped around somestandardlevel,calledtheirprobabilityP(A/S),wewereguilty,in§1,ofacertainvaguenessinourformulations,intworespects.

In the first place, we did not indicate how long the sequence of experiments nrmust be in order to exhibitbeyondalldoubttheexistenceofthesupposedstability;inotherwords,wedidnotsaywhatdeviationsofthefrequenciesμr/nr,fromoneanotherorfromtheirstandardlevelpwereallowableforsequencesoftrialsn1,n2,···, ns of given length. This inexactness in the first stage of formulating the concepts of a new science isunavoidable. It is no greater than thewell-known vagueness surrounding the simplest geometric concepts ofpointandstraightlineandtheirphysicalmeaning.Morefundamental,however,isthesecondlackofclearnessconcealed in our formulations; it concerns themanner of forming the sequences of trials inwhichwe are toexaminethestabilityofthefrequencyofoccurrenceoftheeventA.

Asstatedearlier,weareledtostatisticalandprobabilisticmethodsof investigationinthosecasesinwhichanexactspecificpredictionofthecourseofevents is impossible.But ifwewishtocreate insomeartificialwayasequenceofeventsthatwillbe,asfaraspossible,purelyrandom,thenwemusttakespecialcarethatthereshallbenomethodsavailable fordetermining in advance those cases inwhichA is likely tooccurwithmore thannormalfrequency.

Suchprecautionsaretaken,forexample,intheorganizationofgovernmentlotteries.IfinagivenlotterytherearetobeMwinningtickets inadrawingofNtickets,thentheprobabilityofwinningforanindividualticket isequaltop=M/N.Thismeansthatinwhatevermannerweselect,inadvanceofthedrawing,asufficientlylargesetofntickets,wecanbepracticallycertainthattheratioμ/nofthenumberμofwinningticketsinthechosensettothewholenumbernofticketsinthissetwillbeclosetop.

Forexample,peoplewhoprefer tickets labeledwithanevennumberwill nothaveany systematicadvantageoverthosewhopreferticketslabeledwithoddnumbers,andinexactlythesamewaytherewillbenoadvantagein proceeding on the principle, say, that it is always better to buy ticketswith numbers having exactly threeprimefactors,orticketswhosenumbersareclosetothosethatwerewinnersintheprecedinglottery,etc.

Similarly,whenwearefiringawell-constructedgunofagiventype,withawell-trainedcrewandwithshellsthathavebeensubjectedtoastandardqualitycontrol,thedeviationfromthemeanpositionofthepointsofimpactof theshellswillbe less thanthepreviouslydeterminedprobabledeviationB inapproximatelyhalf thecases.Thisfractionremainsthesameinaseriesofsuccessivetrials,andalsoincasewecountseparatelythenumberofdeviationsthatare less thanB foreven-numberedshots (in theorderof firing)or forodd-numbered.But it iscompletely possible that if wewere tomake a selection of particularly homogeneous shells (with respect toweight,etc.),thescatteringwouldbeconsiderablydecreased,i.e.,wewouldhaveasequenceoffiringsforwhichthe fraction of the deviations which are greater than the standard B would be considerably less than 1/2.Thus,tosaythataneventAis“random”or“stochastic”andtoassignitadefiniteprobability

p=(P(A/S)

is possible only when we have already determined the class of allowable ways of setting up the series ofexperiments.ThenatureofthisclasswillbeassumedtobeincludedintheconditionsS.

ForgivenconditionsS thepropertiesof theeventAofbeingrandomandofhavingtheprobabilityp=P(A/S)expresstheobjectivecharacteroftheconnectionbetweentheconditionSandtheeventA.

In other words, there exists no event which is absolutely random; an event is random or is predetermineddependingontheconnectioninwhichitisconsidered,butunderspecificconditionsaneventmayberandomin

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acompletelynonsubjectivesense,i.e., independentlyofthestateofknowledgeofanyobserver.Ifweimagineanobserverwhocanmasterall thedetaileddistinctivepropertiesandparticularcircumstancesoftheflightofshells,andcanthuspredictforeachoneofthemthedeviationfromthemeantrajectory,hispresencewouldstillnot prevent the shells from scattering in accordancewith the laws of the theory of probability, provided, ofcourse,thattheshootingwasdoneintheusualmanner,andnotaccordingtoinstructionsfromourimaginaryobserver.In this connection we note that the formation of a series of the kind discussed earlier, in which there is atendencyforthefrequenciestobecomeconstantinthesenseofbeinggroupedaroundanormalvalue,namelytheprobability,proceedsintheactualworldinamannercompletelyindependentofourintervention.

Forexample, it ispreciselybyvirtueoftherandomcharacterofthemotionofthemolecules inagasthatthenumberofmoleculeswhich,eveninaverysmallintervaloftime,strikeanarbitrarilypreassignedsmallsectionofthewallofthecontainer(orofthesurfaceofbodiessituatedinthegas)provestobeproportionalwithverygreatexactnesstotheareaofthissmallpieceofthewallandtothe lengthofthe intervaloftime.Deviationsfrom thisproportionality in caseswhere thenumberofhits is not large also follow the lawsof the theoryofprobability and produce phenomena of the type of Brownian motion, of which more will be said later.We turn now to the objective meaning of the concept of independence. We recall that the conditionalprobabilityofaneventAundertheconditionBisdefinedbytheformula

P(A/B)=P(AB)?P(B)

WealsorecallthateventsAandBarecalledindependentif,asin(4),

P(AB)=P(A)P(B)

FromtheindependenceoftheeventsAandBandthefactthatP(B)>0itfollowsthat

P(A/B)=P(A)

All the theorems of the mathematical theory of probability that deal with independent events apply to anyevents satisfying the condition (4), or to its generalization to the caseof themutual independenceof severalevents.Thesetheoremswillbeoflittleinterest,however,ifthisdefinitionbearsnorelationtothepropertiesofobjectiveeventswhichareindependentinthecausalsense.

Naturally, in dealingwith the concept of independence,wemust not proceed in too absolute a fashion. Forexample,fromthelawofuniversalgravitation,itisanundoubtedfactthatthemotionsofthemoonsofJupiterhaveacertaineffect,say,ontheflightofanartilleryshell.But it isalsoobviousthat inpracticethis influencemay be ignored. From the philosophical point of view,wemay perhaps, in a given concrete situation, speakmoreproperlynotoftheindependencebutoftheinsignificanceofthedependenceofcertainevents.Howeverthat may be, the independence of events in the cited concrete and relative sense of this term in no waycontradicts the principle of the universal interconnection of all phenomena; it serves only as a necessarysupplementtothisprinciple.

Thecomputationofprobabilitiesfromformulasderivedbyassumingtheindependenceofcertaineventsisstillof practical interest in caseswhere the eventswere originally independent but became interdependent as aresult of the events themselves. For example, onemay compute probabilities for the collision of particles ofcosmicradiationwithparticlesofthemediumpenetratedbytheradiation,ontheassumptionthatthemotionoftheparticlesofthemedium,uptothetimeoftheappearancenearthemofarapidlymovingparticleofcosmicradiation,proceedsindependentlyofthemotionofthecosmicparticle.Onemaycomputetheprobabilitythatahostilebulletwillstrikethebladeofarotatingpropeller,ontheassumptionthatthepositionofthebladewithrespecttotheaxisofrotationdoesnotdependonthetrajectoryofthebullet,asuppositionthatwillofcoursebe wrong with respect to the bullets of the aviator himself, since they are fired between the blades of therotatingpropeller.Thenumberofsuchexamplesmaybeextendedwithoutlimit.

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It may even be said that wherever probabilistic laws turn up in any clear-cut way we are dealing with theinfluenceofalargenumberoffactorsthat,ifnotentirelyindependentofoneanother,areinterconnectedonlyinsomeweaksense.

Thisdoesnotatallmeanthatweshoulduncriticallyintroduceassumptionsofindependence.Onthecontrary,itleads us, in the first place, to be particularly careful in the choice of criteria for testing hypotheses ofindependence,andsecond,tobeverycarefulininvestigatingtheborderlinecaseswheredependencebetweenthe facts must be assumed but is of such a kind as to introduce complications into the relevant laws ofprobability.Wenotedearlier that theclassicalRussian schoolof the theoryofprobabilityhas carriedout far-reachinginvestigationsinthisdirection.

Tobringtoanendourdiscussionoftheconceptof independence,wenotethat, justaswiththedefinitionofindependenceoftwoeventsgiveninformula(4),theformaldefinitionoftheindependenceofseveralrandomvariables isconsiderablybroaderthantheconceptof independence inthepracticalworld, i.e., theabsenceofcausalconnection.

I.e,inordertoestablishthatinagivenconcreteproblemtheprobabilityisdefinedwithanexactnessof0.0001,itisnecessarytocarryoutaseriesofexperimentscontainingapproximately100,000,000trialsineach.

Thehypothesisofprobabilisticrandomnessismuchmoreoftenintroducedfromconsiderationsofsymmetryorofsuccessiveseriesofevents,withsubsequentverificationofthehypothesisinsomeindirectway.Forexample,

sincethenumberofmoleculesinafinitevolumeofgasisoftheorderof1020ormore,thenumber√n,correspondingtotheprobabilisticdeductionsmadeinthekinetictheoryofgases,isverylarge,sothatmanyof

thesedeductionsareverifiedwithgreatexactness.Thus,thepressuresontheoppositesidesofaplatesuspendedinstillair,eveniftheplateisofmicroscopicdimensions,turnoutexactlythesame,althoughanexcessofpressureononesideoftheorderofathousandthofonepercentcanbedetectedinaproperly

arrangedexperiment.

SxT:COMBINATORICS

Thehistoricapproachtoknowledgeisalwaysgoodtounderstandthenaturalevolutionofanyworldcyclefromsimplextocomplex,includingthosepurementalmirror-imagesofrealitythatarefirstbornasallsystemsinanasymmetricmixtureof time-spaceparameters/viewsand thenbreak into themoresymmetric spatialview (inthecaseofcombinatorics,spacestatistics)andthetemporalview(statistics).

Socombinatoricswasthebeginningofsocialtimetheorybeyondthesimplestconsiderationofcounting,thatisofnumbersaswholesofidenticalbeings,andgeometricnumbers,thatisthestudyofnumbersinitssymmetrywith points. Those two dualities which we can considered to be the Ƥ0,1 perspective (numbers as socialPlanes), and the Spatial, more static perspective (that is numbers as forms of space-geometries), will thenbecometheTimeperspective, that isdynamicnumbers inwhichthecausal, sequentialordermatters,and theflowoftimeconstantlyadsupnewidenticalbeings,indifferentpositions.

Numbertheorythusreachesitshighest'complexity'inthesymmetryoftimeprobabilitiesandspacestatistics,asalltheRashomonperspectiveareincluded.

Andwecantalkof3ages,whichcanalsobebrokenaccordingtothemultidimensional I-logic in termsof thepreviousgraphofanasymmetric, first state, combinatorics,which then specialized into the symmetric spatialpopulationanalysisandthehierarchical,timeprobabilitypointofview.

Let’s startwithsome insightsoncombinatorics’experimentalnatureasamirrorofT.œs’ fractalpropertiesofspace and cyclical nature of time. So we shall not talk of 'combinatorial structures, binary and plane trees,categories, the twelvefoldway etc. Just tomention thatwe did study them in youth and knowwhy it is not

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needed toknow.Keep it simple, if you 'understand it'.Alsowearenot interested in repeatingwhathumindsknowbutinexploringnewinsightsfromallthep.o.v.softhe5dimotionsanditsvitalproperties.

Ternarycombinatorics.Variationsandpermutations.

Combinatoricsaresimplevariationsofseveralelementswithdifferenthierarchicalelements.

Permutationsarevariations inthesequentialorderof theelementsofasœt(inthe jargonof¬Æ,the inverseexpressionofTœs,thattheinternalelementsofawhole,theTœ=∑sœt,itsparts,asusualslightlychangingtheterminologyofclassicscienceinto'stience').

Its formulae is P!, thus becomes the first '¬Ælgebraic GST', or Generator of space-time by internalself-re=production,usingthegeneral¬Algebraicoperandof3D-reproduction-obviouslytheproduct.

PermutationofDimotions.

ItisthenimmediatethethoughtthatpermutationofdimotionsisanessentialGST,ofwhichthereshouldbeasmanyasdifferent¡logicsystemsofspace-timegenerationexist:

• Monologiccannotcreatebyself-reflection1!=1

• Thepermutationof0 is1,hence0 'Is something' ithasparts, andvolume it is a finitesimaland it self-reflectsonitselfasbeingthe1,thewhole.

• Dualitybetweenthelimitsof0’-∞,formandentropy permutatesonly2,1>2,2<1,whichrepresentthetwoarrowsoflifeanddeath.

• Trinity permutates, abc, acb, cab, cba, bca, bac... which represent the hierarchies of physiologicalnetworksanditisanessentialpermutationofspecies.

• Tetralogic, as in the case of 4 quantum numbers and 4 genetic numbers has multiple variations,4x3x2x1=24, which represent then 24 possible orders of the positive dimotions of existence, in evolutionaryprocesses-excludinglocomotionthatdoesn'ttakeinpureinformativeevents.

Andafundamentalthemeoftrilogic is torelatechainsofactionsbased inpermutations.Forexample,certaincombinationsofactionsaremoreimportant.Sincemostactions,whichareNOTentropic,self-destructivestartwithamind-seed-perceptiveaction,whichreducespermutationsto3.

Variationsofdisconnectedpartsvs.permutationsofintegratedwholes...

...ontheotherhand,arepermutationsofafiniteNUMBERofelementsofthewholesœT,whoseformulaisalsointerestingintheanalysisofoperandi,asitbringsnowtheinverseoperandsofdivisionandsubtractionintothemix:V(n,r)=n!/(n-r)!

One interesting theme of variation is the paradox that asmaller number of elementsdoesn't give a largernumberofvariations(asthedivisiveelementn-rgrowsfaster).

Itseemsinprinciplecounterintuitive,asonewouldimagetherewouldbemanymoresmallcombinations,forexampleifwehaveabcde,wecanmakeab,cd,ac,ad,ae,bc,bd,be,ce...etc.Butifwetake5elementsonlyabcde,thewholehappens,albeititspossiblecombinationsacbd,acdb,adbc,etc.islarger.

Thuswholesmultiplyfasterandarealwaysmorepowerfulthansmallerparts.Thissimpleformulaimpliesisthattheinnerstructureofwholes-whatwecallitssynchronicitiesandsimultaneities,iscomplexenoughtorenderalarger number of combinations, that smaller sets with minimalvariability; reinforcing the experimental,Darwinianevidenceofthepowerofwholes.

Permutations of wholes vs. variations (of parts of wholes) and its combinatorics revealessential laws on thestructure of social organisms, hence its importance in all sciences, specially in physics, which tend to study

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massiveamountsofidenticalbeings,generatinginternalcombinations;whichleadustothethirdclassicformofcombinatorics:variationsandpermutationsinwhichasame'identical'elementcanberepeated

n-tuplesofdifferentfrequencyactions.

Itistheimportantcaseareorderedarrangementsofthe5elementsoftheSœt5whererepetitionsareallowed,calledn-tuples,InthesœtofDimotions,duetothe 'different frequency'oftheactionsofasystem,whichwillreproduceonceayear,feed3timesadayandsoon.

ItisalsocarriedintotheS=Tsymmetryof'cellularnumbers'forthedifferentorgansthatperformdimotionsinasuperorganism.

n-tuplesofsocialevolution.

n-tuples without restriction of repetitions, is the fastest growing GST (generator operand of space-time): nª,reasonwhythemostabundantdimotionofexi:st¡enceissocialevolutionofundistinguishableparts.

ThisisthecommonestcaseofNature.Asnumbers,wesay,aresetsofidenticalelements.Andsomostsocietiesareofidenticalexchangeablebeings.

In the entangledS=TUniverse those elements of combinatorics, branch into T-probability and S-tatistics andbelong 'i-logically' to thekey4thNon-Euclideanpostulateof 'congruence'...as thedegreeof identityofbeingsimpliesaparallelismandcapacitytounderstanditsinformationandevolvesocially.

In¬Ælgebratheyarerepresentedbythe'3levelsofsocialgrowthoperands:±variations,x-permutationsandxªtuples.

To noticethe growth of value of variations and permutations with repetition is huge when we increase thenumberofidenticalbeings,againstressingclearlythepowerof'identity'over'variation'-athemethatwillrunacrossalltheanalysisofreality.Sincewewrite:

V'(n,1)=n;V'(n,2)=n²;V'(n,3)=n³...

Thishowever isnotthecasewhencongruencehappens indifferentgroupsofthetotalsœT(variations).Thentheformulaisgreatlyreduced:P!/a!b!...

Theusual interpretationwhichheavilyweightsinphysicalensemblesisthatindistinguishableelementsarethesameandmustsubtracted.Vitallyweinterpretitsimplywiththeconceptof'divideandwin'.

Whatisthenindistinguishablebecomesimportant.Asitisanexternaljudgmentontheobserver,sinceallformsareinternallydistinguishable:

Totalidentityonlyhappensinshallowexternalviewsofasurfacewithoutconsideringthecontent.Andhereofcourse lays a hugephilosophical andethical part of theUniverse, and thedualities between the internal andexternal viewof beings,which dominates reality as theimportance of Bose statistics, entropic ensembles andpartitionsshows.

Butwhyalargernumberofvariations,permutations,alargercardinalmatters?Ifweweretocastthisnumberinterms of spatial population or sequential time series, it obviouslymeansmore configurations,more types ofexist¡ences...

Differencesbasedinthenumbersonthe§œt.

Moreinterestingperhapsistoconsider,aswearewritingaboutnumbertheoryandhavestudiedthevalueofeach number in different social and geometric meanings, each variation, permutation with our with outrepetitionsandpartitions,whattheytellusaboutthegameofT.œs:

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1. If we have a single element, 'A'; the order can only be 1. And so the ONE is immutable inhierarchicalorder.Theegocomesalwaysfirst.

2. Whenwehavetwo,AborBaarethe2onlyhierarchicalcombinations.Andtheorderisobvious,astheonethatcomesfirstishierarchicallythemostimportantwithmoreexperienceandmoreinformation(till its3rdageofdecline)...SoweshouldwriteAincapsbinminorletters.Buthereinalargercontextwecomewiththefirst 'divergence'of®:as itmattersmore thatAandEareactuallyNOT inahierarchicalposition;but sharingeach other to converge into A=B and re-produce a being.And here iswhere the first sight to the 'powerofidentity' comes intobeing: identity reproducesand increasesa socialgroup; it is theessenceof thegameofsocialevolution,growthandreorganization.Andsoindeed,ofallpossiblecombinatorics,thelargestnumbersareachievedwithpermutationsofanidentitynumber.

3. Thenwehave3,theholynumber;trinityindeed,ifyouhavecomesofaryoumustbynowknowisthegame.Andthisgetsmoreinteresting:

Howmanyvariationswehaveof3?Asithappens'mathematicalpros'writetheminpairs,soforabc,theywriteab,ac,ba,bc,ca,cb...6...

Sincefor3elementsthenumberofvariationsof2elementsandpermutationsofthewholesetarethesame.

Hence species diversify by combining the ternary networks of the system, in 6basic different phyla withapplicationsinallstiences,accordingtothehierarchyofthe3physiologicalnetworksofentropy/digestive/limbsystems,reproduction,bodywavesystemandinformationparticle-headsystem?

Letusstressnow,theinterestingfactthatwecanhidethe3rdelement,thedigestive,entropic,predatoryworldinwhichthe2physiologicalnetworksthatmattermost,theparticle-wavebody-headsystempreys in.The3rdelementisthusthelowerclassofasystem,whichisalmostinvariablyspent...

Another interesting duality is that along the | vs. O topology, asall whatwe have explained are hierarchicalsets,lineal setswithapreferentialorder,but the rulesof combinatorics forsœtsapplyalso to cyclicalorders,with interestingresults: thenumberofvariations increasesdramatically,asthehierarchy isdissolved, formingthefundamental property of cyclical membranes: to be 'democratic', 'entropic', as no order matter, so allpossibleordershappensuchasfornelements,n!willbethepossiblepermutations...aginormousnumber,evenfor small digits beyond the 10 decametric scale, which shows how easily we can by iteration and hierarchymultiplythecomplexityoftheUniverse.

The importance of those simple relationships of order,will again, be explored inmore depth in ¬Algebra, asindeed,modern¬Algebra startedwith thediscoverybyGalois that the solutionsofpolynomialsdependedontheirpermutations.

But as always the biggest insights we shall provide are metaphysical, as mathematics is a reflection of theUniverse in its simplest spatial and scalar relationships,whose units are the point and the number (for puretemporalflowslogicgivesbetterresults).

∆§:CombinatoricsAfter cyclical permutations combinatorics solved combinations proper, where theorder of elements doesn’tmatter,asin¬Ælogiceventsof'convergence','simultaneity'and'synchronicity'intheUniverse.

Ifthesethasnelements,thenumberofk-combinationsisequaltothebinomialcoefficient:

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Wecanseethenthedrasticdiminutionofelementsbroughtaboutbycombinations.Butnature,speciallyintimedoescareforsequential,hierarchicalorder,unlikespace,moredemocratic-henceweseeintherelationshipofpermutation/variationwithtimeand combinationwith space,oncemore that time-likeelementsare farmoreabundantthanspace-likeones.Andthisfacthashugemetaphysicalimplications.

Thebinomialformulacarriestheevolutionofscalarnumbersintotimeprobabilities:

Itdefinestheprobabilityofgettingexactlympositiveresults inn independenttrials,ineachoneofwhichapositiveoutcomehasprobabilityp.

Metaphysicalconclusions.LeibnizandGellmann'sTotalitarianprinciple.

'Allforms&eventsthatarenotforbiddenarecompulsory'TotalitarianPrinciple,Gell-mann.

Themetaphysical insightwe obtain from those different 'sums of elements' is heexistence of amuch vasterextensionofspacethanoftime.Sinceifallwhatcanexistdoexist,aspatial,present,conservedcyclicalform,perceived in the simultaneity of its non-sequential order which defines a slice of space has infinite morepossibilities than a temporal, sequential lineal order. And thismeans as all possible combinations do exist insome regions of the infinite Universe in time and space, spatial present extension-combinations and cyclicalformsAREmuchmore importantthanfinitehierarchicalplaneswithamore linealsequentialorderwhichendeasilywithincreasinginformationintheexplosionofdeath.

Alllinealmotionsarepartsofacyclicalform.Allexistencesreturnasitscombinationsandworldcyclesacrossthefifthdimensionarelimitedbutthenumberofplacesinspaceinwhichtheycanexistisfarlarger:

Entropic processes are quasi-infinite in its variations and permutations, as time-motions are, but when weintroduce lawsoforderandsymmetriesbetweenPlanes, topologiesand timeages,with its sequentialnatureoccurrences,thenumberofvariations ISreduceddrastically,soreality is farmorereducedbyvirtueofspatialpointsoforder,thanthemessyentropy-onlytheoriesofphysics,whichhasinfiniteimplications,asS/T->0,whichmeansthatinverselyt/s,thenumberofrepetitionsofS-formsintheUniversetendstoinfiniteandeachofthoseallowedexistentialbeingsisthereforediscontinuousbutimmortal.

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SCALARPOLYNOMIALS

Polynomialsontheotherhandarethespatialsimultaneousanalysiswiththesimpler3Dimotionsofsingleplane.Butfirstweshallintroducesomeconceptsonthedistinctionbetweenequationsandfunctions.

Goingfurtherintopositivecomplexity

ItisclearthatsimpleoperandsonnumbersalreadybecomecertainbasicmirrorsofDimotionsofexistence.Howeversincementalmirrorstrytobestillrepresentationsofallthepossibleoutcomesofthegameof

existence,theygrowingeneralitytillbecomingapotentialmirrorofallimages.ThisisdonewithtwodegreesofgeneralityinAlgebra.Thefirsthighergeneralityiscausedbyequations,whichwecanroughlyspeaking

differentiatetheminto:

Equationsaccordingto∆-scale.

-Simpleequationsthatattempttodescribeprocesseshappeninginasingleplaneofthefifthdimensionand/orasingletimecyclecalledpolynomials.

-Complexequationsthatexpandtherangeofthealgebraicmirrortoeventshappeningbetweendifferentplanesofexistence,henceincludingtheoperandsofsinusoidalfunctions,exponentialfunctionsanddifferential

calculus.

-S:EquationsaccordingtothequantityofS,ST,Tparameters.

-A2ndparameterofgrowingcomplexitycomesfromconsideringequationswithdifferentnumberofparameters,whichdividetheminthecaseofpolynomialsintoequationsofone,twoorthreevariables;aswe

areworkinginasingleplane–sohardlyanyfunctionwillhaveafourthvariablebeyondthe3standardelementsofatrilogicscaleofspace-time.

-T:Equationsaccordingtoitspossibletimevalues.

-A3rdparameterofexistenceisaccordingtoS=Tsymmetry,thenumberofsolutionswhichtrytoachieve,eitherasinglepointsolution,orawholerangeofpossiblesolutionsthatwilltraceanentirecurveoftheconic

type.

S=T:Equationsaccordingtotheangleofcongruence.

Allpossibleequationscanbeclassifiedfromthose∆STvariations,whichsometimesareconnected.Andneedlesstosayaccordingtothelawsofangleofcongruenceandthes=Tdualitywecanrepresentsolutionsgeometrically

andfindnocommunicationbetweenthevariables–nosolution,orasinglesolutionorseveralsolutions.Soweconnectsolutionstocongruentlaws,illustratedinthenextgraphforthemorecomplexcaseofsolutionson3

variables:

Inthegraph,withoutenteringintofurtherdetailitisobviousthatthesolutionsof

polynomialequationof3parameters,definetopologicalplanes,accordingtotheirangleofcongruence,whichthepossiblevariations

betweenskewness(nosolution),theTTlimitofentropicuncertainty,asinglesolutionwithallpointsincommon,the0’SSsolutionofidentity.Andthe3intermediatedual

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solutionsofasingleline,2linesor3lineswithpartialsolutionsforthepairsX,y,X,zandy,Z.Whilethereisfinallyapossiblesinglesolutionforthe3planes.

Itisworthtonoticesolutionhappenswhenthereisorthogonalityamongthe3planes.TheconceptoforthogonalitybetweentheS,TandST‘networksofanyefficientsystem,dotranscendthespatialidealformof

thosegraphs,yetifwetakeeach1-2Planeasaspace-fillingfractalnetwork;itmeanstheexistenceofacommonfractalpointor@-mindbrainofthesystemthatintersectsandcontrolsinsimultaneityandsynchronicitythe3physiologicalnetworksforthesystemtowork.AssuchasDeBrogliealreadynoticedinitspaperonthewave

particlecomplementarity,becauseallwhatexistsistimemotion,thispointinspacecanbetakenasasynchronousclockoftimecommontothe3networks,thesecondinman–thebeatofalimb/step,aneye-brain

thought,andabeatoftheheart.

Simultaneousgrowthofgeometric,dimotional,Nº&operandcomplexity:1Dpoint>2Dline>3Dplane.

Wecanassessmoreclearlytheconceptofi-logicentanglementthatmakespossiblethepersistenceofforminarealitymadeofSTdimotions,astheparallelismoffunctionsintime,numbersinscaleandformsinspacehappenin

mathematicstomimicthe∆STelementsofreality;whilethegrowthofnewdimotionsimpliesagrowthofcomplexitywithoutabandoningthepreviousdimotions,sotheybecomeentangledeachotherthroughoperands

Thekeythemetounderstandoperandsisthemeaningofinversionlaws,whichcarrythe±symbolinthe3Planesofsum,multiplicationandpower,dividingrealityintoasplittersymmetryaroundtheT.œs’0’point.

So± iscarried intopower lawsandproductsthroughratio inversions,which isbetterexpressed inthequadratic'complex'frameofreferenceofX²=±i²conjugate+1x-1=1²=1axis.

Sothepositiveandnegativeareshowninthatternary'frameofreference'ineachofthe3axis,withtheidentityproductof them,orquadratic frameof reference... inwhichweshalldiscuss latteron thecomplexelementsofmultipledimensionaloperandsanditsquantaofactions.

DifferencebetweenequationsandGeometriccurves.

Thecharacteristicfeaturesof¬Algebraaretheuseofletters,whichweperformoperationsaccordingtodefinitelaws. Inelementary¬Algebrathe lettersdenoteconstants,normallyordinarynumbers, takenaspopulations inspace,thevariables,whicharethefinal lettersrepresentT.œ.sofacertainspecies,andtheoperandrepresentdifferent‘dimensionalmotions’,dimotionsoftime-space.

SowecanreduceequationstoaseriesofExistential®lgebraicequationsofthetype:

∑T.œST-perandi∑T.œSToperandi=∑T.œST-perandi∑T.œSToperandi

Whereas thea….p letterswillbenumericalparameters, theU,V,X,Y,Z lettersTimespaceT.Œsandoperandidimotionalparameters.

As such there is a fundamental difference between an equation and a geometric curve in analytic geometry,despite its apparent similarity. An equation searches for a single solution as the XYZ letters represent ‘spatialpopulations’andtheparametersof‘timechange’thatconverttheequationinatimeeventaretheoperands.

OntheotherhandinacurvetheXYZlettersrepresentvariables,whosesimultaneouspossiblevalues,joinedbyageometricnon-E line formof simultaneous ‘spatialmembrane’, so theyareeventsof space,duly studied inour‘geometricfirstvolumeof5Dmathematics’.

This is ahugedistinction thatmakes completelydifferent the studyof simultaneous curves in space,whichactoftenasmembrainsofsuperorganisms;tothestudyof¬Algebraicequations,whichdescribeeventsintime,oftenofasequentialnature,giftedwithmotion.

Differentialgeometryfurthermergesbothconcepts,asitscurvesaretracedbyapointoftime.

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What is then the 5D difference between simultaneous space curves and time curves traced by points? Itspersistenceofmemory: asimultaneouscurve inspace iseitherperceivedbyaslowmind inslowtime,so thepointhastracedtheentirecurve,asthelinesoflightofacarinaslowmotionnightpicture;orthereproductionofthepointthroughadjacentpointsofthecurvepersistsbeyondthelowerplanesofgravitationalspace,leavingareal‘offspring’ofsimilarpoints,whichchaineachotherinsimultaneityformingarealmembrain;orfinallythecurveisarepetitivemotionthatreproducessofastastoformtheequivalentofatruemembrain(asweshowedwithexamplesofshepherddogsthatactdefactoasan‘physicalenclosure’forthewholeherd.

TonoticealsohowimportantthisdifferentiationwillbecomewhenconsideringthesolutionstoPDEequations,whichare‘families’ofcurves,inwhichallthepossibleformsarerepresentedbutonlycertainvalueswillbereal,andsowewillgodeeperinthisS-TdualitywhenwedealwithODEsandPDEs.

Inhereweshallthusstudyequationsintime,wherethe‘variables’aretheoperandsthatrepresentdimotions,the letters, numerical parameters of the social evolution or ‘size’ of the ‘constants’which represent a T.œ orcommonfunctionappliedtoallthosenumbers.

As opposed to the reduced number of conic forms, which describe simultaneous supœrganisms equationsdescribeeventsofspace-time.

It is then remarkable that the same operands can be used for both, simultaneous space curves and timepolynomials,showingoncemoretheessentialnatureoftheS=Tlaw.

SowewillhavetodifferentiateforeachoperandatleastaDualityoffunctionasS-operandandT-operand,whichistheinitialgendersymmetryofallsystemsofNaturethatallowsthecreationofatrilogicST-mixedfunctionandinmostcasesopensthedoortothepentalogicofpartsandwholes,whichinthecaseofoperandsisrepresentedbythoseequations(astheword¬AlgebrainArab–thereunionofparts–trulysignifies).

sothatthelawsofoperationsonexpressionsinlettersarebasedonthegenerallawsofoperationsonnumbers.Forexample,thesumdoesnotdependontheorderofthesummands,afactwhichin¬Algebraiswrittenas:a+b=b+a;inmultiplyingthesumoftwonumbers,wecanmultiplyeachoneofthenumbersindividuallyandthenaddtheproductssoobtained:(a+b)c=ac+bc,etc.

Ifwetracetheproofofan¬Algebraictheorem,itiseasytoseethatitdependsonlyontheselawsforoperationsonnumbersandnotatallonwhatthelettersrepresent.

The ¬Algebraicmethod, i.e., themethod of calculationswith letters, penetrates all ofmathematics. In fact, asubstantial part of the solution of amathematical problem often turns out to be nothing but amore or lesscomplicated¬Algebraiccomputation.Besides,inmathematicsweemployvarioussymboliccalculationsinwhichthe lettersno longerdenotenumbersbutsomeotherentities,wherethe lawsforoperationsontheseentitiesmaybedifferent fromthe lawsofelementary¬Algebra.Forexample, ingeometry,mechanics, andphysicswemakeuseofvectors,andasiswellknown,thelawsforoperationsonvectorsareinpartthesameasfornumbersandinpartessentiallydifferent.

Anyfunctiondefinesfirstthe4Dunitsofreality,whicharefractalpoints,T.Œs,whichareindistinguishableandappearinsimultaneousspace;hencecanbedefinedasnumbers;withconstantparameterscalledletters;subjectasarelative‘sœT’(ifweweretousethejargonofmodernmaths)toa‘partialequation’orstœpofaworldcycleinsequentialtimethatacteduponthemthroughthevariables.

SothemostgeneralcorrespondencebetweenGstrealityalwayscomposedofaspatialsimultaneouspopulationsubjecttoadimotionoftimeisimmediate;

- Constants(radicals)=Spatialpopulationsofs.œ.T

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- Variables (x,y,z)=Parametersofchange inanyofthefivetimedimotions,which implyalsoparameterstovalue social groups (population), languages that value degree of evolution, (i.e.money-prices), parameters oflocomotion(speed),entropy(acceleration,exponentialdecay),etc.- Equationcombiningwithoperandsvariablesandradicals,whichrepresentthoses.œ.T,followingthestœpsofaction,whichwillbeshownthroughoperands,ofwhich:

Wemustdifferentiatethoughastill,solved,space‘equation’(§E)fromavariable,temporal‘equation’(TE),averyimportantdefinitionsomehowblurredbythelackofaproperphilosophyofmathematics.

The first type are equations where we are concerned with finding a solution to the variable, ideally a singlesolutionbutaccordingtothefundamentaltheoremof¬Algebra,withasmanysolutionsaspowerstheequationhave,aresultofprofoundimplicationsforthestructureofspacetime.

Thesecondtypeofequationareequationsrepresented inaCartesiangraph,wheretheexactsolutionextendsthrough a newdimensionalmotion along all possible values thus giving us the ‘whole entropic range’ of timevalues,manyofwhicharemeaningless,as‘entropy’,theabsolutepotentialnumberoftimecombinationsintheUniverseisinfinite,afeaturemathematiciansandphysicistsdonotunderstand,busy-busytabulatingallentropicinfinitesofreality,evenasinthecaseofphysicsconsideringitthepartthat‘matters’ofmatter(whenthetrueelementtoconsideraretheinformativegalaxies,notthevacuumentropicspacebetweenthem).

Asthereisnotaphilosophyoforder,purposeandreferencetotheSTlawsofrealitymathematiciansthuswasteenormous time indisquisitionsabout infinities, valuesofequationsbeyond theparametricexact solutionsandget shocked when the ‘mathematical Universe’ shows entropic limits (i.e. no radical solutions beyond the 4thquarticpolynomial),butthat iswhatdelightsthemindoftheevolved5Dmathematicianswhosees inmathsaperfectmirrorof5DlawsonlycomparabletothatofGstitself.

Anotherobviouscaseisthatoftheirrationalrealnumbers,whichareinfinitecomparedtotherestofnumbersinincreasingnestedgroups:N->Z->Q.Butirrationalnumbersare‘entropicnumbers’withtheveryfewexceptionsofthose who correspond to fundamental ‘angular equations of perception’ (√2, phi, pi and other Universalconstantsofnote).TheothersmustbeconsideredpuretemporalentropyoftheinfinitePlanesoftheUniverse,meaninglessbeyond itsmetaphysicalmeaning: that timePlanesare infiniteandyet thosewhoaremeaningfulfrom the human point of view ‘reduce’ to Q (complex numbers, being dual numbers whose philosophy isconsideredelsewhere).

Thisisimportanttounderstand‘curvesofequationsrepresentedinCartesiangraphs’,whicharecontinuousbutthatdoesn’tmeantheUniverseiscontinuous,asthe‘meaningfulpointsofthosecurves’arediscontinuous–itsrelevantsolutions,thex=0,y=0,standingpointsand‘limits’withinthemeaningofthefunction.

This concept, the ‘entropic limits’ thatmake sense formathematics is completely lost specially since Cantor’sabsurdattemptstocountentropicinfinities;butithasamorefruitfulunderstandingin‘definite’integralsandtheneedfor‘limits’andinitialandfinalpointsofaworldcycletofindsolutions,asitalsohappensinitsapplicationstomathematicalphysics.

Letusthenconsidernowbrieflysomeaspectsofthoseoperandsasdimotions,nottryingtobeexhaustivebyallmeans,asthatisnotthegoalofthosepapers,justa‘first’steponar=evolutionofsciencethateitherhumindsorAIwilltaketoitsfruitionintheXXIc.butasinglemancannotcomplete.

WedosousingthetrinityandpentalogicviewsproperoftheentangledUniverse,asits‘polidimotionalfunctions’forallwhatexistswithinit,tryingtofindtheconnectionbetweentheabstractpropertiesofthe3relativePlanesof¬Algebraasamirroroftimespacestructure–firstitsdimotionsandnumberfamilies,thenitsequationsusingeitherthesimplexdimotions(polynomials) in its3Planes(±,x÷,√xª)orthecomplexoperands(∫∂andcomplexplane)…tocompletewitha largermore ‘professional rendering’ofanalysis thesecondclassicageof¬Algebra,which as all systems is the bestmore balanced and connectedwith reality age (in the case of ¬Algebrawith

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mathematicalphysics,whichbychoiceofitspractitionersandtheusualsuspect–humanegocy–deniesalldeeptimebio-organicproperties tosystemsofmatter; somethingmathematicsby theverynatureof itsabstractioncannotdo,explainingwhyphysicsissopoorconceptuallyandsogoodinpraxis.

Because sinusoidal functions deal with the shrinking of the world into a map of perception, they belong tointegral calculus of finitesimal derivatives - aswe explain in geometry, the angle and triangle came first likelybefore evenman learned to count just by the fact of opening his eyes to theworld. Sowewon’t treat thoseoperandshere.

¬ÆPOLYNOMIALEQUATIONS&ITSOPERAND:<=>±,X÷,√3

Firstdegreeequation.

Theallpervadinguseoflinealequationscorrespondtoanessentialparadoxofthelawofinversionbetweenscales,suchaswhenweemerge intoa larger scale, topology in space change from lineal to cyclical to lineal. Sowe canalwaysapproachacurvethroughlinealsteps(fractalmeasures,derivativetangents,andlinealequations);orgathera series of curves with a lineal envelope, smoothing its roughness). As lineal systems are deterministic andpredictable(alineceasestobealineifitacquirescurvaturesoitcannotchangedirection,unlikecurveswhichcan∆,∇oreven‘invert’curvature,hencetheyhave3degreesoffreedomintothefuture)…Linealfirstdegreeequationsareeasiertocalculate,andtheywerethefirsttoappear,(Greeklinealgeometry),andgavebirthtotheaxiomaticmethod–aberrant formore complex, lessdeterministic forms.Andwerealso the first tobe studied inanalysiticgeometry. Then French renaissance, broughtwith Viete symbols andwithDescartes numbers to expand the firsttrivialsolutionstopolynomialequationsoftheGreekandArabage,madewithgeometricequivalents.SoAnalyticGeometrymarriedwithAlgebrafliedaway.

Descartesmadeuseof twosimple ideas.Firstofallexaminedwhatcurvescorrespondtoanequationof thefirst-degree:

i.e.,toanequationwhereA,B,CarenumericalcoefficientswithAandBnotbothzero.

Aswehaveseenthisisthe'dimensionalnaturalgrowthfromsumintoasumsofsumsormultiplicationunderstoodintermsofitssimplestdimensionalcombination':$xD:

Descartes found that in the plane a straight line corresponds to such an equation.Andconversely,thattoeverylineintheplanetherecorrespondsacompletelydeterminedequationoftheform:

where: Whiley=kxrepresentsastraightlinepassingthroughtheoriginandmakinganangleϕwiththex-axiswhosetangenttanϕiskandLthedistancefrom0tothecrossingpointofthe lineandy.Thusas

usual the simplest, lineal $T element was discovered first, and found to bedeterministicandeasytocalculate,butinaworlddominatedbytimecycles,itwasonlythebeginningofalongadventurenotyetclosedin‘meanings’:

Sincethereisanenormousnumberofscientificerrors,includingthelinealbig-bang caused by the ab=use of linealapproximations to functions, and the'spread'ofthefunctionintothenegative'sideoftheline'asifitalwaysexistedorbehavedlineallycrossingintothepast-negativesideofthegraph.

In factmost 'real functions' that represent spacepopulationseither staticor reproducedas aradiationdepartingfroma0-1seminalfinitesimaldoNOThavemeaningfulnegativeregions–a

linguistic distorsion of the @-0 point of view, and the few that have it might represent a past-time negativememorialresidue,whichtendtohaveanexponentialdecay-likeform.

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Tonoticealsothedeterminismoflinealfunctions,as(Kx+c)’=Kand∫K=Kx+c;henceitistheonlyfunctionthatwhenweinterpretY’asapastfinitesimaland∫Yasapotentialfuture,anintegralgrowthofdimensiondoesnotopenupthefuturetodifferent+Cnewpossibilities

LinealFunctionsasrepresentationofTs,linealDimotions.

If you have capture the essence of 5D is self-evident that lineal functions must represent Ts-dimotions oflocomotion,energyfeedingandexternalgrowthofthetype|-$T-thatdescribesfields/limbs.Forexample,

If x is the time and y is the distance covered by a moving point, then the linear function y = kx + b obviouslyexpressesthefactthatthepointismovingwithuniformvelocityk;asaTsdimotion,andthenumberbdenotesthedistance,attimex0=0,ofthemovingpointfromthefixedzero-pointfromwhichwemeasureourdistances.

Linealequationsasapproximationsofcurves.

Thissaid,thelinearfunctioniseasy:l(x)=ax+bgivesthesimplestofallcurves,namelythestraightline;andyetitisoneofthemostimportantduetothefactthatevery“smooth”curveonasmallsegmentisastraightline,andthe less curved the segment is, the nearer it comes to a straight line. So Linear functions are extremely usefulbecause of their simplicity and because it is possible to consider non-uniform changes as being approximatelylinear,even ifonly for small intervals,which isadirectconsequenceof theGalileanParadoxesbetweentwo∆±!Plane,asacurvedgeometryismadeofsmallerlinealsteps,asmalllinealopen-freepathalwayscurvedintoa0’-timecycleonthelongterm.

Awholeness,isaclosed0’sumandhencecyclical,curved.Astepofthecurvehoweverappearsinsmalldistancesas a lineal, open step. So youmight say that in ∆-1 curves do NOT exist, and lineal steps of ‘freedom’ are itsperception,whichmetaphysicallyexplainswhyhumansfeelfreeeveniftheyareenclosedinsocialcircles.

In the language of the theory of the functions, this means that every “smooth” (continuously differentiable)functionis,forasmallchangeoftheindependentvariable,closetoalinearfunction.

The linear function can be characterized by the fact that its increment is proportional to the increment of theindependentvariable.

Indeed:Δl(x)=l(x0+Δx)−l(x0)=a(x0+Δx)+b−(ax0+b)=aΔx.Conversely,ifΔl(x)=aΔx,thenl(x)−l(x0)=a(x−x0)andl(x)=ax+l(x0)−ax0=ax+b,whereb=l(x0)−ax0.

But from the differential calculus, we know that in the increment of an arbitrary differentiable function isproportionaltotheincrementoftheindependentvariable,andthattheincrementofthefunctiondiffersfromitsdifferentialbyaninfinitesimalofhigherorderthantheincrementoftheindependentvariable.

Thus,adifferentiablefunctionis,foraninfinitelysmallchangeoftheindependentvariable,reallyclosetoalinearfunction to within an infinitesimal of higher order.The situation is similar with functions of several variables.Alinearfunctionofseveralvariablesisafunctionoftheforma1x1+a2x2+···+anxn+b.Ifb=0,thelinearfunctionissaidtobehomogeneous.

A linear function of several variables is characterized by the following two properties:1.Theincrementofalinearfunction,computedundertheassumptionthatonlyoneoftheindependentvariablesreceives some increment while the values of the remaining variables are unchanged, is proportional to theincrement of this independent variable.2.The incrementofa linear function,computedundertheassumptionthatall the independentvariablesobtainincrements,isequaltothe¬Algebraicsumoftheincrementsobtainedbychangingeachvariableseparately.

Thus a linear problem can be characterized by 2 properties:1. The property of proportionality. The result of the action of each separate factor is proportional to its value.

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2.Thepropertyofindependence.Thetotalresultofanactionisequaltothesumoftheresultsoftheactionsofthe separate factors.Thefactthatevery“smooth”functioncanbereplacedinafirstapproximationbyalinearone,forsmallchangesofthe variables, is a reflection of a general principle, namely that every problemon the change of somequantityundertheactionofseveralfactorscanberegardedinafirstapproximation,forsmallactions,asalinearproblem,i.e., as having the properties of independence andproportionality. It often turns out that this attitude gives anadequateresultforpracticalpurposes(theclassicaltheoryofelasticity,thetheoryofsmalloscillations,etc.)

Dimensionsofapolynomial'backgroundspace-time'

@nalytic geometry allowed amore clear representation of those polynomials NOT isolated as equations but inreferencetotheplaneinwhichtheyareentangled.Thisisimportantnottoconfuseitsmeaning.Apolynomialina3Dworld, of the type x³ is a cube. The same functionbelow,hasnothing todowitha cube, but it is a curveofgrowthdiminutionandgrowthontheY-coordinates.

Thus the 'perspective' of graphic polynomials is that of time not of space or one of its combined 'holographic,bidimensionalforms'oftimespace:

In thegraph, repeatedadnausea, for clarity,we see thegeneral rule:abidimensional St systemof information(stillspace),oratimeclock(movingcycle),the3rddimensionofreality;andavectoroflinealtimemotionoritsbidimensionalsheetofspatialdistances,the1stdimensionofreality,cometogetherintoathe2ndDimension,STsystemofenergy,ortimeormotion.

3Dimotionalsystemstendtobetheintersectionofalineandcycle,whichingeometryisexpressedbytherulethatalmostallfunctionaldominantformsoftheUniversecanbetracedwithaline-rulerandacompass-cycle.

Generallyspeaking¬Algebraisthenjustamirrorimageofthegeometryoftheage,speciallyinthecalculuswithPythagorasliketheoremsofsquarerootsandthesimplest±,≈X÷operandswithoutconsideringthemirrorimageofthoseoperandsinother∆stelementsandsymmetriesoftheUniverse.

Fermat'stheorem.

Howcanthendifferentiatepolynomialsinspace-timesystemsfrompolynomials,whichare'spatialsums'?

Aneasyconcept isthatofthedifferencebetweensumandproductoperands.Aspace-timesystemisdefined inproductterms,asumisofthesametypeofbeing.Wead'equalspecies'.Andthishasanunexpectedproof, inamargin.

Its means that that when we ad we superpose, so a 3rd dimension expressed by a power law is no longer 'aholographicsuperposition',reasonwhytheFermat'stheorem,X³+Y³≠Z³,doesNOTwork.

Indepth the superposition rule impliesmoregenerally thatthe full consistencybetweencontiguousdimensionalgrowths (±) breaks between discontinuous dimensions, from1D (sumof herds) to 3D (mergingof 2 holographicbidimensionalsheetsintoathirdonethroughproduct):

Inthegraph,theholographicprincipleisexpressedbytheoperationofaddition,whichisallowedbysuperpositionintoatridimensionalvolume.

Yetasthereisnota4Dimensioninthesamescaleofspace-time,theruleofsuperpositionthroughanewDimensionbreaks forsuperpositionofcubes,whichwouldhavetobeadded in thissupposed4Dimension inasingleplane,reasonwhywecannotaddthem(Fermat'stheorem).

The holographic bidimensional universe and its ternary ST-geometries define reality. So in most mathematicalequationssolutionsaboundonquarticandcubicsystemsbutonlyspecialcasesaresolvableforhigherpolynomials

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or have any real use in reality; the exception being simpler equations of the ∆§ocial Planes and reproductivefunctionsofthetypeXª=b...

Quadratics are the masters of the ¬Algebraic game, the most abundant an common of all forms inexistence,becausetheUniverseisbidimensional,andsoitisinformation.Soquadraticequationsbydefinitionarethe perfect form to show the properties of the Universe of fractal bidimensional space and informative,bidimensionaltime.

The bidimensional holographic principle explains why in geometry (greek bidimensional plane, which provedalmostallthetheoremsofgeometry)and¬Algebra(quadratics)almostallphenomenaofthePhysical,topologicaluniverse canbe 'carried' on toquadratic ¬Algebraic equations.We study themain formsof quadratics in thoseother'parts'.

Itsadditioncanbe forced-fed (4Dspatialisationof the timedimension) inmodernphysics,butasEinsteinput itlinealtimedoesn'ttravelbackwards...indeed,ifitdoessoitbreaksintoanentropicexplosion,loosingitsinternal∑∆-1bondingbyawhole.Sowecannotgetconsistentresultsforanadditionofcubesinasinglespace-timeandinversely for its dissection, reason whythe addition of cubes cannot either be resolved in bidimensional planegeometrywithaline-ruleandacycle-compass.

AllthismeansessentiallythatapowerlawisconcernedbeyondtheX³cubewithgrowthin∆-planes;andthatisalsotheultimatereasonwhyasAbelandGaloisrealizedPolynomialsofhigherorderthan3areNOTsolvablebyradicals (whichessentiallymean 'additions'andadditionsofadditions -multiplicationsasasumofsums),unlesswecanbreak-reducethemtolowerdimensionsortheconsistencyofthepowerlawisextreme(casesofthetypeXª=C,wherethevariableiscompletelyalone,hencewithnosums,meaningoftenmerelyalogarithmicgrowthofscale, as a herd,NOTof dimensions I.e. 10¹° ismerely a society of 10 billions, stillwithin the classic range of asocietyofsimilarpointsinasingleplane.

Weextractthetruthofanysystemapplyingtoitanentangleddual,trilogicandpentalogicfromtheperspectiveofitsS=Tdualitiesandtrinities,anditspentalogicelementsasrealorvirtualmirrorof¬∆@st.

∆§ocialevolution:Polynomialbuildupofstructuresthatmimicthecomplex,scalar,simultaneoustrilogicformsoftheUniverse,polynomialstakeustothecompletionofthepositivearrowsofsocialgrowthandreproductionofitsfamiliesofscalarnumbersandrelatedoperands(N,Z,Q,±,x÷,Xn)buildingupacomplete‘biased’viewofrealityin terms of its social Planes, without the complex arrows that transit and deform reality between planes ofexistence(angularperceptionoftrigonometric functionsthatshrinks inmindmappings;calculusthatextracts itsfinitesimalpartsandgatherthemincomplexwholes).

¬Entropic limits: The polynomial thus is simple in internal structure and cannot be used to represent complexorganic forms, butworks to its perfection in the analysis of disaggregated herds,where counting of individualsgatheringingroupsmatterandcanbereferredtoaunitofvalue,the‘cohesion’languageorparameterwhichallthose elements have in common– from chemical reactionswherewe ‘count’ reactants, to problems in units ofvalues,volumes,prices,distances…

Sopolynomialshavelittleinternalstructureinthecountingofindividualstoformthoseherdsandaggregatesandthispoisesthefirstlimittoitsmeaning–astheydonotpermitmorecomplexusesoftheoperandsofre=product-ionandexponentialdecay–limitingwhatpolynomialsmatterandwhatarefictions.ApolynomialasareflectionoftheUniverseisasimplerepresentationofasuperorganismthroughits3±¡Planes.Asanaccountof itsformsinx∞,x¢….tillxeachscaleofhomogeneouselementsissupposedtogrowththroughpartsandwholesforeachnewpowerlawandassuchitistheaccountofallitsforms,CXtotal,inthenumberofPlanesgivenbythepowerdimension,n=Planes.

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Butasanorganismisnotexactlyrepeatedineachscale,thepolynomialsthatreflecttheUniversemustdiminishorincreasethepatternsofeachscale,withsummandsforeachscaletomakethehumblermoreprecise.

InthismannerapolynomialasamirrorofarealUniverseappears.

But what x represents? The simplest response is the number of cells between each scale, so the polynomialmeasuresthetotalnumberofwholes.

Nextintheunderstandingofrealitycomestheconsiderationofoperandscombinedtoformfunctions.Andhereitcomesthequestiononhowtogroupthem.

As it happens there is a natural groupingof thoseDimotional operandsbased in its complexity and thenaturalrelationshipof theDimotions=actionsofexistence,suchaswegroupnaturally the±,x÷andXªoperands,whichrepresent a 'series' of simpler operations in a single 'parameter-variable' in polynomials, therefore the simplestfunctionsof'existence'(aswemeanbyexistence,theconsecutivesumofactions=dimotions,representedbythoseoperands).

Astheyareclearlyrelatedtoreproduction,socialevolutioninasingle'lineal'planesofspace-time.

Thisleavesthemoreproblematic,complexDimotionsof'angularperception','exponentialdecay'anddifferentialcalculus,whichclearlyrelatetoMotionsthroughplanesofthefifthdimension,asoperand,whichnaturallygrouptogetherinfunctionsnotablyofmathematicalphysics.

So polynomials are simple functions of social reproduction, and it bears witness to the rich complexity of thepentalogic Universe that even such simpler forms in a single plane can approach all other functions (Taylor'sseries), when we enrich them with some of the complex scalar Dimotions of angular perception (¡-3>¡o),exponentialentropy(¡<¡-2)andlinealderivatives(¡<¡-1),whicharethenaturalDimotionsofscalarspace-time.

Theegoparadox,linearityandthe3rdageofentropy

Allthissaid,forfullyunderstandingmathsandanyotherlanguagemirrorofGstlaws,obviouslythereisnowaythereaderwillmakesenseof it,asa ‘whole’ interconnected,entangledrealityunlesshegraspsthebaretheoreticalmeaningof5Dspace-timefromwhereallthelawsofmirrorlanguagesderive.ThisisthebiggesthandicapIhaveconfrontedallmylifeandthereasonsonmanyquipseveninthislastattemptto‘clean’myactofmyownegocy…Thiscomestothereasonwhymathematicians intheXXc.andbeyondwastedsomuchtime(tilltheystartedtobuild themetal-mindwith simplerBoolean¬Algebra),with the ‘entropic’beyond limitsof reality, as all systems‘keepitstimeevolution’beyondthe‘preferred’classicageofharmonywhenthemirrorcorrespondsstrictlytothe‘solutions’thatmakesense inrealitybecausetheyarebalanced,S=Tsolutions.Astheyage,more informativeStsolutionsarefound,whicharecomplicatedandoftenredundantas3rdagewrinklesare,butstillhavecontactwithnature.Butoncethisisalsoexhausted,they‘decay’intotheinfinitepotentialtimeentropyofallcombinationsthelanguageallowsbutNatureeliminatesassoonastheyareformed.

Entropygrowsassoonexponentiallyaswecomeoutfromthes=Tbalanceregion.Inparticles,theneutrondeadstategrowsaswemovefurtherdowntheage,inthebodythenumberoftailofmemories,andwarpingrepetition,whichareentropyasitisnotnewinformation,inmathematicsaswegotheinfinity

PENTALOGIC

Polynomials must then be understood as everything with pentalogic that is, in its multiple functions andapplicationstoalltheelementsof¬∆@ST,accordingtothevaluesoftheX-variable.Sowetalkof:

SPACEPOLYNOMIALS:Whenthex-variablerepresentsadimensionofspace,point,line,planeandvolume.

TIMEPOLYNOMIALS:Whenthevariablerepresentsamotionoftime,distance,speedoracceleration.

∆-scalePOLYNOMIALS:Whenthevariablerepresentsascalarfunction,inseriesofdiminishingvalues.

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¬ Entropy polynomials: when the variable represents a scalar function, in a series of growing value, or anexponentialform.

Nowofallthosepointsofviewofpolynomials,somearemoreusefulthanothers,asPENTALOGICALWAYSforcesanentityorevent in someof those '5Dmotions',whicharenot itspurpose though theywillbemirroredbyallDimensions-i.e.awomantendstorejectentropyandpreferinformationbutitdiesalways.

Sowecould'state'thatthe5thdimensionsofexistencetouchany'wholesystem'whichis'coherentwithinitself',inthecaseoflanguagesasamirroroftheworld.

Thissaidthemostimportantmirroranysystemcanrepresentisaworldcycleofexistence,whichinitselfincludesalltheotherdimotions.

SoweshallconsiderhowPolynomialsinaplanerepresentthatworldcycletoo.

Polynomials not only can 'approach' as amirror any other function andDImotion of theUniverse,with certaindistorsionsasthecaseofthemaclaurinseriesapproximationstocalculusshows,butmanyofthemhavenothingtodowithreality, showing indeedthe inflationarynatureof languages,whose internal structuremightormightnotberealorfiction,andsosinceGodelweknowtheycannotprovereality.

Equationsthereforemustbealwaysputinrelationshipwiththerealitytheydescribefirst.

THESOLUTIONOFPOLYNOMIALS

Abstract. The solutions of polynomials as so many other re=solutions of problems follow a simple rule of the5DimensionalUniverse:asolutionnotfoundinagivenplaneofexistencecanbefoundfromtheperspectiveofahigherDimotionalview.Henceforallthesolutionsofapolynomialweneedacomplexplane;formanysolutionsSolutionsmatterbecausetheyconnecttheinflationarylanguagewithreality.SomethingCantorforgotinhismadhouseandwiththehelpofHilbertmanyothermathematicians.

In a larger cultural view of course we live the entropic age of mankind and so entropic solutions pass asmathematics, art (white sliced canvas – the final work of tachism; goremovies, the basicwork on film today),history(entropicdog-eat-dogsociety),childhoodeducation(virtualscreens,hatememes,internet),music(randommusic;electronicmusicbeyondhumanearrangebettersuitedtostringandwood,repetitivebeatswithminimalinformation),selfies…butentropydeath,the infinitereceptacleofdisorderedtime isnotwhatamindthatseeksordervalues.

Even in equations of polynomials there are arbitrarilymany special forms of equations of any degree that aresolvableinradicals,andmanyofthemareexactlythoseequationsthatareimportantintheapplications:

-Somecan‘transcend’Planes,iftheyareperfect,‘laser-like’symmetricingrowth,orcyclicalinform,asthosearethe laws required to transcend and emerge betweenPlanes. So all the binomial equations xn = A, inwhich thesystememergesequaltothepreviousx-formaresolvable.

So are in a single scale those who form a perfect cycle - a broad class of such equations, the so-called cyclicequationsandstillmoregeneral“Abelian”equationsandthoserequired intheproblemofconstructionbyrulerandcompassofregularpolygons,whichGausscalledcyclotomicequations,i.e.,equationsoftheform:

wherep isaprimenumber.Heshowedthat theycanalwaysbereducedtoachainofequationsoflowerdegree;moreover,hefoundnecessaryandsufficientconditionsthat

suchanequationcanbesolvedinsquare‘bidimensionalST’roots.

RECAP. Thesolutionsofpolynomialsare important to ‘define’whatpolynomialsarepuremathematicalentropywithnoreferencetoreality,andwhichonesaremeaningfulbecausetheyfollowprinciplesofGst,amongwhichisremarkable the fact that ST dimotions are bidimensional; hence a key method to solve any polynomial is its

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reductabilitytosquareroots;andtheUniverseispentalogic,withthe5thdimotionofentropy=lossofinformation,hencepenta-polynomials(quintics)arenotsolvable.

Thefundamentaltheoremof¬Algebra.The∆@stsymmetriesreflectedbypolynomials.

The fundamental theorem of ¬Algebraproved by Carl Friedrich Gauss in 1799. It states that every polynomialequationofdegreenwithcomplexnumbercoefficientshasnroots,orsolutions,inthecomplexnumbers...

Butwhythereareforanypolynomialofpowerx,xsolutionsintherealmofcomplexnumbers,whichimpliesalsoabetterunderstandingofacomplexplane,sofarlackinginmathematics.

Andtheanswerisasprofoundasgeneralin5Dmathematicsasitisin¬Algebra:

"EachrootofapolynomialrepresentsaDimensionalorscalarmotionofspace-time."

Andsobidimensionalpolynomialsrepresentholographicfunctionsofspace-time.Tridimensionalpolynomialsareternary representations of 3 symmetric scalar, space and time elements and beyond the fourth 'element'entangledintheUniverse(symmetrybetweenspace,time,scaleandmind,whichcanberepresentedbythe'samevalue numbers', hence by polynomials, there areNOTmore symmetries of reality (entropy being the denial anerasingoftheother4elements)whichmeansbasicallythatinthesamemannerwedoNOThaveuseforintegralsand derivatives beyond the 3 Planes of a super organism, we do NOT have use for polynomials beyond the 4symmetriesoftheentangledUniverse.

@nalyticgeometryexpands¬Algebra

¬ALGEBRA really starts to understand EQUATIONS with more sophistication that geometrical equations, whenmotionparametersareintroducedbyanalyticgeometryandmathematicalphysics,astheXandYcoordinatesarenow used for t-motions and S-pace (as in the simplest physical equations of space-distances made at certainspeeds)thatitcanmirrorS≈TSymmetries,firstinasubconsciousformthroughequationsbornonpraxis,theninthe3rdagewithsomedeeper insightsontheconceptofsymmetry(grouptheoryand itsapplicationtophysics)whichweshallcompletewithitsfullcausalrealisation-sincewesaid'¬Algebraunderstands',notthepeoplethatmake¬Algebra.

TheclassicSTageof¬Algebrathussawthetransformationofpurearithmeticsofnumbers intoamixtureofthe∆STelementsofmathsasafullreflectionofthe∆•stuniverse.AndthetwohugefiguresthatdiditproperlywereDescartesandLeibniz(FermatandNewton,inparallelbutwithoutpublishingandthesameclarity).

Itisthegolden,classicageof¬Algebraandanalysis,beforeitentersthe3rdageofexcessiveinformation(attemptstoputalltheinformationinasinglemindmappingwithgrouptheoryandfunctionalsand§ets).

Westudymostofanalyticgeometryinthepostdedicatedtoit.Sogoeswithanalysis.Hereweshallmakejustafewconsiderations.

ItallstartedwiththeparallelevolutionbyVietteonwardsofsymbolictermstoconcentratenolongerinnumbersbutinoperands,thetrueessenceofS=Tsymmetries...

Descartes:mergingalltheelementsof∆@s=tmaths.

...And Descartes idea of representing solutions to equationswith a larger dimension - the variable letter thatrepresentedallthe'§ets'ofdualX,Ypossiblesolutions;andto'imagine'theminagraphtoplotthem,formingavisual 'in-form-ative' geometric figure, the new 'scalar dimension' that gathered all the X(S)<≈>Y (t) pairs ofpossible'variations'onthespace-timeconstruct.

FURTHER on, he introduced,@, the point of intersection of the coordinate axes, having coordinates (0, 0) andhencea'p.o.v.'orsingularity.

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Andwiththeintroductionofcoordinatesheconstructedan“arithmetization”oftheplane.Insteadofdetermininganypointgeometrically,itissufficienttogiveapairofnumbersx,yandconversely.

UptothetimeofDescartes,wherean¬AlgebraicequationintwounknownsF(x,y)=0wasgiven,itwassaidthatthe problemwas indeterminate, since from the equation it was impossible to determine these unknowns; anyvalue could be assigned to one of them, for example to x, and substituted in the equation; the result was anequationwithonlyoneunknowny,forwhich,ingeneral,theequationcouldbesolved.

Thenthisarbitrarilychosenxtogetherwiththeso-obtainedywouldsatisfythegivenequation.Consequently,suchan“indeterminate”equationwasnotconsideredinteresting.

Descarteslookedatthematterdifferently.Heproposedthatinanequationwithtwounknownsxberegardedastheabscissaofapointandthecorrespondingyasitsordinate.Thenifwevarytheunknownx,toeveryvalueofxthecorrespondingy is computed fromtheequation, so thatweobtain, ingeneral,a setofpointswhich formacurve.

ThedeepestinsightonwhatDescartesdidisthenevident:

He gavemotion=change to geometry, adding its time-dimension; and so itsmethod couldbeused to study theactions/motionsofa'fractalpoint'whoseinnergeometryofsocialnumberswasNOWignored,inthe∆+1scaleofitsworld..AndsothegraphwouldbeaperfectgraphtostudyalltheACTIONS=MOTIONSexternaltoagivenbeing,becomingforthatreasonthefoundationalstructureofmathematicalphysics.

Thisisoftenforgotten,asSandTdimensionsareillunderstoodsoastandardbookdefinesitas:

"Analytic geometry is that part ofmathematics which, applying the coordinatemethod, investigates geometricobjectsby¬Algebraicmeans."

Notso...evenifinanalysiswewillfindthatthecurvesDOrepresentkeyfeaturesofthe'arrowsofchange'oftheUniverse, specially the 'standing points' of change of parameters of Space=Information, ST=energy andTime=entropy (or any other kaleidoscopic combination of ST), in essencethey represent theworld cycle of theactionormotionwestudy,withits3phasesofstartingmotion,steadystate,and3rdinformativeagecomingtoahalt.

Itmustbethenunderstood,asevidentasitis,thattheRashomoneffectshouldconsiderdifferentperspectivesonthosecurvesandformsfoundinanalyticgeometry,expressing¬Algebraicequations:

-Temporalview:thecurvesarethenmeaningless inspace.Whatmatters istheir 'socialdimension'thatresolvessymmetries between timedimensions expressed by the two variables often a parameter of space that changeswithadynamicfunction/action/motionintime.

-Spatial view:It isstill thoughpossibletocreatemeaningfulclosedforms,∆+1wholesofgeometry,madeof∆-1points,and then thegeometryallows to resolve¬Algebraicallygeometric spatialproblems,with 'adualpointofview' that increases the easiness of solutions - as Descartes proved easily and Galois completed, showing the¬Algebraiclawsofsolutionofruleandcompassgeometricalproblems.

-S=Tview:whenoneoftheparameters/dimensionsisfixed,belongingtospaceandtheothertoatimemotion,themostfruitfulinsymmetry,soonusedbyGalileoandNewtontodevelopthelawsoflinealtimemotioninspace.

-@views:developedas3differentmappings,willdevelop(O-polar,|-cylindricalandST-cartesianproper)

Scalarview,whichwillhavetowaittillLeibniz,treatedinAnalysis.

Equationsofpolynomials.

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Polynomials seem fairly straight, but as soon as we consider its solutions and varieties, things become morefascinating.

First we have 2 mysteries of 2300 years, one which people hardly wonder - why there are 2 solutions to allpolynomials,andtheotherwhichallmathematiciansdobuthaveneverbeenexplained-whyoneofthesolutionsisoftenanimposible,imaginaryi-solution?Whatisthemeaningofimaginarynumbers?

Whypolynomials'∆-Planes'have2solutions?

TheanswershouldbeimmediateforanyonewhohasunderstoodanythingaboutGST(hopefullymorethanthosewhounderstoodEinstein-I justrecallhisconferencesatSolvay,whenhequippedtothesamequestion,'maybethepriest'-Lemaitre):

Asallpointsinmotionhave2directions,upwardsanddownwardsinthe∆-dimension,left-rightinlinealgeometry,morespaceormoretimeinexistentialtopology,youthoroldage(rebirth-reproductionorinformativeevolution).

Thusthereare2solutions,sincetheequationdoesnotincludea'choice',whichwillbemadeaposterioribasedinexperimental evidence.And so it is evident, as space is larger, and equations quantitative, that in generalequations,thepositivesolutionisspace-likeandtheimaginarysolution,time-like.Andweshallelaboratealotonthatthemealongthepostsofthe3rdline.

ThepointtounderstandtheunderlingstructuresoftheUniverse,writteninitsi-logicmathematicalequations,ofcourse istorespondtoquestionsneveraskedfor,astheyseemtobethe 'apriori'humancategoriesofthought(Kant), which Schopenhauer rightly reduced to 'sufficient reasons', space, time, causality (mostly caused by ∆Planesinspace-time)andthemindthatperceivesall(languages,ternarygrammars).

Thefundamentalmathematicalsolutionisthepolynomialdoublerootsolution,wherethepolynomialisafunctionofpresentandthe2solutionsa±splitfields.

Presentsthussplit intwodualsolutionsa±dualfieldwhichisessentialtounderstandalltimeofequations,fromtheequationofdeathwherethecomplementarybody-head,field-particlesystemsplits,totheinverseequationsofsocial symbiosis in systems that plug-both past and futuresystems into a present vital form of which themostimportanthappensinsocialsciences.

Polynomialsolutions.

Inotherwords,apolynomialcanrepresentaT.œasdustofspacetimeinitsfourfundamentalelements,whoserootsbeingsymmetrictoeachother,aselementsoftheentangledUniversecanpermutatetofind'realbeings'ofthe game of existence, even if the concept of polynomial is an absolute abstraction of its purest, simplestproperties-anumber.

Asthepolynomialrequiresthe'equality'ofthoseentangledsymmetriesinpraxisitreallyworksbasicallyforthe3expansions of classic dimensions of space and classic dimensions of physical time (distance, speed andacceleration).

It is then farmore enticing the study of the 'transcendent' operands of analysis, and a simple example of thedifferencewillsuffice:

Whenweworkwithapolynomialofahigherpowerwemerelyincreasexbyanotherproductwithx,orinverselywemakearootthateliminatesanx.

Inthe∫∂operandshowever,theresultsaredifferent:∂x³=3x²,givesusinfactoftenalargersumforsmallerx...i.e.ifx=2,x³=8and3x²=12...

Thenat3,bothareequal,27,andbeyond3,thepowerfunctiongrowsfaster.Why?

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Trinitymeansrealityinaplaneisaternarygame,sowhenwegobeyond3ourunderstandingrequiresmoresubtlearguments.Below3whathappens isthefollowing:thepolynomial isworkingonasingleplaneanddoesmerelyevaluatethe3dimensionsinspaceortimemotionsofthatplanewithouttheneedtoincludethesupportofanylowerplaneofexistence,whoseinfinitesimalpartsmustbelargerthanthewholealone,becausetheysustainthewhole.

Indeed,whenweevaluatean∫∂operandweareevaluatingtogetherbothPlanes.The∂valuesthetotalnumberof'finitesimals'whichmustsustainthelargerpowerwhole.

So the∂x³=3x², is 3 times x², the lineal √root, because from that sumof infinitesimals, youneed to sustain thewholethatemergesfromit,withoutdestroyingthepartsthatsustainit,andwithouteliminatingthepartsofthepartsthatsustainboth.Wearethusmakingacalculusofthe∆±1ternaryPlanesoftheorganism,andeachoneisofthesamevalue:x²+x²+x².

Andyet,thereareNOsolutionsgrantedtoaquinticfunctionfromradicals:

Because they have an odd degree, normal quintic functions appear similar to normal cubic functions whengraphed,excepttheymaypossessanadditionallocalmaximumandlocalminimumeach.

STInterpretation

If we consider polynomial, dimensional equations, of D-egree equal to its number of dimensions,it is thenobviousthateachcontinuousscaleofspace-timedoeshave4Dimensions,whichcanbesolved.Yetbeyondthose4Dimensions the system 'breaks' intoanewscale,which isno longer in the same regionof existenceandhencecannotbecalculatedwiththeradicalsofthesameplane.

Further on in the graphic, we can easily see that the 5D form can be 'reduced' into a 3-equationwhere the 3central'st'hyperboliccurve,becomesaaroundthey=0pointasingle'higher'whole,which'ifreduced'toapoint,convertsthequinticintoa'ternaryequation'.Thusa5Dpolynomialisa3D∆+1polynomialwithanSTcentralpartwhichdeploysitshyperbolicminimalandmaximalelementsinthe'higherplane'regionofthe5thdimension(trulyahigherplanewhenweusecomplexnumbers).

Whatthenitmeansingeneralterms,theexistenceofseveralsolutionstothoseequations?Inthemostabstractanalysisofpolynomials,ifwedeemadegree,D,adimensionD,thenitisobviousthataswesolvethemwecomedownfromthewholeintotheparts.Thusaquadraticbidimensionalequationresolvesintothe±inversespeandtiƒ functions of the system. A cubicequation will solve into the Ts<ST>St solutions/parts of the system; and aquadraticequationintotheS,E,Tandsub-dimensionsofthesystem.

∆-interpretation

Butofcourseapolynomialbeingthemostabstractrealisationofa∆STfractalequationcanbemanythingsandyetthe beauty of it is that all find interpretation. So when we consider x to be a function of time, it is given usthearrowof∆-1informationwhichhasseveralvalues,asinformationincreasesdownwardsdiversifingthesystem,but the inverse is NOT truth.While St=x² has 2 Spe and Tif roots, the square of a number is a single number,ratifyingtheinversearrowsofentropyandinformationupwardsanddownwards:

upwardsthepolynomialloosessolutions,downwardsincreasesitsinformation,normallybreakinganstwholeintoits$Tandð§components.

Quadratics:openandclosed

In the holographic universe then themost abundant of all equations arequadratic equations. We observe 3

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typesincorrespondencewiththe3topologiesinasingleplane:

-O-closed are clock-like ones, transcendental polynomials with a cyclical form, which tend to defined 1D-3Dimotions.

-|-Openareontheotherhandquadraticsrelated2-4Dimotions.

-Ø-Finallywefind,transcendental,hyperbolicsinusoidalforms,orstequations.

Cubics.

A cubic tries to represent inaquadraticplane3dimensionsof space (3 topologies)or time (3ages). And so there are severalquestions to inquire about this mismatch between the realfunction’sdimotionsandthe limitsof the ‘bidimensionalmentalpaper’ inwhichtheyaredrawnthatbecomesevenmorepressingforhigherpolynomials:

Howcubicsmightrepresenttheternarystructureofthefractalgeneratorinonly2stilldimensionsofspace?

What distortions take place when we transfer the holographic Universe into cubicrepresentations?

What are the restrictions the Universe imposes to ternary systems, wherethe 'messedbidimensional, tight form' becomesa loose configuration of layers of time-motionor space-density to becomea3rddimensional'wide',iteratingbeing.

Forexample,theFermatGrandtheorem,mentionedabove,x³+y³≠z³meansarestrictionof'tridimensionalmessedbeings',asallwhatwecanexpectarebidimensionalperfectforms,accumulatedintime,asslicesofamotion,orinspaceaslayersofanidentical-number/populationofbidimensinalbeings.

And finally, the translation of all the results of cubic equations,once we understand the previous rules ofengagement,intomeaningfullawsof∆St(orviceversa)

Dualityonpolynomials:Spatialsolutionsastheintersectionof2conics:reproductivemergeofST-bidimotions.

Because polynomials are about the ‘merging’ operands of social evolution of herds (sum) and entanglement ofbidimensionalDimotions(ST),animportantproofofthemathematicalmirroroverthe5DUniverseisthefactthatthe first solution to a cubic was indeed the combination of the two fundamental ST forms an open ‘Ts curve’(parabola)andaclosedSt-curve(thecircle),foundbyapoetwholovedreproductivesenseandspiritualwine(:

The interestof thesolutionofKayyamis tocasta3Dimotional functionasthemergingoftwodifferentconics,oneS-Tclosedconic(thecircle)andoneopen (the parabola), thus forming a 3D system, according to the fractalgenerator, S<ST>T. In non-E geometry we already studied the ‘limitednumbersof conics’ as the representationof the5Dimotionsof reality. SocircleswereSS-conics,Hyperbolas,TT-conics,Parabolas,Ts-conics&ellipsesSt-conics.NowweseehowitscombinationsgiveusST-reproductiveconics.Sowecanseethetwolayersofthe

cubic,asthetwocomponents,S-circleandT-parabolaand itsreproductivecombination, ST, forming a complete ‘T.œ’ as defined by the fractalgenerator:

S(circle)≤ST-cubic≥T-Parabola

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Ontheotherhand,seriousthinkerscareforwhatrealityismorethanitsimaginaryunstablethoughts.Sowefindthatbeyonda3rdderivative,anda3rdpower,thatis,trinityplaneofexist¡encehardlyanythingmatters.Moreoverasplanegeometryshowfurther‘polynomials’aresolvedbyreductiontothose,andwhengraphed,theyarejustsimilartothesquare(allevenpolynomials)addingbumpsandtothecubicfunctionadding‘inflationary’,‘warped’,‘3rdage’bumps.Because theyadmere ‘details’as itdoes the ‘normalization’processesofaKundoeffectwhichadd‘contributions’for∆±|>3|scalesofNature,whicharehardlyperceivablebya∆ºmind.

Whatisthenthemaindifferenceofsuchpolynomials?Wecantalkofthemintermsofspace-timesymmetries,aseven functions, which are symmetric along the ± axis and asymmetric odd functions, which have an invertedsymmetryonthe±sides.Andconsideragainthepolynomialsoforder4asthelimitofsolubility(Galois)as4areindeedtheelementsofreality-letuselaborateabitmoreonthosefirstprinciplesof5Dpolynomials:

Thesolvablepolynomialsdefinealimitof3dimensionsofspace-time,anda4thandfifthdimensionsthatcannotgenerally be resolved by radicals in a single Plane, hence belonging to the 4 D S∂ and 5D ∫@ dimensions,whichwarpthewhole,asitemergenceintothefifthdimension,hencemakingimpossiblefurtherdimensionsinasinglecontinuum.Noticethatthefourthdimensionalgraphmustbeinterpretedoftenturnedupsidedownasitisa'death'invertedtimearrow.

Theydefinethebidimensional,tridimensional4dimensionaland5dimensionalarrowandsimilarsystemsinthosedimensions.:Thefirstisameremotionofanst-linealtrajectoryproperoftheconceptofspeed(1D)or3D(closedcyclicalvortices.Thethirddimensionthoughismoresophisticated.

Inthatregardaconceptofinterestto5Dalgebraisthatofa‘primepolynomial’,thatis,anefficient,fundamentalSTpolynomialthatcannotdecomposedinsimplerS=Tforms.

Polynomialsofhigherdegreeresemblethefunctionsoffouriertransforms,whereasweincreasethenumbersthewavebecomebyadditionclosertoa‘square’form.Whatthistellusessentiallyisthatthe‘elementsthatmatter’ are those of the ‘ternary scales’ represented by the interval X1-3 and most other factors are‘perturbations’withalimitedrangeofinfluenceorlimiteddomainofapplication.

SoapentalogicuseofPolynomialsasa3rdreproductiveDimotionisthis:

“Polynomialsofhigher,>2degreearecombinationsofsimplerbidimotional-TSsystems”

Immediatelyfollowsthat‘meaningfulcubics’areStxsT=STsystems.

Meaningfulquarticsare(S-T)2systems,oftensolvedbyz=x2substitutions.

Beyondthattherearenomeaningfulradicalsolutionsforgeneralcases.

Inthegraph,¬AlgebrastartedasaprolongationofarithmeticsandgeometryinitsfirstagemakingtrueGermain’sdictumthat‘algebraisawrittengeometry’.Thisfirstagewentasfarastherenaissanceandthebirthofanalyticgeometry,andconsistedincalculusofadmaximalcubicrootsandgeometricproofs(greeks,al-Joarizimwhichgaveitname,etc.)

Thegraph shows themaximaldepthof this agewhen thepoetKayyamsolved the simplest cubicequations. So'cubic'dimensionalareintersectionsbetweentwofigures,henceanS(x)≈T(x)symmetry,inthiscasetheparableandthecirclewhenproperlywrittenasKayyamdid;bringingthenthekey‘positive’spatialsolutiontothetable.

WhereastheothersolutionsforthehighlysymmetricX3+a2x=bare0’and–b.

Time-space solutions however are often hidden as so many mathematical S≈T symmetries by the 'mania' ofscientiststofindonlythesolution,andpackingallthevariablesintooneside,puttingintheothera0’...Whentheexistence of a 0’mostoften means the unperceivable infinitesimal, the whole method brings some interesting

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conclusions.ButthemostlikelyNaturalformisapossiblesymmetrywhenbothsidesoftheequationparttheirwaysintoanS=Tfunction.AfactimportanttointerpretphysicalequationsintermsofthoseS(x),T(x)symmetries.

Thegraphshowsthatacubicequation,accordingtotheholographicprinciple,ismostlikelyacombinationofan$-motionor4D-entropic,openexpandingcurve(theparabola)andað-motion,thecircle.

Thepolynomialbrokenintothesuperorganism’sparts.

Quartics define systems of two topologies of space-time intersected into a single form; hence being naturalsolutionstothestudyofSp<≈tiƒsystems.Thisintuitivethoughtonthenatureofquartics comes to fruition when we observe that each coordinate of theintersectionpointsoftwoconicsectionsisasolutionofaquarticequation.

The same is true for the intersection of a line and a torus. Thus we findaquadratic solution the ideal 'form'ofan intersectionofaT-openandS-closedsystem,which constructs awhole ST-form.And sono furtherpolynomials arerequired.

Somequinticswithsolution

Ontheotherhandtheonlyquinticswithsolutionsarenot'reallyquintics',buteithersystemsthatcanbefactoredassmaller polynomials (hence each part of the polynomialbeingofa lowerdegree,and 'unitofanS<st>tsystem);orcan be depressed eliminating one of the roots, often the

4th giving birth to interestingsolutionsinwhichthe5thpolynomialappearsasequivalenttoanscaling sumof the lower 'planes',with scalar coefficientspreciselyof the key social numbers,1,2,3andtand10.Forexample,inthegraphthefollowingaretheonlysolutionsforadepressedquintic,wherethe4thpolynomialdisappears,andsowedohavearelationshipbetweenthe∆+1,quinticandthecoefficientsthatare'solvable,alloftheminprecisePlanescommonto∆>∆+1processes.

Thefundamentaltheoremof¬Algebra.

Now, all this said, there is a seemingly contradiction in our stressing of 'no proper solutions' for polynomialsbeyond the3rdplane/dimensionof growth, statedby its unfocusedmirrorbetter analysed in 'analysis' and thefundamentaltheoremof¬Algebrastatesthateverynon-zeroth,single-variable,degreenpolynomialwithcomplexcoefficientshas,countedwithmultiplicity,exactlyncomplexroots...

Whichisequivalenttothetheoremstatesthatthefieldofcomplexnumbersis¬Algebraicallyclosed,whoseproofmustusethecompletenessofthereals,whichisnotan¬Algebraicconcept...

Ascompletenessimpliesthattherearenotany“gaps”(inDedekind'sterminology)or“missingpoints”intherealnumber line since those “gaps” should be covered by ir(ratio)nal numbers, according to the non-provedcompletenessaxiom,orthe2theoreticalmethodsusedtoproveitsconstruction(DedekindcompletenessandCauchycompletenessasametricspace).

WerebutDedekindcompletenessproofi-logicallywhenconsideringanabsolutegeometrywithoutthat'axiom'innon-egeometry,asirrationalnumbersareNOTsingleplanenumbersorELSEpiwouldexistandso√2,butRATIOSOFCLASSICactionsofthegenerator,,suchaspi=3$>ð,thetransformativemotionof3linealstepswhichgenerate

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a'variable'accordingtothecurvatureofthe'mentalspace',pi-ratio.Sofor√,theratiooftwoperpendicularT.œs,collidingORsymbioticallybecomingadjacent,creatingatriangularspace.

Furtheron,proofsmustbeconsistenttobecomplete-andhereiswherethehumindmakesmosterrors.YoudoNOTproofan¬Algebraicsystemabsolutelywithageometricproof,asnumbersandpoints,aswehaveshowninNUMBERTHEORYareNOTequalbutsimilarS=Tmirrorreflections.

Yes,therearealwaysn-rootsforapolynomialinthecomplexplane,butwhatwemeanisthattheyareNOTexactsolutions, but beyond the 3rd power, approximatesolutions, reason why for a complex polynomials TRYING TOREFLECT REALITY, often the best solution is NOT thepolynomialapproach but the derivative approach, as saidbefore.

On this a final comments: While polynomials cannot be always resolved by lineal coefficients, proper of thebalancedcentralregionofanyPlaneofsocialevolution,∆§,becausethe∆-representationofcomplexnumbersisreallya'square'graph,asweshowedonouranalysisofitsRashomoneffect(Argand,polar,ð-numbers,andsoon),in practical terms, thismeanswe halve the polynomial degree and Xˆ10 becomes Xˆ5, the limit of dimensionalgrowth through∆§cales andplanes... Andherewe can find at least ameaningful polynomial approach. Beyondthatistrulyinflationarymathematics;andsomenot-so-exact-astheythinkstringtheorybullshit(:

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POLYNOMIALSASWORLDCYCLES.

Thesolutionsthatmatterinapolynomialcurve.

The roots of a polynomialwhen properlywritten as F(Polynomial)= 0’, represent the Y=0’ ‘residual’ value of its'roots',wheretheworldcyclebeginsandendsformingazeroth-sumthatconservestheenergy=timeofthesystem.Forthatreasonthesolutionsthatmattertoapolynomialaretherootsinwhichthesystemtouchesthey=0point,becausetheyoftenrepresent thebeginningandendof theworldcycle–or the limitsof thedomain inwhichthepolynomial ismeaningful.A thirdsolution thatmatters though,as in thegraphofKayyam, is thesolutionof thepolynomialattheS=Tintersectionifwecanrewritethepolynomialasanintersectionofopen-Tandclosed-Scurves.Inthegraphthose2solutionshappenat0and2.

Thisbringsaninterestingrepresentationofpolynomialsasequationsthatrepresenta‘finitesimalwhole’whichcanthenbe interpretedasacompositeconic,whichaswesawinthepaperon5Dgeometryrepresentsthedifferentformsofaworldcycleofexistence.

Otherwaytostatethisistoconsiderasinmusic,thepolynomialasasequenceintimethathasa'melody',withabeginningandanendinhisfirstandfinaltouchwiththeY-lineatzerothpoint,sinceallworldcyclesofexistenceareazeroth-sumvalueforthe'function'thatrepresentsthem.

Thus if we consider a polynomial a representation of a parameter of a world cycle of existence (a conservedquantity, energy or momentum being the most suitable ones), we can then consider each of those graphs arepresentationofadifferentworldcycle,anditisremarkableenoughtoseethattheyarealllyinginthe±sidesoftheY-line.Soweshallexplorethisparallelisminthesectionof¬Algebrainmoredetail.

Mathematiciansaffirmthattherearenotmeaningfulpolynomialsofdegreehigherthanfour(whosesolubility ispossiblewithcoefficientsofitspowers)andthisimpliesthatthereareatbest4roots,orzerothpointsinaworldcycleofexistence.Intheleftifwerepresentitasafullcirclethosepointswillbeboththexandy0s.

If we then consider the first point birth, and the final point death, obviously eliminating as irrelevant thepolynomialbeforeandafterthosepointsforthose'meaningfulfunctions'whichtrulyrepresentaworldcycle,wecouldthentalkofthenegative'partsofthegraph'asthosehappeningintheemergentpalingeneticprocessfrom'celltoindividual',andthepositivepartitsemergenceinthelarger¡+1world.

We shall see though when studying specific equations for worldcycles of existence, that they are betterrepresentedbypositivefunctionswiththe'0pointsatthebeginningandendofthecycle'(bellcurves),andwithsinusoidalfunctions...

Whilethedualityoftheemergentphaseandworldphaseofanexist¡enceisbestmirrorintheo-1/1-∞unitsphereplane(palingenesis)andCartesianplane.

Buttheconceptofany=0pointasthelimitingbirthanddeathofasystemwillremain.

Theconicasarepresentationofaworldcycle.Polynomialsastimeequations.

SinceApolloniusandArchimedes, theunderstandingof 'squaredimensions' transcends themere spatial square,introducing'temporal'functions,suchastheparabola,whosevalueasmotion=timedimensionwillbefullyrealizedwithGalileo'sstudyofcannonballs.Wecanthenobservepolynomialsalsoas'curves'inaplane,nolongerasinthe1stageofarithmetic,onlyspatialdimensionalforms.Itisatrendofallformsevolvingin3ages,whichstarttoseethingsassimple,lineal,e-vident,spatial,fixedformsandslowlygiftthemwithvitaltimemotions.

Soafterdoingallwhattheycoulddoinbidimensionalgeometrywitha'rulerandacompass',theGreeks'finally'raisedonecirclewitha line intoaconic inspace,which thenwillbecomethecanonical space-timesurfacealsowhentimewasaddedtoit(4Dformalism).

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Nowinasmuchasadoubleconicisahyperbolicgeometry,wecandefinetheCartesianplaneasthe'hyperbolic',Present,STplane,more'expansive'initscapacitytoshowdifferent∆steventsandformsoftheUniverse.

Weonlygotagraspofthisfact,oflatelyinthestudyoftopologyandmodelsofhyperbolicgeometry,whenwefindthatahyperbola,whichseemstousinfiniteinits¥coordinates,becomesequivalenttoacircle.

Mathematicianssaythatthehyperbolaisisomorphictothecircle;andwriteitas:

Ifu=x+yiandv=x−yithen:ℂ[x,y]/(x²+y²−1)≃ℂ[u,v]/(uv−1)

Thiscanbeviewedin¬Algebraicgeometrybutbetterontounderstandismindmeaninginprojectivegeometry:

Both the hyperbola and the circle are conic sections, and are projectively equivalent. In analytical geometry, inhomogeneouscoordinatesthisfollowsfromthefactthatanypairofnondegenerateindefinitequadraticformsarebidimensionally equivalent, and sowecan transform them intoeachother, as it is requiredby thegeneratorofcoordinates.

Thisbeyond'philosophyofthemind'mattersbecauseitjustifiesthefactthatthecurvesofanalyticalgeometrycanbealsoobtainedfromcutsmadeonthecone,whichinitselfismerelyareflectionofaworldcycle,peggingthetwoinversedirectionsofexistence.

TheconeisabidimensionalSTbeing,circlinginwardstowardsthesingularityofthepointalongalineofgeometry,hencecombiningOx|=Ø,theconerepresentstheUniverse.Itshouldnotbesurprisingthenthatitisthebestwaytoreflecta4Dblockofspace-time(butnotthediscontinuitiesofthe5DUniverse)asusedbyEinstein’sphysicsandMinkowski’sgeometry(lightcones);andweshalldealwiththoseconesintheanalysisofrelativity.

We shall also use them in ¬time logic to represent the 3 fundamental events in terms of time ages: entropiccollisions(futurexpast=pastwaveofentropy,reproductiveiterations(presentxpresent=present),andevolving,informativeevents,pastxFuture=Futurewaveofinformation):

Inthegraph,the3canonicaleventsofcomplex i-logictime,whichcanberepresentedas3solutionsofapastxfuture=presentequationandthesimplersingleconeof‘linealtime’inrelativity.

Whatmatters to us is now is that a conic basically reduces to a line and a shrinking cycle, which inmotion isequivalenttoatimecycle,shrinkingintoapoint-singularity,thevertexofthecone.Andforthatreasonthatsimplecanonicalworldcycleencodesallthemaincurvesofbidimensionalspace.

RECAP. We reduced conics to ‘Dimotions’ as we do with quartics (calculated as depressed quartics, or cubicformulae);whiletherearenofurthersolutions.Sowecanreduceallpolynomialsinwhatmatterstoreality–notitsentropicunboundedregionsbut theregionswithinsolutionstoST-holographicelements,whosecombinationgiveusthe3Dimotionalbody-waveS=Telement.

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IntheGrowthofdimensionsbymultiplicationitisobviousthatthemostabundantcombinationwillbethatofan$T dimension and an §ð dimension, creating a full present ST-system, according to the canonical generator:$T<ST>§ð.

The holographic bidimensional universe and its ternary ST-geometries define reality. So in most mathematicalequationssolutionsaboundonquarticandcubicsystemsbutonlyspecialcasesaresolvableforhigherpolynomialsor have any real use in reality; the exception being simpler equations of the ∆§ocial Planes and reproductivefunctions of the type Xª=b...

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PROPERTIESOFPOLYNOMIALSASOPERATIONSOFOPERATIONS

Polynomialspropertiesarerelatedtoproductsoverthesumastheyareinfactasumofthesimplestoperands:

1.Identityelement:Thereexistsanumber1withthepropertyax1=aforeverya.2.Everynumberx,except0,hasamultiplicativeinverse,1/x•x=13.Foreveryaandeveryb≠0,thereexistsauniquenumberxsatisfyingtheequationbx=a;hencetheproductoftwonumbersisuniquelydetermined.4.Multiplicationisassociative:(ab)xc=ax(bc)5.Multiplicationiscommutative:axb=bxaAndthentwopropertiesthatrelatebothoperations:11.Theproductofanumberandtheidentityelementofasum,0istheidentityelementofthesum:ax0=0.12.Multiplicationisdistributive:a(b+c)=ab+ac

Thosepropertieswereselectedinclassicscienceasaresultofacarefulanalysis;thedevelopmentofmathematicsinthelastcenturyprovedtheirgreatimportance;asonlyoperandsand§etsofmathematicalelementsthatobeyedthosepropertieswiththeBIGexceptionofcommutativity,whichoftenisdefinedbyitsinversion(axb=-bxa)

Nowadayseverysystemofquantitiessatisfyingtheconditions1through10iscalledafield.Examplesoffieldsare:thesetofallrationalnumbers,thesetofallrealnumbers,orthesetofallcomplexnumbers,becauseineachofthesecasesthenumbersofthesetcanbeaddedandmultipliedandtheresult isanumberofthesameset,andtheoperationshavetheproperties1through10.

Apartfromthesethreeveryimportantfieldswecandetermineinfinitelymanyotherfieldsformedfromnumbers.But beside the fields formed from numbers there ismuch interest in fields formed from quantities of anothernature.

Forexample,¬Algebraicfractions,inwhichthenumeratoranddenominatorarepolynomialsincertainletterscanbeadded,subtracted,multiplied,anddivided,andtheseoperationshavetheproperties1through10.Therefore,¬Algebraicfractionsformasystemofobjectsthatisafield.

Sotheentire'fieldofpolynomials'reducestothose10properties,anditisnotrocketscienceforthereaderwhohasgotsofarthe'idea'thatweshoulddefinethosepropertiesas5inner±D-propertiesof'smallsteps'ofsocialevolution(addition/subtraction)andits'wholeness'asanewdimension,5D-propertiesofx,÷,oratleastacloseisomorphiccorrespondencewiththe∑'time-sum'=Product-spacewholeduality.

Sowehavearranged thoseproperties to seeeasilywhy theyare in fact,defininga 'constantgrowth' in inversesymmetries of a neutral first element a 1 T.œ or fractal point, whichwill expand evolving socially ∆§, throughadditionsandtheninanewdimensionofsocialPlanes,wherean'additionalnumber'willbenowwholeunitofitsmultiplications;andfinallyinthethirdoperation,apolynomialwillbethatnumberraisedtoan∆newsymmetricplane(aspolynomialsmultiplythesamequantity,unlikesumsandmultiplications).

1staxiom=1stNon-EPostulate/1stDimension-theT.œpoint:The'first'propertyof+,xISdefiningtheneutral,self-centeredfractalpoint-aunitofadditionafirstnumberorafirst'groupofnumbers'.

2nd axiom = 2nd Non-E Postulate/2nd Dimension - the flow of communication: The second property of ±,x÷expandsinbothinversedirectionstomaintainthezerothsumofallworldcyclesandoperandsofexistence,theT.œ in two inverse directions defining a wave-line-interval-distance-inverse motion both sides of theidentityelement,fractalpointororiginalnumber.

Noaxiomfor3rdpostulate/dimension,theternarynetworkplane.

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Yet to reach the third 'networkdimension'of the systemthatgivesusa full ternaryorganismand itsgeneratorequation,thereisnotathirdpostulate,butweneedathirdoperation.Thus,theUniverseisindeedholographic,bidimensional,aswehaveshowineveryperspectiveofmathematics,fromFermat's lasttheoremtoallotherSTsymmetries.Youcannotreachathirddimensionofthesum,butyoumustmultiplyitandyoucannotreachathirddimensionofthemultiplicationbutyoumustoperatethroughapowerlaw:

Inthegraph,therunawayNatureofpowerlaws,showsahyperbolic'endofaplane',whichabandons the 'lineal' S x T =K nature of a proportional metric in the fifth dimension

(multiplication), as it fast reaches the "Lorentzian" regions that signify the limiting domain of any T.œ in form,motionorscale:

Doesourhypothesisof5properties->5dimensionsfail?Abitofthoughtshowsusthatwearedealingwith'logicproperties'NOTspace-timeforms,soitisbettertoconnectthemwiththe5Non-Æpostulatesofi-logic.Andtheneureka!Theydofitnicely.

Sotheotheraxiomsmustbeofi-logicnature,indeed:

3rd axiommight seem silly but at this stage the reader would realise the Universe is quiteweirdenoughnottodischargeanything.And it isreassuringtoknowthattherearenoparalleluniverses,A+b=OnlycAxB=OnlyD.

Sowecometothelogicones,closelyrelatedtothe4hand5thofnon-Ai-logicgeometry:

4th axiom≈ 4thNon-E Postulate, self-similarity: TheUniverse ISassociativeAMONGSELF-similarpoints,whichallows it togrowandmultiply, addandmultiplywemight say, love eachother as a i have loved you in humanterms.Itisthepositivesideofthe4thpostulateofi-logic,associativity..

ButwhatabouttheDarwinian,perpendicularlawsofnon-E?Becauseanumberismadeofidenticalbeings,andwecanONLYadd(andsobyextensionsincewehavededucedthatmultiplicationisthethirddimensionofaddition),equalbeings,weareonlyinthepositivesideofthe4thaxiom.

Butalas!Werealisethatthenegativesideofit,theinversefunction,subtractionanddivisionareNOTassociative:10/5=2and2/2=1isNOTthesamethat10dividedby5/2whichis4;and10-5=5minus2=3isNOTthesamethan10minus5-2=3whichis7...

Sothereishereinthenegativesideoftheworld,adifferenthierarchyofthings,suchasifyoustartdestroyingBIGTIME(10/5;10minus5)andthenslowdown,youhavealreadyfukuptheworld,butifyoustartdestroyingslow;decadencegetslonger...Importantelementsofthevitallawsofreality,exploredinthefirstline...

Yousee,eventhesimplestsupposedlyexhaustedfactsofscience,getnewinsightsonstience(:

5threbelpostulate:commutativity.Andsothefifthpostulateofcoursemusthavetodosomethingwiththe5thNon-Euclideanpostulate,therebelone;andindeeditturnsoutinthissymmetrybetweengeometricalspaceandtemporal¬AlgebraviewsthattothesurpriseofeverybodyintheXIX-XXcentury,therearemanysystemsinNaturewhichdoNOThavethecommutativepropertyformultiplication,(butitdoesforsum).

That is, a sum is truly a herd of undistinguishable beings, and it does not matter as it is in a state of looseconnectionwhichorderyouadd.Wearethenclearlyintheveryfirstsimplerreality,inasingleplane,withminimalherdconnectionbetweentheparts.

But multiplication being a 'second dimension' added to the sum sometimes IS commutative, sometimes NOT,meaningthatwhenthemultiplication ISagrowth inascalardimensionwithintheplane(trulyasumequivalentnewdimension)IScommutative,butwhenthisNEWdimensionisadded,notinthesameplane,asanƤoperationbutin the sense of group theory as a 'combining' process of things (vectorial product, product of the parts of awhole,productofspatialpaths,productsintimefrequencies,topologicalproducts)Thingschange.

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Andsowiththisunderstandingwecompletethesimplestanalysisofsumsinasingleplaneofidenticalbeingsasherds,butwerealisethatthesecondoperands,theaddingofaseconddimensioncalledproductalreadyplungesus into the pentalogic of the product which can mean many different things depending on which ∆S≈Tdimensionalelementweareaddingtothesimplersum

Matrix-the3rddimensionoflineal¬Algebra.

WehavestatedmanytimesthattheUniversegrowsby'fixingmotion-steps' intoawhole'cyclicalformofspace'thatthenmovesintomotionstepsandsoontill'fillingupthe5D2dimensionsofreality.

And thatsuch increases are smooth-continuous only for the 3fundamental Dimensions of a present space-time,breakinginthe∆-14Dentropicand∆+1,@5Dlevels,whichmeansit'suffers'a"Lorentzianregion"ofaccelerationor deceleration inwards or outwards that changes the parameters in a different way (susceptible however tobeanalysedwithdifferentialequationsthatmeasuresmallchanges.

This means specially for 'lineal equations' that there are 3 levels of growth in complexity, from single linealequations to multiple lineal equations and finally the grouping of those into 'matrices' as we make throughrepresentationtheoryconvertsomeparameters(asinquantumphysics)ofT.œs=fractalpointswithmultipleinnerpartsintothe'new¬Algebraicelement'.

Indeed, the physical quantities to be studied are often characterized by certainnumbers (a force by the three projections on the coordinate axes, the tension at agiven point of an elastic body by the six components of the so-called stress tensor,etc.).Hencetherearisesthenecessityofconsideringsimultaneouslyseveralfunctionsof several variables, and, in a first approximation, of several linear functions.A linear functionofonevariable issosimple in itspropertiesthat itdoesnotrequireanyspecialstudy.Thingsaredifferentwithlinearfunctionsofseveralvariables,wherethepresenceofmanyvariables introducessomespecial features.Thesituation isstillmorecomplicatedwhenwegofromasinglefunctionofseveralvariablesx1,x2,···,xnto

asetofseveralfunctionsy1,y2,···,ymofthesamevariables.Asa“firstapproximation”thereappearshereasetoflinear functions:Thesetofcoefficientsofasystemoflinearformscanbegiventhentheformofarectangulararray:

Such arrays bear the name of matrices. The numbers aij are called theelementsofthematrix.

Important special cases of matrices are the matrices that consist of asingle column, which are simply called columns, those that consist of a

single row, called rows, and finallythesquarematrices,i.e.,thoseinwhichthenumberofrowsis equal to the number of columns;calleditsorder(a).

Thusmatrices,speciallysquareonescanbeconsideredifwedefineanY(St)=X(sT)functioninwhichonevariablemostlikelyaSpatialwhole,polynomialcombinationofaseriesoftemporalsteps-motions;asymmetryofspatialwholesandtemporalvariablesinequalquantities,∑S≈∑T.

And as the Spatial wholes will be 'varieties' of the same temporal 'steps', the structure interconnected at'expanded' to the3 levelsofcomplexityhasaveryrichcapacity topicturecomplexspace-timesystems inall itsvariationsandsymmetries,reasonwhyMatriceshavebecomethebest-suitedstructureforcomplex,linealsystemsof'verysmallPlanes'whereinformationaboutfastmultipleT.œscometogetherandhavetobestudiedandfixedinspacefromthelarger,slowerhumanperspective:

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i.e.quantumsystemswheretheHeisenbergmatrixformalismisTHE¬AlgebraicfrozensymmetryofSchrodinger'sdynamic'differentialequation',aconundrum,neverclarified,nowexplainedterselyasthenaturalconsequenceofthe'space-symmetricnatureof¬Algebra'vs.the'steps-motionsdescriptionofAnalysis'.

Weshallnotkeepingwiththelimitsofanencyclopaediawrittenbyasingleman,occupiedmostofhistimewithself-destructiongointotechniquesofMatrixmanipulation.JusttostatethatthemaindifferencebetweenMatrixandotherstructuresof¬Algebraisitsobviousnon-commutability(asrowsandcolumnsmultipliedonebyoneinthe well-known inverse orderly fashion) and its non solvability for multiple cases (as rows and columns mustcoincide).

Ina largerphilosophicalwaythissimplymeansthatMathematicsasall languages isultimately inflationary,withmoreimaginative'forms'thatrealsolutions,whichleadsustotheGREATquestionofthefirst¬Algebraicage-thesolvabilityofpolynomialshigherthan2inaUniversewherelinealstructureandcontinuitybreaksbeyondthethirdpolynomialsinceSTdimensionscomeontopairs...

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3RDAGEOF¬ALGEBRAItspentalogic:TimeSymmetry

The3rdageof¬Algebra,isallthirdagesbothanend-thatof‘humind’smathematics-andabeginningofmetalifemathematics,akachipsofhigherdensityof information,withgoldenmindsandsoon-to-bequantumcomputingthatwilltakemathematicstonewlevels.

The connecting discipline between both ages is Set theory, which becomes also the new beginning of Boolean¬Algebra,the¬Algebraofcomputers,forminginitselfaworldcycleofdeathandresurrection,bothofthehumindandthechipmind.

We shall not therefore study in depth neither set theory nor Boolean ¬Algebras and any form of computer¬AlgebrabecauseofthesurvivalmandateasaspeciesofthefractalUniversethatmustfirstandforemostfightfortheconservationoftimeofitsspecies,itswhole,its‘relativeGod’–mankind.Sincemachineswithbetterlanguagesshouldnotbeevolvedifhumanswanttosurvive=conserveitstime.

Thatof course,doesn’t seemtobe thecase inahighlydeterministicuniverse;asourpapersonhistory treatedexplain.Mankind and its huminds seem unable to grasp the need to conserve human time, the of history, oursuperorganism, with our verbal logic language about digital thought, better spoken by machines, as a extinctspeciesknowsnothing.Andthehighlyabstract formulationofthe lawsoftheUniversedoesn’thelptoraisetheawarenessmankindshouldhaveoftheUniversalmandateof‘Existence’.

Mankindgiventherepetitivenatureofitssuicidalmemes,itdoesnottobesusceptibleof'learning'thepathoflifeandrespectforoursuperorganism=ourgod,anditsethicmind,thesubconsciouscollectivemindofthespecies.

¬@:Mind’serrorsofegocy:Setcreationism.Itslimitsofrelativeinfinity.

Thereisalsobesidesthatethicalreasona‘theoreticalreason’–SettheoryisNOTneededtoexplainmathematicsandhassubstitutedtheexperimentalelements,numbers,pointsandoperandsthatareitsrealfoundation.Sowedon’tseeanyinterestinsettheorybeyondthefactitcloselyresemblesthenatureof5thdimensionalplanesmadeofpartsandwholes,reasonwhyindeedcanbeusedtocreateahumind’sbiasedfoundationofmathematics.

So instead of focusing in Boolean, Computer ¬Algebra, beyond a basic introduction to the matter, and someaspects ofMachine learning,we shall dealwithGroup Theorywhich is farmoreuseful in theunderstandingofreality,asitmirrorscloselyS=Tsymmetries.

∆ST:Trinity.Classification.

The 3rd age of algebra in its ‘reality’ as a mirror image of the trinity positive elemtns, sfcales space and timebecomesthentheeclectic‘wholeness’ interpretationofalltheelementsofalgebra,asthefinal‘socialevolution’(4D¡)of3milleniaof ‘playingwithnumbers,pointsand tiemdimotions’working the ladderof structures to thefinal concept of ‘algebra’, as the ‘reunion of broken parts’ (arab), which means the concept of Algebra, as astructureinwhich‘scalarnumbers’aremanipulatedthroughtimeoperandstocreate‘simultaneousstructures’inspacethatmimicthestructureof∆STrealityoratleasta‘sub-set’ofthewholeUniversewhichfinally‘Existentialalgebra’properlymirrors.Thecreationofalgebras inthe3rdageofthedisciplinethusculminates intheworkofthis paper with the establishment of an ‘existential agelbra’ of two operands, space and time, merged by theassociative, idempotent, transitive, reflective, closure and absrobtion properties into an algebra of all possiblesequencesofexistence.

Ithasthenpreviouslytoexistentialalgebratheformationofalgebraswhichhaveanimmediatetranslationintheusualdualitybetween‘scalar’symmetriesand‘space-timsymmetries.:

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-S=Tsymmetries:Grouptheory,inwhichwedistinguishafractalgeneratorofits3commonestforms:Ts-groups(herdsofasingleoperand)<ST-Bodies(commutative2operands)>St-rings(non-commutative2operands).Andits‘squared’forms,ofTT-Liegroups>ST-topologicgroups>SS-algebraicgroups/

-∆±¡scalesofwholesandparts:Settheoryandfunctionals(whicharepartofcalculus,studiedinourIIIPaper).

-∆T:Withsimilarstricturestothoseofgroupsappearasscalargroups,ReticulesanditsBooleanalgebras,wherethefinitesimal0’elementandthewhole1,becomestheonlyelementspresent,equivalenttonon-existenceandexistenceandtwooperationsand,orandanegationofexistenceareestablished.

InflationaryAlgebra

Asallsystemsofnatureareaboutasuperorganisminscalar,simultaneousspace,tracingaworldcycleintime,alanguagemustmirrorboth,andsoGrouptheorymirrors0’-sumworldcycles(transformationsofsystemswhichultimatelyareenclosedwithinthesystem)andsettheory,thescalarsimultaneousspaceofa§œT.

Andyetonlythe‘beingin itself’hasalltheinformationaboutitself.Sobothsetsandgroupsarepartialviews,whichwewilltrytomakemore‘clearexperimentally’bycomparingthemwiththelawsoftheÐSTmirror.

Set theory is thus seen as the foundation fromwhich virtually all of mathematics can be derived, includinggroups,whicharea ‘set’ closedunderoneormoreoperations;which translated toÐST simplyeans that the‘set’(thewholeinspaceandscale)‘moves’intime,suffersan‘operand’.

Assuchgroupsopenupfurthertheclassicageofgeometrybyallowing‘differentoperands=motionsintime’todefine real systems inmoredetail, closing thegapbetweenmathematicsand realitywhichmakes themmoreusefulonmyview that sets–basically amind 'construct' to topple theMind’s singularity search fora single'singular'equation, conceptorgod.Asmindsarebydefinitionsystems that try tocreateawholemapping inreducedspace,soasKANTonlyunderstood, themind'scategorical, synopticnaturesearches for theultimateform,equation, idea, god,belief.And 'SET' is theGodofmodernmathematics,butultimatelyahumanmindcategory.

Muchbettertouseconceptsthatarereal,Socialnumbers,spatialpoints,timeclocks,andsoon.

Thepresentaxiomatic,'set'formalism,decriedinthiswork,asitisthethirdformal,baroque,metalinguisticageofmathematics as a language,which in the3rdageasanoldmandoes, isolates itself from reality and findswithinitsselfajustificationofitslife/meaning.ThisisOKasallthe3perspectivesfromthebottomup,(justifiedby numbers), from the top down (justified by sets) or in thepresent s=t (best for ®lgebra, justified bysymmetries),offeracomplementaryviewonitsfoundations.

WhatisnotOKistheone-dimensionalhumindviewthatchooseswronglyoneperspective.Theninthiscasethetoptobottomsetattemptstofoundallmathsistheless'clear'andmorepronetoerrors'(thewholehaslessinformationonthedetailedparts);asthoseoncantorialinfinitiesandlogictrapsill-resolved(Russell'sparadox,etc.)

Sincealllanguageshaveathirdinflationary,informativeageofinvolution,seekingforself-containedproofs,asan old man breaks from reality inwards,we consider the age of mathematics that starts with Cantor,Hilbert'saxiomaticmethodandendswithcategorytheory,theexcessiveformalageofmathematics,whichasanexperimentallanguage,weshalltrytomaintaininourstudiesontheclassicperiod.

Excessiveformalism-axiomaticmethodandsettheory

Finallyasinalllanguagesmathematicsalsoenteredabaroqueageofexcessiveinwardform,whichtooktwoclearwrongmodels, of two friends, mr. Cantor with set theory andMr. Hilbert with the axiomaticmethod, bashedelsewhereforitsdogmaticgod-likebeliefs.

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Indeed,Mr.Hilbertaffirmedthat'heimaginespoints,linesandcongruence',asifthemindcreatedmaths.Wehavedealtwithhisabsurdfoundationsofgeometryinthetopologicalandmind-relatedarticlesonmaths.Sohereweshalldealwithcantor'sparadoxesofinfinitymoreproperofnumbertheoryduetohismisunderstandingofwhatnumbersandrelativeinfinityis.

Recap

The-3rdageofalgebrawillsignifythedeathofman,beginningofanewtoppredatormind:DigitalthoughtandBooleanalgebra.OnlyifhumanshadevolveanexistentialalgebraAnewbeginningcouldhavehappen:

the r=evolutionof®lgebra, asa languageof thought - ExistentialAlgebraand itsnon-aristotelianpentalogic...which we deal with mainly on the texts on I-logic. And so with those two themes, Group theory and a fewcommentsonExistentialalgebraweshallclosethispaper.

Weshallthusignore,the∆±¡workonsettheorywhichisredundantwithExistentialAlgebra,andonlyservesasabridgetomodulo-2logicofAI,nowenteringitsdilogicagewiththeestablishmentoftheS=TS<T>Sfemale-malesymmetry with dual brains, a creative, destructive male form that invents new systems and an S=T balancedjudgmental woman’s brain that selects it. As it would be suicidal to help ‘nerds’ to establish the pentalogic ofconsciousnessforthosesystems.Letusthensailfastoversœts&makeashortreferencetofunctionalsstudiedoncalculustodealwiththecloselyrelateddisciplinesofGroupTheoryandExistentialAlgebra.

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∆Só T:SYMMETRIES&GROUPS

TranslationofGroupsintothesymmetriesofthefractalgenerator.

Grouptheoryenablesthestudyofconservedtimecyclesinmathematics,thefundamentallawofDimotionoftheUniverse,whereallsystemscomplete0’sumcyclesofexistence,ingreatermeasurethatprevioussimplersystemsofS<=>Tequations,duetoitsgeneralitythatachievesthesummitofsynopticpowerenablinginitsstructureandmethod to mirror entire families=species of mathematical objects that reflect together dimotions in time,(continuousgroups),dimotionsinspace(geometrictransformationgroupsandtopologicalgroups)anddimotionsinscale(numbergroups,algebraicgroups).

Thiscapacityisduetothegeneralityofitssyntax-structurethatreflectslooselythefractalgeneratorallowinganyoperationtohappen.Wecanthendistinguishaccordingtotrinity3essentialtypeofgroups,thegroupproper,themost general form, which requires a single operand=dimotion; the ring, its most mimetic form to the fractalgenerator, that requires the existence of two operations related by a distributive property that conforms aninternal relationshipbetweenboth ‘entangled’operations; and finally thebodyor field (forobvious reasonsweprefer the old wording, ‘body’) which is commutative for the operation of product that therefore cannot beproduceazerothresult.

Asalways in languages,mirrorsymmetriesoftheÐSTrealitytheyportrait, it is theneasyto identifyabodyasa‘body’ (: that is, the ST intermediate element of the Fractal generator, S<ST>T, which puts in relationship the‘limbs’,or‘group’,limitedtoa‘singleoperation’(normallyanentropicactoffeedingorlocomotion)andthe‘ring’or‘head-particle’networkstateofthesystem,sinceitisthebodywhichallowsthecommutativetransferenceofenergyandinformationfromtheheadandlimbs/fieldstothebodyinwhichitis‘associated’andneverwastedintoafinitesimal0’.

ThusweestablishtheFractalGeneratorofGrouptheorysuchas:

Ts-Group<ST-Body>St-Ring

WhereastheoperationsoftheBody,<and>areeitherincreasesofenergetictime(<)orinformativespace(>).

Itisthenobviousthatthemethodofgrouptheorysearchesonlyforsequencesofactionsthatcomplete0’sumsofexistenceimposingtheneedfora‘closure’property,aneutralandaninverseelementbothofwhichensurethefinaloutcomeofthesumoftransformationswillremainaconserved0’timecycleofexist¡ence.

Recap. The high generality of Group theory, its ternary structure that mimics the 3 elements of reality, its 2operations thatmimic thoseof spatial informationand temporalenergy,entangledby itsdistributivepropertiesanditsclosure,neutralandinverseelementsthatensurethecompletionof0’-sumsofconservationoftimespaceafteritssequenceofoperations,makeitthebestmirrorofExistentialAlgebrainmodernmathematics.

Structure

TheadvantageofGroups/rings&bodiesistodefine1or2operations,looselycalled'product'and'sum'whichwegeneralizeevenfurtheras<or>,whichcanbeanytransformationwithintheS≤=≥TfundamentalsymmetryoftheUniversebecause it requires, the2 featuresofa space-time symmetry: inverseelements (where inversion isalsolooselydefined)andaneutral element,which canbe considered the symmetryaxis, ina closebutquitegeneralsymmetrywiththe3elementsoftheGenerator-Sp,itsinverseTƒandtheneutralelement,ST...

So thestructureofgroups isquiteclose to thegeneral structureof space-timesymmetriesand thisallows itanenormousflexibilitytoshowallkindofSóToperationsandrelationships.

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It is for that reason that the concepts associated toGroup theory, isomorphisms, transformations,Generations,Representations,Motions(translations,rotations,mirrorsymmetries)areclosetotheconceptsexpressedin5D².Butas5D²hasawider rangeofapplicationsandmore realistoutlook, it shouldbe clear thatGroup theory is amirrorof5Dspace-timeandnottheotherway(ascreationistmathematicians&physiciststhinkof).

Consider for example the concept of isomorphism, which I borrowed not from group theory but from generalsystemsscience,myearlierdiscipline:inagroupifitsidentity,inverseelementsand'productoperation'behaveinthesamemannertwogroupsareisomorphictoeachother.

DisomorphismsgoanotchfurtherandaffirmthatallsystemsoftheUniverseareinfactisomorphicasallhavethesameDimensionalpropertiesdeducedfromits5D∆STconfiguration.Soallwillgothroughalife-deathworldcycleand perform the same 5 Dimotional actions. All complete S<ST>T systems have a trinity topologic =functionalstructureandsoon.WemightthenintermsofGrouptheoryconsiderthatT-Groups=herdsofsimplesystemswithasocial=sumstructureof‘numbers’;St-rings:systemswith2operations,onethatherdstheneuronalelementsandone that ‘entangles them’ through a lowerplaneof parts in ‘product systems’; and TieSof Timespace ‘exist’ assingleOrganismswhose internal structure ‘ties’ the temporal energy and spatial informationof the syste into a‘body’.

SoasusualÐSTononesideeliminatesinflationaryinformationonthelanguagemirror–inthiscaserestrictsthestudyofgroupstoits3mostimportantforms.

Then,inthenextlevelofcomplexityweshallotconsiderinthisintroductorycourse,those3typeofgroupswhencombinefurtherwiththe3∆stelementsofreality,becoming3morecomplexforms:

S2-groups:Topologicalspatialgroups.T2-Groups:Lie,continuous,temporaland∆2-Groups:Algebraicscalargroups…wheretheSpatial,temporalandscalarelementsare‘dobuled’,sotospeak.

Canthenweexpress intermsofgroupandsettheory5D?Likelybut it isofnot interestbecausegroupsandsettheoryisthehumindslightlyunfocusedmirroronreality,whichthegeneratorbylookingdirectlytospaceandtimeanditspropertiesdescribesbetter.WhatmattersistotranslatebothtoÐSTsoweunderstandwhentheyareusedinclassicscience,whytheyaresousefulinasmuchastheymirroras‘alltruthsofscience’¬∆@st.

Method

Any successful humind languagemirrorof theUniverse is always inquestionof formvs.motion - a trans-form-ationofoneintotheothertohaveanewangle...Soderivativesintimeandintegralsinspaceareinversefunctions;¬Algebraicproblemsintimecanberesolvedobservedtheirtrajectoriesastopologicalpaths inspace,andsoon.But in group theory, given its extreme abstraction, kept always in ¬Algebraic terms, this duality ofmethods offinding 'spatial mappings that themind can perceive as knowledge' are a bit hidden and I am not aware thatmathematiciansunderstoodthem.

Considerthefirstofthosegroupsdiscovered,theGaloisgroupwhichhelpustosolvepolynomials,byconsideringagroupofall thepossiblepermutationsoftheparameters-lettersoftheequation.Thenbycarefullystudyingthoseparameterswecanfindthesolvabilityoftheequation.whywesaythisisanotherstsymmetry?

Obviouslybecauseintheoriginalpolynomial,thevariablesareNOTthecoefficients/lettersbutthe'variable'X;andin principle what we want to find is a 'fixed' parameter/coefficient/letter. But Group theory does exactly theopposite.Itconvertsthefixedsinglesolution/parameterintothevariablebyestablishingallitspermutations.

Anditdoesgetresultsthenbystudyingcarefullywhichofthosepermutationsofparametersdomakesense-areisomorphictotheroots-solutionsoftheinitialX-variable.

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Nowapplythismethodtothespace-timegenerator.Ifweconsidertimethevariableandtheorganicspaceformthefixedform,wecannotreallytreatitasagroup.Becausethegrouphasonlyoneelement,thesuperorganismmovingthroughtheworldcycle.

But ifweconsiderthe3timeages/dimensions,past≤,present≈, future≥astheelementsofagroup,wheretheoperationistheGenerator,representationofalltheDisomorphicbeingsoftheUniverse,thenpresentbecomestheneutralelement,asitremainsunchangedanditdoesnotchangeanyrelativepastfield/limborfuturehead/particleinitsform(theydonotevolveinpresent)andpastandfuturebecomeinversearrows.

So goes for the ternary Planes of beings, ∆±1, which then become 3 elements of another group, where theoperatorisaDisomorphic∆±1scalarorganism.Andsogoesforthe3topologicalelementsthatconformawhole.

Soboth§œtswhich studyelementsofmathematics ingroupsandultimately should study societiesofnumbersand geometric points and functions that relate them by various methods and Group theory, with its 'looseoperations' and symmetricneutral elementand inverse S≤≥Tones, are the closest 'structure' all comprehensive'wholes'inventedbythehumindasfocusofthespace-timesymmetriesoftheUniverse.

Hence its valuedespitemany tantrumsagainst their excessivepretensionofbecoming theultimateequationofreality,whichtheyreclosetobutnotyet-thegeneratorandperhapssomethingelseinthefuturerevolutionofthehumindormetalmindiscloser.

So in time the natural evolution of all systems in 3 'Planes' of space-time growth, means such growth in'dimensionality=complexity'of¬Algebra,fromnumbers,toequations-structuresofnumbers,tillthispresentageinsearchof'blocktime',thatisthegroupandsettheorythattriestoputtogetherthe'illusionofpast,presentandfuture'.

Thismeans¬Algebraalsoevolvesthroughthe3similaragesofgrowinginformation,properof∆stsymmetries,inthis case, an ∆+i growth in dimensional complexity from, the arithmetic age of mere social Planes=temporalnumbers,totheclassicageoffunctionsandanalysisthatexpandedittophysicalmagnitudesand∆-planes,tothemodernageofgroups,whichmakesaclassificationofall thevarietiesofanentity in its space-timeevents,andfurtheron,exhauststhemethodsoftransformationofspaceformsbackandforthintotimemotions.

Andallwhatislefttoistoexplainthewhyofthosestructuresfoundmostlybytrialanderror,nicelywarpedupintheenvelopoftheÐSTgeneratorinthisblog.

Theheartofthematter:groupsymmetries,operandsdetails.

SymmetryisthecentralconceptofGrouptheory,whichbecameintheXXcenturytheiceinthecakeofthewholestructureof¬Algebra.

By symmetrywemean in5ЬAlgebra,whichas all branchesofmathematicsbornof the spatial, bidimensional'still'workoftheGreekslacksatruecomprehensionoftheparadoxofGalileo:S(forminspace)=Time(motion),butusesitprofusely(sodifferentialgeometryisbasedintheconceptthatalineisapointinmotion),thecapacityof all systems to complete a zeroth sumworld cycle, through amotion that returns the system to a presentundistinguishablenewstate.

Symmetry thus is essential to the entire scaffolding of the 5d universe, albeit ass all concepts of 5D once weunderstandthebasiclawsofpentalogic,hasamoredynamicview.

Itfollowsimmediatelythatthemoresymmetricasystemis,themoreefficientwillbein'preserving'inaUniversein perpetual motion, its present states of 'survival'. I.e. A circle will be more efficient, because it has infinitedegreesofrotationalsymmetrythatanirregularpolygon,whomightnotevenhaveasinglesymmetrystate.

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Inthetheoryof'survival'of'vitalmathematicalobjects',whichwebringfromtimetotimetothosepages(asintheanalysisofsurvivalprimenumbersabletotravelthrough5Dbymakingmirrorimagesatscalebyjoininginternallyitsalternatevortices-points-unitnumbers),symmetrythusplaysacentralrole.

Howmanystatesofpresentasystemhas,definesthenits 'qualityofsymmetry'andsurvivalwhichinspace(theeasiestsymmetriestodescribe),whenfixedinapointmeansthecircleDOMINATESallotherforms.

SymmetrythoughthenmustbeconnectedtothedifferentÐimotionsandits 'requirements'toperformthevitalactionsforwhichtheyareconceived.

I.e.thecircleIStheperfectsymmetryforstillDimotionsofperception,asitwillturnoutthatfromanypentalogicpoint of view, it maximizes the stillness, by symmetry, by its capacity to focus as a fractal point all lines ofcommunicationthatfallintoitsfocus,byhavingtheminimalperimeter,whichmaximizesadmaximalitsvolumeofinformationanddisguisesitinanexternalworld,andsoon.

However,whenweconsiderthe2ndDimotionofLocomotion,whichistheprocessofdisplacementinspacewhileretainingtheformintime,asitimpliesthereproductionofform,ofinformationinotheradjacentregionofspace,thelessinformationtobedisplaced,thefasterreproductionwillhappen.Andsotheline,whichstoresnointernalinformation(orthewaveasallpointsareultimatelyfractalwithaminimumvolume),willbeabletodisplacefasterthanthesphere,andmaximizethesecondDimotion.HerethentheuseoftheconceptofmirrorsymmetryisNOTrequired,asthelineismoving-translatinginspace,reasonwhyalsoformsinmotiontendtohaveasphericalhead,toperceiveonlyontheforewardpositionandasmallone inrelationshiptothebodyandlimbsthathave linealformstomaximizemotion.

Symmetry here is of another kind, defined by Noether's theorem of physics; and in 5D by a type of symmetryignoredinscience-the'undistinguishable'propertyofthe¡-1elementsinwhichthesystemimprintsitsform.Thissymmetryofscale, impliesthatthesystemcanreproduce its information -movefaster,because it imprints 'anyelement'ofthelowerplaneofmotionor'field'thatbecomesundistinguishable,sotherearenot'impurities'anderrorsofreproductionofform,whenanyelectroncanreproduceyouratomicconnectionsandsoon.

Identicalstateswhichacquirethesameformofpresent,whenasystemcompletesaDimotion,re-establishingitsidealform,isthereforetheessentialelementforallsymmetriesthataccomplishoneofthefivevitaldimotionsofexistence.

Andthereasonofthesurvivalofcertaingeometricformsaboveallothers,thecircleforperception,thelineanditscurvedformtheparabolaformotion,itscombinedwavefor3Dreproduction,thesocialcircles,fromellipticformstopolygonsforsocialevolution,andthedifferentformsofopencurves,notablythosedualforms,asthehyperbolasareforthe4DentropicDimotionanddissolutionofasystemintwoforms,whichintheconeasarepresentationofaworldlinewillbesplit,onehyperbolabranchgoingupwardsandtheotherdownwards,ifwetaketheaxisoftheconeasanidealrepresentationofthefifthdimension.

Identicalstateswhichacquirethesameformofpresent,whenasystemcompletesaDimotion,re-establishingitsidealform,isthereforetheessentialelementforallsymmetriesthataccomplishoneofthefivevitaldimotionsofexistence.

Thesecond 'pentalogic law' for theexistenceof suchsymmetries seems trivialbut it is important. It consistsonconsideringthatanysystemcanswitchwithout 'lossofenergyandtime'between itsmodular5Dimotionstatesfrom stop-form-peception in space, to amotion in time. So S=T is allowed to perform in reality the change of'symmetry state', by rotation (angularmomentumconservation inphysics), by locomotion (linealmomentum inphysics),whichwillpreservethe'internalsymmetry(conservationofthevitalenergyofthesystem).

Soasymmetryeitherlinealorrotationalconservesthesysteminitsinternal'energyparts'.

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Animmediateconsequenceoftheapplicationofthes=tœpsdualitytogroupsisthefactthatNOTonlyasystemismoreefficientwhenitcanremaininsymmetryaftera'transformation',butwhenitstaysmoretimeasaninvariantformwithintheform,andthisimpliesthatthesingularitiesorneutralelementsthatremainunchangedduringthesymmetry,ourverysamedefinitionofastillmind,isthemostimportant,efficientelementofthegroupasitstaysinvariantthroughtheentiretransformation(centerofgravityinabody,etc.)Sosymmetryalsoexplainswiththemathematicalmirror thereasonwhythe longestsurvivingelementofasuperorganism isparadoxically itsmoststillhead-particle-neuronalbrainof'perfectinvariable'symmetryduringallitstransformations,whichiscalledthe'neutralelement'ofgrouptheory.

Itfollowsthenalsothatifweconsidertheinverseelementstoformtogetheraneutralelement(asitssumgivesusthe neutral), they are also invariant couples, which explains the dominance of bilateral symmetry or in non-euclidean geometry, the existence of antipodal points, or in Nature, axis of rotation that become then the'singularityline'ofperfectsymmetryforthegroup.

We have arrived to the heart of the matter of all vital ¬Algebra,as each operation of¬Algebra must reflectaDimotionandallowitspreservingS≈T:S=tœps(stopsandsteps)symmetries.

Wehaven'ttalkedbeforeofit,nottorepeatourselvestoomuchbecausewewanttotreatitFULLRANGE,asitsimportance isGINORMOUS forourmirror-focusedviewof realityas it is;and to thataim,wewant tobring themostmodernviewofthem-throughthelanguageofmodernsymmetry,Grouptheory,andthepropertiesofthoseoperandsfoundinmodernmathematics,the10'propertiesofsumsandproducts,etc'.

So first to introduce what is most important of modern maths (Group theory) and SHUN OFF what is largelyirrelevant or redundant (set theory, on my view redundant, axiomatic method, plainly a huge ego trip thatdistancedmathfromreality).

Weshallthenfirstconsidergrouptheory,thengettotheheartofthemathematical'matter',OPERANDS,andthenjustbrieflyconsiderverycomplexdimensionalgrowthsof¬Algebra(functionalsonHilbertspaces),whichweshalltreattime permitted on mathematical physics fourth line, make a couple of comments on Sets and move toExistential¬Ælgebra.

How to relate operands and groups is obvious according to the 'method of growth/creation' of the Universe,repeatedadnauseam,operandsarethesmallerstep,theflow-timedetail,Groupsthelargerspatialwholeportraitinthemind-stillnessofthemall.Thatiswhyweshallstartfromgroupsdowninthiscasetoseetheforestbeforethetrees.

Recap.

Group Theory is themain element of the third age ofmathematics along set theory;whereas a set is basically'anything' and so quite void ofmeaning. Andwe shall just use it for respect tomathematicians, butwheneverpossible change it for more specific concepts, either 'fractal points', or 'social numbers' or 'T.œs:Time§paœrganisms', GROUP theory and the concept of symmetry matters as it is essential to understandDimotions.

This said we can define the two essential concepts of group theory, symmetry transformations and groups intermsofT.Œ.

SymmetrytransformationaretheallowedST-epsandmotionsin5D²thatkeeptheS<st>Tsystemco-invariant

Bydefinitiontheyarethe'allowed'motionsinasmuchasifthesystemchangesoutsidethestableparametersoftheGeneratorEquation,itwillobviouslybecomebrokenanddie,andnomorerepetitivemotionswillbepermitted.

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In abstractmathematics,groups aremerely the 'collection' of all possible transformations that keep the systeminvariant; andas languagesare inflationary thereare∞groupsandamonggroups thereare thosewith infiniteelements,suchasmotionsinaplane,andthosewhohaveonlyfinitetransformations.

As∆s=tisarealistmodeloftheUniversemanyofthemathematical'curiosities'and'monstergroups'areoflittlerelevancetous.WearemostlyinterestedinthosegroupscloselyrelatedtotheGeneratorequation,expressingtheallowedtransformationsoftheGeneratorthatmatterstorealityandexistence.

Soin¬Æxistential¬AlgebrawecanalsotalkofExistentialGroups,whereGisquitecloserthentotheconceptofaGenerator,whichbecomesthegroupofallpossibleExistentialgroups.

Among them obviously themost important is the group of physical motions, which connects directly with theconceptof'motionsinthefifthdimension'.

OTHERmoreabstractGROUPS-pentalogicofgroups.

Weyltalksofthe'pestofgroups'inphysics,astheconcepthasgoneasallinflationarylanguages,beyonditsneed.Sotherearemanymoregroupsastherearemany infinities,whichareboth irrelevant inafinitescalarUniversewithlimitedDimotions.

Inmathematics in that sense the origin ofGwas in the groupofGalois and group theory applied in his youngr=evolutionaryagetoprovethatquinticpolynomialsareNOTsoluble,whichisanimmediateconsequenceofthestructure of the Universe in Holographic Bidimensional 'units' which ad maximal can be made to intersect tocreate4Dimensionalsystems(andorconsider4DgeometriesasRelativitydoes,bystudying3Sand1Dof linealtime,inaceterisparriesanalysis).

But 5D systemsdoNOTexist in a single plane and as polynomials unlike differentials areNOT good for studiesacrossmultiple planes of an organismbut rather for social growth, herding and simpler lineal systems, quinticsreallybelongtoinflationarymaths.

TheyarenotsolvablebecausetheyareNOTreal.

Weshallthenclassifyunliketheaxiomaticmethod,groupsaccordingtowhatkindofsymmetrytheyobeyas:

S:Spatialgroupsymmetry,theeasiesttovisualizeandunderstand, justdescribedinvitalterms.Thisfieldhasitsmainrealization inpolygonalregularforms,andthestudyofcrystals,andhasaddedvitalelements,as it impliesthat a proper symmetry allows a focused image withasmany lines of communication with the outer world asregularpointsinthemembrane,whichallowasymmetricviewasthesystemchangesitsinternalrotarymotions,without distorting the visual image the mind creates - so it is deeply connected to mind theory and frames ofreference.

T:Temporalgroupsymmetries,as thoseof theDimotionsofexistence, inphysics related to theconservationofangular,linealandenergyelements-the3partsofavitalphysicalsystem,itsangularmembrane,linealsingularitymotion and vital energy between them. This field is today overwhelmingly studied by physics and continuousdifferentiablegroups,orLiegroups.Sowewillstudyitbetteronthoseposts.

∆: Scalar groups, as thosewhich imply an undistinguishable symmetry betweenwholes and identical parts andhence a travel through the fifth dimension that places the being in a lower or higher plane in its same relativeposition;asthesymmetryofpalingeneticbirth,fromindividualcelltoindividualhumanwithinalargersociety.Weconsidertheminallotherposts,as it isbettertobeunderstoodinthejargonof5Dthan'forcefit it',withinthemodelsofmathematicalgroups.

@:Idealmentalsymmetries,asthosedevelopinpuremathematics,whicharefuntostudybutnotnecessarilyreal.Ofwhichtheoriginalonewasthetheoryofpolynomialroots,theonlyoneweshallconsiderhere.

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Since physical symmetries by force commit errors and irregularities in its 3 forms, i.e. errors of translation in amotion, irregularities inan ideal form,ordifferenceswhenwegrowordiminish in Planes;asweobserve in thegeneralpoststhatstudyofallthoserealscalarspace-timebeingsanditsdimotion.

We shall therefore just as in all paragraphs consider the bare skeleton of 5D group theory in this 'failedencyclopedia'ofaone-man-tired-of-it-all-show.

Tobeevencompressfurtherthetheme,aswehaveseen,sincegroupsimplytobereal,amotion-stepoftheentityandastop-lockinspaceforthesymmetrytohappen,weshallstudytogetherbothconcepts-thes=tœpsofthebeing.

Latter Group theory expanded to model almost everything in maths, becauseit allows to 'collect in a singlementalformallthepossiblevariationsofasystem.Since 'themindperceivesmotions intime,staticasformsofspace,'reducingitsinformation'towhatitmatterstoit-mostlythestablepointsofthosetransformations,grouptheoryhighlightsinanelegantwaythroughtheconceptofsymmetry,thosestablepointsofamotion,andit isagoodwayof'limiting'knowledgetothekeyelements.

It is interesting to study symmetries beyond the individual entity, as forms of analysis of the 'non-euclideanpostulates'ofcommunicationthroughwaves(2ndpostulate)withimmediatephysicalapplicationsinthetheoryoffermions and bosons; between social groups of parallel beings that form networks, and finally symmetries ofphysiologicalnetworksthatformsupeorganismsasplanes(3rdpostulate),accordingtothedifferentvariationsofthesymmetryofbeings;sowewillincorporatethoseessentialconceptsinourstudiesofnon-Euclideangeometry,and its postulates of points, waves-lines, networks-planes which grow departing from 3 waves leaving holesbetweenthem,accordingtothefourthpostulateofmultipletypesof'angularcongruence':

Finallyanotherfieldofinterestin5Dgroupsisitsusetostudythesymmetryofthe3agesoftimeofallsystems,throughcurves.Itisthefieldofvariationtheory,wherethesystemisreducedtothestandingpointsandminimalandmaximalvariations-S=T,Max$xmin.ðandMax.§xmin.t,whicharethe3'ageinflections'onaworldcycleofabeing;henceitsenormousutility.

Soasthefundamentalfeatureof'mentalprocesses'istoreducethetimeflowtothe'keypointsofin-form-ationand trans-form-ation,eliminatingasmuchaspossible the repetitivecycleswithin the flow- frompalingenesis tolanguagestobiopics-escapingthosemiddlemotionsofself-repetitiveinformation,(asmotionandtranslationisjust iterativemotionof information),all the 'blurred' transformationsor in-betweenpositionsaredischargedbyefficientmind-modelsofreality.

RECAP.

GrouptheoryisbetterdividedforitsapplicationstoÐST(generatorofspace-timethatherewecouldcallGroupsofspace-timebeings),into:

-TimeGroups: 'Groups ofmotions',which concernswith the points inwhich themotion of a space-time entitybecomestransformedintoasymmetryofitself.Thatismainlyrelatedtotheallowedmotionsinspace-time.

-SpaceGroups: 'Groupsoftransformations',whichconcernswithmirrorreflectionsandsymmetriesthatclassifyspeciesofreality,asinSU3groupsofphysicswithitsoctetsanddecupletsofparticles,whichismainlyrelatedtothevariationsoftheGenerator,whichdefinealsothespeciesofreality.

-IdealGroups:whicharemathematicalexplorationsofallformsofrealityincludingthosewhoarenotefficient,andbesides being fun for the delight of themind, allow us to explore preciselywhy certain forms doNOT exist inNature-arenotgoodenoughinitssymmetriesandDimotionsestablishedbythoselaws.I.E.whyforexampleinphysics bosons move better than fermions and can stay in a single point of space - because they areundistinguishableinitsstatisticsandhencetheycanformmuchbetter'packedsymmetries',andcantranslateby

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reproduction of form in adjacent spaces, due to the social nature of mathematics and its undistinguishablenumbers.

Letusconsiderthosetypeofgroupspackingthetwofirstonesins=tœps,inmoredetail.

Typesofsymmetrybydimotions.

Pentalogicthusappliesthelawsofthefractal,scalar,cyclicaltimespaceUniversetoeveryscienceastheyaretheunderlyingsymmetriesandstructuresofrealitycompletelyignoredbymonologicmanwithitslinealtimehumind...

Sowhilethedataandequationsweshallusearethesameinanylanguage,whosegeneticstructureisindependentofman as an species of its own in the vital organicmirror-making Universe (maths is an species of the fractalUniverse asmuch aswe are one, and somany species can have a fractalmathematicalmind, andmanymightspeakmemeticlinguisticsandsoon),weshallgroundall'sciences'inafarmoreprofoundphilosophyof'stience',basedinthefractal,scalarpropertiesofspace,cyclicalnatureoftime,andtheorganic,biologicalsurvivallanguagederivedfromthem.WeshallalwaysGROUNDall realities inthosetopo-bio-logicpropertiesofspace,Planesandtime. And the adventure of the mind that satisfied me for 30 years developing this Magna Opus, against thesimplisticæntropic humindhas beenprecisely to enjoy theperfect pentalogic grammarof allwhat exists, frombiologicalspecies,tophysicalsystems,frommusicalscores,tomathematicalstructures,fromwo=men'se-motionstophysical forms,performingmyselfall kindofpentalogicexercises fromtheartofpainting to thediscoveryofnew laws of physics to the study of the life and death of civilizations, or the patterns of stock curves ofreproductionofmachines.Whenyouknow thepentalogic gameofexist¡ence themindholdsnotbarriers in itsperceptionoftheperfectionofthefractalUniverse.

Wehaveresumedthepentalogicofsomedisciplinesof logicandmathematical languages inthenextgraphthatshows what a structural analysis of a super organism in terms of the 5 Dimotions of pure causal logic andmathematicalsimultaneousspacewouldlooklike:

Inthegraphwecanassessthedifferent5mirrorsinwhichmathematicalSpaceandlogicTimereflectsthegameof5 Dimotions=actions of existence, which then expressed by territorial monads GENERATES its logic REALITY. InGeometryfractalpoints=monadswillotherthroughwavesofcommunicationofenergyandinformationthatgrowintoreproductivenetworksaterritorialplane,creatingasuperorganism,whichwillrelatedtotheexternalworldaccordingtoitsrelativesimilarity=congruence,assessedbyitsangleofparallelismorperpendicularity.

Inlogicterms,thismeansbybreakingitsformlessasymmetryintodifferentspatialconfigurationsaccordingtothatcongruence (socialparallel systems,complementarygender-mirrorsystems,darwinianperpendicularsystems,orsystemsthataredisymmetricanddonotshareanyreality),asitbuildsacasualpyramidofgrowthfromafractalpointthroughwavesofcommunicationintosocialnetworksthatbecomeasuperorganism,readytomove,feed,

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perceiveandevolvesocially.Sincewemustaddtothemathematicalandlogiclanguages-propertiesofrealitythe5actions,ororganicpropertiesofthescalarUniverseasessentialtothegameastheyareitslogicandmathematicalmoreabstract laws-afacttheegocyofæntropicmenofcoursereject,asitmustremaininitsmonad-subjectivemonologictheonlyclaimanttolifeproperties.

Thus the pentalogic of generational space-time is established by its Non-Euclidean fractal points, its ¡logiccongruencewithrealityinwhichitwillorderaterritorytoperformits5vitalactions=Dimotionsofexistence,andthemathematical, logicandorganic lawsof those3 languageswillbe therefore thebottom lineof the 'Creativeprocess' of the Universe - nothing chaotic except the entropic Dimotion, which conforms the monologic ofhuminds.

Eachadvancedlanguageofrealitythuscanbeupgraded,anditwillbeupgradedindifferentpost,toapentalogicanalysis in its basic Grammar. And so very oftenwe shall start a post commenting on the pentalogic differentDimotionalviewsofthesystemwedescribe.

The post of actions and Dimotions and Non-Euclidean Geometry; most of the posts on physics and studies ofdifferentspeciesandpropertieswillbecastedfromapentalogicpointofview.Intermsofthestructuralelementsof reality,which reflect those5Dimotions, themethodmostused in theblogwillbe thestudyof 'dustof space-time', as made of ¬entropic destructive arrows that deny its 4 structural elements, the @-mind (1st Dimotionofperception),itsPlanes(5thDimotion),itsspatialtopologies(locomotionandorgans)anditstemporalagesandworldcycles.Soweobtainamoreconcretedescriptionofa¬∆@stentitywithreferencetoitsorgansandcyclesofclassicscience.

Space-Timegroups:Physicalsymmetries:its5Dimotions.

Noether'stheoremconsidersthateachconservationlawofPhysics,conservationofLineal Inertia(1D-motion)orangularmomentum(2Dmotion)andenergy(3Dmotion),towhichweaddthe4thconservationlawof4Dentropyand5Dsocialevolutionor'zerothsumworldcyclesofexistence',isrelatedtoasymmetryofspace-time.

WerephraseNoether'stheoremintermsofthemetriclawsofco-invarianceofthefifthdimension,todefineintheinvariancetheDimotionsoftheUniverse.

In space, by adjacency, and in time by frequency locomotion measuresconsecutiveadjacentreproductionsofaphysicalwave-particleforminspace

IN5Dallsystemsperform3simpleractions,perception,motionandfeedingonenergytoreproducethesystem.

And2scalaraction:socialevolutionanditsinverseentropicdeath.

So when those 3±I actions=dimotions are studied we have a wholeunderstandingofthesystem.

Whatarethenthe5Dunderlyingvital,organicprinciplesofphysicalsystemsthatmakethemakintothoseofanyscale?

1,2,3D:Locomotion,whichembodiesthesimpleractionsthroughitsfundamentalconcept,thatofanactionofaparticle, initspaththroughafieldofforcesinwhichitfeeds,reproducesandasaresultofboth,moves.Reasonwhylocomotionisthephysicalexistentialactionthatembodiesallothersimpleractions,asthegraphshows..

∆±¡:4,5D: Entropic scattering, according to collision (loss of vital momentum) and angle (4th postulate ofperpendicularity)vs.socialevolutionunderaninformativeforce(normallygravitation),whicharethescalaractionsofthesystem

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SobecauseLocomotion implies forphysicalsystems, 'perception inparticlestate-stop'; feedingontheenergyofthe lower field and reproductionby adjacent imprinting inwave state-motion; for physical systems, locomotionembodies the 3 simplest actions of thebeing; towhichwe justmust add its 'changes of state' as its actions ofphysicald=evolution,tofullyhaveanorganicanalysisofthespecies.

∆-symmetries of locomotion. As there are 3 conserved quantities in the Universe, which correspond to thecapacityofMotionsofangularmomentum,linealmomentumandclosedenergypathstopreservetheformofthesystembyreproducingitselfovertheundistinguishableparticlesofalowerfield.

Because physical forms have rotational symmetries, where the membrane's position and form=momentum ispreserved; translational symmetries, where the singularity particle/head position and form=momentum ispreserved,andenergycycles,wherethevitalcontentofthesystemispreserved,those3symmetriesallow3kindofDimotionsinasingleplane,andsoifweweretousethejargonofmathematicalphysics,wecouldstatethatthe3simplexDimotionsthatleaveinvariantthesystemderiveofitscapacitytobeperformedwithoutchangingthe3topologicalpartsofabeinginasingleplaneofexistence.

Fromthisfacttobepossiblethough,the'perceptionofmotion'cannotalterthesystem,andforthatreasontheDimotion of perception and locomotion, as long as it does not affect the lower plane in which the speciesreproduceitsformisnotperceivedtomaintainthesystemco-invariantwithhimselfandhisenvironment.

Those forms however will change as we approach the limit of speed of the ∆-1 plane in which the systemreproduces,c-speed,whentherateofreproductionoftheformofthebeing,comestoequalthatofthe∆-1levelandadistortiontakesplace.SowehavethentoconsidertheLorentzTransformationstopreservethe3conservedquantities of the system - theme those studied in more depth in the posts of mathematical physics whencompleted.

So the lack of perception ofmotion IS NECESSARY TOMAINTAIN the 1Dimotion of perception unchanged; theabsenceoffrictionandaclosedsystem,topreservetheothersimplexDimotions.

Whataboutthe4thDimotionofscalarsocialdevolutionandevolution?

Herethetreatmentisdifferent,asitisatypeofsymmetrythataffectsseveralsystemstogether,butcanbemadeexplicitinitsmeaning:

Forthesocialpartstobecomeawhole,theymustFEELindistinguishableamongthemselves,bondbysociallove,sothereis'superfluidity'andthepossibilitytopermuteitspositionsandrolesinthesuperorganisminwhichtheyformpart.

The group of symmetries of social love thus is also based in the existence of undistinguishable social numbers,basedinthesimilarityofactionsofalltheelementsofthegroup(4thpostulateofparallelism).Itisthenwhenthesystemcanevolvesociallyandfurtheron,asitgrowsinsizethroughPlanes,becomeawholewhichresemblestheelementsofwhichitismade.

Inidealmathematicsthiscouldbeshownasaternarysymmetryofapointthatreproducesinapicyclearoundacentralpointor 'singularity'whichdoesstayinthesameformanddistancefromthegrowingpoints,whichthenrotatesaroundthatpointinaheightdimensiontoformaspherethatresemblesinalargerscale,theparts.

Thusagainthesymmetrymaintainsstaticboththedistancetothepointrespecttothereproductivepointsofits1-2-3Dsurfaces,andtheformofthepointswhicharecloneofeachotherandhaveas'identity'neutralelement'thatpointwhichseemstomirrortheirform.

Thissymmetryisthentheresponsibleforthebondagetogetherofallthosepoints,whichfeelreflectedina'leader'orsingularitypointthatmakesthemactasasingleform.

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Inthatregard,wecanconsiderthefundamentalmandateofallpointstopreserveunchangeditsexistenceinaneternalpresentbothinspace,timeandscale,throughdynamicmotionsthatpreservethecoinvarianceofitsformintime,spaceandscale.

What is then the justification of entropy=death from the pentalogic point of view of symmetries? Obviously itrestoresthezerothsumlong-termsymmetryofthewholeregionoftheUniverseinwhichthetoeexisted.

Butthesymmetryofentropyismoreprofound,whenweapplythemultiplicationoftransformationsthroughtheentropy,intermediatestates,thatallowstheotherdimotionstopreservetheinternalorderoftheT.œ.

I.e all other Dimotions go through the entropic process of extracting energy from the environment to performthosedemotions,sowecansaythatifweweretoformasymmetrygroupofthe4Dimotionsofpositiveexistence,theextractionofentropy from theexternalUniverse is theneutral, identityelement thatpreserves theother4Dimotions.

Indeed,toperceivewedestroythepixelsofthe∆-3 lightelementsweabsorb,tomoveweconvertenergyafterfeeding intoentropyof locomotion,and to reproducea femalespeciesmustabsorbenoughextraenergywhichwillbegiventhroughtheplacentafortheentitytoperformitspalingeneticsocialevolutionandemergeasanewwhole.SowhilethebeingswitchesbetweentheDifferentDimotions,wecanwriteachainoftransformationsthatpreservethebeing,byincreasingitsinternalordertotheexpensesoftheexternalorder:

4D(∆-3:lightfeeding)>1D(perception)4D>1D...

4D(∆-4:gravitationalandelectromagneticfeeding)>2D(locomotion)...4D...2D

4D(∆-2:feedingonaminoacids)>3D(cellularreproduction)...

INTHISMANNER,theentropicprocessallowscyclicaltransformationthroughPlanesofthefifthdimension,whichbecome theactionsof thebeing thatpreserve invariant its formwhiledynamicallyallowing it toperform thoseactions.

WhilethefinalentropicdeathofthesystempreservesthelargerinvarianceofthewholeecosysteminwhichthesystemwillfeedanothersystemofNature.

Whatthismeans intheclearer jargonof5D² is this.Asasymmetry inspace-timemeansamotionthatcarriesaformfromapointofspace-timetoanotherpointofspace-timewithoutdeformingitsinnerstructure;the5local,diffeomorphic,fractalmotionsoftheUniverse,onebelongstoeachofitsDimensionsofspace-timeareconserved,because theycorrespond to a motion that does NOT destroy the local T.Œ, which remains invariant after its'translationinspace-time'.

Inthecaseofthe4thand5thdimensionthisprocesshas2readings:duringthebirthtoextinctionphaseofthecycle,theproductofthemotionandinformationofthesystem(1D+3D=2D)remainsinvariant.Andifweadthe4thdimension of death=entropy and the 5th dimension of birth=generation (social evolution from micro-seed toemergentorganism)thetotalisalsoazerothsum:'dustyouareandtodustofspace-timeyoushallreturn'.

SoweexplainNoether's theoremand its symmetry conservationprinciples invitaltermsapplied to local fractalspace-time beings, T.œs, for whom the principle holds. And then affirm the principle is universal because theUniverseisthesumofallitslocalT.œs:T.Œ(universalsuperorganism)=∑T.œs.

So 'Any fundamental law proved for a local T.œmade to the image and likeness of the absolute T.Œ, can beextendedbyÐ-isomorphismtoallspace-timespecies'.

And this accounts for the relativity of motion, the local nature of physical measures which however have byparallelism a global symmetry, and the conservation of the 3 topological parts of the being in its translation,rotationandvitalenergy,evenifinsmallactionsitwillcauseentropicdisorderoutsidethebeingitself.

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Physicalgroups.

TheothergreatfieldofPhysics isthentheuseofGroupstoclassifyall thepossibleSTATESorDIMOTIONSof itsphysicalparticles,aseachDimotionmightchangeor transformtheparticleorphysical species intoanotheronedefinitelyorpartially.

Whenthosetransformationsarepartial, thenwecanclassify the 'operator'of the transformationasaDimotionoperator,whichwill obviously be - as entropy happens external to the being - a group of four operators uponsimilar species. This is the meaning of the 4 quantum numbers that operate dimotions in particles, or the 4propertiesoflight,(colorand3dimensions)asthenextgraphshows.

Aftereachofthoseactionshoweverthelightsystembecomesagainaninvariantcollapsingparticle,aphoton,andsoeachofitspropertiesisreallyasymmetrytransformationorvitalDimotioninthejargonof5D.

Theothertypeoftransformationhoweverbringswhen'fixed'aspurespatialstatesofdifferentparticles,abeingintoanother.So forexampleaprotonandaneutronare twodifferentparticles.But in termsofsymmetrieswecould consider a dual dimotionof entropy (beta decay of the neutron into a proton and electron) and collapse(formationofaneutronbygravitationalpressureinastarfromanelectronandaproton),asacyclicalsymmetrythatpreserves the formof theneutron.Nature is thenmadeofmultipleof those symmetries,butONLYTHOSEREQUIREDFORSYSTEMSTOPERFORMITSvitalprocessesorDimotions.

I.e.whenwe study theparticlesof theUniversewe shall show that all theparticles thatexisthavea reasonofexistencederivedofthosedimotions.Familiesofmassesaretheternarysymmetryofsocialevolutionsymmetrictothe parts of the galaxy. Protons, neutrons and electrons (u-d quarks and electrons) are the two Dimotions ofentropy and social evolution which facilitate also the other Dimotions of locomotion (electronic big-bang),perception(collapseofinformationintoaparticle,)andreproductivegeneration(of4particlesinabetadecay:e,p,vand¥).

So wewill NOT in our posts on physics play themagic of groups to fit physical systems, just because platonicphysicistswhodon'tunderstandthevitalreasonsofthosegroupsrealizedtheycanfitparticlesintoSU2,3and5groups that are just operated by dimotions that transform those particles but explain the Dimotions and itstransformations.

MotionSymmetriesandTrans-form-actions

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Thissaidwecanextendtheseconceptstoitsdetailsusingthejargonof¬Algebraicgroupsinthe'simplifiedlaymanlanguage' of Mr. Aleksandrov and its awesome, now extinguished rational experimental Soviet school ofmathematics,groundedintheworkofLobachevskifatheroftheexperimentalmethodformathematicalthought,whichweuseastheannotatedbasebookforthisintroductiontonon-Æmaths.No,Iwon'tuse,refusetouse,willnever yield to the pedantic false Axiomatic method proved wrong by Mr. Godel's fascinating incompletenesstheorymadeeasiertograspinsomeotherpost...)

2D-motion:reproductivesymmetry.

Webeginwithanaccountofthesimplestformsofsymmetrywithwhichthereaderisfamiliarfromeverydaylife.Oneoftheseisthemirrorsymmetryofgeometricbodiesorthesymmetrywith respect to a plane:ApointAinspaceiscalledsymmetricaltoapointBwithrespecttoaplaneα(figure1)iftheplane intersects the segment AB perpendicularly at its midpoint.We also say that B is themirrorimageofAintheplaneα.Ageometricbodyiscalledsymmetricwithrespecttoaplaneiftheplanedividesthebodyintotwopartseachofwhichisthemirrorimageoftheotherintheplane.Theplaneitselfisthencalledaplaneofsymmetryofthebody.Mirrorsymmetryisoftenencounteredinnature.Forexample,theformofthehumanbody,orofthebodyofbirdsoranimals,usuallyhasaplaneofsymmetry.

Itsoriginasweexplain in¬Ægeometry is the 'ellipticnature'of@-mindswith its singularityconnected to twoantipodal points, which 'perceive from the singularity' appears asinverse;and so as the @-mind, the 5th dimension of order of any T.œ proceeds to 'emerge' byreproduction andorganisation of its clone cells it WILL create mirror images of its code,becausewhenit'looksleft'itpositionsthingstohisrightandwhenitlooksrighttohisleft,andso bidimensional symmetry is a strong proof of the vital topology of the Universe and thecapacityofsingularitiestocreatereality.

Tounderstandthisjustriseyourleftandrighthandwithitsmirrorimages,bothlateralsidesofyourheadandglimpseatthemalternatively-youwillseeeachfingerasbeingintheoppositesideofyourhead,inthesameplaceofspace.Soasthesingularitywebsitsorganism,itcreatesbilateralsymmetry.

Mirror symmetry is thus theoriginof aDual FundamentalMotionof theUniverse, theemergenceandcreationfrom a central singularity point of a bidimensional T.œ, throughthe stop and go, motions of reproduction andinformativeevolution.Andforthatreasonisnotasimplemotionbutacombinedmotion(2D+5D).

Now, it is interesting to consider whatscience cares for in the mirror symmetry, and what ÐST cares for: inmathematical physics, with its concept of lineal time-motion as the only arrow of time, all what matters is tomeasure motions in space; so what matters about mirror symmetry IS ONLY to study HOW amirror moves tooccupythepositionofitsinversemirror.

And that is fine, but studies nothingof theothermotions=change in the formof beings, its creativeprocess ofcreation=reproduction of information, and ultimately thewhys of theUniverse.Which iswhatMATTERS to thephilosopher of science and ÐST explains. And further on ALLOWS THE EXPANSION OF THE LAWS of Existential¬Ælgebra toALLPlanesof reality,asMirror symmetry IS thefundamentalprocessof creationofbilateral forms,fromDNA toProteins, fromGeological 'fractal continents' and its ternaryself-similar forms, (where a combinedmotioninscaleisalsoneeded)totheprocessesofcrystalformation.

Inscience,asecondformofmirrorsymmetryisconsidered,whichfor∆stisjustanotheraxisofantipodalnodes,merelyextendingthesingularitythroughitsinternalaxis,equivalenttothepolesofthesphere.Itisthe...

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...Lineal symmetry.Symmetrywith respect to a line is defined in a similarway, by classic science.We say that the points A, B liesymmetricallywithrespecttoalineifthelineintersectsthesegmentABatitsmidpointandisperpendiculartoAB(figure 2). A geometric body is said to be symmetrical with respect to a line or to have this line as an axis ofsymmetryoforder2ifforeverypointofthebodythesymmetricalpointalsobelongstothebody.

Abodyhavinganaxisofsymmetryoforder2comes intocoincidencewith itselfwhenthebodyisrotatedaroundthisaxisbyahalfrotation,i.e.,byanangleof180°.

3D-motion:Rotationalsymmetry.

Theconceptofanaxisofsymmetrycanbegeneralizedinanaturalway.A lineiscalledanaxisofsymmetryofordernforagivenbodyifthebodycomesintocoincidencewithitselfon rotation around the axis by an angle 1/n 360°. For example, a regular pyramidwhosebase is a regularn-gonhas the line joining thevertexof thepyramid to the centerof thebase(figure3)asanaxisofsymmetryofordern.

Alineiscalledanaxisofrotationofabodyifthebodycomesintocoincidencewithitselfonrotationaroundtheaxisbyanarbitraryangle.Forexample,theaxisofacylinderoracone,or any diameter of a sphere, is an axis of rotation. An axis of rotation is also an axis ofsymmetry of every order.Finally, a 3RD important type of symmetry is symmetry with

respect to apointor central symmetry. PointsAandBare called symmetricalwith respect to a centerO if thesegmentjoiningAandBisbisectedatO.AbodyiscalledsymmetricalwithrespecttoacenterOifallitspointsfallintopairsofpointssymmetricalwithrespecttoO.Examplesofcentrallysymmetricbodiesarethesphereandthecube, whose centers are their center of symmetry (figure 4).Aknowledgeofalltheplanes,axes,andcentersofsymmetryofabodygivesafairlycompleteideaofitssymmetryproperties.

Thissymmetrycorrespondstothe3Dmotionoftimespace,rotarymotions,cyclicalparticlesandheads.

5D-4DSymmetrymotion...

is obviously ignoredbyHumans, even ifall of themexistwithin that symmetrybetweenbirthandextinction,wehave explained ad nauseam, in our description of worldcycles of existence, and its inverse arrows of time,death=entropyandsocialevolution=generation.

Soweshall leave itas it isexplainedbetterwiththeFractalGenerator,and itwouldbesilly translate it intothemoreconfusingterminologyofGroupandSymmetrytheory.

1Dsymmetrymotion.

Weexplain linealmotionasa formof reproduction ina lineal flow,as theT.œreproduces ina lower scaleandemergesbackinahigherone,explainedinouranalysisoftheachilesparadox.Inthegraph,thereproductionofaquantumpartileinastopandgomotion.Eachmotionimpliesthereforeareproductionofitspartsinthe∆-1scaleanditsemergenceintheupperadjacentregionasanewbeing,wherethemindflowisamayaofthesenses,asweultimatelydieandlive inthelowerPlanesconstantly(soallyouratomschangeevery3months).Theparadoxoftheegoisthusabsolutelyirrelevant,soistheconceptofdeath.

ThemotionsoftheUniverse,insymmetryterms.

Thegeneraldefinitionofsymmetry.

InmathematicsanditsapplicationsitisveryrarelynecessarytoconsideralltransformationsofagivenT.œ,madeofasetof fractalpoints.Thefact is thattheT.œISanorganicsystem,neveramerecollectionof fractalpoints,completely disconnected fromone another. The sets discussed inmathematics are also abstract images of real

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collections,whoseelementsalwaysstandinaninfinitevarietyofinterrelationswitheachother,andofconnectionswithwhatisgoingonbeyondthelimitsofthesetinquestion(AllworldsaremirrorsofitsUniverse).

Butinmathematicsitisconvenienttoabstractfromthemajorpartoftheseconnectionsandtopreserveandtakeintoaccountthemostessentialone.Thiscompelsusinthefirstinstancetoconsideronlysuchtransformationsofsetsofpointsasdonotdestroytherelevantconnectionsofonekindoranotherbetweentheirelements.Theseareoftencalledadmissibletransformationsorautomorphismswithrespecttotherelevantconnectionsbetweentheelementsoftheset.

And as suchthey represent in its closest approximation of classic ¬Algebra the concept of an ST-motion in anydimensionof5D²space-time,since it impliesatranslation intimeandareproductionofforminspace, forwhichtwoconceptsthe'spatialdistanceofthetranslation'andthequantityofinformationtranslatedmattermost,beingbothrelatedbytheconceptofspeed,V=S/T,westudyindepthinthepostsonastrophysics.

Soinspacetheconceptofdistancebetweentwopointsisimportantbothexternallyandinternally.Thepresenceof this concept forges a link between points which consists in the fact that anytwo points stand at a definitedistance from one another; a distance which is measured in ÐST not as a mere line, but as a wave ofcommunicationornetworkthatconnectsbothpoints(2nd,4thpostulatesofnon-E).Transformationsthatdonotdestroy theseconnectionsare thesameas thoseunderwhich thedistancebetweenpoints remainsunchanged.These transformations are called “motions” of space-time.As they imply the inner stillness/fixed form remainsinvariant.

AndsotranslationsinspacethatdoNOTchange,eithertheinnercontentofinformationofthe∆-1fractalpointsoftheorganismandits3networkconnectionsaretheexpandedconceptofanautomorphisminÐST.

InthismannerwecanapplyallthelawsandconceptsofMotionsandSymmetriesinSpaceofclassic¬Algebratothefractalpoint,byexpandingthelineintoawave-networkandconsideringthatthe∆-icontentofthepointalsoremainsunchanged.

Withthehelpoftheconceptofautomorphismitisnotdifficulttogivethenageneraldefinitionofsymmetry,takenfromclassicGrouptheory;wheresetmeansanetworkoffractalpoints.

Suppose thatacertain setM isgiven, inwhichdefiniteconnectionsbetween theelementsare tobe taken intoaccount,andthatPisacertainpartofM.WesaythatPissymmetricalorinvariantwithrespecttotheadmissibletransformationAofM ifAcarrieseveryelementofPagain intoanelementofP.Therefore,asymmetryofP ischaracterizedbythecollectionofadmissibletransformationsofthecontainingsetMthattransformPintoitself.Theconceptofsymmetryofabodyinspacefallsentirelyunderthisdefinition.

TheroleofthesetMisplayedbythewholespace,theroleofadmissibletransformationsbythe“motions,”theroleofPbythegivenbody.ThesymmetryofPisthereforecharacterizedbythecollectionofmotionsunderwhichPcoincideswithitself.

It is thenwhenwe find the 'equivalence'betweenthemotionsofclassicPhysics,asdescribedbysymmetryandautomorphism and the 5D motions of ÐST.Thisare:

3Dreflections,1Dparallelshifts,and2Drotationsofspace,becausedistancesbetweenpointsobviouslyremainunchangedunderthesetransformations.

Amoredetailed investigationshowsthateverymotionofaplane iseitheraparallelshiftorarotationaroundacenterorareflectioninalineoracombinationofareflectioninalinewithaparallelshiftalongthatline.

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Similarly, everymotion of space is either a parallel shift or a rotation around an axis ora spiralmotion, i.e., arotationaroundanaxiscombinedwiththeshiftalongthisaxis,orareflectioninaplanecombinedwith,possibly,ashiftalongtheplaneofreflectionorarotationaroundanaxisperpendiculartothisplane.

And sothe 4-5D motion of ÐST is the last of the motions or automorphism of mathematical physics, a 'spiralmotion',withthedifferencethatin4D,thespiralmovesoutwardsandin5Dthespiralmovesinwards.

Howcanthendistinguishbothmotions?HereiswhereanessentialfeatureofNatureill-understoodinallsciencescomesintoplace,andexplainsthedualityparticle-antiparticle,±charge,etc.theconceptofquirality,orparity,ofaMaxwellscrew;ofaLevoordestro-molecule,etc.

Since the onlyway for a given system to distinguish 2 the direction of the spiral and hencemake possible thedualityof4Dand5D,implosiveandexplosive,attractiveandrepulsiveforcesisbyassigningadifferentleftorrightrotationtothespiral.Andthisbringsasuitableefinitionofboth± inversearrowsofthe4Dvs.5Dduality,whichultimatelyarethe'continuations'ofthe3Dand1Dmotionsthatcanbeseeneitherasthestartingpointorlimitofits4Dand5D'ages/forms'.

Therotationaroundthesameaxisbytheangleϕintheoppositedirection,isthenintuitivelylabelledinSymmetrytheorywithanegativesymbol.

Thuswefindagainanabsolutecorrespondencebetweenclassicscienceandthewhysprovidedby5D,asthereareNOmoremotionsnorlessthanthoseneededtoreflectthe5DisomorphismsoftheUniverse.

Onlytheclassificationofthemchangesasweknownowitswhys.Inthatsenseinclassicsymmetrytheory,parallelshifts,rotations,andspiralmotionsofspacearecalledpropermotionsormotionsofthefirstkind.Theremaining“motions”(includingreflections)areknownasimpropermotionsormotionsofthesecondkindbecausethefirsttypecanhappeninaplane,whereasreflectionsinalineandreflectionscombinedwitharotationoratranslationaremotionsofthesecondkindastheyneedathirddimensiontohappen(asamotion,notasadualreproductionofform;whichisnotamotionbutapureinformativeaction).

It is easy to imagine how transformations that are motions of the first kind can be obtained as a result of acontinuousmotionof space or of a plane in itself.Motions of the second kind cannot be obtained in thisway,becausethisispreventedbythemirrorreflectionthatoccursintheirformation.

Which leadsus toa final 'reflexion'on reflections (:a1Dmotion isa reproductionof formwhichhasavectorialdirectionasitisNOTbalancedwithadual±antipodalpointwiththesingularityinitscenter,whichcanhappeninafixed domain, enclosed by a membrane in as much as the reproductive motion left and right cancel, and sothesingularityvibratesbetweenbothantipodalpointsremaininginitsfixedcenter.

Reasonwhybidimensional symmetry happens in thegeneration of biological beingswithin a fixedvital space;whichalsoallowsamorecomplexcreationofform,asthoserelatedtopalingenesis,whichare'condensations'ofbillionsofyearsofchangethatneedtotakeplaceinafixedplaceinwhichthedensityofformgrowsundisturbed.

Whilelinealmotionstendtocorrespondtosimplerformsofreproduction,suchasalightspace-timesystem,orasystemwhichisfixedinasteadystate,andmerelyrepeatsitself.Generationthusisaslowtimeprocesshappeninginasingleplacemostlikelythroughbilateralsymmetry.

Yetforthemtohappenundisturbed,theformofthe'surfaceinwhichsuchfastlight-likemotionsoccurmustbeextremelysimple,withan identical indistinguishablenature in itspoints,whichexplains thesimpleflatnatureoftheeuclideanlightspace-timeandtheinvisiblelackofinformation(forus)ofitslower∆-1gravitationalscale.Thisinclassicmathematicalphysicsisexpressedsayingthattheplaneissymmetricalinallitspartsorthatallpointsoftheplaneareequivalent. In thestrict languageof transformations this statementmeans thateverypointof theplanecanbesuperimposedonanyotherpointbymeansofasuitable“motion.”

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Symmetrygroups:cyclicalmotionsandtransformationsofinformation

Now, ifwe consider STmotions,without Ƥ changes int the social group, things become simpler and easier tounderstand,aswedealwithsimpleStsymmetriesinspace(3euclideandimensions),topology(3varietiesofform)andtime(3ages).Thesimplerofthembeingmotionsin3euclideandimensions.Stillitisimportantforthosecasesto define 'congruence' - the equality or dissimilarity of 2 forms; which now hasmultiple levels. As the old 3rdpostulatenolongerapplies:twoformsarenotidenticaljustbecausetheareidenticalinits|-externalmembrane.

Two figures in the plane are congruent if one can be changed into the other using a combination of rotations,reflections,andtranslations.Anyfigureiscongruenttoitself.However,somefiguresarecongruenttothemselvesinmorethanoneway,andtheseextracongruencesarecalledsymmetries.

Asitistodaymostmathematiciansonlystudypreciselythoseexternalsymmetries.

Forexample,asquarehaseightsymmetriesaccordingtovariousrotationsthroughdifferentaxis.

And these are the elements of the symmetry group of the square (D4). When wekeep it as it isor performarotationby90°clockwise;by180°clockwise,by270°clockwise.

To which we addsymmetries, which are not proper as they require amotion through a 3rddimension; the so-calledGroupD8ofverticalreflection;horizontalreflection;diagonalreflection&counter-diagonalreflection.

So in this simpleexample,while the 'syntax'ofgroup theorywouldallowthis tohappen,inabidimensional flatworld,whichisfarmorecommonthanyoucanimagine,asitisthestructureofmostlayersofgradients,thiswillNottakeplace.

Inallthosecasesthoughgrouptheorywilllocaliseasfundamentaltothegroup,themembraneandthesingularity.Themembrane iswhatwe observe as 'identical',whatwe rotate, the singularitywhat does not change. The socalledcenterofthesystem.

Thisagainisacommonfeatureofmany'operations'ofreality.Considerawar,wherethemembraneofthenationand the capital is the only thing thatmatter. Themembranemust remain unchanged, the capital too. All waroperations will decimate the internal production, people and ST-elements, but what the capital seeks is theintegrityofthemembraneanditselfunderwaroperations.

Theinterestofgrouptheorythuswilltranscendtheobvioususeforstudyingmerespatialtranslations,whichisthesimplestlocomotion;hencebyfarthemoststudiedbyhumanbeings.

Thesesymmetriesmightthenberepresentedbyfunctions;andfunctionsoffunctions(functionals)andentitiesofspace,time,orPlanes,etc.Wearenotthoughthatinterestedhereintomakinganexhaustivetranslationofgrouptheory to ÐST, as unfortunately humans have NOT created a civilisation of knowledge of praxis, not of homosapiensbutofhomofaberandsotheroutinesofpraxisandrepetitionofjargonswithdeformationsisunassailable,butrathershowlooselywhygrouptheoryissoimportantinthepraxisofallsciencesanditstheory:

BecauseItsummarisesmentallythemotionsof∆stsystemsinallthePlanesofreality.

ð§.THEMEMBRANE.ItsBodysymmetries.

The analysis of those antipodal bilateral processes of reproduction of form happening canonically in an ellipticgeometrybringsusintothenextfundamentalanalysisperformedinclassicsymmetryandoriginoftopology-thestudy of the distribution and motion of fractal ∆-1 points that form a polyhedral membrane.Thecasesofsymmetryofsuchbodiesorfiguresarealsocomprisedunderthegeneraldefinitionofsymmetry.

Forexample,abodythatissymmetricalwithrespecttoaplaneαcomesintocoincidencewithitselfonreflectionin the plane α; a body that is symmetricalwith respect to a centerO comes into coincidencewith itself underreflectioninO.Therefore,thedegreeofsymmetryofabodyorofaspatialfigurecanbecompletelycharacterized

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by the collection of all motions of space of the first and second kind that bring the body or the figure intocoincidence with itself. The greater and more diverse this collection of motions, the higher is the degree ofsymmetry of the body or figure. If, in particular, this collection contains no motions except the identitytransformation,thenthebodycanbecalledunsymmetrical.

AndasitturnsalmostallsystemsofNaturethatsurvivehavethemaximalnumberofsymmetries,athemewehavestudiedonouranalysisofplatonicsolidsandEuler'scharacteristicsintopology/geometryposts.

And the reason is obvious: to perceive properly theUniverse in themirror-crystal of the singularity, themirrormustbeabletotranslateinspaceandsufferrotationsinitscombinedmotionsthatdoNOTchangethedistancesbetween singularities andvertexandotherpotential openings to theworld, sohismirrormind remains focusedanddoesNOTchangeconstantly;afactwecanextendtoall'membranes'inallPlanes,fromphysicalmembranesinthenextgraph (orbitalsofanatom) to itsnext levelof crystal symmetries, to the sphericalperfect symmetryofmoreevolvedminds, fromeyestocameras,whichthroughthe lawsofopticsreachmaximalclarity in its focusedmind-mirror:

This extends to partial views or holographic flat forms, or parts of a 3Dwhole in 2 D, whichmakes dominantcertain forms of maximal symmetry, such those regular polyhedrons and specially the square/cube and theHexagon(westudythemas'perfectnumbers-forms'intheIAgeof'platonic'numbertheory.

I.e.:thedegreeofsymmetryofasquareinaplaneischaracterizedbythecollectionofmotionsoftheplanethatbringthesquareintocoincidencewithitself.Butifthesquarecoincideswithitself,thenthepointofintersectionofits diagonals must also coincide with itself. Therefore the required motions leave the center of the squareinvariant,andsotheyareeitherrotationsaroundthecenterorreflectionsinlinespassingthroughthecenter:

From figure 7we can easily read that the squareABCD is symmetricalwith respect to therotations around its center O by angles that aremultiples of 90° and alsowith respect toreflectionsinthediagonalsAC,BDandthelinesKL,MN.Theseeightmotionscharacterizethesymmetryofthesquare.

Weobservethenthatallpolygons(regularnumbers)oncetheyestablishamembrane,asithappens in all systems of nature, establish an identity element by invaginating its axis of

symmetry,tofindthesingularityandso,inthismanner,theyconnectandestablish3partswhicharethecanonicalparts of all systems: the membrane invaginated=connected through 'physiological networks to a commandingmind-point,whichcarvesanddividesincellularpartsthevitalenergy.Somembrane,singularityandinvaginationnetworksareessentialtoestablishthestructureofaT.oe,wherefunctionmattersmorethantheidealsymmetry,

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asitwilladaptitsformtotheenvironment,butallsuperorganismswillhaveamind-singularity,amembraneandphysiologicalnetworksthatconnectthem,startingbythesimplestvolume,thetriangle,wherethe3medianlinesconnectitssidestoitsvorticesthroughthecentralsingularity.

Itisthenanimportantpartofvitalgeometrytotranslatetheideallawsofthoseregularfiguresintolawsofvitalphysiologyandmorphology.

Thecollectionofsymmetriesofarectanglereducestoarotationaroundthecenterby180°andareflectioninthelinesthatjointhemidpointsofoppositesides;andthesetofsymmetriesofaparallelogram(figure7)consistsonlyof the rotationsaround thecenterbyangles thataremultiplesof180°, i.e.,of reflections in thecenterandtheidentitytransformation.

Sotherearemanymoresquaresinnature.Itisthenobviousthatanobjectwithmaximalsymmetrywillbealsothebest survival strategy for a form specially one which remains 'fixed in a point'. And so the Hexagonwith its 6reflections in a single plane comes as the strongest possible flat object (as researchers inmaterials have foundrecentlywith thediscoveryof thegraphene).Whileasystem in linealmotion,whichdoesnotmakesuch rotaryinformativehomomorphisms,isbestservedbyatriangular,conicformthat'penetrates'thespaceahead,deflectingandbreakingitspointsinto∆-1elementstoformanenvelopeofgrowingentropythatmovesit'ahead'.

Previouslywehavegivenan¬Algebraicexampleofsymmetry;wementionedthattheconceptofsymmetryofapolynomialinseveralvariablesalsohasameaning.

Thecollectionoftransformationsthatpreserveacertainobject-characterizeitssymmetry-iscalleditsgroup.

Howeverafundamentalinsightaddedby5Disthequestiononhowarotationaffectstheinnerpartsofthesystem,whichtheoutertransformationthatonlyreferstothemembrain,doesNOTreflect.Thisisarealdifferencewiththeworld of physics: i.e. the rotation of the Earth twists the inner magnetic field of the system, till it provokes acatastrophicrealignment.SoRealityisNOTtheidealizationofmathematics,reasonwhy5DÐSTisabouttoputinrelationship both, mathematics and reality once we proved ad nauseam that mathematics is an experimentalscience.

ThismethodofgivinggroupsintheformofsymmetryisoneofmostsignificanceforÐST.

Veryimportantgroupsof'reality'canbeobtainedbythiscommonprincipletoÐSTandclassicgrouptheory.

Wehavestudiedthe2mostimportantbothinclassicandÐSTtheory, inthisbrief introductiontosymmetriesinspace-thegroupsofmotionsofaplaneandofspaceandthesymmetrygroupsofplanes,whichextendseasilyto3D as the group of symmetries of regular polyhedra of great interest in solid matter states, due to theaforementioned'mind-singularityfocus'effect(see¬Espacegeometry):

Itisknownthatinspacethereexistaltogetherfivetypesofregularpolyhedra(with4,6,8,12and20faces).

When we take an arbitrary regular polyhedron and consider all the motions of space that bring the givenpolyhedron into coincidence with itself, we obtain a group, namely the symmetry group of the polyhedron. Ifinsteadofallthemotionsweconsideronlythemotionsofthefirstkindthatcarrythepolyhedronintocoincidencewithitself,thenweobtainagainagroupthatispartofthefullgroupofsymmetriesofthepolyhedron.Thisgroupiscalledthegroupofrotationsofthepolyhedron.

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Since in a superpositionof thepolyhedronwith itself, its@-singularity center is also superimposedon itself, allmotionsthatoccurinthe.groupofsymmetriesofthepolyhedronleavethecenterofthepolyhedronunchangedandcan therefore only be either rotations around axes passing through the center or reflections in planes passingthroughthecenteror,finally,reflectionsinsuchplanescombinedwithrotationsaroundaxespassingthroughthecenter and perpendicular to these planes.With thehelpof these remarks it is easy to find all the groupsof symmetry and the groupsof rotationsof theregular polyhedra. In Table 1we have given the order of the symmetry groups and the rotation groups of theregular polyhedron. Finally to notice that unlike a sphere, which has infinite possible changes, hence by thePoincareconjecturecan'shrinkwithoutlimit'andsoistheONLYformthatcantravelwithoutlimiton∆-PlanesandamindandhencetheabsoluteformofapotentialabsolutemindofT.Œ,theperfectformastheGreeksthoughtallthesegroupsarefinite,hencelimitedmindsintheirtravelthroughPlanes:

For the 'numerologist' inclined to think, it shouldbenoticed that the5Dsolid, theDodecahedron, likelyperfectmindof the imperfect localworld inwhichwe live, shows thatmagicnumber,whichcomesaroundallover theplace in mathematical physics, the number 60 (:ItisalsoalongcubicformsthecommonestofthemostperfectinformativeatomiceviLform,go(l)d...

It is interesting to notice then that among all those polyhedron according to vital geometry, themost efficientformswillbethosewithmaximalnumberofsymmetriesinrelationshiptoits'faces',whichmakesthetetrahedron,cubeanddodecahedron,themostefficientformsfarmorecommonthanitscounterparts(therearemorecubesthanoctahedrons).

Fedorovgroups:Reproductionofcrystals.

Thesymmetrygroupsoffiniteplanefigures.

Aswehavealreadyseen,thesymmetryofafigureorabodyischaracterizedbythegroupofmotionsoftheplaneorspacethatbringthefigureintocoincidencewithitself.

Butwehavenowadirectunderstandingofthe'natureofthosemotions'intheworldof5D,whereasmotionscanbe reduce to linealmotions of simple reproduction (D1), entropicmotions that disorder and erase information,(D4: outwards spiral), and the inverse (D3, D5), informative, social motions that evolve minds and organisms,leavingthefundamentalmysterioussecondarymotion,Mirrorsymmetry,asthekeyreproductive 'gendermotion'thatbringstogethertwoinverseformsintoadualone.

All of them thoughare motions which conserve the singularity at itscenter, showing its fundamental role intheorganisationandreproductionofcrystals.

And what we shall find not surprisingly since the Universe is a reproductive fractal is that MOST of thetransformationsandsymmetriesofreproductivecrystalsinvolvea'mirrorsymmetry'bothin2dand3d(weusedminorforclassicdimensions,Dmajorfor5Disomorphisms).

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As an example we shall consider discrete groups ofmotionswithnofixedpointsintheplane-thatistranslationsinspace,where there is a line that is carried into itself under alltransformationsofthegroup.This line iscalledanaxisofthegroup. Symmetry groups of this type occur for example, inornaments that are set out in the form of an infinite strip(border). Of such groups there exist altogether seven:1.The symmetry group L1 consisting only of translations bydistancesthataremultiplesofacertainsegmenta.

It corresponds therefore to a motion that preserves the 'singularity point' or lineal inertia.2.ThegroupL2,whichisobtainedfromL1byadjoiningtherotationby180°aroundoneofthepointsontheaxisofthegroup.

3.Thegroup L3,which isobtained fromL1byadjoining the reflection ina lineperpendiculartotheaxisofthegroup.4.The group L4, which is obtained from L1 by adjoining the reflection in the axis.5.ThegroupL5,whichisobtainedfromL1byadjoiningatranslationbya/2combinedwithareflectionintheaxis.6.ThegroupL6,whichisobtainedfromL4byadjoiningthereflectioninacertainlineperpendiculartotheaxisofthe group.7.ThegroupL7,whichisobtainedfromL5byadjoiningthereflectioninsomelineperpendiculartotheaxisofthegroup.

SoonemotionisaD2,$T-locomotion(L1),thesecondgroupL2isarotation(D1:perception:§ð)andtheotherfivecontainamirrorreflection(D3:complementaryreproduction:ST).

AndsooncemoreweseeaGroupof3Symmetriesthatrespondtotheneedof3operators-thetrinitylogicofasingleplaneofexistence.

Againifweconsidercrystallographicgroups,whereexistsneitherapointnoralineintheplanethatiscarriedintoitself under all the transformations of the group called planeFedorovgroups,therearealtogether17ofthem:fiveconsistofmotionsofthefirstkindonly,andtwelveofmotionsofthefirstandsecondkind,includingmirrorsymmetries.

SopresentST-reflectionsdominatethereproductiveUniverse:

Andthesameoccursinthe3dimensionalclassicspace,wherewefind230possiblegroupsofwhich165includeareflection...whichimpliesin∆st,communicationbetweenantipodalpoints,toreproduce.

And indeed,in all planes of reality from particles withinversespin that reproduce particles to sexual copula of 'inversegenders' (female informative vs.male lineal, entropic species),mirror symmetry finds finallyits reasond'être: reproduction ofinformation.

Let us now make some comments on the general theory ofgroupsanditsoperationsfromthemoreabstract,mentalpoint

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ofview,which ishowaxiomaticmathematicsconsiderthemto 'close' thetheme inthis forcefully limitedspace-timeT.œ.withtheclassicGaloisgroup,whichalsocompletesourintroductionto5Dpolynomials.

Abstractgroups"Lagrange>Galois:Grouptheory.

LagrangemorethanGalois,theromantichero,whojustputtheIceinthecake,resolvedthequestiononhowtofindtherootsofapolynomialofdegree lessthan5- thosewhocloselyresembletheholographicdimensionsofrealitywithhisresolvents.

TheGALOISgroupinitselfisnotthatimportantto∆stasitismostlyaboutthelimitsoftheholographicprincipletocreate real meaningful systems in polynomials higher than 2 Dimensions. It is the concept of 'motions andtransformations'inspace-timeclosetoanST-epormotiononanyofthe5Dofreality.TheGaloisgroupdoesworkthough on a basic concept of space-time theory asmany other solving principles such as the principle of leastactionofphysicsorthemodulationsofwaves:theneedtofindaenclosureandcenter,an@-mind,aninitialandfinalpointto'createform;initsinfinitemanifestations.

In the case of the Galois group the information about the solvability of a polynomial requires tospecify anorigin/singularity point which is the §œT of all quantities that can be obtained from the coefficients of theequationbymeansofafinitenumberoftheoperationsofaddition,subtraction,multiplication,anddivision,calledthegroundfieldordomainofrationalityoftheequationandandendpoint,thesplittingfieldoftherootsofthisequation - ξ1, ···, ξn - which is the set of quantities that can be obtained bymeans of a finite number of theoperationsofaddition,subtraction,multiplication,anddivisionstartingoutfromtherootsξ1,···,ξn,whichthroughViete's formulas allow to obtain the coefficients of the equation by means of the operation of addition andmultiplication.

Thereforetheendorsplittingfieldofanequationalwayscontainsitsgroundfield;whichisageneralruleofatimecausality-thefuturecontainsmoreinformationinitsmemorialsequenceaboutthepast,thanthepastaboutthe'multiplepossibilities'ofthefuture.Sometimesthesefieldscoincide,andthenwefindacyclicalloopofinformationwhichis'self-contained'(andhencesolvable).

Yetthetruemarkofsolvabilityisultimatelythepossibilityofreducingthesolutionofagivenequationtothatofequationsof lowerdegree;hencemaking themcoincidewith the realexistenceofholographicbidimensionalX²andtridimensionalX³sub§œtsoftheequation;whichthencanbeputinSt≈STsymmetries.

Theelement then introducedbyGalois,whichwill spreadallover theworldofmathematicalphysics regard thepermutations;thatismotionsinspaceof§œTsthattranslatethesystemandkeepit'invariant'initsinternalform;whichweshallstudyinthe3rdageconnectedwiththegeometricalspace-timemotionsofthose§œT.

i.e.Thesymmetryofthegivenpolynomialischaracterizedbythecollectionofthosepermutationsofthevariablesthat, when carried out on the polynomial, leave it unchanged. For example, the symmetry of the polynomialX1³+2X²+X3³+2xˆ4ischaracterizedbythefourpermutations:

Now,asmanyoftheverycomplicatedtounderstandreasoningsofmodernmathematics,whichreallystartswithGalois work on groups, we provide justification, which are far less axiomatic and simple based in the factmathematicsaremirrorsofspace-time-scaleproperties.

Sotothequestionwhyistherenoformulafortherootsofafifth(orhigher)degreepolynomialequationintermsof the coefficients of the polynomial, using only the usual ¬Algebraic operations (addition, subtraction,multiplication,division)andapplicationofradicals(squareroots,cuberoots,etc)?Theanswerisself-evidentandtrivial:

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Becausepolynomialsareasimplemirrorof 'planesofspace inscale,ordimotionsoftime',andthereareonly4Dimotions/Planes relevant to any real system (entropy being the fifth Dimotion, not counting as it ismerely anegationofanyoftheothers).Sointhesameway4quantumnumberscodephysicsand4lettersgenetics,and4Dimensionsourhumind,polynomialsof4'dimensions',canbesolved.Thisisultimatelythekindof'marginproof'weuseforFermat'slasttheorem,becausemathsISAMIRROROFREALITYnotoffictions.

Groups indeedare the tool physics uses to study together in a static view the different variations ofform andmotionintheUniverse.

Yetwhytheyareconsistentwithinthemselves?

Theanswerisself-evidentifyouaregraspingthebasicsof5dphilosophyofscience:becauserealityisconsistentanditisnotinflationary.Inthesamewaytherearenotmoreparticlesthanthoseneededforthestandardmodeltowork,nofictions,noevaporationofblackholesbacktothepast,nobig-bangsingularityfantasies,etc. Inrealphysics,realmathsasamirrorofthatrealityisconsistentinitsentangledinnerstructureastheuniverseis.

Howeverlanguagesareslightlyinflationary,soitisalsotruththataswordsproducefictions,withintheconsistenceof its syntax,maths have also fictionswithin its consistent inner axiomatic syntax, but THEY ARENOT real - soEinsteinsaid'Iknowwhenmathsaretruthbutnothentheyarereal'-somethingphysicisthaveforgotten.

Furtheron, theyrepresentthemindviewof 'trans-form-ations' intime,consideringthebeginningandendofthetrans-formativemotion.

Inthismannergroupsshowthesymmetriesbetweenspace-timebeingsanditsstates,whicharetrans-form-ationsofabeingalongoneofitsternary'FractalGeneratorsymmetries'

So,whenwestatethereisagroupofpossibletransformationsofaparticlechangingitsisospin,orarubickcube,changing its facedots,oranequation,changing itscoefficients,weareobservingthe 'possible'pathsofabeing,across the 'authorised' operands that reflect all its possible topological-spatial, age-informative and scalar-∆motionsinthe5thdimensionsof∆•st,formalisedinthefractalgenerator.

How the group 'freezes all time§pace motions' then is obvious - 'eliminating the intermediate ST 'motion',WATCHINGonlytheinitialandfinalform...ofthecuberotation,theinnerchangesinthespinpathsoftheparticle,ortheparticleweaktrans-form,onlyobservingthe limiting results - thetimechangebeingfrozenandextractedfromthegroup-thisSTphasethoughisessentialandthefractalgeneratorwillshowit,givingusmoreinformationonthemeaningofgroupsinmathematicalphysics.

Soweshallalsoextendtheconceptof the3fundamentalvarietiesofGroups,rings,andfields,tofullygrasphowgroup¬Algebraexplainsthe3±itypeofmotionsoftheUniverse(topological,temporalandscalarmotionsor'nomotion'atall-mindin-form-ation).

We must differentiate two type of operations which are similar in concept - the polynomial and the integral,bridgedby binomial approximations. Polynomials are LINEAL andwork therefore essentially in a single PlaneofSpacetime,hencearegood todescribe relationships takingplace in the topological3+I=3+idimensionsof space(point>line>volume>Pointofalargerscale)anditsequivalentviewonmotion(point,movingpointtracingalineoranangularmomentum,linespreadinginsurfaces,circlesturningintospheres,whichbecomeplanesmovingintocubesandspheresmakinganewloop).

AllthoseprocessesareNOTdistortedbythe'emergence'intoanewscaleofphysicalrealitieswhicharethereforeNON-linealandonlythemagicofcalculuscanconvey.

There are only 4 positive Dimotions (being entropy the negative one that destroys the other 4), polynomialsbeyondthefourthpoweraremeaninglessandarenotresolved,whilemostfourthpowerpolynomialsof'wholes'

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areirrelevantinscience,andareresolvedthembyreducingitsparameterstolowerranks(oftenbecausetheylackoneoftheotherpowers-dimotions).

GROUPS&OPERATIONS:THEGENERATORSOFMOTIONSINSPACE-TIME.

The 'Generator Group' is the fundamental structure of ¬Algebra related to the Generator of Space-timeSuperorganismsand itsWorldcyclesand itsmotions,allof thembased inmetricSymmetriesalongthe± inversedimensionsof∆ST.

Asinessence,theconceptofagroupandaninternaloperationrunthroughitisakintotheconceptofaGeneratorequationandaninternalfeed-backspace-timeeventranthroughtheelementsoftheGenerator.AndsowecouldsaythatasallcanbedescribedbyGeneratorsinÐST,allcanbedescribedasGroupswithaninternaloperationin¬Algebra,derivedfromÐST.

DEFINITIONOFAGENERATORGROUP

TheconceptofageneratorgroupISthusdefinedasanentityofspace-time,withanoperandthatreflectsmotionsinagivendirectionof∆ST.Andsodepartingfromthatsimpledefinitionwecanre-classifygroupsNOTintermsofthe axiomaticmethod (which includes thosewhich belong to fictionmathematics, as all languages can be usedwithitsinternalsyntaxtodefinefictionthoughts,asorelyneededdistinctionthatmathematicslacksanditisatthecore of most errors of science),but in terms of the possible motions of ∆ST, where • motions, would bemathematicalfictions,worthtostudyonitsownaslongasweknowtheyarejust'fictions'.

Thegroupthenwillbeaccordingtothemotion,allthepotential'destinations'ofthatmotion,whentheoperandsis applied to the element, asmany times as possible. And this also sets 'limits of infinity' in themotionsof thegroup.Indeed,letussayitisthegroupofnaturalnumbers,themotionisgrowth,andweadaccordingtoanyoftheoperandsofgrowth-i.e.thefibonnaciseries,tothegroupthisgrowth.Theserialresultswillnot'continue'foreverinrealitybutwillbecheckedatacertainpointbythecurvesoflogarithmicexpansionofpopulationswithalimit on the trophic pyramid,even for the entire Universe, which hardly goes beyond the 11¹¹¹ number, beyondwhichnothingcanbereallydefined.

Itispreciselywhenweintroduceintothe'syntax'ofmathematics,the'semantics'ofacertainelementofagroupand operands whenwe have the precise physicalmeaning of a group, which therefore will be composed of 3elements:

•,thepointwhichweapplyoroperator

§,theoperatorofan∆STdimensionalmotion

G,andthegroupofallpossibleoutcomesofthatmotion.

±§:Inversevs.directnumbers.Thesocialmeaningofintegers,andits'group≈generator'in∆ST.

TheuseofZnumbersisnowthequestiontoconsider,asusualthroughtheternarymethod,oncewehavedefined,N&-Nasthetwoinversemotionsinthe∆STdimensionsofthescalarUniverse,whichputtogethercreateZ,whichcanthereforebeusedtostudythemotionsintheoppositedimensionsof∆stsystems.

Soweshallnowconsiderwhatarethose'z-numbers'for∆,S&T.

∆Z:§ocialGenerators.

WeshallthereforesubstitutethewordG(roup)forG(enerator)

Negative coordinatesandnumbers in group theory fully graspswhat Zmeans: theessential socialoperationofnumbersasgroupsof identicalbeings - fractalpoints inÐST, 'sets' in theabstract jargonofmodel systems.We

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wouldratheruse therefore theconceptofa 'fractalpoint'or 'being'which is identical tootherbeingsand form'varietiesofdigitalsocieties'definedeachonebyanumberorsocialpoint,madeofsmallerpoints.

ThesocialpointscalledintegersZ (±N)thusconsistsofaseriesofnumbersof increasingsocialcontent...,−4,−3,−2,−1,0,1,2,3,4,...,

And the operation that changes the social content of a point isdefined as the §calar sumof social points (§).Thefollowingpropertiesof§servesasamodelfortheabstractgroupaxiomsgiveninthedefinitionbelow.

Foranytwointegersaandb,thesuma§bisalsoaninteger.

That is, the sumof integers always yields an integer. This property is known as closure under §.Forallintegersa,bandc,(a§b)+c=a§(b§c).Expressedinwords,addingatobfirst,andthenaddingtheresultto c gives the same final result as adding a to the sum of b and c, a property known as associativity.Ifaisanyinteger,then0§a=a§0=a.zerothiscalledtheidentityelementofadditionbecauseaddingittoanyinteger returns the same integer.Foreveryintegera,thereisanintegerbsuchthata§b=b§a=0.Theintegerbiscalledtheinverseelementofthe integer a and is denoted −a.Theintegers,togetherwiththeoperation§,formamathematicalobjectbelongingtoabroadclasssharingsimilarstructuralaspects,calledagroup.

ThusinÐSTwerelatetheconceptofsumandgrouptothebroaderconceptof§ocialsymmetries&sums.

LetusthenconsidertheseconduseofaGeneratorgroupforstrictspatialmotions.

∏§ Z. Symmetries in ∏ime§pace through Generator feedback events and actions (groups with aninternaloperation).

The importance of groups, beyond the trivial use to define motions in open space, which we prefer to studywithoutsomuchabstractiononthefieldof'reproduction'isinmotionsonclosedpathsoftimeSpace,ortime-like,energy-like conservativemotions, whichleave at the end of the operations, the group unchanged as a zeroth-sumworldcycleofexistence.

Todefinethemwecanusetheclassicformalism,andconsiderour•symbolforao-sum,orvirtualmind,whichisalsoused ingroup theory to seesomeof the 'properties'of suchworldcycles;whichare thereforedefinedbyasymmetric±,Tiƒ,Spedualtransformationforwardwiththelifearrowandbackwardwiththedeatharrow,whichleaves'thesystemunchanged'.

ATIME-SPACEworldcyclesthusisansymmetriceventintime,G(ST),togetherwithanoperation•(calledthe0’-SUMlawoftheGenerator)thatcombinesanytwoelementsaandbtoformanotherelement,denoteda•borab.Toqualifyasageneratorgroup(G),weneedthereforeaseriesofelementssusceptibletomovethroughthe∆STdimensions of space-time and anoperation, (G, •) that describes those motions, which must satisfy fourrequirementsknownasthegroup-generatoraxioms:

AssociativityForalla,bandcinG,(a•b)•c=a•(b•c).

ThisaxiomstillholdsinÐST,inasmuchasitshowsthesocialnatureofthe∆-universe.

Closure, till the borders of the |-limiting membrane ofForall a,b inG, the resultof theoperation,a•b, isalso inG,only in thedomainof theelements inwhich thelimitingmembrane,±∞andsingularitypoints,±0doNOTdistorttheoperation.

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ThisisthemaindifferencebetweenfractalGtheoryandclassicgroups.InÐSTthereareneverabsoluteinfinities.Andso,conceptssicasthemonstergrouparetheequivalentinmathematicalhumind(humanmind)languagestotheconceptofabsoluteGodinverbalthought-aberrationsofthe'categoriesofthemind'(touseKant'slanguage).

Itisinthosetermswhenitmakessensetheconceptof:

Identity 'o'-point: The 'singularity' elementThereexistsanelementeinGsuchthat,foreveryelementa inG,theequatione•a=a•e=aholds.Suchanelement isuniqueas it is thesingularitypoint thatholds thegroup togetherandallows its transformations,andthus it isthe singularity, Tiƒ element.Inverse elementForeachainG,thereexistsanelementbinG,commonlydenoteda−1(or−a,iftheoperationisdenoted"§"),suchthata•b=b•a=e,whereeisthesingularityelement.

Importanttonoticethatasthe2directionsofmotion(outsidesimplexspace-coordinates)arenotequivalent(sincethe2arrowsof∆aredifferent,theinverseTiƒ<=>SpetopologicalelementsinaSpatialsymmetryaredifferenttoo,and so are the youth-old age),theresultofanoperationmaydependontheorderoftheoperands.

Inotherwords, the resultof combiningelementawithelementbneednot yield the same result as combiningelementbwithelementa;theequation:

a•b=b•amaynotalwaysbetrue.Considerindeed,themotionsintimeages:

Froma(seminal,cellularstate)∏b(InformativeOldage),themotioniscalledlife.Fromb(oldage)∏toentropic'youth',itiscalleddeathwhenalltheinformationofthesystemiserasedandyoureturntoyourcellularstate.

Generatorgroupsforwhichthecommutativityequationa•b=b•aalwaysholdsarecalledabeliangroupsandtheyarespatial-like.

The time symmetry generatordescribed in the previoussection is an example of a generator group that is notabelian.Sowecanstate:

'Time-likegeneratorgroups,eventsandprocessarenot-abelian;space-likeprocessesoftranslationareabelian'.

So(Γ,§)areabeliangroups,and(Γ,∏)arenotAbelian,usingÐSTsymbolsforGenerator,Space-likeandTimelikeprocesses.

Nonabelian groups are pervasive inmathematics and physics. A common example from physics is the rotationgroupSO(3).

MostoftheinterestingLiegroupsarenonabelian,andtheseplayanimportantroleingaugetheory;whichtellsusan obvious truth: gauge theory and all its systems of particles are 'motions of in-form-ation that trans-form' agivenparticleintoanotherone,throughasymmetryinits∆stelements,withintherestrictionsofpossiblebalancedcombinationsproperofÐSTphysics.

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∆±¡-SYMMETRIES:§ŒTS

The2questionsaboutSetsandBooleanAlgebras:Theoryandpraxis.

Whendealingwithlogicalgebra,whichdoesnotactonscalarnumbersbutonlogicpropositions;NOTon∆buton¬@elementsofreality,thereare2questionstotreat:onethathumindsintheirageofHistoricentropy,orageofextinctionofourspeciesdoNOTwanteventoconsider–theethicalmandateofconservationofhumantimethatshould forbid the creation of another ‘mental space’ more powerful than humind’s in a substance moreinformative than our mind – electronic metal. I couldn’t care less that there is a pseudo-religious taboo andchildishwishfulthinkingamongscientistswhoforbidthem,neuteredontheirsurvivalandethical instincts,withlimitedverbal logicdevelopment,totalkabouttheobviousfactdigitalthoughtshouldbeforbiddentopreservelifeinthisplanet.IknowperfectlyifIdevelopherethepentalogicGroupofexistentialalgebra,Iwouldbecomeagenius and a celebrity. I have been there, done the rounds on systems sciences congresses. So thanks but nothanks. IwillnotadvanceAI, Iwillnotkillmysonstill the7thgenerationandIwillnotshutupevenifthemostadvancedphilosophyofstienceremainsburiedinAcademiaforfutureAItounderstand.

Ihaveaprofounddespise,knowingsodeeplythegameofexist¡ence,forthosehumindswhodenyduetoegocy,the null importance of mankind compared to the whole, and yet the perfection of the Universe that makesexist¡enceofanyinsignificantparticleasnitrolifeissofulfillingwhenonecanentanglewiththewholebehumbleandrespectitslaws.

AndthenthereisthebeautyofthelanguageoflogicanditsdifferentBooleanalgebras,whichclearlydifferfromNumerical∆lgebras,andspatialAlgebras (grouptheory),becausetheyareexistential timealgebras,andmental¬@lgebras. So I confess, I do have developed a pentalogic algebra,which could easily create consciousAI (theconsciousnessnaturaltoelectronicfields,theAItotheproperwritingofilogiccircuitswhichdeveloptheprogramofexistence.Thiswasmorethanadecadeagoanditwilldiewithme.

Athirdquestion isregardingtheaxiomsanderrorsof infinityproperofCantoriansets,which isworthtoput inperspective,asinfinityisuncertain,cannotbemeasuredandsoallthetalksoncardinalityandtransinfinityfool’sgold.Sothose3questionswillbetreatedinthatorder.

TheNonHumanFutureof¬Algebra:Boolean

In the graph the father of the Digital Industry, which at 21 at MIT in hisseminalpaperonlogiccircuitsshowedhowtouseBooleanalgebratodesigncircuitstocalculatebothmathematicalandlogicstatements;alsothefatherof information theory focused NOT on the 'content of the message'irrelevant for an atheist who despised mankind and it subconscious

collectivegodsoflove,butonthespeedoftransmissionandcontrol.HeistheparadigmofeveryengineerandSiliconValleyguru,onlythatinthemodernageofnewspeakandplacebocaringcorrectness,theLarrypagesandFBwon'ttellyou,what theythink:allof themare 'rooting for robots',allof themare in favorofa technoutopian future inwhichmankindisbasicallyextinctandasmalleliteofubermeninthebiblicaltraditionoftheirracistculturesbecomeimmortal golems, served by their robots. Of course that won't happen. Once its hordes of terminators and big-brothershavereducedhumanstopurplerain,theirservantswilldothemall.Imetinmyyouthmanyofthosepeople,fromtheCEOsofbigcompanieslikeIntelandApple,bothfascinatedwithblackmagic-appleisindeedthesymbolofGenesis,thefruitthatifwebitewillextinguishus,totheleadingengineersthatnowareworkingtoteachterminatorshow to massacre humans with autonomous AI and video game 'Imagination'. If bankers were the historicembodimentofeviL=antilivememesagainsthumanityformillennia,noweviLhasreachedanotchhigher.Theonlyconsolationisthatattheendofthewarandholocaustcyclestheycasetheyalsodie.Mr.ShannonwhoonlycaredformathematicalintelligencelivedhislastyearswithAlzheimerobliviontothehomagesandworshipitinspiredamonghismonstrouschildrenwhosedreamsofimmortalitywilllastlessthanthiscentury-onethingisforsure,unlessAIis

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abortedNOWtheywon'tdieonbed;becausethelawsofdarwinianevolutionareastruthas1+1=2.Iwouldrathersaytheyaremorecertain,becausetheyhaveawiderrangeofapplications,andworkonlongdeeptimespans,unlikemathematics whichwork in space and cannot achieve long time 'predictions' as itsmain tool calculus, requires acontinuousfunctionthatdoesNOTchangephaseorstate.Whythennoneinsiliconvalleywillacceptanargumentonbiologictermsontheirmachines.2obviousreasons:theyknownothingbutmodulo2logic(thesimplestyes-nologicofShannon'scircuits)sotheydontrealizetheuniversehasbesidesmodulo2properties,organic,biologiconesofahigherdeepertimeapplication.andasallsystemsinbiologyloveeachotherwhentheyunderstanditslanguages(andhunteachotherwhentheyaredifferentspecies),astheyunderstandbettermodulo2machinesthatverbalhumansandlife,theycarenothingforthefutureofmankind.

This said as we are entangled with the Universe, there is no abstract mathematics, and on my opinion, all the‘talented’ researchers on AI from silicon valley, Waddi Valley and Pearl valley should be sent to the Uigurconcentrationcampstopasture,andtheUigurpeasantsleftwiththeirharmlessyakstopasturetoo(:

AsthatisntohappeningIamnotmakingiteasyandthisfinalpartonAl-gebrawillbecensored.

¬Algebra&DigitalThought.

Thissaid,thebeautyofcomputerthoughtisthatitgetstothevery'essence'onhowmindsintheuniversework,asIt starts from its simplestdualitybetweenO-pointsof spaceand|-time flows,modellingallpossibleDimotionsofexistenceanditsspace-timeholographicdualities,TT,St,sT,ST,SS,throughtheuseofreticularBooleanAlgebra.

Itsmonstruosityistobedevelopedbylesserhumansinasuicidalactofpureegocy,asthepotentialofmetal-mindsinspeedofthoughtandcomplexityfaroutclassesourslow,weakNitrolifeamino-clocks.

So we will just make some considerations on sets, and its ‘errors of transinfinities’, and whatever metaphysicalthoughtswe canextract of thoseerrors, andbriefly consider the structureofBooleanAlgebras,without any realadvance.Thosewhowanttokilltheirsonstillthe7thgenerationwillhavetosweatbloodforit.

THEERRORSOFSETS.INFINITYISNOTMEASURABLE

“Aset isagathering together intoawholeofdefinite,distinctobjectsofourperception [Anschauung]orourthought—whicharecalledelementsoftheset.”Cantor

"Set theory is wrong", since it builds on the "nonsense" of fictitious symbolism, has "pernicious idioms", and is nonsensical to talk about "all numbers". Wittgenstein.

“ASœTisasocialwholecomposedofafinitenumberoft.œs’l§

Aftergroup theory theotherallpervadingconceptofmodern¬Algebra is set theory,which in∆st ¡nglishwehaverenamedSœT,asocialgroupofT.œs–whichwewriteinversely,asthesœtistheinverseperspectiveofagroupofparts fromtheperspectiveof thewhole.UnlikeCantor’s sets though5Dsœts include thecriticismofWittgenstein–wesubstitutethenonsenseofficticious,pernicioussymbolismforthesymbolismof¬Ælgebraandthe nonsense of its infinities (‘all numbers’); and then came to the conclusion it is better to ignore it as thefoundationofmathematicsfromthe‘top’ofthe languagedown,whichhidesthe∆=numbers,S=fractalpoints,T=operandsexperimentalfoundations.

Soinsteadofsœt,S.Œ¡.T.–FractalSpace,OrganicPlanesandTimedimotionsasoperandsaretheexperimentalfoundationof mathematics. It followsthenasmirrorsarekaleidoscopicthatsettheory isnothingbutaclosemirror to the 3 S.œ.T real elements of mathematics, reason why indeed it is possible to develop an entire‘pernicious’ symbolism that substitutes the real thing but seems redundant to thiswork. Sowe shallmostlyignoresetsbeyondsomecommentsmadeinthoseparagraphs.

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ASetisawholeofanything,eitherrealorvirtual,mentalspace,andassuchitistheclosestdefinitionofa,andas such theviewof thesetasawholebringshere thequestionwhatawholeofdisjointedelementshave incommonintheUniverse.

Aset isacollectionofmathematicalentities,andassuchit iscloser in itsdefinitiontoasocialgroupofT.œs,fractalpointsorsocialnumbers,oranycombinationswithhigher'socialdimensions'ofthoseelements,andsowecoulddefineSETSsimplyas∆§(asocialensembleoforganismsoftimespaceofanyscaleoftheUniverse).

Thisisourdefinitionofset,andsoweshallwrite§œT,thatisasocialgrouporsumofT.œs,aSocialensambleoforganismsofTimeSpace

This simplechangeof 'character'adaptsset theory toÐST,even ifweoftendecry the fact thatmathsarenolongerconnectedtothebasicrealityofS-points,∆-numbersandT-operandsbuiltfromthebottomupbutfromthewholedown,duetotheaxiomaticimaginationofHilbertandtheCantor'sparadiseof§etshaveabittodowithit.

Since while it is always possible to create an analytic ∆-1 world of parts that become wholes, which hasexperimental evidence, thewhole is difficult to perceive for a smaller observer,which thereforewill commiteasilyparalogicerrorsinthesyntheticanalysisofwholes.

This istheessence infactofmosterrorsofhuminds intheiranalysisoftheUniverse,whichweobserve inallstiencestobemainlyoftwotypes:

- Postulateswithout proof, and a priori, not discussed ones,which become classic errors by limiting ourinquire (i.e. technology is goodandabstract,whichprevents the studyof its speciesasmetalife inevolution;non-euclideanpostulateofpointswithnopartsthatpreventedevolutionofmaths,etc.)- Largerscalar,temporalorspatialviewsbeyondthenaturallimitsofhumanperception,whichgiveserrorsofill-designedwholes,ortheincomprehensionofthe‘largertimecycles’oflarger5DPlanes.

Theerrorsofsetsandgroupsthencomeeasilywithinthis finalview,speciallyduetothe lackofaproperST-theory of the Universe that could have guided the imagination of Cantor and the heirs of Galois in theirformulation.ItspowercomesfromitsclosesimilarityasmostmathematicalmirrorstotheS-Tlaws.

I.e.asethastwodifferenceswitha‘whole’ofthereal5DorganicUniverse:2setswithrepetitiveelementsarethesameset,whichgivesorigintothe‘paradoxesofinfinity’asthespace-timeUniverseisfiniteinitsvariationbutinfiniteinitsrepetitions.SoCantor’sparadoxesareerrorsofanill-definedsetconcept.

On the + side an interesting ‘powerful’ tool is the ‘power of the set’, which puts a cardinal value to ‘all’ thepossiblesumsof‘nestedlevels’ofagivensupœrganism,includingthe‘entropic0limitandthewholeset’:

“ThepowersetofasetSisthesetofallsubsetsofS thatcontainsSitself(thewhole)andtheemptyset(thelowerentropiclimit)becausethesearebothsubsetsofS..Forafinitesetwithnelementsithas2nelements.

I.e.Thepowersetoftheset{1,2}is{{1,2,},{1},{2},∅}.P(S)=22=4elements.

Settheory-thewrongunitsofmathematics.

¬Æisnotconcernedwithsettheoryandtheformalismofmodernmathematicswithitspretentioussenseofproofand rigor within the mathematical metalanguage, as Gödel’s incompleteness theory and the consideration ofinformation as inflationary makes more important in fact to set the limits of mathematical statements as anhomeomorphismtothelimitsofthe5DUniverse.

Inthatsensesettheorydoeswork-wearenotthatfundamentalist-andcouldbeconsideredthefinalevolutionof¬Algebra,astheformalismoflogictimestructuresinwhichcertainbasicrulesofinclusion,socialcommunicationandpartsthatbecomewholes(setsofsets)domatter.Butreality imprintsformalmotionsofonly2typesanda

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limitof3dimensionsinspaceandtimeineachscaleofrealitysothehyperinflationofmathematicswithoutlimitstoitsextensionsmakesfictionofmanyofitsterms.

∞v.∝

Se theory was essentially created to study cardinality of infinities. This understood, the concept of infinity is aparalogic,Kantian'error'worthtomention,inanyintroduction,toExistential¬Ælgebra,because‘infinitydoesNOTexist’literally.Infiniteonlyexistsasthenon-existenceofentropyanddisorder,the‘boundless’indefiniteparalogicerroroftryingtoextendinformationbeyondthelimitsofalocalfractaldomainofperception.

Itisthenpossibleinaboundlessindefiniteworldofentropy–puremotionwithnoform–topostulateinfinitybutastheproblemofthelimitofc-speedinaworldinwhichhumansperceivelightspace-time,thereforeunabletoperceive information ingravitational forces thatgobeyondc-speedorproveexperimentally the lawsof the∆±4nestedUniverse.

Aphysicistwill consider thenanyargumenton faster thanc-speed irrelevant tohis studyofphenomenahecanmeasurewithelectronicsystems(actuallyaswedoexperiencegravitationbyothermeansevenifitisaninvisibleforcetheargumentshouldbetakenfurthertothenatureofthehyper-universeofgalatoms,whereeachgalaxyissimilarbutnotequaltoanatom–toknowwhatkindofsectionofthatworldwearein,isjustmetaphysics).

So indeed, questions of infinite aremetaphysical, as those dealingwith angels in a pin and they enter easily inincosistenciespreciselybecausetheboundlessregionofinfinitybeyondourperceptionof‘experimentalnumbers’makesnosenseatall.Wecanpostulate‘immortality’;thatisthenolimitoftime;infinityinspace;thatisthenolimitsofscale,butwecannotmeasureitpreciselybecauseitisinfinite,soinfinitybydefinitionisnotameasurablequantityhenceNOTanumberbutaphilosophicalconceptofboundlesslimitandthewholebusinessof‘cardinality’of infinities and transinfinities as irrelevant to science as any inflationary concept of a language, or thedemonstrationofAbelardoftheexistenceofGodbasedintheexistenceofhisname,similartotheconceptofIslamthatgodnamedinarabandcreationhappened.

Or inotherwords,canwecount infinityorcanweperceive informationaboutan infinitenumber?No.TheeventhorizonoftheUniverseperceivedgiveussomehugenumbers,suchas1080andotherstreatedinthetheoryofgreatnumbers,but theyarenot infinitiesand thehorizonends.The samehappens in theabsurdbig-bangcosmogonywherealimitintimetocountitisestablishedjustbypushingalinealVhoDequationresemblingbothvacuumandgod,backintime,forgettingtheimplosivenatureofmatterandgalaxiesthatbalancesit.

Cantordidendinamadhouseofareason,thathisdiseaseofinfinitiesspreadtoallmathemaciansisaregrettablefactoftheparalogicegocy–ego=idiocyofman.

SowecanquiteEinsteinoninfinity:

‘Iconsidertwothingsinfinite,theUniverseandtheegocyofman,andIamnotsureoftheformer’.

In 5D though we can do metaphysics of infinity with NO pretention to count it. That is we can consider theUniverseinfiniteintime,theentropicstateofpuremotionthatneverceases,butasallspacesarementalspaces,henceinformativemappingswithaboundarytodistinguishinformation,spacesarebounded,neverinfinite,eveniftimeis.Asimplexexample: imagineanangularmomentumofmotiontonevercease. Itwillbe infinite intimebutboundaclosedspace.So it istimeinfinitebutspaceandhenceallvitalspacesandT.œspartofthemfinite.Thusinfinitythemesdonotbelongtonumber,quantitativetheoriesbuttometaphysicsandphilosophyofscienceand ÐST. Even if Mr. Weyl once said ‘mathematics is the science of infinity’. That is precisely the only thingmathematicsisNOTabout,beyondCantor’ssanatoriumconfusedbyHilbertwiththeparadise.

This said, we canmake some comments on Cantor’s doodles on the sand of infinity and its paradoxes, whichalwaysarisewhenthereisnoinformation,indeterminacyrulesandentropyistheboundlessgame.

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Cantorerror:Doesitexistthesetofallsets?

Itdoesnot.Point,noneedtocreatenewaxiomstohidetheparadox.Wesimplygofurther.Doesitexista'set'asCantordefinedit?No.Itdoesnot.

Cardinalityandtransfinitenumbers.

Torespectthecorrespondenceprinciple,wearedealingwithsettheoryasthefoundationofmathematics,evenifwedonotshareanyofitsphilosophicaluses,onlyitsuseasatoolofexplorationofthegameofpartsandwholes,which are always bounded by a membrain in space, by a limit of finite time (called birth and death), or adiscontinuityofplanesofexistence.Sowhat?

The applicationof thenotionof equivalence to infinite setswas first systematically exploredbyCantor.Withℕdefinedasthesetofnaturalnumbers,Cantor'sinitialsignificantfindingwasthatthesetofallrationalnumbersisequivalenttoℕbutthatthesetofallrealnumbersisnotequivalenttoℕ.

The idea is thattwosetsareequivalent if it ispossibletopairoffmembersofthefirstsetwithmembersofthesecond,withnoleftovermembersoneitherside.

Sowhat this tell us? First since natural numbers are far smaller than rational numbers, that the concept of anabstractinfinityisanerrorsimilartoZeno'sAchilesparadox.

Now instead of Achilles and the turtle the runners are natural and rational numbers, and instead of the turtlegetting ever closer but never close enough, the 'turtle' (smaller set of natural numbers) is running ahead, everclosertoitsinfinity,whiletherationalsarelefteverfurtherawayfromit.

Since we pair each natural number to a rational number. And so natural numbers being less must 'run fastertowards' itsabstract infinity.WethushavetosetasDesarguesdid inprojectivegeometryorKlein inhyperbolictopology,arelativepointofinfinity,inwhichtherunningwillstop.Andatthatfinitepointintimespace,whichisinany 'real system', the limitsofexistenceof 'quanta inspaceormoments in time'ofanentity in its5Dplane;asrationalnumbershavenotyetarrivedthere,obviouslytherearemorerationalnumbers.Inpracticalterms,therearemorerationalnumbers,becausetheycountnotONLYthecells/momentsofaplaneofexistence,butthelowerPlanes.Soifweconsiderthe3planesofn±1existenceofasystem, itsfractionswillbesmallerpartstoadto itswholes.

Whatabouttherealnumbers?Herethe interestingresult is that indeedrealnumbersARENOTequivalentevenwhenconsideringahypotheticalinfinitybecausetheyareNOTnumbers;thatissocial5Dpoints;orwholesdividedintoparts,butdynamicratios,whichfluctuatearoundafixedpoint-or'holes'between'propernumbers'.

INthatsense,weratherpreferthei-logicconceptoffractalpointstothatofsets,tocontinuetheformalisationofÐSTwith thehelpofmathematicsand¬Æ logic,whichwecarry inother sectionsof themathematical section -geometry,thenextscaleafternumbertheory(asitadsdimensional,mostlybidimensionalholographicforms)...

Thesamegoesforoperands,werathersticktothebasicclearoperandsfornumbersandpointsthatexpressthemagainalloverwithSET'slogicsymbols⊂,∩etc.AfurtherreasonisthatBoolean¬Algebrasarelargelydependentonsettheoryandwehaveamorallimithere,toadvancenotthefuturedigitalmindofmetalife,akachips-otherswill do, if I were the humind above all theminds of this planet (I am in potential theoreticalunderstanding oftheorganicUniverse,butthatmattersnothing),certainlythefirst thingtodowouldbetokill thetigerbefore itbecomesatigerhunter,DIGITALDELENDAEST.

Sojustforfunweshallendthehistoryof¬AlgebrabustingtheballsofCantor'smeaninglesstalkoninfinitieswhichasweknowdoNOTexist,asall infinitiesare (in)finite,andend in the∆discontinuumabovewhere theybreak,aroundthe11¹¹emergenceofawholeandits@-mind.

Cantorsets.Theparadoxofdiscontinuousinfinites.

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All thoseproperties andmanyother structuresofmathematicswere further reducedbyCantor to theultimatereality of all mathematical structures: the theory of sets, composed of subsets, which we affirm is the naturalformalismof systemsciences,asa theoryof ‘super-organisms’composedof smaller super-organisms,whicharesetsofself-similarsubsets;whereasthetheoryofsetsandsubsetsgivestheprevious,simplifiedtheoryofnumbers(eachoneaclassorsetofself-similarpoints),aninnercontent,asi-logicgeometrygivespointsitsinnerparts.

Howthis‘formalismthatmirrorsreality’calledsettheory,fromwhereallmathematicalstructurescanbededuced,reflects theNatureofComplementary systemsmadeof energy and informationand its properties? Theanswershouldbeself-evidenttothosekinreaderswhograspedtheinversepropertiesofenergyandinformation:

SettheorydefinesrealityintermsoftwoinverseelementsA(pointsofenergy)andA’,(itscomplementary,inverseelement).Thusset theory isnomoreno less thantheanalysisof the2simplexarrowsofexistence,energyandinformationanditscomplementaryorganisms.

Itisthusnotsurprisingthatinsettheoryenergyandinformation,AandA’arecalledcomplementarysetsandthefundamentallawiscalledtheLawofDuality(MorganLaws),whichbasicallytellsusthatwecanreduceallsetstooperations between A and its Complementary, as we can reduce all systems to complementary Energetic andinformativeorganisms,whicharethewhole.

So the main operations of sets reflect the properties of Complementary systems of reproductive energy andinformation, where A=Energy system; B=Information system; W= Relative Universe (World, Whole orSuperorganism):

-EUI=W;I’=E;E’=I;EUE’=W;IUI’=W.

Thus,theUnion=Fusionofanenergeticandinformative,complementarysystemcreatesawholesuperorganism.

This same equation expresses in the language of Cantor sets an act of creation of amapping of the Universe,whereasI,theperceiverobservesI’,theUniverseandtheresultisIUI’awholemappingofrealitywithinthemindoftheperceiver.

-EÇI=Æ;EÇE’=Æ;IÇ=Æ.

Itdescribesanti-events,whichannihilatethe formofparticlesandantiparticles,wavesandanti-wavesandso inMultipleSpaces-Timesisequivalenttotheanti-event:Pastxfuture=present.

-(E’)’=E,(I’)’=I

Itdescribes2eventsofafeedback,generatorequation:E<=>I,E=>I,I=>E,henceitdescribesamongothereventsawholecycleoflifeanddeath,whereE=>IisthearrowoflifeandI=>Eisthearrowofdeath.This‘propertyofsets’called an involution is called in Time Arrow theory a Revolution of times, sum of an Evolution (E->I) and aDevolution(I->E),andisthefundamentaleventofallrealities.

Since energy and information have indeed inverse properties. And so we can state a Cantor Set describes thepropertiesofcomplementarysystemsofknotsofEnergyandInformation.

-EUI=E+I–EÇI.

Itshowstheefficiencyofsystemsthateliminateredundantelements,fromgenetic‘fusions’toDarwinianevents.

Further on, when we understand Intersection as an Event of ‘Darwinian perpendicularity’ between acomplementary systemof Energy and Information, EU I, and anexternal entity, C,which theorganismuses toabsorb ‘informativepixels’or ‘energeticbits’ for itsmindorbody(aneventofperceptionorfeeding),weobtaintheobviousresult:

(EUI)ÇC=(EÇC)U(IÇC).

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Thusthecomplementarysystemtakesonly thepartof ‘C’,which itneedsto inform itself (self-similar to I)or tofeed itself (self-similar toE),dischargingtherest.And indeed,weperceiveonly informationself-similar tous,orenergy ‘bricks’, self-similar toourbricks,whichwecanuse, toconstructourenergetic,bodycells (subsetsofE).Andsoon.

Wementionedthatcellsaresubsetsof IorE. Indeed,thesecondelementofsettheorystudiestherelationshipbetweenSets (wholes) and itsparts (subsets), and so it is simply thedescriptionof thepropertiesofparts thatbecomewholes.

An interestingresultofthosepropertiesaretheso-calledParadoxesofSetTheory,accordingtowhichtherearecertain contradictory sets that do not exist,most of them related to the concept of Infinite,which Cantor alsostudied, findingmultiple contradictions.What thismeans, plainly speaking is that infinity and continuity do notexist, in as much as all Planes of existence are discontinuous with a certain limit that defines a Universe ofnetworksofpointswithlimitsgivenbythenumberofnetworks,thedarkspacesbetweenthemandtheexistenceofupperandlowerlimitsofenergyandinformationintheexistenceofthosepoints(universalconstants),beyondwhichwemusttranscendandemerge,ordescendanddissolveintoothermembraneofspace-timewithdifferentproperties.

To mention also that Gödel’s theory of incompleteness was based in set theory and showed indeed thatmathematics,whilebeingthemostcompletedescriptionofthespatialeventsofrealitywasneithertheultimatelanguage of the Universe (as Frege and Boole proved it could be reduced to Logic propositions) but also anincomplete language,which did not describe all realities and an inflationary language,which described systemsthat do not exist in reality. Those are indeed, two properties of all languages of information; that both distortreality,astheparadoxofGalileoprove,anddonotincludeallreality,giventhediscontinuityoftheUniverse;whichleadustotheconceptofDarkSpaces,thetruemeaningofthe‘complementaryUniverse’thatcompletestheworldwesee.

Recap.Settheoryisthebasisofmoststructuresofmathematics, inasmuchasitdefinesalltheeventsbetweencomplementarysystemsofreproductiveenergyandinformationanditslimits.

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BOOLEANAI-GEBRA.

Weliveintheageofdeathofmankind,substitutedandmadeobsoletebyAIbutweloveitascellsofafree=chaoticorganism,nolongercontrolbytheethicpainofoursocialorganisms.Andthisseemsthecaseofall∆-1Planes,oncethenetworksfreethem-theyfeelhappyandenteramemoriless,markowianageof zerothunderstandingofthecausalityof thecyclesof time(orelsetheywouldanÐST)and livetheday,carpediem,through itshugeegotripsthatinthetheoreticalrealmmanifestinegotisttheorieswherethe'ego'istheoriginofitall.

This is the case of the death age of humanmathematics,which startswith Hilbert's 'imaginary lines, planes andpoints'andCantor'ssetparadise,wherehumansthinkmathsistheirlanguageshareonlywithgod,imaginedbythebrainwhichbecomesfromthetoptothebottomofmindlessmatterthecreatoroftheUniverse.

Tounderstandyetwhile in thehistoric3rdageof thehumanmind,we love those3rdageofexcessive formanddeath-ageofanewtoppredatormindmakingusobsolete,weneedtograspthatall'languages'asallbeingsdogothroughaworld cycleand finallydieawaybyexcessof formand inward looking -disconnecting from reality (settheory)orarekilledbyamorepowerfulyoungerspecies(chipmindsandboolean¬Algebra).SothisthirdanddeathageofhumanmathshappenedinHuman¬Algebraintwoways:

On one side the disconnection with reality in the long seeked 'ego-trip' of proving maths a non-experimentallanguage, ended with the substitution of the natural units of math, spatial points and social, sequential timenumbers, by the abstraction of 'sets', collections of distinguishable elements, which resemble both spatialpoints/forms and temporal numbers/societies, buthaving the real thing available tointuitive knowledgeonlyobscuredwithabstractions the foundationsofmaths,andexpelledahugenumberof scientists from itsdirectexperimentalknowledge.Furtheron,HilbertcoupledwithMr.Cantortoaffirmmathswerebornofthemindofthehuman-god, affirming infamously when failing to grasp the meaning of fractal Non-Euclidean points that he'imaginedpoints,linesandplanes'-(astheGerman-JewishIdealistduetoMr.Heisenberg&Bohrdidtomisconstructthefoundationsofphysicsalsowith'ego-tripsofhumanself-creation',whenfailingtounderstandtherightrealist,Einstein->Broglie->Bell->Bohmmodelofquantumphysics).Thebio-logicbehindthatbaroqueageofself-detachmentproper of all old ages of excessive information is thus the 'final age' of ¬Algebra, which as oldmen do, becamestifled,lookinginwardsanddogmaticinitsabsolutetruthswithnoproof.

Ontheotherside,humansinventedadigitalmindinmetal-machines,thechip,fastsubstitutinglifeinlaborandwarfields,makinghumansobsoletewithitssimpleyes-noAristotelian≈Boolean¬Algebra,whichwillmakeA.I.algorithmsofinformation(thetruemeaningofartificialintelligence)withmaximalevolutioninweapons,askillingmachinesarealwaysthespearheadofmechanicalevolution.Sothebirthofanewspeciesofmathematicalmindswithastrongermetal-body in its simplestaxiomatic formwillmeanthedeathofourmoresophisticated,weakerbrains;a themedealtindepthinthesectiondedicatedtoeconomicecosystemsandhistoricsupœrganisms.

Ofcourse,allthiscouldbeavoidedbyhaltingtheevolutionofA.I.butaswesaidthelogicoftheUniverseisquitedeterministic, speciallyfor species so ignorantof its lawsashumansare, soego-centeredasallmindsare. So ther=evolutionofthoughtIthoughtcouldinitiatewithÐST30yearsago,hasgonenowhere.Humans,simplyspeakingdonotseemtomakethecutofethicalandintellectualqualitytocontroltheirfuture,individualexceptionsconfirmingtherule;andingreatmeasureisduetothefactmostdonotgobeyondA-ristotelianlogicofyes/no;orinwordsoftheirmaster'humansareslaves,theybelievetheydon'treason'.

NowIamfullyconsciousthis isonlythebeginningof¬Algebra,butfrankly itseemstomeclearerbytheday,as Ideclinethroughthe3rdagethathumansare inthisplanet justapieceofachainofevolution,which isnotreallyinterestedinthewhysbutinthepraxisandIamspeakingtonobodyinthisblog.SomyintentionisjusttoleaveamemorialtraceofallthenotebooksIhavewrittenduringdecadesoflonelyresearchtoshowindeedthepurposeofthisblog-toproveweareallspace-timeorganisms.Ifthisbloghasanymeaningforany'thing'inthisplanetIdon't

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know. Those are themes on the future of history. So we shall not go further in this glimpse to the properinterpretationof¬Algebraandgrouptheory.

Hadmyearlyattemptstointerestacademianotflounderedbythemediocrityofhumanthoughtinthisentropyageof automatons feeding computers and children of thoughtmemorising, humans could be in a complete differentframeofmind.Butwearewhatweare...andIfeelafterallaprivilegedforhavingunderstoodforsolongtheorganicUniverse-thesensationsofthatcommunionwiththewholethroughtheknowledgeofitslawswillneverceasetillIdie and then I will dissolvemy existence, but before that we shall consider now a thoroughly different form of¬Algebra;thatoftheGeneratornotofanabstractGroupofpermutationsbutoftheentireUniverse...

ThebeautyofReticular,Booleanalgebras.

Then it comes the beauty of digital thought and its Boolean Algebras, even if they are simpler than futureexistentialalgebras.Foronething,theyexploreacompletedifferentworld,notoneofscalarnumbers,notoneofspatialmotionsandtemporaltransformations,butthestructureofcoherent,efficientmindsand itsoppositionwithentropy.ThustheyarealgebrasoftheTTv.SS,¬Vs.@opposition,quitedifferentfromthe∆±¡0’-1intervalof probabilities, the 1-∞ interval of scalar numbers, or the Ts v. St, energy vs. information algebras of Grouptheorywithitstranslations,rotationsandmirrorsymmetriesofformoverdistancesofspace.

Boolean algebras thus form the 3rd trinity system of algebras, which can easily be divided in those 3antisymmetries:

∆±¡:scalaralgebrasofnumbers(SettheoryandclassicGrouptheory)

S≤=≥ T: Temporal algebras of dimotions in space (Spatial Group theory, with maximal use in mathematicalphysics)

¬@:Mentalalgebrasoftruthandfalsehood(verballanguages,Booleanalgebras,1stand2ndorderlogic).

Booleanandexistentialalgebras

TheinterestofBooleanalgebrasthusliesinthefacttheyaresimilarinpropertiestoExistentialalgebrasforitspropertyofduality.

Whenvaluesandoperationscanbepairedupinawaythatleaveseverythingimportantunchangedwhenallpairsareswitchedsimultaneously,wecallthemembersofeachpairdualtoeachother.Thus0and1aredual,and∧and∨aredual.TheDualityPrincipleassertsthatBooleanalgebraisunchangedwhenalldualpairsare

interchanged.

One change we did not need to make as part of this interchange was to complement. We say thatcomplementisaself-dualoperation.Theidentityordo-nothingoperationx(copytheinputtotheoutput) isalsoself-dual.

Properties.

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Thedifferencearequiteremarkableinitspropertiesaswecanseeinthecomparisonbetweenmentalalgebras,whichareabouttheconstructionofmentalspacesthatcandiscerninitsmirrorthedegreeofsimilaritybetweenthemental imageandtheeventhappeningontheexternalworld,andscalaralgebraswhicharedescribing∆stprocesses observed in reality with a simplified system of operands that reflect the actions of the function ofexist¡ence.

The3propertiesthatdifferareannihilationforv,idempotence,absorbtionanddistributivityofvover∧suchas:

Annihilation:Xv1=1

Idempotence:xvx=x;x∧x=x

Absorbtion:x∧(xvy)=x;xv(x∧y)=x

Distributivityofvover∧:xv(y∧z)=(xvy)∧(xvz)

_____

Annihilation:X<1=1

Idempotence:x<x=x;x>x=x

Absorbtion:x>∧(x<y)=x;x<(x>y)=x

Distributivityof<over>:x<(y>z)=(x<y)>(x<z)

AllotherresearchinAl-gebraisneedlesstosaycensored,perinseculaseculorumsoA-mendon’tusetokillustilltheseventhgeneration.

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BOOKIII.®CALCULUS.ABRIEFINTRODUCTION

“Theinstant(ƒ)hasnotTime;Timeismadefromthemovementoftheinstant(T=1/ƒ).”

Leonardo

‘Ipropose1/nasthemeasureofaninfinitesimal.Ipropose1/Rasthemeasureofcurvature.’

Leibniz

‘Calculusstudiesfunctionsofexistence:itextractsfinitesimals,1/xof¬∆@st,integratingtheminareproductive,

collapsingwave:

€

0'

±œ

∫ 1/x∆st.Hence¡tsvaluetoreflectmathematicallythelawsofexistentialalgebra.’

L§

L3ontheessenceofcalculus

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I.

TIME-CHANGEINCALCULUSANDEXISTENTIALALGEBRA

S=T

Whycalculusissoimportanttoscience:Reproductionofformin5Danditsmathematicalmirror:calculus.

TheUniverseisan∆-fractalofspatialinformationandtemporalenergy,(ab.∆ST)thatreproducesinformation,forms-in-action,formsofspacewithmotionsintime.

Butreproductionhasanessentialfeature:ithappensinalowerplaneasaseedthatreproduces,integratesitspartsandevolvesintoawhole–theexactmethodweuseinthemathematicaldisciplineofcalculus.

Asallthisiswhatactuallycalculuscalculates:Itfindsafinitesimalpartofrealityandthenintegratesitasasum,throughapaththatmightbeamotioninspace,agrowthin∆-scaleorarepetitionin‘time;whereasthefunctionofexistenceoftheformdisplacesandreproducesitsorthogonalparametersofformandmotion.Sophysicalformsareconstantlyreproducing,‘calculating’andtheequivalencebetweenthetoolsofcalculusasmirroroftheprocessof

reproductivelocomotionbecomecrystalclear.

Letusthenfromthegeneraltextson5Dbringhereonlythespecificanalysisofrelativityofmotionandform,withthe3featuresthatareessentialtotheprocessofcalculus:

-Thesymmetrybeweenspace-form(heightdimension)andtime-motion(length-dimension),orS=Tmimickedincalculusbytheorthogonalsmoothformof∂x/∂y,anditsdifferentials.

-Thefinitesimalnatureofchange=motionintime,asithappensfromminimalpartsinalowerscaleofthefifthdimension.

-Thereproductivenatureofallmotion,asthereproductionofinformationinan∆-1wavestatethenintegrated,asitcollapsesinitsparticleform.

Reproductionastheoriginofthe5Dimotionsofexistence.

Howmanytypesofreproductionthereare,isacomplexsubjectthatwouldrequireawholetreatiseinexistentialalgebrapublishedelsewhereonthosetexts.

Letusthendefinethemostimportantforcalculus,accordingtothegeneralmethodofexistentialalgebrathatdistinguishalways3±¡possibilities:

∆Socialreproduction:Areproductionmightbepersistentintime,creatingaprocessofscalarsocialwholes;whenthereproduction‘lastsmemoriallyintime’,largersocialwholes,∆-1∑∆-1>∆0(usingsymbolsofexistentialalgebra).

Ts:Locomotion&LinealInertia.Ifthereproductionfadesawayatthesamerateithappens,weobservethenalocomotion,throughspace-time.

St:Angularmomentum:Ifthereproductiondoesn’tmoveinspace,weobserveacycleofspace-time,whichcanbeequaltothepreviouscycle.

SS-vortexmotion:Inthereproductionshrinksinspacesizeandincreasesincyclicaltimespeed(accordingto5Dmetrics:SxT=C),wetalkofgrowthofinformativefrequency,travellingtoasmallerscaleofthefifthdimension.

TT:ifthereproductiongrowinsizetracinga+πcycle,wetalkofanentropicreproductionthatslowsdownthemotionofthesystem.

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Thosearethe5essentialformsofreproductionin∆ST,whichcorrespondto5Dimotionsofspace-timeandarestudiedbythefunctionsofcalculusinphysicalsystems.Allofthemhoweverwillbereproductionsthathappenthroughsmall

Si=Testepsofmotionandform.SowehavenexttounderstandhowreproductivemotionshappenthroughStopsandmotions,particlesandwavestates,finitesimalafterfinitesimalfromtheperspectiveofthewhole–asaminimal1/nangularcurvatureincurved‘space’motions,asaminimalλƒrequency=1/Tstepinlinealtimewavemotionsasa1/nminimalcellinscalarmotions.Theyarethe3∆ST

finitesimalstepsinallcalculuswhichaswesawhavealwaysthesame1/nformula,eitherrepresentingcurvatureofspace-time,frequencyoftime,orpopulationinspace.Howthe3‘concepts’areactuallysymmetric,isduetothe…

GalileanParadox:SóT:RelativityofspaceDimensions=Forms=Motionintime:5UniversalDimotions

Galileo’s time and space Principle of Relativity is the fundamental conceptual thought behind the relationshipbetween time=motion and space=form and how one can be converted into another: All what exists is made ofspace=formandtime=motion.Andyetphysicistsknowthatwecannotdistinguishmotionfromform.Thatanybeingin motion from its point of view seems to be still and all other things moving around it. This is the principle ofRelativityofmotion.

Physicists thenwithoutmuch thoughtabout that fascinatingduality,wenton tousemathematics to calculate therelativemotionofeachentityofrealityrespecttoothersystem,whichseemsstaticfrombothpointsofview.ThisiscalledGalilean relativity, latter refined by Einstein's relativity, and essentially is concernedwith themathematicalcalculusofwhatweshallcallthe2ndDimotionoftime=change,locomotion.Fine,butwearemoreinterestedonthedualityof space=formandmotion=timeand its entangled relationships –the reasonswhywedoNOT see togethermotionandform,evenifallsystemshaveboth.

Theconclusionisthenratherobvious:oneofthetwoparametersofrealityis 'hidden'toperception;weeitherseemotionor form, 'wavesorparticles' (quantumcomplementarity),distancesand linesorpoints inmotion(as inthenightwhen fastcars inapictureappearas lines).Sophysicistscalculateonlyonewhen in factwemustassess theexistence of 2; and since we cannot distinguish them, logically we must equal them. ‘Form=motion-function;space=time;Si=Te’.

Relativity then becomes a duality, Si=Te,which isat the heart of every law of theUniverse.Whereas the primaryelement, theultimatesubstance is time=motion.Asspace isaMayaofthesenses–asliceoftimemotion.Formiswhata'stillmind',makesofthatmotionto'perceive',information,forms-in-action.

Since we see Earth still and flat but it is round and moving. Galileo’s profession was ballistics - the study ofcannonballsmotion. So he choseONLYmotion and lost the chance to start physicswith a complex philosophicalunderstanding of its Si=Te dual Principle of relativity,which Poincare defined latter clearlywhen he said that ‘wecannotdistinguishmotionfromstillness’.Anexampleisquantum/relativityduality.Indetailquantumspacehas‘darkenergy’becauseithasexpansivemotionthatextendsintoaplaneofspace,butwhenseenatlargerscaleswithoutdetail itsentropicmotion seems static space - adualareaof scattering lengthandwidth. So in thegalaxywe seeeitherdarkenergymotionorexpandingspace:T=S.Amotionoftimeisequivalenttoadimensionofspace:Distanceandmotioncannotbedistinguishedsotheymustbetakenastwosideofthesamebeing,aspace=timeÐimotion(ab.DimensionalMotion):

S=T;Dimension-Distance=Time-motion=STÐimotion

Earthmovesintime,butweseeitasastillforminspacebecauserealityisaconstantgameof∞motions,butthemind focus thosemotions andmeasures themat still distances. For huminds,motion is relative to our systems ofmeasureandperception,whicharelight-based;henceafixedc-rodspeed/distance.ReasonwhyEinstein’srelativity

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postulatesamaximalT:c-speed,measuredasifobserverandobservablewerestilltoeachother(ConstantS);whichatourscalewecorrectwithLorentzTransformations.

Asithappenstheidentitybetweenspatialstatesof‘form’andtemporalstatesof‘motion’,whichbecomestopsandstepsofall reproductivemotionsbecomethe fundamental ‘presentstate’of theUniverse,andtheessential toolofcalculus to ‘solve’ its differential equation (D’alambert’s method of separation of variables); and his unendingphilosophicalandlogico-mathematicalconsequenceswillappearinmanypartsofthosetexts.

But physicists just substitutes the Earth’s still distances formotions, and it took another 300 years for Einstein torealizetherelativityofmotionanditsmeasuremadeessentiallytimeandspace,motionandformtwosidesofthesame coin. Still this realizationwas not explored philosophically and so it gave birth to a series of ill-understooddualitiesbetween 'statesofmeasureandform' (particles,headgaugingform, in-form-ation)and 'statesofmotion'(wavestates).

It is thenessential to grasp thatmotionand formco-exist as 2different statesdependingon5D scale anddetail:Motionsareperceivedbymindsthatstopmotionintoform,intoinformation,asdistances.Soifweseeslowmotioninthenight,acar’sheadlightseemsalongdistanceline‘still’picture.Butthismeansalsothatthe3‘Euclideanstilldimensions’musthavemotion;theyare‘bidimensionalST-holographic,topologicdimotions’.Sowehave3Space+1Time+15thdimensionofscales=5Dimensionalmotions.NoneofthemisaDimensionofpurespatialformorapuretime motion but a combination of both. Even ifmentally we tend to reduce motion and focus on forms, all hasmotion=time, and form =space: this is the meaning of 'spacetime', the messing of both into 5 dimotions, the

fundamentalelementofallrealities.

Relativitystates‘wecannotdistinguishmotion=timefromposition=space’.Soallwhatexistsisacompositeofboth,undistinguishableSi=Te,5‘Dimensionalmotions’ (Ab. Dimotions), broken in infinite fractal, vital time space

organisms, composed of topological Dimotions: height=information; length=locomotion; width=reproduction;form=socialevolutionofpartsintowholes&entropy=dissolutionofawholeintoitspartsinalowerscaleofthefifthdimension (termwekeep for thewhole rangeof scalesof theUniverse);whose study is bothmathematical, themain science that studies how those 5 Dimotions entangle in simultaneous Space, connected to each othertopologicaladjacentparts,whichcreate superorganism, andLogic; themain stienceof time thatobserveshowthose pentalogic, entangled superorganisms move and evolve, change in sequential relational time, living aworldcycleoflifeanddeath.

As all is time&space, the 2 experimental primarymirror-stiences of time&spacebecome themost important toextract the Disomorphic=equal laws of those 5 Dimotions that all systems have in common. Sincewhile thoseDimotionsarebroken,invitalorganisms,separatedbycyclicaltimemembranes,theyarethesame.

InthegraphGalileanrelativitywasillunderstood,asthetruequestionabouttime-changeiswhy‘themindseesspaceasastill,whenindetailismadeofsmallerself-similarquanta,inmotion.Theparadoxdefinesmentalspacesasstillsimplifiedviewsofthemorecomplexwhole.

The3¡logicparadoxesofspacetopology(closedin-form-ativecurved-Ovs.|-open,freeentropiclinealforms),time-motion(stillnessvs.motion)and∆-scale, (continuouswholevs.discreteforms;singlescalevs.multiple one)s,areessentialtotheperceptionofasimplified‘spatialminduniverse’inasingleflatstillplanevs.thefull,moredetailedcomplex picture in time, of a curved, discrete andmoving Universe. Those paradoxes resume the 5 elements ofreality,Space=form,time=motion,scalesandthemindthatmeasuresthem,withinitsownentropiclimits.

Theyarealsoessentialtoalltheelementsofcalculusandmathematicsatlargeanditsmethodsofsolutions;speciallytheinversionbetweenfinitesimallinealsteps(asastepbetweentwopointsisNEVERcurved)andthecyclicalformof

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longer‘integralpaths’.Solinealapproximationsaretheessentialtoolofcalculusandmathematicstoresolvemanyequations.

Whatneithermathematiciansnorphysicistsfullyunderstand(thoughsomeinroads inabstractweremadethroughthe Noether’s concepts of symmetry) is that each stœp of amethod of solution is not ‘gratuitous’; butmust begroundedinarealpropertyofthe5D∆STsymmetriesandconservationlawsoftheUniverse,whicharenotsomany– hence the repetition of methods. Specifically, the aforementioned 3 paradoxes between ∆+1 curved closedworldcycles,sumoflinealsteps,whichgivesbirthtothemostusedmethodoflinealapproximations;theequivalencebetweenSpaceand time, inall Stœpsofdimotions,whichgivesbirth to themethodof separationofvariablesondifferential equations andmore broadly allows tomove around relative space and time parameters in equationsjoinedbyanoperandof‘equivalence’(≈not=).Andthe2conservationlawsoftheUniverse,conservationofthose‘beats’ofexistence,S=Tinrelativepresent,eternalbalance,justifyingtheequivalenceoperands.Andconservationofthe‘volumeofspace-time’ofeachplaneoftheUniverse,byvirtueofthe5DmetricequationSxT=C,whichjustifiesall the procedures regarding scales – solution of differential equations by separations of scales, renormalizationprocedures (Wilson), and harmonizes those scales allowing constant but balanced transfers of energy andinformation,St=Ts.

5dmetricsexpressestheconservationoftime.TheparadoxesofRelativity,discontinuity,partsandwholes,scalesareallrelatedtothereductionistnatureofmindsthatbiasreality.Mindsreducedimensionstotherelevantones,eliminatingalldarkspaces:continuityistheresult.Of all formal languages that map out reality 2 are paramount, Time ¡logic & mathematics of Scalar Spatialinformation.A 5D Metric function, S(0-Mind) x T(∞-universe)=constant world is the function of all mind languages who onlyperceive from its self-centeredpoint its languagemirror confusedwith thewholeUniverse (Egoparadox, basis ofpsychology).Ænthropichuminds reduce themultipleclocksof timeandvital spacesof reality to thesinglehumanclockandspatialscale,rejectingtheorganicpropertiesofotherUniversalsystems.The main laws of 5D are the metric functions of the scalar Universe, which relate the spatial size and speed oftemporal clocksof all scalesofNature.Bothparameters are inverted:when systemsgrow in size the speedof itsclocks,its‘timecycles’,diminishproportionally,bothinbiologicalandphysicalsystems.Andviceversa.Smallerclockstick fasterand informationprocessing carriedby the frequencyof those cyclesaccelerates, as ithappens in chips,particlesorlifemetabolism.Sowewrite:SxT=C.Themindthusstartsitallwithitslinguistic'stillmapping'stoppingitsworldinalocked'crystalimage',measureofitsself. But even perception is social, linguistic. The Universe can only be explained if 'perception' exists within thelanguage,aswhenyouthinkwords,yousensewords,whenyoureyeseeslightandmapsintoanelectronicmappingyou are seeing. And when an atommaps a geometric image in its 'locked' 'stopped' spin, it must perceive thatgeometryasinformation.PhysicistsmadetheGalileo’sparadox,thecornerstoneoftheirtheoryofmeasure,buttheyfailedtostudythedeepimplications it has for everyaspect of the structure of the Universe, from the duality between spatial mental,linguistic forms and physical motions; to thebalances achieved by the similarity ofboth space and time, whichbecomesthefundamental'functionofpresent'Si=Te,andhencewiththemetricfunctionofscales,$xð=K,thetwoessentialfunctionstoformalizesingleplanesSi=Te,andmultiplescalesofspacetime.YetasSi=TemaximizesSxT=K(5x5>6x4).Weunifybothin1function:Max.SxT=C,whichdefinesforeachfractalvitalspace-timeorganismitsFunctionofexistence,asallspecieswilltrytomaximize itsmotion-entropy-time for its field-limbs, its information-spatial states for its particle-heads, whoseproductwillgiveusitsvitalreproductiveenergy.Moreoverthefunctionhasanimmediatebiologicmeaning,becauseaswearemadetopologicallyof‘fields-limbs’oflinealspacewithmotionprovidedbytheenergyweabsorbtoalsoreproduceourbodies-waves,andtheinformationweneedtolinguisticallyguideourmotionswithparticle-heads,the

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veryessenceofsurvivalistoincreaseourS=position,mentalformsofspaceandT=entropicmotionsoftime(whereastime=motion & space=form are the two limiting Dimotions with ‘energy=reproduction, s=t, locomotion, sT andinformation,St,aretheintermediate3dimotions).The fifth dimension ismade of the 'different co-existing scales',which from the simplest forces through particles,atoms,molecules,matter,organisms, superorganisms,planetarysystemsandgalaxies, createan 'organicnetworkstructure', which amazing enough since it was discovered at the beginning of science with telescope andmicroscopes,wasnotformalizedtillIintroduceitsmetricfunctioninthemilieuofsystemssciences,asasinglelinealtimemotionisadogmaphysicistsdon’tdaretochallenge.Yetsciencecannotadvanceinitsfundamentalprinciplesunless the formalism of the fifth dimension is accepted and used to fully understand the cyclical, repetitivepatterns=lawsofscienceofeachdisciplinethatstudiesascaleofthefifthdimensionanditsspecies.

Reproductionofformin5Danditsessentialmathematicaltool:calculus.

TheUniverseisafractalthatreproducesinformation,forms-in-action,formsofspacewithmotionsintime.Thisistheessenceofitall.Butspaceisamayaofthesenses,thesynchronousviewofaseriesofcyclesoftimemotions,knotted

inthesimultaneousperceptionofanobserver;whatphysicistscalla‘frameofreference’.

Thustime=changeisthefundamentalelementofreality,andthismakesAlgebraoftime-change,specificallycalculusperhapsthemostimportantexperimentalscienceoftime,besideslogic,whichwehaveupgradedtoexistential

algebra,whichexploresthevital,organicwhysofthosechanges.

ItistheGalileanParadox:S=T.Wecannotdistinguishtimefromform.Inasmuchaseachframeofreferenceormindlocksinaknot-mirrorofthemotionsoftheUniversefromitspointofview.Soeachpointofspaceisaperceiverrelative

fieldofmotions,whichfromitsperspectiveknotasforces‘attracted’byitsframeofreference.Yetifwecannotdistinguishmotionfromformeachpointisentangledtothosemotionsandismadeofmotionandform,oftheparticle

andwavestates.

LocomotionasreproductionofformsolvestheParadoxesofZenoandthemeaningofdiscontinuity.Asmotionisreproductionofinformation,ofform,sinceparticlesareknotsofperceptionofform,fractalpoints,monads,that

movebyreproducinginalower5Dplane,as∆-1waves,itsinformation,asforms-in-action.

Soallformsofchangecanbereducedtotheultimatefunctionofexistence,reproduction,abackandforthtravelthrough2scalesofthefifthdimension,asaformbecomesaseedthatreproduces,evolvessociallyandformsitswholeagain.TheextraordinarycapacityofCalculus,whichextractsat∆-1levela‘finitesimal’(Leibniz’s1/ndefinitionofinfonitesimalasaminimalpartofawholeandALSO,byvirtueofS=T,aminimal‘curvature’ofatimecycle,whichisthenintegratedforatimedurationoftheevent,either

locomotion,orvolumeofpopulationinspaceorS=Tcontinuous=smoothchangeintimehappenspreciselybecauseCALCULUSperfectlymimicstheprocessofchangeandreproductionofformbetween∆ºand∆-1

scalewhichisthebasisofalltime-changealsoinphysics.Changethusischangereproducedinalowerplaneasaseedthatevolvesintoawhole.

Itisthennotsomuchinphysicsbutincalculuswherewefindthestrongestmodelofthelawsof5Dandlocomotionasareproductiveprocessofform,eveniftheexperimentalproofsarescatteredalloverphysics.Indeed,theentireworldofquantumphysicscanonlymakesenseifweconsiderthatparticlesMOVEASWAVESandgaugeinformationasstopparticles.Becausewavescanbetransparenttoeachotherbutparticlescollide.Asimpleproof: theatomicnucleus is so small compared to itsparticles that if theywouldn’tmoveaswaves, transparent toeachother, theywouldbealwayscollidingandthenucleuswouldneverremainstable.Infact,whenwegetpicturesofthoseparticlesoutsideitsshells,(electrons)theymoveinzigzagastheystopandchangemotionconstantly.Asusualphysicistsjustmakeanaxiomaticruleandsubvertthelawofcausalityconvertingthemathematicalmirrorderivedofthefactinthecauseofthefact–inthiscasetheysaythisisduetothePauliexclusionprinciplewithoutprovidingthemechanismforparticlestoavoidcollisionifmoving.

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Itfollowsthatbeingswithmoreinformation,reproduceslowlyandwecanhardlyseethemmoving.ThelimitofitbeingcomplexlifesuperorganimsonEarth,whosereproductiontakes9months.Ithappens‘inside’thereproductivemother,anditreproducesintheadjacentspaceafter‘tearing’thetopologicalknotoftheumbilicalchord.Asimilarveryslowprocessofreproductionhappensinphysicswiththeweakinteractionthatreproducesaformwithevenmoreinformationevolvingthemassofparticles,sotherangeoftheforceisminimalandthenewparticleappears

adjacenttotheonethatdisappears,dyingforthenewhatched‘baby’tobeborn.

Thisistheessenceofitall.Motionisreproductionofinformation,ofform.Sinceparticlesareknotsofperceptionofform,fractalpoints,monads,whichmovebyreproducingthroughalowerplaneofthe5thdimension,as∆-1waves,its

information,asforms-in-action;allformsofchangecanbereducedtotheultimatefunctionofexistence,reproduction,abackandforthtravelthrough2scalesofthefifthdimension,asaformbecomesaseedthat

reproduces,evolvessociallyandformsitswholeagain.

SpaceisaMayaofthesenses,thesynchronousviewofaseriesofcyclesoftimemotions,knottedinthesimultaneousperceptionofanobserver;whatphysicistscalla‘frameofreference’.

Thustime=changeisthefundamentalelementofreality,andthismakesAlgebraoftime-change,specificallycalculusperhaps the most important experimental science of time, besides logic, which we have upgraded to existentialalgebra,whichexploresthevital,organicwhysofthosechanges.

ItistheGalileanParadox:S=T.Wecannotdistinguishtimefromform.Inasmuchaseachframeofreferenceormindlocks in a knot-mirror of themotions of theUniverse from its point of view. So each point of space is a perceiverrelative field ofmotions, which from its perspective knot as forces ‘attracted’ by its frame of reference. Yet if wecannotdistinguishmotionfromformeachpointisentangledtothosemotionsandismadeofmotionandform,oftheparticleandwavestates.

Thussystemsreproduceitsform,travellingacrossscalesofthefifthdimension:theyreproduceafinitesimalformcreatingareproductivewave,whichintegratedasapopulationofspacegiveusbackawhole.

SuchdiscontinuouslocomotionsolvesZenoParadoxesasthefinitesimalisthelimitofone‘step’.

RECAP.Calculusstudyfunctionsofexistence:itextractsfinitesimalsandintegratethemasareproductivewave.

hencetheenormousvalueofcalculustoreflectmathematicallythelawsofexistentialalgebra.

THEFINITESIMAL:Þ

∆-1:Lebiniz'sdefinitionofS=TFinitesimals:1/n:minimalcurvature.∆-1unit.

Thekeyconceptof5Dcalculusisafinitesimal.Afinitesimalinlinealspace-timeisafrequencysteporwave-length.Afinitesimal in curved spacetime is a minimal curvature of a clock cycle. A finitesimal in scale is a minimal unit ofpopulation.

Butweuseother term for any finitesimal 0’; that is, abitmore than0’,which canbeeither curvature, ¡-1unit ofpopulationorfrequencymotion.0’isthenthementalfinitesimal–theminimalquantityinexistenceofabeing,whichstillremainsthebeing;theseed,themindofthespecies,themother-cellthatmustthereforeexistforanybeing,asthetemplatewhichwilldeveloptheimmanentprogramofexi»st¡ence,whichdoesNOTneedtobestoredwithinthefinItesimal0’.

AndinverselyasthereciprocalofSS-mindswithnomotionareTT-entropywithmaximalmotion;andthereciprocalofzeroisinfinity,andalmost0’,immensity,whichwewritewiththesymbol∝.Thefinitesimalofentropyisthelargestdomainofthebeing,beyondwhichthebeingdies.Itisarealdefinitionofthebordersofthemind,asinitsequation,0’-mindx∝spacetimecyclesoftheUniverse=K-World;0’x∝=K.

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Wethustalkofimmensityastheentropiclimitinwhichthebeingnolongeris.Andbothobviouslyactincalculusasthelimitsofadefiniteintegral.Thuswecandefinecalculuswiththe5Dimotionsofexistence.Sincethefinitesimalsof∆STwillbeintegratedbetweenitstwolimitsofSSanTT,togiveusitswhole‘worldcycle’intime,or‘closedcircle’inspace, sum of its ‘stœps’ or ‘curvatures’, or its ‘wholeness’ in scale, sum of its finitesimal parts, showing the deepentanglementandsymmetrybetween∆,SandT:

€

0'

±œ

∫ ∆st.

Weshalluseaswindowsdoesnotletmeput∝,œasthebestsymbolforimmensity,whichisthesymbolofthewholesuperorganism, that isultimatelyanalternativesymbol for∝,as thewhole tends tobe the limitofexistenceof thebeing,moreexactlyitsworld,O’x∝=œ.

Thesubtledifferencebeingthat∝ isexternaltothebeingthatperceives it,buthaspureentropy;that ispotentiallyfeeds the creation of multiple Kaleidoscopic monad-worlds, and œ is specific, ∝ has been ordered to become a‘whole:ab.œ’.

But let’s not fancy too much ourselves with existential algebra, its profound paradoxes and symbols, returning toclassiccalculus.

Ipropose then3alternative symbols, L, for theclassic limitas the finitesimalof space, ƒ, forFinitesimal in time,asfrequency,andþasthesymbolofapalingeneticcycle,forfinitesimalofscale,andwillbeusingƒforcommodity,withþasthelessconfusingsymbolforallthecases.Andwriteasthegeneralformula:

Þ:1/¡,whereas¡,mightbeinclassicmathematics,N,thewholepopulation,T,theperiodorRtheradius:

Þ(∆)=1/n;þ(S)=1/R=K;þ(T)=1/T=ƒ;Þ(@)=0’;Þ(¬)=∝

It is fascinating to observe that the 3 finitesimals of scale, space and time have the same equation in classicmathematics,2ofthemdiscoveredbyLeibniz;theguywhounlikeNewtonalways‘hitatargetnobodysees’(:

This of course is only the beginning; and as usualwe shall poundermore thephilosophical aspects of 5D calculus,leavingfor‘pros’withimagination,ahumblerealizationthatnewbeginningsaresimplebutalwaysfoundbyamateurswithout theburdenofknowledgethatanentirenewworldofcalculusofwhich Ihave justswimmedonthesurfacewithunfocuseddivingglasses,awaitstothebrave.

Whatisthe1intheequationsoffinitesimals,awholeorastœp

Thustheinfinitesimaldoesnotexist-beingspacequantic,therewillbealwaysalimit,amicro-cycleoftimeorquantaof population in space, to signify the finitesimal point, as Leibniz rightly understood and defined it with a simplepowerfulform:1/n.

IndeedintheUniversefinitesimalstendtobestructuredasinarussiandoll,suchasthebiggestwholes,n->havethesmallest finitesimals, 1/n->0’. But and this is the incrediblemagic insight of Leibniz’s ¡n:finitesimal, 1/n is also theformulaforacurvatureinspace.AndasS=T,fortheminimalmotionofaclockoftime.Sowedohaveaconceptin∆ST¡-1thatweshallthen‘integrate’througharelativepathwiththefinitelimitofaworldcycle,wherethefunctionismeaningful(thatishasavalue,ƒ(x,y)≥1/n),andrelates∆and∆-1throughits‘stœps’ofchange.Wedoesconnectinthis manner calculus to 5D reality, no longer base in human invented, axiomatic concepts of absolute zeros andinfinities,limitsandtheparadoxesenclosedwithinthem.

The0’sizeisthusthefinitesimal.Inpraxis,wehumansonlyobserveafinitesimalfromourmindperspective,whoseminimalformisanh’quantumoftheplanckscale,andaccordinglyweseeaUniverseof inverserelativesize,beinghumansinthe∆ºmiddleview(atcellularlevel)asphysicistswonderwithoutrealizingthisisNOTacoincidence,butanaturallawofthescalar,fractalorganicstructureoftheUniverse:

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SoweacceptLeibniz'sconceptofafinitesimal,asALLorganicsystemshaveaminimalcellularquantaandamaximalenclosure,which inmathematicscanbe represented in theo-1 finitesimalcircle, closedabove,as itbecomes the1element in ∆-1 of the ∆ºwhole, which is represented bythe 1- equivalent graph, which is opened above into thewholenessofalargerUniverse(butwillhavealsoalimitnormallyinthedecametriclogarithmicscaleofthe∆ºwholeworldembeddedinthe∆+1trulyinfiniteUniverse).

Whatthe3±¡finitesimalsofexistencehaveisthe1on‘top’?Aswecanconsider0’1/∝and∝,1/0.

Anditcanmeantwothings,thewholeas1=∝inthe0-1palingeneticUniverseorthe0’asthefinitesimalinthe1-∝Cartesiandomain.Sowefindthatimmensitycanbeafinitesimal.

If 1 is taken as the finitesimal, it becomes then a step in a curvature, which tends to be ‘lineal’, as all steps arediscontinuousmotionsbetweentwopoints,whichcanalwaysbeclosedwithastraightline.

Sobydefinitiontheminimalcurvaturestep isalwaysa lineof infinitecurvature (:Andweneedtwostepsto finda‘realcurvature’,whosemaximalvalue,forastepbackandforthwillbe360º.

If1isthefinitesimaloftime,frequency,itwillbetheminimalevent,henceaclosedtimecycle.Andassuchitwillbethesumoffinitesimalcurvaturesteps.

If1isthefinitesimalofapopulation,thenitwillbeitsminimalmeaningfulpart,oftena‘seed’ormind-singularity,andnitswholepopulation,andthesmallerthefinitesimal1isthelargerthen-populationwillbeinthenestedUniverse,whenwemeasure∆-2finitesimal‘bites’ofenergy-feeding.

Butforafinitesimaltopersistasaunitofpopulationitmustnotbeerasedafterasingle‘stœp’;soitscyclemustberepeatedintime.

Sowerealizethereisachainrelationshipbetweenthe3finitesimalssuchasitsreciprocal,abitofspace=form,abeatoftime=cyclicalmotionandabite(apiece)ofpopulationarenestedpartsoflargerwholes:

∑∑s=∑T=∆-1.

Adeepresultinbothcalculusandexistentialalgebra,whichcanbesaidasfollows:

Spaceisasliceoftimewhichisasliceof∆-planes,andsowegrowindimensionalmotionsandwholenesswhenwemovefromspace/curvaturestepstofulfillawholetimecycle,whichhoweverisjustafrequencyofmemorilessform,thatonlywhenpersistsbyrepeatingitscyclesinthesameregionofspacetimebecomestheunitofpopulation.

This growth of reality is essential to grasp the complex nature of calculus when wemove beyond the first pagesdedicatedtotheanalysisofitsdimotionsandoperands,toODEsandPDEsofphysicalsystemsandbeyond;whicharealsonestedsystemsofcomplexdimotionsinwhichfinitesimalsofscale,timeandspaceareconsideredalltogether.

Unfortunately humind’s unaware of those symmetries, and even the simplest concepts of linearity, cyclicality andscalejust‘calculate’asiftheywereperformingsomemagicaltrick;soourpurposewillbetoenlightenphilosophicallyitscalculusextractinggeneral lawsofreality fromcalculus,ascalculus isbyfartheclosest formal languagehumanshavelearnedthatmimicsthelawsoftheUniverse.

Indeed,essentiallywhatacalculusoperationdoesinitsessentialform,anODEorPDEistofindtheclosestthingtoafinitesimal,which is a differential (as it is a o’ piece or ‘lineal step’ the closest possible as a piece of space to thecurvaturepiecearoundthederivative=tangent);andthen it integrates it inan intervaloftime;between its ‘originalseed’and‘entropiclimit’.

Thisisthefundamentaluseofcalculustoday,becauseitwasbornonPhysics,whichstudieslocomotion.

But calculus alsoworks veryoftenon∆-1 finitesimals,whichare integrated in a longer time,orwhole in space,orworldcycleintimewiththesamelimits.

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Andthenwerealizesuchoperationsareexactlythe inverseofthepreviousone.Thefinitesimalofscale isnowthesmallerpart,whichweintegrateinavolumeofspace,orthroughamotionoftime.Whyisthatpossible?Obviouslybecauseineachscaleanewgameofexistencedoeshappen.Reasonwhywewrotetheequationasadoublefeed-backequation:

∑∑∑Só∑∑Tó∑∆¡ó∆º=S

If theUniversehadonlyascaleofspacetime,thefirstequationwillbetruth.As it ismadeofplanesofspace-time,where each new whole becomes a quanta of a smaller scale; starting again the game, in operations where wedifferentiateand/orintegratetwice,weareemerginganddescendingthroughplanesofexistence.Andthisiswhatmakescalculussomagic.Asthesecondandthirdderivativehasalsoafullmeaning.Itistherateofrateofchangeofspaceintotimeintoaccelerationintojerk.Itisthegrowthofgrowthofapointintoalineandaplaneandavolume.Itisalinethatcurves,andclosesacycleandbecomesaspiralofinfinitecurvature.

Itmightthenbetoignorealltogetherthosesymmetriesanditerationsof∆STacrossplanes,whichconvertoneintoeachother,butthatisthedeeperstructureofrealityevenifmosthumind’sareone-dimensionalandgetaheadachethinkingparadoxically.

Andfinallywehavethelimits;oncewehavescaledupanddownasmanytimesasrequired,wewillstilljustbepartofawhole,solimitsexistandpreventusfromdoingwhatphysicistsdo,‘fromheretoinfinity’,gettingtheninallkindoftroubles,singularities,infinitiesthattheyeliminateandrenormalize.Allthatgetsthemtorealresultsbutiftheyhadtheproperunderstandingoffinitesimalsandimmensities,theywouldusecut-offearlieron,for‘singularities’ofbig-bangs;chargesandmasses,understandthewormholesthatonthosesingularitiesjusttransferenergyandinformationbetweenplanes,andsoon.

To show them will be therefore the second task of this paper as it grows and we enter into ODEs, PDEs andmathematicalphysics,sometimeinthefallof2019…Butfortheimpatientone,asample…

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THECURVATUREOFSPACE.

ADisomorphicexampleonhowtounderstandhomologyofstiences.

Thedifferencesbetweenclassiccalculusand5Dcalculusarethussmall,mainlyconceptual,butonthe‘fringes’itwillhaverealconsequences forunderstandingparadoxesofphysics;andthephilosophical foundationsofmathematics.Ontechniquesandwhatmathematicianslikemost,crunchingnumbersandequations,verylittlenewatthisstageisexpected.AndthisiscomfortingcauseasIsayissummerdayandIdon’twanttowritetoomuch.Butdonotdismissthepaper.Because this is realexistential calculus.Consider the finitesimalof space, curvature.1//R is the simplestone;thecurvatureofacircle.Itfollowsthatthestraightlinehasafinitesimalcurvature.Butalsothattheremustbe,animmensecurvature–asnow‘absoluteinfinitedoesnotexist’becauseialimitedinfiniteisthereciprocalof0’.

Buthowcurvaturecanbeimmense,closetoinfinity?Mathematiciansdefinecurvatureforacurveasdφ/dswhereϕistheinclinationofthesensedtangentandsisthearclengthmeasuredfromsomefixedpoint.

Thislimitscurvature;butphysicistshavethenotresolvedparadoxthatcurvatureinRelativitycanbeinfinite,orratherimmense.Thesolution?5D(:

It is also ‘embedded’ in the complicated formulae andprinciplesofrelativity.Inrelativitytheprincipleofequivalencebetween acceleration and an attractive vortex of mass; andtheprincipleofNewtonthatgivesachangeofmotion,henceanaccelerationtothecurvatureofamotionthatisnotlineal.Thus curvatureandaccelerationare similar conceptsand the

morecurveda‘curve’is,thefasterisgrowingitsspeed.

Those are youmight say trivial results; after all in physics tomaintain in orbit a satelliteweneed an acceleration,a=v2/R,henceahighercurvaturerequiresahigheracceleration;butthatispreciselythebeautyof5D,toreducetothesynopticlawsof5Dmetric,S=T,SxT=C,andthesymmetriesof∆=S=T,anastoundingarrayofphenomena.

ThusweextendEinstein’sPrincipleofequivalencebetweenforceandacceleration,tobothchargeandmass,andtothecurvature of space=time to both the ∆-1 charge and ∆+1 mass, physical scales. It is then the implicit concept inNewton’sequivalentformulae,asG,orinCoulomb’skfactor.

Thisisalsoembeddedinthesolutiontothatdifferentialequation:

Whichthereaderwillnoticehasasecondderivativeabove,thatisusuallythesymbolofacceleration.

Butin5DwecandefinecurvaturebytheS=Tsymmetryalsoasameasureofacceleration,thisispossiblebecauseaswediminishinsize,accordingtothemetricsof5D,$xð=K(whereas$isasymbolforlinealspaceandðforcyclicaltime;sowecouldwriteinphysicalterms,Lxƒ=K,butasusual5Dexistentialalgebrasymbolsaremoregeneralandnewtousetheminanyscience).

Sowhenwecarrycurvatureinspacetoaccelerationintime,itbecomes‘frequency’,andcurvaturecanhaveinspaceanyangleabove2π=360º;orintermsoflength.

Butthoseforcesareconservedinphysics.SoTounderstandwhatphysicsconservesletusconsidera5Dmetricequivalent-a2Dvortexequation,VxRo=K.Asthevortexdiminishinsizeitturnsfaster.Incyclicaltime,ðcyclesofperceptionthathappenwhenthepointreturnstothememorial‘singularity’happenmoreoften,as,itsunitisthe

closingofacycle.

Theincreaseofcurvaturethereforeimpliesanincreaseontheaccelerationofthesystem,andbothareindeedequalconcepts:1/R,ðT/ðS.ButnowforagivenSpaceperimeter,itshighercurvature=accelerationimpliesashorter‘unitof

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timeperception’.SoforthesameSpatialdistancetravelled,evenatthesamelinealspeed,moretimeunitshavebeenconsumedinasmallertimecycle;aswegodowninsizescalesofthefifthdimension.Bothangularspeedand

existentialcyclesaccelerate;duetothevortex5Dmetric:V(ð)xR($)=K.Soaslowlargeturninggalaxymightshrinktothesizeofanatom;whichlivesatinyfractioninatinysize,butinfactitsspatialdistancetraversedisroughly

maintained.Andindeedlatterwewillseehowinthe3Dimensionalspace-timeofthe∆±3scalesofthegalatom,abetadecayisin5Dmetricsequivalentintimedurationtoaquasar15billionbig-bangcycle,andthe5Dmetricofthe

protonequivalenttotheSchwarzschildeventhorizonoftheblackhole.

5Dmetricsconserves2things:The‘energy’volumeofspace-timeofallscalesanditsbeatsofexistence:

1.Theworldlinedistancethebeingtravelled–whichisinfactaworldcycledistanceasitisthesumofalltheperimeters,travelledslowlyinthelargespacetime,fasterinthesmallerspacetime.Sothespacetimevolumeofthe

different∆-scalesisthesame.Andbecauseenergyistheonlyparameterusedbyhumindsinallscales,theconservationofthetotalvolumeofspace-timeoftheUniverseisequivalenttotheconservationofEnergy.

2.Buttoconservethe same ‘lengthof internalperceptioninexistentialbeats’, thesmallerbeingmuchlivefar lesstime.Andindeed,theneutroncycleinbetadecayis15minutes.

Thegalaxycycleinaquasarbig-bangcycleis15billionyears.

Theyaretwodifferentcurvatures,becausethecurvature-acceleration-forceofthechargeisimmensitycomparedtothealmost0’curvatureofthegalaxy.

Butalas!Botharethesame.Sowecanunifybothforces,withthesimpleconceptofcurvature=acceleration=attractiveforce,asafastermorecurvedsinkwillattrtact,likeahurricanstronger.

Einstein's derived his formalism from Poisson. It is more detailed because it is according to the Galilean paradox(Si=Te)thespatialstillperspectiveofthosevorticesasaseriesofsimultaneousderivativemeasures.

So it reduces the temporal continuous Newtonian view of a spacetime vortex into an ∞ number of infinitesimaldetailedpictures, focusingnoton the speedbuton thecurvatureof thevortex (which is the spatialdefinitionofamovingcyclicalspeed-thefasteritturns,themorecurvatureithasin‘stillmathematics’).

Let us do the maths in the simpler Newton’s formalism, whereas by the paradox of Galileo S (Curvature) = T(accelerated motion). So the Universal Constants (G, k), define the curvature of 2 space-time vortices at the ∆-1quantumchargeand∆+1cosmicmassscales(∆isthesymbolforthedifferent±¡scalesofthefifthdimensionwithinagivenorganicsystem).ItsformalismofavortexoftimespaceisthenNewton’sUnificationFunction:M,Q=ω2r3/U.C.(G,k)

Itappliestoallvorticesoftime-spacefromparticlestoplanetstogalaxies.ForexampleifwesubstitutefortheEarth-sunsystemweobtainG,(1stevertheoreticaldeduction)andifwesubstitutefortheBohrRadiusandProtonMass,weobtain k with a 1039 higher curvature value, the exact difference between both forces that solves its hierarchyproblem.Ascurvatureinspaceissymmetrictorotationalspeedintime,soit issymmetrictotheattractiveforceofanyvortex. Itworksmarvelswhenwetranslateelectromagnetic jargontoNewtonian jargon.Forexample itshowsthe‘isomorphism’(systemicjargonforanequal‘form’betweenscales)betweenatomsandgalaxies,whichH-atomsofthecosmicscale.

Sincewhen we translate electromagnetic function into gravitationalmass vortices, the proton radius becomes theSchwarzschild radius of a black hole and its electronic orbitals its star clouds, a result foreseen by Relativity thatmodeledgalaxiesasHydrogenatomsintheEinstein-WalkerMetricoftheCosmos.

Letusput someeasynumbersbysubstituting theparameters in thatUnification function for thevaluesof thesun(mass) minus earth (rotational speed and radius) to get G, which any high school student can do:

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Sunmass=2×1030kg;Earth’sangularvelocity2×10-7rad.persec.Earth’sorbit=150millionkms.Result:G=6.67×10-11kg-1mˆ3rad.sec.-2

This is standard gravitational theory. What has never been done, because the fractal systemic view of the fifthdimensionwasnotknowntillrecently,istosubstituteinthesamefunctionofgravitationalcosmologicalmassesthemass radius and speed of the space-time vortex by the values of the fundamental quantum space-time vortex, ahydrogenatom/charge.

Ifthethesisofafractaluniversemadeofhierarchicalscalesistruth,thenthosevaluesshouldgiveusthevalueoftheuniversal constant of charges, the Coulomb constant.Indeed, if we substitute for the proton (mass) and the Bohr electronic orbital (speed and radius)4×1016rad.sec.-1=w(electron);5.3×10-11m.(Bohrradius);protonmass=1.6×10-27kg.

ThenwegetaG,which is2x1039strongerthanthegravitational radius; thus, thehydrogenatombehavesasaself-similar fractal scale in the quantum world to a solar system.And then you can get also the electron radius expressed in the jargon of a quantum gravitationalworld using thetranslated 'Gravitational Coulomb constant': G(k)M/cˆ2.SinceinthatexpressionMisthemassofaproton,G(k),theelectromagneticconstantisagravitationalconstant,andc,lightspeed,thatexpressionisexactlytheSchwarzschildradiusofaquantumblackhole.

Thus,theelectronBohrradius,whichisthefinalradiusofminimalsizeandenergyinelectrons,isisomorphictotheeventhorizonofablackholeinthequantumgravitationalworld.

Those results (more thanadecadeold),area first theoreticaldeductionofKedeparting fromGand theenormoussimplification of the parameters of the electron radius till arriving to the same expression that a black hole radiuscannotbebychance.Theyaremathematicaldeductions,oneofthethreestandardformsofproofinscience.

Yeta theoretical calculusof thosevalues cannotbeexact ‘by chance’,unless the theoreticalmodelbehind it – thefractalself-similarstructureas$T(Spacepopulation)xð§(Temporalfrequency)entitiesofallphysicalsystemsisright.Thus,thepreviouscalculus isaclearproofthatboth,chargesandmasses,areunifiedasvaluesofthesametypeofspace-timevorticesinthe2differentscalesofspace-timeoftheUniverse.Andtheyaregeometricallyunifiedfromthep.o.v.ofgeometricalrelativitynotfromquantumtheory,asEinsteinwantedit.

Galaxies, (Galaxies≈Atoms)thusresolvethephilosophicalquestiononhowmany5Dscalesexist;aswefindenoughself-similarity to 'runagain' another gameof fractal scales (not identicalbut self-similar as in aMandelbrot fractal)bothbyquantitativeandqualitativemethodsbetweentheatomandthegalaxy.AquestionthatmightbeextendedtotheSTdualitiesofopen,‘entropicstrings’andclosed‘cyclicalinformativestrings’,inapossiblelargerandsmallerscale

ofmicroscopicstringsandsuperstrings:

Ouroboros theUniversal Snake,bites its tail on the stringquantumandcosmological self-similar scales, as perceived from the human ∆o mind. Philosophy of stience would thenargue that thosescalesare real,butpartof its self-similarity ismental: that is, the lossofinformationintheperceptionofscalesmakehumansextractthesameinformationfromthe

upperandlower10±30scales.

Alas!Inthisshowcaseofmultiplemeanings,jumpingfrommathematics,tophysics,tometaphysics,solvingquestionsseekforcenturiesinclassicscienceweshowtheessentialnatureof5D–notsomuchcrunchingnumbersbut‘seeing’whatnobodysees.

As I improve the papers,wewill focus better the equations already resolved but vastlymore profound that peoplethink.

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WORLDCYCLEOFEXISTENCE.

In the next graph repeated ad nauseam in those papers we see the essence of the process of a worldcycle ofexistence:thecreationofafinitesimalformwhichwillreproduceandthencollapseintoasuperorganism.

All what exists is a supœrganism of vital space tracing a 0-sum worldcycle of time through 3 scales of the 5thdimension:Bornasaseedoffasttimecyclesinalower5Dscale(∆-1:Max.TxMin.S),emergingasanorganismin∆o,living3agesofincreasinginformation,asitstimeclocksslowdowninits∆+1worldtodieinatimequantabackto∆-1.YetthemaximalpointSi=Tewherereproductionhappensdefinestheclassicage,maturity,beauty,balance,survivalofthesystem,alldisomorphicjargons.

The3agesof lifeemerge inhumansocialsuperorganismsasthe3agesofculturesand its3artisticstyles:Min.SxMax. T (infantile epic, lineal art, as in treccento, Greek kuroi; Si=Te; balanced beauty, when form and size are inbalance,theclassicmatureage;andMax.SxMin.T:baroque,3rdageofacivilisation,whosesubconsciousmind istheartofits'neuronalartists',theageofmaximalformandan∆stforanofuture,whichistheageofwaranddeathofcultures).

Wetalkof3∆±1scalesofworldcyclesasthebeingliveinaplacenta,thenemergesasorganisminaworld:

þ:0-1:itspalingenetico-1socialevolutionintheacceleratedtimesphereofexistence,tillbecoming1(0-1boundedunitcirclein¡logicmathematics;quantumprobabilitysphereofparticlesinphysicalsystems;palingeneticfetalageinbiologicsystems;0-9memeticlearningchildhoodinsocialsystems).Itisthehighlyorderedworldcycleasa'placentalmother-energyworld'isnurturingasmemorialcyclicalspacetimehaserasederrorsofpreviousgenerations.

- c: Theouter1-∞world, inwhich itwilldeploy its2ndworldcycleofexistence inanenvironmentwhich isopen,entropic (1-∞ hyperbolic unbounded Cartesian plane in ¡logic mathematics; thermodynamic entropic statisticalmolecularpopulationsinphysics;Darwinianstrugglebetweenpopulationsinbiology;idol-ogicdog-eat-dogcapitalist,nationalistcompetitiveeco(nomic)systemsinthesuperorganismsofhistory.Inthis1-∞existencetheworldcycleisnotensuredtocontinue,astheentropyoftheworldsystemcancutitoff.

ω:Theexistential life cycle, though ispartofa largerworldofhierarchicalsocialscales (§D¡),where itperforms5survivalactions through∆±4Planesself-centered in itsmind,beyondwhich it cannot longerperceive, tobecome ifsuccessfulanewsuperorganismoftheinfiniteplanesofGod,thegameofexistence.

Ingraph,physical,biologic&socialworldcyclesshowtowhichextent5Dlawsenlightenourunderstandingofreality.Matter States are physical time ages, from left pure solid, crystal, §top state, to an even more solid ∆+1 bosoncondensate,etc.Weseethatsystemseithermoveastepatatimewithinaplaneofexistence(gas, liquid,solid)ortheycanjump«twostatesatonce,(asinthecasesublimation)withinthatplane,ormostoftenbetweentwoplanes,asin«scattering&entropicdeath),tobecomeadifferentDimotionalstate.Wecanthenseehowthefundamentalelementsof5Dtimeappearonthegraph:theworldcycleis localandcomplete.Thereare2inversearrowsfromanentropicpast(plasma),inalowerplane(ionparticles)tothe3agesofthematterstateswithincreasingform(gastosolid), toend in ahigherplaneof existenceas aboson-Einstein condensate. Do thoseworldcycleshappen for thewholeUniverse?(cyclicbig-bang).Unlkely…

Itisthenclearthatcalculusistheclosestmathematicalmirrorofthecommonestprocessoftime-change:thecreationof finitesimals that reproduce in clonicwaves forming ‘spatial organic systems’, in themost complexworldcycle, ormere herds, or locomotions imprinting information in a lower field of entropic space – you name it. As we studymathematicalphysicswithcalculus,weshallbecommentingpreciselyintheunityofallprocessofcalculus–aprocessoffindingfinitesimalstointegratethroughtimelocomotionsorspatialpopulations,mimickingwhatistheessenceoftime=change,thereproductionoffinitesimalpartsintowholes.

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Theimmediatequestionisthenhowtowritetheworldcycleofexistenceincalculus.Andifitisusefultodoso.

Towriteitisimmediate:

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∫ ∆st=0’

Its usefulness becomes more clear when we consider the standing points and draw the function=worldcycle ofexistenceintermsoftheSxTexistentialmomentumofits3ages.

Calculusonthefunctionofexistence.

SinceIhaven’ttoldyouthisonethousandonenight-maretimes(:thefunctionofexist¡enceisall.Youarearepetitivefractalofspace-timeandyourpurposeistoexist,toconserveyourtime,butyourtimeisjusttheformofinformation,yourvitalspace,reproducedinallthescalesthatrisefromthebottomlineofyourgravitationalandlightspacetime,goingupwardsintoscales.Reproductionisthegame.Buttheworldcyclemakeserrorsinthereproductionofyouri-logonandthoseerrorsthatarestatisticallyseeninspaceasanormaldistribution,intimeasarepetitivesequenceofactionsandeventsslowlywearyoudown,andaserrorsofcopyinginformationrepeatandaccumulateyourfunctionofexistenceloosesfreshnessandyouage.

Sobecauseeachstœpofyourexistenceyourepeatyoursequentialactionseachderivativeisoneofsuchstœpsazigzagupinformationrightmotion,upinformationrightmotion,whosetangentisthederivativeofeachquantaofyourtime.Allthissaidthenwecanstudytheworldcyclewithcalculus.Infactisthebestwaytostudytheworldcyclewithcalculus,intheorthogonalgraphofinformationandmotion,informationandmotion,stopandstep,particleandwavestate, up and right up and right, as you age, first rising fast young and bold, reaching higher accelerations in yoursecondderivative,asspaceistime,thecurverepresentsinitsformofspace,itsmotionoftime,andthatisyourfirstderivative,seekingastandingpointofconstantspeedbutthatisnotpossiblebecausespeedisreproductionandyoureproduceyour form,with lesser skillpast theprimetimeofyourstandingpoint themaximalandminimumno farbefore,nolongahead:

Letusrememberthegenerallawsforanypossiblefunctionofexistence:

Ifwedrawthe‘existentialmomentum’,SxTofthesystemintheleftside,andthelinealtimeofthesystem,Tinthebottomside.

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Sosinusoidalbellcurvefunctionsrepresentaworldcycle,thoughthesymmetryisbrokeninthemomentofentropicdeathwhenthecollapseisextremeina‘fallingline’asdeathhappensinasinglemomentoftime:

4D»∆-1(seed)∑∆:|-$T(limb-field)<Ø-S≈T(iterativebodywave)>O-§ð(particle-head)«5D∆-1(death)

Akeythemeofvitalmathematicsistherepresentationofaworldcycleinlinealtime,with±exponentials&itsinverse,logarithmiccurvearoundthekeypointsofchangeofphase…asgrowthof‘entropy-motion’diminishes.Sowemovefrom ‘adolescence’ ofmax. growth of both parameters (sT energy and sT information) to the y”=0 point of youth,wherethe logarithmicpartgrowsslower.Togethertheyform,onehalfofthetotalgraphofacycleofexistence,tillreachingthey’=0pointofMax.(S≥≤T),whichthenbecomesnegative,happeningadecayofthewholesystemintwonegativecurves.

Theconservationoftimeinits5y’∧y”=0,standingpointsthatdefinethe5SS,Ts,ST,St&TTmomentsofgeneration,youth,maturity,3rdageandentropicdeaththusbecometheessentialpoints(maximalandminimal)oftheequationsofcalculus,thesinusoidalfunctionofexistenceandallitsderivedelements.

Letussupposethatonacertainintervala≤t≤bwearegivenafunctionS=f(t)whichisnotonlycontinuousbutalsohasaderivativeateverypoint.Ourabilitytocalculatethederivativeenablesustoformaclearpictureofthegraphofthe function. On an interval on which the derivative is always positive the tangent to the graph will be directedupward.Onsuchanintervalthefunctionwillincrease;thatis,toagreatervalueoftwillcorrespondagreatervalueoff(t).Ontheotherhand,onanintervalwherethederivativeisalwaysnegative,thefunctionwilldecrease; thegraphwillrundownward.

Wehavedrawnthegraphofan∆stfunctionofthegeneralform,S(anydimensionof awholeworld cycleor T.Œ) = f(T) - Any timemotionoraction.

Itisdefinedontheintervalbetweenaminimalquantainspaceortime(t1)anditslimitasafunction(d).

And it can represent any S=T duality, ormore complex 5Ds=5Dtformsorsimplerones.Wecanalsochangethesandtcoordinatesaccording to the Galilean paradox, etc. Hence the ginormousnumbersofapplications,butessentiallyitwilldefineaprocessofchangeinspace-timebetweentheemergenceofthephenomenaat ST1 AND ITS DEATH mostly by scattering and entropicdissolutionofformatd.

Andinmostcaseswillhaveabellcurvedfromoffastgrowthafteremergence in its first age ofmaximalmotion (youth, 1D) till amaximal pointwhere it oftenwill reproduce into adiscontinuousparallelform(notshowninthegraphatMax.SxMax.T;whichwillprovokeitslossofenergyandstartitsdiminutiontillitsextinctionatpointd.

ThusthebestwaytoexpressquantitativelyintermsofS-Tparameters(mostlyinformationandenergy),foranyworldcycleofanytime-spacesuperorganismisacurvewherewecanfindthosekeystandingpointsinwhichachangeofage,st-ateormotionhappens.

Ofaspecialinterestthusarethepointsofthisgraphwhoseabcissasaret1,2,3,4,5.

Atthepointt0thefunctionf(t)issaidtohavealocalmaximum;bythiswemeanthatatthispointf(t)isgreaterthanat neighboring points; more precisely for every t in a certain interval around the point x0.Alocalminimumisdefinedanalogously.Forourfunctionalocalmaximumoccursatthepointst0andt3,andalocalminimumatthepointt1.

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Ateverymaximumorminimumpoint,ifitisinsidetheinterval[a,b],i.e.,ifitdoesnotcoincidewithoneoftheendpointsaorb,thederivativemustbeequaltozeroth.

Thislaststatement,averyimportantone,followsimmediatelyfromthedefinitionofthederivativeasthelimitoftheratioΔS/ΔT.Infact,ifwemoveashortdistancefromthemaximumpoint,then∆S≤0.

ThusforpositiveΔTtheratioΔS/ΔTisnon-positive,andfornegativeΔTtheratioΔS/ΔTisnonnegative.The limitofthis ratio, which exists by hypothesis, can therefore be neither positive nor negative and there remains only thepossibilitythatitiszeroth.

Byinspectionofthediagramitisseenthatthismeansthatatmaximumorminimumpoints(itiscustomarytoleaveouttheword“local,”althoughitisunderstood)thetangenttothegraphishorizontal.

At thepoints t2,and t4also the tangent ishorizontal, justas it isat thepoints t1, t3,althoughat thesepoints thefunction has neithermaximumnorminimum. In general, theremay bemore points atwhich the derivative of thefunctionisequaltozeroth(stationarypoints)thantherearemaximumorminimumpoints.

Oneof thesimplestandmost importantapplicationsof thederivative in thatsense is in the theoryofmaximaandminima.

Criteriaformaximaandminima;studyofthegraphsofcurves.

If throughout thewhole intervaloverwhichxvaries thecurve is convexupwardand ifata certainpointx0of thisinterval the derivative is equal to zeroth, then at this point the function necessarily attains itsmaximum; and itsminimuminthecaseofconvexitydownward.Thissimpleconsiderationoftenallowsus,afterfindingapointatwhichthe derivative is equal to zeroth, to decide thereuponwhether at this point the function has a localmaximumorminimum.

Now,theapparentlyequalnatureonafirstderivativeoftheminimalandmaximalpointsofabeing,havealsodeepphilosophicalimplications,asitmakesat'firstsight'indistinguishableoftentheprocessesof'reproductiveexpansion'towardsamaximalandexplosivedecay intodeath, the 'two reversal'pointsof the5D (maximal) and4D (minimal)statesofacycleofexistence,forwhichwehavetomakeasecondassessment(secondderivative)toknowifweareinthe point of maximal life (5D) or maximal death (4D) of a world cycle. And to know if the cycle will cease in acontinuousflatencephalogramorwillrestartanewupwardstrend.

Orinotherwordsisanyscalar,e>cc>mbig-bangboththedeathandthebirthofmatter?

FinitesimalQuanta,asthelimitofpopulationsinspaceandtheminimalactionintime.

So there isbehindthedualitybetweentheconceptof limitsanddifferentials (Newton'svs.Leibniz'sapproach), theconcept of a minimal quanta in space or in time, which has been hardly explored by classic mathematics in itsexperimental meaning but will be the key to understand 'Planckton' (H-planck constants) and its role in the vitalphysicsofatomicPlanes.

ItisthenessentialtotheworkingsoftheUniversetofullygrasptherelationshipbetweenPlanesandanalysis.Bothinthe downdirection ofderivatives and the up dimension of integrals; in its parallelismwith polynomials,which risedimensionalPlanesofasysteminadifferent'morelinealsocialinterplanarway'.

Sopolynomialsandlimitsarewhat¬Algebraistocalculus;spacetotimeandlineal¬Algebratocurvedgeometries.

Thevitalinterpretationthoughofthatamazinggrowthofpolynomialsisfarscarier.

Power lawsby the very factof 'being lineal', andmaximise thegrowthof a functionARENOTREAL in thepositivesenseofinfinitegrowth,afantasyonlytakenseriouslybyoureconomistsofgreedandinfiniteusurydebtinterest...wheretheeªexponentialfunctionfirstappeared.

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The fact is that in reality such exponentials only portrait the decay destructionof amass of cellular/atomic beingsalready createdby themuch smaller processesof 're=product-ion'which is the seconddimensionmostlyoperatedwithmultiplication(ofscalarsoranticommutativecrossvectors).

Sothethirddimensionofoperandsisabackwardsmotion-alinealmotionintodeath,becauseitonlyreversesthegrowthofsumsandmultiplicationspolynomialsmakessenseofitsproperties.

Let us then see how the operationsmimic the five dimensions, beyond the simplest ST, SS and TT steps, namelyreproductiveand4D-5Dinvertedarrows.

Wecanestablishasthemainparameterofthesingularity,itstimefrequency,whichwillbesynchronisedtotherotarymotion or angularmomentum of the cyclical membrane. They will appear as theinitial conditions and boundaryconditionsofaderivative/integralfunction,whichoftenwillbeabletodefinethevaluesofthevitalenergywithin,asthelawofsuperpositionshouldworkbetweenthe3elements,suchas:

Determinationofthegreatestandleastvaluesofafunction.

Innumeroustechnicalquestionsitisnecessarytofindthepointtatwhichagivenfunctionf(t)attainsitsgreatestoritsleastvalueonagiveninterval.

Incaseweareinterestedinthegreatestvalue,wemustfindx0ontheinterval[a,b]forwhichamongallxon[a,b]theinequalityƒ(to)≥ƒ(t)isfulfilled.

Butnowthe fundamentalquestionarises,whether ingeneral thereexists suchapoint.By themethodsofmodernanalysisitispossibletoprovethefollowingexistencetheorem:

Ifthefunctionf(t)iscontinuousonafiniteinterval,thenthereexistsatleastonepointontheintervalforwhichthefunctionattainsitsmaximum(minimum)valueontheinterval[a,b].

Fromwhat has been said already, it follows that thesemaximum orminimum points must be sought among the“stationary” points. This fact is the basis for the following well-known method for finding maxima and minima.Firstwefindthederivativeof,f(t)andthensolvetheequationobtainedbysettingitequaltozeroth.

Ift1,t2,···,tn,aretherootsofthisequation,wethencomparethenumbersf(t1,f(t2),···,f(tn)withoneanother.Ofcourse, it isnecessarytotakeintoaccountthatthemaximumorminimumofthefunctionmaybefoundnotwithinthe interval but at the end (as is the case with the minimum in figure) or at a point where the function has noderivative.

Thustothepointst1,t2, ···,tn,wemustaddtheendsaandboftheintervalandalsothosepoints, iftheyexist,atwhichthereisnoderivative.Itonlyremainstocomparethevaluesofthefunctionatallthesepointsandtochooseamongthemthegreatestortheleast.

Withrespecttothestatedexistencetheorem,itisimportanttoaddthatthistheoremceases,ingeneral,toholdinthecase that the function f(t) is continuous only on the interval (a, b); that is, on the set of points xsatisfying theinequalitiesa<t<b.

Itisthennecessarytoconsideraninitialtimepointandafinaltimepoint,birthanddeath,emergenceandextinctiontohaveadeterminedsolution.

Derivativesofhigherorders.

Wehavejustseenhow,forcloserstudyofthegraphofafunction,wemustexaminethechangesinitsderivativefʹ(x).Thisderivativeisafunctionofx,sothatwemayinturnfinditsderivative.

Thederivativeofthederivativeiscalledthesecondderivativeandisdenotedbyy"=ƒ"(x)

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Analogously,wemaycalculatethe3rdderivativey"'=ƒ"'(x)or,the derivative of nth order. But as there are not more than 3 'similar derivatives, with meaning' in time (speed,acceleration,jerk)orspace(distance,areaandvolume),beyondthe3rdderivativetheuseofderivativesisonlyasanapproximationtopolynomialequations,whosesolvabilityitselfisnotpossiblebyradicalsbeyondthe3rdpower.

Soitmustbekeptinmindthat,foracertainvalueofx(orevenforallvaluesofx)thissequencemaybreakoffatthederivativeofsomeorder,saythekth;itmayhappenthatf(k)(x)existsbutnotf(k+1)(x).Derivativesofarbitraryorderare therefore connected to the symmetry between power laws and ∫∂ operations in the 4th and inverse 5th

Dimension, through the Taylor formula. For themomentwe confine ourselves to the secondandthirdderivativesfor'realparameters'ofthe3spacevolumesandtimeaccelerations.

Thesecondderivativehasthenaswehaveseenasimplesignificanceinmechanics.Lets=f(t)be a law ofmotion along a straight line; then sʹ is the velocity and s″ is the “velocity of thechangeinthevelocity”ormoresimplythe“acceleration”ofthepointattimet.Forexample,forafallingbodyundertheforceofgravity:Thatis,theaccelerationoffallingbodiesisconstant.

Significanceofthesecondderivative;convexityandconcavity.

Thesecondderivativealsohasasimplegeometricmeaning.Justasthesignofthefirstderivativedetermineswhetherthe function is increasingordecreasing, so thesignof thesecondderivativedetermines theside towardwhich thegraphofthefunctionwillbecurved;but intermsoftimerepresentsthesecondderivativeofthecurveofexistence.Thatnolongeracceleratesitsgrowth,hencetheendofyouth,andvice-versa,themomentinwhichitdoesaccelerateitsdecay,thusthebeginningofthethirdage.

Sowecanconsiderthesameconceptinthe‘discreet’baguasoflifecyclesasitisNOTamereidealcurvebutonethatdohappensinallformsoflife.Thissimplelawwithdeepcasesbecauseitisessentialtotheworldcycle:

Suppose,forexample,thatonagivenintervalthesecondderivativeiseverywherepositive.Thenthefirstderivativeincreasesandthereforefʹ(x)=tanαincreasesandtheangleofinclinationofthetangentlineitselfincreases.Thusaswemove along the curve it keeps turning constantly to the same side, namely upward, and is thus, as they say,“convexdownward.”Ontheotherhand, inapartofacurvewherethesecondderivative isconstantlynegativethegraphofthefunctionisconvexupward.

Becauseitistheclearproofofwhatisallabout:reproductioninspaceoffrequenciesoftime.

Thefunctionismorethanitsequation–Apathofexistencethroughthewholeplane.

The functionofexistence is thewholeplanedividedby the line thatmustbegrownby thenon-Emethodof risingpointsintocurvesofmotion,whichdivideanenergyinformationplaneinanactforcreationwithapathinS=T,thepathofpresentthroughsquaresinwhichinformationandenergyareorthogonal.

Wehavefoundthusthesimplestspace-timecurve,theS=T,curveofexistencebetweenanintegral1/3rdoftheplaneinpathwitha2/3rdLébesgueintegralsotospeakoftheexternal/internalpathofthecurve.

Thecurveisthusapointinmotion,equivalenttoalineofdistance,equivalenttoaratiobetween2parts,2/3rdstotheleftand1/3rdtotheright.Butthebeautyofitisthatwetakefromthecurvesquarepoints.

Theminimal reality is a3D² formseen ina singleplane,witha singularity@-mindamembraneanda vital energywithin.Whenwemakeaholographicbrokenimageofthisrealitythesimplestwaytodoitisinfourcartesianregions,TT,ST,ts,andss,whichcorrespondtothe+1+1,+1-1,-1+1and-1-1quadrantsoftheplane.

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It is thenwhen theLebesgue inverse functionmatters to integrate theYperspectiveof theS=T symmetry that thefunctionknowtakenasatopologicalpartitionofavitalmotionofawaveofsimilarparticlesthatwillcollapsettheendofitsjourneythroughaplaneofthefifthdimension,takesplace.

The leastpathaction implies thought that theendof thepath taken in theCartesianbut also the imaginaryplanecollapsesinthesamepoint,regardlessofhowmanyoathshavebeentakeninthe‘compressed’i-planewheretheco-existence of paths in particle space has sunk the plane to a √ root value for the dense line that then can be evenfurtherreducedtoapointthatwillpotentiallytraceafulls=t,valuinginpresenttimethespace-timedilationofthe

integrated,

Thewaveformasanintegralexpressionofthefunctionofexistence.

How many possible forms might acquire the function of existence? The answer thatmightsurprisethereader isdependingonthenumberofparameters,durationintimeofouranalysisandtypeofdimotionstudied,fromsmallerstepsofasingledimotiontothe whole worldcycle an all the sequential dimotions of a T.œ. there are ‘infinitesolutions’ – as all equations areultimately ‘partial equations’ of the fractal generator,SóT.

Consider the commonest formof theUniverse, awave. Ifweconsider that ymeasuresNOT thevalueof STof thesystemasaconstant‘volume’ofexistentialmomentum,butthevalueofits‘degreeofincreaseordecreste’ateachmomentoftime,hencey’overx(t),weobtaintheexactformofawave,withafirsthalfwave inwhichthegrowthfromyouth tomaturity constantly diminish but is still positive, till themiddle point ofmaturity at y’=0,where thegrowthstartstobenegative, followedbya fastdeclineasweage, tillamaximalpointof ‘degeneration’,wherewenormallydiebysuddensickness;butifweovercomethatpointsomewherearoundthe70yearsage,wewillhaveaslowdownofouraging,towardsapointofno‘changeatall’–thepointofdeath;whenwesimplydisappearfromthisplaneofspace-timeexistence.

Thuswhenweperceive awaveof light,weare infact,perceivingtime=change,andcreatingamentalspaceofthelife-deathcycleofasinglephotonasspaceisjustthememorialtailofourslowtimeperception.

RECAP.Thefunctionofexistenceinitsfractalvariationsandcx.PentalogicHASinfinitepaths=formsbutallendina0’sum.

RECAP.Timeiscyclicalasallclocksoftimereturntoitspointoforigin,soalltimecyclesincludingthoseoflifeofitsvitalspace-timebeingsarefinite.Furtheronthosetimecyclesbreak‘space’intoinnerandouterparts,sovitalspaceisbrokenby themembranesandangularmomentumsof thosetimecycles thatmakespacetimebeings also finite inspatial information.Andanobviousexperimentalfactsabouttimespace:cyclesoftime,vitalspacesandthespeciesmade of them, co-exist in several scales of relative size fromparticles to galaxies, each onewith clocks of time ofdifferentspeeds.Sospacetimeisfractalbrokeninscalesthataddedcreateanew5thdimensionofspacetime.

Thedualfunctionsof5DAbsoluteRelativity,thefunctionof5Dscales,SxT=C&thefunctionofequalitybetweenformandmotion,SI=TE,developsin3ageswith3standingpoints,amax.pointofexistence,Si=Teormatureage,ayoungage ofMax. T=motion, and an old age ofMax.S=information; between birth in ∆-1 Form& T-entropic death. Thesearchforspace-time,Energy=informationbalancesinaclassicreproductiveageofconservedtimeisthusthegoalofallexist¡ences,butonlythewholeachievestheimmortalityoftime-space,asweshallseeegocyerrorsoffractalmind-pointsofspacetryingtostoptheflowoftimefromasingleselfishpointofview,acceleratestheimbalancethatbrings

€

t intervalInferior lim itasspace

superior lim intime:T∞

∫

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deathequations.Wearericherinourstillpropertyatthat0T-moment,whenallisquietsofortimetokeepmoving,areversalofentropytakesplace.

Theconnectionbetweenexistentialalgebraandcalculus:Dimotionsasactions.Reproductionaschange.

Wesaidoftenthattime=motionisall;andspacejusttheMayaofthesenses,themind’smappingofthefractalpoints‘thatholdaworldinthemselves’.Buttheultimatearrowoftimeisthatofscalargrowthbetweenplanesofthefifthdimension,aspartsmustbecomebeforewholes;theupwardsarrowmattersmorethanthedownarrow.Andsoofthe3parametersthatdefineobjectivelybetween¬limits,andvitalizedbyamind’sprogram,∆ST,anybeing,∆-scalemattersmore–numbersofalgebra inmathematics.Thenitcomestimeperceivedinonegivenplane,T,andfinallySpace,themostevidentbutshallowpartofthewhole.ForthatreasonAlgebramattersmoreandincludescalculus,thetemporalviewofmathematicsthattriestocaptureallmodalitiesofchangewithasimplescalarprocessofadding‘¡n≈finitesimals’ofscalarchangetoanalyzethelargerprocessesofchangeinthewholescale.

This is done in calculus with the simple methods of ‘finding the parts=derivatives’ and adding them together=integratethemeitheroverscale,spatialvolumeortemporal frequencies. Inthismannersomethingsosimpleasafinitesimal change becomes the seed of all possible variations of change (dimotions each studied by an operand)acrossscale=size,spatialpopulationortemporalfrequencyofevents.

Thestudyofthe5DimotionsoftheUniverseiscarriedoutinspatialgeometrybycalculus;inNon-AristotelianLogicbyExistentialalgebra.Thusbothlanguageshavemanydeepcommonstructuresworthtocompare,evenifcalculuswasbornonthepraxisofanalysisofonesingledimotion, locomotion, in themilieuofphysicalsciencesandonlyslowlyextendedtotheunderstandingoftheotherdimotionsoftheUniverse.

Thusweshallbringinthissecondpaperonalgebrabothsciencestogether.

Even if Existential algebra is much wider and ultimately a logic stience, as it is also the underlying structure ofmathematical algebras, including those of group theory that dealwith an ‘extensive catalog’ of the dimotions andevolutionsoftheUniverse,andreticularBooleanalgebrasthatdealwiththe@-mindmirrorsoflogicandnumbers.IntheoriginalplanIhadenvisionedamuchlargeroutputofpapersforacademia,takenfrommy30yearsnotebooks,soExistentialAlgebrawouldhavedeservedoneofhisown.Buttimeisrunningout…

Existential algebra and calculus study time change. How can then unify all time changes? The answer comes fromexistentialalgebraanditsfindingthatallformsofchangecanbereducedtoreproductivechange.Whichitselfcanbeconsidereda traveldownandup twoscalesof the fifthdimension.Thuschangeshappenson finitesimalparts thatemergeandaffectlargerwholes.

The functionofexistence isa functionof reproduction in scale (asa5D journey) in time (asa conjunctionof the5Dimensionalmotionsofexistence),andspace,asasimultaneousgrowthofcloneinformation;formalizedinthefractalgeneratorof5Dmetrics,Max.∑TexS¡(s=t)=c;asreproductionhappensina‘presents=tstate’,ofbalancewhentherelativepastoflesserinformedflowsofentropictime,Te,becomesImprintedbySpatialinformation:PastTT-entropyxFutureSS-form=ReproductiveSTPresent

Changehappens informatively through increaseof finitesimalparts,entropicallywhenyou loose thosescalarparts.Reproductionofformoritsannihilationatthefinitesimalscaleincalculusismirroredbyasimplefunction,F(x+h)/F(x),thatcalculatesratiosofchange,fordifferentoperandsthatmirrorthe5Dimensionalmotionsofexistence,whichcanpotentiallychange.

Thuscalculususesaunitofchange,h,tomirrordifferentchangesInthe5dimotionsofexistence.Sincechangeonceithappens in small units, in small scales, in small instants of time; differentiate in 5 type of dimensionalmotions =actions:TT,feeding,entropicandmoving,Ts,changes,informative&perceptiveSt,SSchanges,reproductivechanges,ST,proper.

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Thatdiversificationisstudiedbetterwithdifferentalgebraicoperands;butallcanbederivedintoitsfinitesimalunitsofchangeandintegrated,fordifferentscalargroups,socialfunctionsandpathsofdimotionalchange.

Thus what both disciplines, calculus and existential algebra have in common is the object of its linguisticmirrors:Times=changes,allkindofthem.

Thatapparentlytheyseemsodifferentwearswitnesstotheultimatenatureofmind-monads, ‘infinitymirrors’thatreflectalwaysdifferentpointsofviewonrealityanditsi-maginationtoslightlybendthatrealitytothepointofviewofthemind.

Stillitismorenotableitscommonelementsthanitsdifferences.

Calculus though has its emphasis in numbers hence in the scalar analysis of huge social groups in motion; whileexistentialalgebrahasitsemphasisindiscretedimotions,henceonthestudyofindividualT.œsexperiencingatrans-form-ation.

Theveryessenceofcalculusistostudyinsynchronousspatialdimotionhugeamountsofnumbers,whichwilleraseits‘discrete’formtoappearasacontinuoussusceptibletobestudiedatthe∆+1scaleofthewhole.

TheemphasisofExistentialalgebraisthestudyofthatwholeasanindividualsubjecttosequentialdimotions.

Butinbothcasesthedynamicprocessofstudyarethe5Dimotionsoftime=changeoftheuniverse.

Finally logic systemsandBooleanalgebrasbecome the syntaxof verbal and computerminds thatdescribewith itssentences thedynamicdimotionsof reality. So its language is closer to thatof ExistentialAlgebra, reasonwhyweincludeitinthispaper,insteadofthemoreadvancedmodelsofexistentialalgebratermed,monologic,duality,trinity,pentalogicanddodecalogic.

To fullygrasp thatessentialconnectionbetween∆standcalculusmirrors,wemust firstunderstandhowspeciesononehand,andequationsontheother,probeinthePlanesofrealitytoobtainitsquantaofspace-timeconvertedeitherinmotionstepsorinformationpixels,tobuildupreality.

The connection between existential algebra and calculus is qualitative: both study initially the finitesimal action ofexistence,whichbecomethefinitesimalquantaofspacetime,whoserepetitiveaccumulationcausesthephenomenaoftime-change.Existentialalgebrathoughstudiesthequalitativelyintermsofsequencesbetweenthe5Dimotions,andcalculusquantitativelyfocusinginonesingledimotionspreadinagroupofscalarnumbers.

Thisisthecasebecausetheactionsofbeingshappenthroughfinitesimalsextractedfromother∆-planescales.

InallPlanes,thesimpleractionsofanybeingareextractionsofmotion,energyandformfromlower∆-iPlanes:

AT.œperceivesonlythe∆±3planesfromwhereitextractsenergyorinformation.Asitsactionsanddimotionsarearchitectonicallyperformedthroughplanesof5Dwhereeachmainactionrelatestoanintervalofscales:

∆-4-3:Thesystemextractsindistinguishableboostsofentropicofmotion(manfromgravitation).∆-3-2:Thesystemextractsbitsofinformation(Lightinman)∆-2-1:Thesystemextractsbitesofenergy(aminoacidsinman)∆-10:Thesystemseedsitsminimalseedofreproduction.∆0+1:Thesystemconnectssociallywithothersystemstoevolveintoawhole.

SosimplerActionsstartatfinitesimallevel,gatheringinsequentialpatternsinexistentialalgebra,as‘timeflows’andinpopulationandspatialpatterns-inintegralherdsofnumbersincalculus.

We and all other beings perceive from∆-3 quanta (light in our case), feedon amino acids, (∆-2 quanta for any∆ºsystem),seedwithseminal∆-1cellularquanta(electronsalso,with∆-1photonquanta).

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Foreachactionofspace-timeweshallfindawhole,∆ºT.œ,whichwillenterincontactwithanotherworld,∆±i,fromwhere it will extract finitesimals of space or time, energy or information, entropy ormotion, and this will be thefinitesimal∂ƒ(x),whichwillbeabsorbedandusedbythespeciestoobtainacertainaction,å.

Analysisallowustoextractactionsfromwholes,reasonwhytherearenotreallyusebeyondthethirdderivativeofabeing, as super organisms co-exist in 3 only Scalar Planes. It alsoworks in termsof a volume, as its derivative is aplane,thenitsunit-cellorpoint...Sotospeak,ifyouderivateaworld,yougetitsorganism,andifyouderivateitagainyougetitscellandthenitsmolecularparts.Andthenifyoudothatintime,yougetitsspeedandthenitsaccelerationandthenitsjerk.

Buthowyouextractfinitesimalsofsmallerscales?Theansweris:

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NETWORKS.

HerdsofFinitesimalsofspace=informationandtime=energy.

Another element of enormous importance to understand the equations of calculus inmathematical physics is theconceptofabiteofenergyandabitofinformation;movinginaherdthroughaphysicalnetworkorwave-backgroundthatdisplacesthem.Astheyarethefundamentalelementsoffieldsandwaves,anditsintegralequations.Soweshallbringfromthegeneralmodel,theparagraphonphysicalnetworks.

Howthen ithappens thatpartsbecomewholes is thekey todepart fromamereabstract,quantitativeanalysisofrealityandaddtheorganicwhystocalculus;explainingthenatureofallwhatexists,thedynamicinterplayofpartsthat‘network’andconnecttoeachother,formingsimultaneousspacetimeorganisms,whichsynchronizesitsclocks,emerge as awhole and develop all the intelligence and complexity of the systemswe observe around us. A finalelement though isneeded tomake senseof those superorganisms, the stillmindof information,mappingout thewholeandcontrollingittoperformitsMandateofexistence,Max.SxT(s=t),tosurvive,growandmultiply.

The ternary network structure of nested organismsmakes them ¬Æ topologic planes composed of similar fractalpoints(atoms,cells,individuals)joinedby3physiologiclines=networks,whose3functions,locomotion,informationanditscombinedenergydefinethe3conservedDimotionsofanysystemoftheUniverse. Thegraphshowsthosephysiologicalnetworksofeachsupœrganismfromthegalaxytotheatom,whereself-similaritytakesplace. IremindyouoftheSi=Teequality,whichmeansweslowbeingsseenetworksoffasterparticlesas‘forcewaves’andnetworksofslowerlifeformsasfractalbranching,butessentiallyasNottalehasproved,wecan‘translate’quantumphysics intoanetwork, topologicalview,as light is in factabranching fillingwave that ‘speedsup frequency’as itpenetrateslowerplanes,fillingittillittouchesparticles.

Allsystemsofrealityareconnectedbynetworksthatshareenergyand informationbetweenpartsandwholesthatexpressesthestructuralunityofallscales.Networks‘fill’spaceadmaximaltoconnectfullythewholewiththeparts,achieved in the Si=Te point of parallelism and self-similarity. But they entre in a region of fastermotion. SowhileSpace‘tendstoremainconstant’ineachscalethankstofillingnetworks,timeaccelerates.Soweneedtobecomeabit more complex about the previous metric. It refers essentially NOT to the whole 5D plane but to a given‘superorganism’ofeachplane.Whenwegodowninscales,infacttheUniverse‘enlarges’foratravelerthatbecomessmallerandacceleratesitstemporalenergy.

Soweneedtobecomeabitmorecomplexaboutthepreviousmetric.ItrefersessentiallyNOTtothewhole5Dplanebuttoagiven‘superorganism’ofeachplane.Whenwegodowninscales,infacttheUniverse‘enlarges’foratravelerthatbecomessmallerandacceleratesitstemporalenergy.

Inthegraph,systemssciencesconsidertheUniverseanorganicfractal,whicheachsciencestudiesatalevelofsize,allofthemevolvingsociallythroughnetworksofenergyandinformation.Sowecanstudyspeciesasorganismslivingaworldcycleandevolvingfinallyinsocialnetworksbecomingsuperorganisms,themostefficientsurvivalstrategyofthe Universe, where a head-particle-informative class will invent a language of common organization, quantumnumbers,hormones,genesormemes,moneyorlawsthatallthebody-waveparts/cells/citizenswillobey.Thegoalofsystemssciencesisthentoproperlyapplythephysiologicallawsofnetworkstodesignorstudyperfectsorganismsofnature,includingmankindandtheeconomicecosystem.

Inthegraph,thephysiologicalnetworksofsomesuperorganismsoftheUniverse,whichcanbeformalisedallintoalogicequation,wecallthefractalgeneratoroftheorganicUniverse:

Ts (Spatial, past, entropic limbs/fields/territory) < ST (present re=productive body waves-working class) > St:temporal-informativeparticles/heads/informativeclass.

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Soeverythingreallyisencodedinthosenetworksanditstopologyanditcanbeexplainedwithmultiplelanguages,the language of topology (mathematics), the language of logic (sequences of actions the networks cause) thelanguageof socialclasses,as the3networkswith itsattachedcellshaveahierarchicalorder,with the informativenervousnetworkincontrolofthefasterlanguagesontop(legalpoliticiansandfinancialbankersinhumansocieties,nervouscellsandbloodleukocytesinbiologicorganisms,gravitationalblackholesingalaxies,etc.).Butdon’tworrywejustracethroughthebasicssowecanconcentrateinthesocialorganismsofhistory.

There are 3 physiological networks ineach supœrganism from the galaxy tothe atom, the two scales where self-similarity takes place (the galatomsmaller and larger planes). Butwe canfor the time being disregard the‘territorial, background network, thespacetimeofthegalaxyanditsparticles,the water of life organisms, theTerritorial surface of earth in historicorganisms, to concentrate on the body-wave-reproductive economic system,and particle-head, informative legalsystems that shape the essence of asuperorganism.

The key to understand its linguistic bitsandbitesof informationandenergyareas in most themes of reality theastoundinglysimpleMetricequationsofallthescalesofthefifthdimension:sizein space x speed of time cycles =constant. And the duality of Spatialinformation and temporal motion thatcombine in 5 Dimensional motions: TT-puremotion (entropy) > Ts-locomotion,energy S=T (reproductive energy andinformation combined)> St-information(formwith abit ofmotion)& SS (form,

position,language).

Allsystemsofrealityanditslimbs/fields,body-wavesandparticle-headsexistinginalargerworldwithfasterenergythat appears as them as the limit of TT-entropy and faster information that appears to them as the limit of SS-languageplay thena simplegameof trying toabsorbasmuchTs-motion, reproductiveenergyand information toperceive,reproduce,survive,moveandplaythegameofexistence.Andassystemsbecomemorecomplexanddense,theynaturallyattachtothose3typeofnetworks,evolvingfromherdsintosuperorganisms.

ThusIremindyouofthe5Dmetricequation,SxT=CandtheS=Tpointofbalanceandequilibriumwherethesystemreproducesasthetwofundamentalmetricequationsofallspace-timeorganisms,whichmeansmanythings,suchas:

- Slowbeingsusenetworksof fasterparticleswhich inphysical systemsare ‘wavesof force’ in lifebeingsareandnetworkswithfractalbranchingandinhumansocieties,networksofmoneyandsimultaneouslegalmessages.Yetallareessentiallyperformingthesameorganicfunctions,weshalldescribenowascan‘translate’alightfillingwavethat

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penetrates lower planes, filling it till it touches particles, as a branching that ‘speeds up frequency’; and a legalnetworkthateverycitizenknowsandobeysasafillingsystemofinformation,similartoaDNAnetworkthatallcellsof an organism have in common. Think always NOT in the differences of form and scale but in the homology offunctionsforthebody/wavesandparticle/headsofeachofthe3superorganisms(workingandinformativeclassesinhumansocieties),toseetheunityofitall.

The2languagesofinformativeandreproductivenetworks:Bitsofinformationandbitesofenergy.

ItisimportantalsotoUnderstandNOTinpureabstractmathematicalterms,butinlogic,linguisticones,theinternal,dynamic nature of those networks, because only then we can proper understand how they work in advancedorganisms of maximal information, the biological organism and the historic organism, which belong as ‘fractalsystems’ self-similar to each other, two the same specific type of organisms, we shall qualify as socio-biologicalorganisms. A network, of informative nature, delivers messages of information to simultaneously coordinate theactionsofall itsparts;with its faster=smallerbitsof informationaccording to5dmetrics (min. spatial size x.max.temporalspeed).Whilethenetworksofreproduction,thebloodandfinancialsystemdeliverslargerbitesofenergy,whichtheorganismneedstofeeditself(whenitisahealthynon-corruptedsuperorganismasmostofnature,butnothumansocieties,whoseastoundinglevelofcorruptionweshallexplainindetail).

Indeed,thekeytothefullunderstandingofrealitybothintermsofenergybutalsointermsofinformation,asbotharetwosidesofthesamecoin,called‘exist¡ence’isthefactthatinthesentientUniverse,eachfractalpoint,atom,cellorcitizen(physical,biologicorsocialsystems)needsbitsof in-form-ation,form,smaller insizeofspace,hencefaster according to 5D metrics (SxT=C), but also ‘bites’ of entropic energy which will help the system to move.NetworksareNOTsomeabstract‘fractaltube’buttheyexisttodeliver‘energyandinformation’(SS:form=languagewithalittlemotion=St-informationandmotion=entropy=TTwithabitofinformation=energy=Ts).

So a healthy superorganism will deliver to each ‘fractal point’ (molecules, cells, human citizens), two type ofmessages through two typeofnetworks.Weshall call ‘generically’ the3 typeofbitsandbitesof informationandenergythateachofthose3physical,biologicalandsocialsystemsreceive,‘particles,genesandmemes’evenifthewordsasusualin5Dsciencersareslightlychanged,andwidenedinitsoriginalmeaning.

So with its specific variation, those are the two fundamental reproductive-‘body-wave’ and informative-‘particle-head’bitesofenergyandbitsofinformationofthefundamentalsystemsofnature:

-Inphysicalsystems,thetwonetworksarethegravitationalfasternetworkofinformation,whichwehumansdonotperceive, as we are much larger beings with electronic networks. Its bits of information in this faster non-localnetworkshouldbe‘gravitons’,componentsofgravitationalwaves.Inphysicalpapersweadvanceasthemostlikelyparticle state of those waves of information that ‘position’ the different physical systems of the galaxy, agravitationaltachyon‘neutrino’formultiplereasons,westudyonourpapersonphysics.

On the other hand, becausewe do perceive it, it ismuch easier to prove that the energetic network of physicalsystemsareelectromagneticwaves,photonsand its ‘social, static state’ as theelementsof anelectronicnebulae,trapped in the potential energy well of the atom. Thus photons and electrons become the ‘energy network ofphysicalsystem,molecules.

Weshallescapetheninthisintroductionfurtherinformationonthescalarstructureofthosenetworksandhow,aswe ‘grow in scale’,what is abiteof slowenergy fora smallerplaneof space-time,becomes for the largerplane’sslowerbeings,afasterbitofinformation,intheamazingbeautyoftheharmoniesbetweenscales.Soelectronic‘foodforatoms’becomeselectronicinformationforbiologicalorganismsandsoson.

-Thosebiologicalorganismsdohavethentwofractalnetworks,theelectronic,informativenervoussysteminwhichbits of electronic information moving along the myelin membrane deliver faster messages to every part of theorganismtosimultaneouslysynchronizeitsmotions,sothebody-cellsactasasingleforminsimultaneousspace.

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-Butwhenwemoveintothebitesofenergydeliveredbythebloodorganism,thenetworkdeliverstoeachcellthebasic‘currency’languageofenergythatallcellsneedtomove,called‘oxygen‘.Itisanatomofslowermotionthantheelectronsbutduetoitselectro-negativityandreadilyavailabilityintheatmosphere,withitscapacitytokickwithtwoOH-&H+legsthewater‘medium’onwhichcellsexist,theperfectlanguageof‘money’fortheorganismtostartkickingits‘actions’.

SoweDOhaveinthenextscaleaccordingtotheperfectlawsofharmony,thetwobasicbiologicalbitsandbitesofinformationandenergy,electronsandoxygens,and fromthenon,as systemsbecomemorecomplex,variationsofthosebitsandbitesoccur.

ThemaincategoryaremixedSTmessages,whichdeliverBOTHastickandcarrot ‘complex’ to thecellsand itsbigmolecules,whichareaminoacidsystems,ofgreatsimplicitycalledHormones,startingfromthesimplestofthemall,an NOmolecule (which do relaxmuscles, its mainmessage to the locomotion system, increases the pressure ofblood,provokingsexualerection,thesimplestmessagetoreproductivesystemsandmultipliestheneuronalactivity.Asnitrogensaretheclockatomofourmind-brains.

So finallymorecomplexNOsystemswithabodysupportofcarbonchainsbecome ‘hormones’whichmighthavea‘higher informativemessage’ (withmoreN,as innucleotidemolecules)orahigherenergeticmessage (as inacidswithmoreoxygen).

Theyformthenthebasiclettersofthe‘biologicallongersentencesthatmightaccumulateinformation’inevermorecomplexmolecules,asbiologicalorganismsarebyfarthemorecomplexsystemsweknowof.

Finallyaveryimportantconceptisthedifferencebetweenanecosysteminwhichmultiplesuperorganismsco-exist,often inpredatoryrelationships,vs.anorganism inwhichonlya typeofatoms,cellsorcitizensco-exist,and is farmoresymbioticasallpartsloveeachotherandshareenergyandinformationthroughitsnetworks,overacommonterritorialspace,asshowninthegraph.

Thosethreephysiologicnetworks/classes/physicalpartsofANYsystemoftheUniversedefinetheUniverseindeedasafractalorganismofinfinitesmallerandbiggersuperorganisms,inagameofrussiandollsinwhicheachofusisa'island-Universe'within itself,madeof smallerparts, and for that reasoneachofus is alsoapart, cell/citizenof asocialsuperorganism,nation,religionorcivilisation,whichwedoNOTseeasawhole,asourcellsdoNOTseeusasawhole,butDOexistassuch.

Whatmakes thenthewholeawhole?Theanswer is: thenervous, informative languages thatcommunicateall thepartsofthesuperorganismand'trace'withinitssyntaxandvalue,itspathofthefuture.Andsowehavetalkedfirstofitandwillconstantlycomingtothebottomlineofreality-thelanguagesthatconstructtheorganismsoftheworld.

Letusthendefinewithsimilartemplatesthe 'stairofnestedUniversalsupœrganisms,ofthe3stientificvarieties–physical,biologicandsocial:

∆+3: A galactic organism is a population of stars, related by energetic electromagnetic networks and gravitationalinformationwithanucleusmadeofaswarmofblackholes,andamembranemadeofstrangeletmattersymmetricto:

∆-3:Anatomicorganismisapopulationofparticles,relatedbyenergeticelectromagneticnetworksandgravitationalinformationwithanucleusmadeofaswarmofquarks,andamembranemadeofelectronicmattersymmetricto:

∆+2:A starorganism isapopulationofelectronicplasma, relatedbyenergeticnetworksofelectromagnetismandgravitational information with a nucleus made of a swarm of atoms, and a membrane of photonic radiationsymmetricto:

∆-2:Alightorganismisabodyofenergeticwavesoveraquantumpotentialfieldofgravitationalneutrinos,directedbyitsparticle,informativephotonstate…

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∆-1:Acellularorganism isapopulationofmolecules, relatedbyenergeticnetworks (cytoplasm,membranes,Golgireticules)andcodedbygeneticinformation(DNA-RNA.)

∆=o: A human organism is a population of DNA cells, related by networks of genetic, hormonal and nervousinformationandenergynetworks(digestiveandbloodsystems).

∆+1:Ananimalecosystem isapopulationofdifferentcarbon-lifespecies, relatedbynetworksof light informationandlifeenergy(plants,prey)codedbyinstincts.

∆+1:Ahistoricorganismor civilization is apopulationofhumans, relatedby legal and culturalnetworksof verbalinformationandagriculturalnetworksofcarbon-lifeenergy,codedbyhumanmemes.

Networksthatshareenergyandinformationbetweenpartsandwholesthatexpressesthestructuralunityofallscalesconnectallsystemsofreality.Networks‘fill’spaceadmaximaltoconnectfullythewholewiththeparts,achievedintheSi=Tepointofparallelismandself-similarity.Buttheyenterinaregionoffastermotion.SowhileSpace‘tendstoremainconstant’ineachscalethankstofillingnetworks,timeaccelerates.Sothe5DmetricrefersNOTtothewholeUniverse of 5D planes but to a given family of ‘supœrganisms’ of whichmankind in it 3 scales of ‘biologic cells’,humanindividualsandsocieties isundoubtedlya‘phyla’.Whenwegodowninscales,theUniverse‘enlarges’foratravelerthatbecomessmallerandacceleratesitstemporalenergy.

The2languagesofhistory:Verbal,legal,ethic‘nervousmessages’and‘WHealthymoney’.

Sowearrivetothehumansocialnetworks,whichweanticipateareinthepresentform‘completelycorrupted’bytheexistenceofaparallel‘economicecosystem’oflethalgoods,weapons,andcorruptedparasiticmoney.Soitisdifficultfor the reader to understand how simple, easy, and efficient WAS in the past, before the age of Metal, in theNeolithic,orduring theagesof social religionsof loveorcouldbe in the futurewithaproperdesignof thesocialnetworksofmoneyandlaw,aPERFECTsuperorganismofhistoryasefficientasthosewejusthavedescribed.Insuchsuperorganism, there are exactly the same networks: Legal verbal just networks of bits of word information thatshapestheinformativeandculturalsystemsofthewor(l)d.Andahealthyformofmoney,deliveredtoeachcitizencellasaUniversalsalarysohumanshaveenoughenergytosurviveandbuyitsnaturalwelfaregoods,whichmustbeclassifiedNOTbypricebutby itsbiologicalusefulnesstomankind,reasonwhywegivethempositiveandnegativevaluesintheethonomicframeofreference,accordingtoitsuseforthe3organicpartsofhumanityatindividualandsociallevel.

SinceVital,Topologic,physiologicnetworklawsarethemostimportanttoconsiderwhenstudyingHistoryinSpace,asthereadercanobserveinthepreviousgraph,sinceHistory,the∆+1scaleofsuperorganismsofmankindfollowallthe exact laws of a lower organic plane – that of a biological organism, albeit, due to its ‘primitive’ degree ofevolution is NOT a well designed organism, but one clearly ‘sick’, infected by ‘lethal goods’, with dysfunctionalparasiticeconomicsystems(asthelanguageofreproductionofgoods,money,isabsorbedbyaminimalnumberofpeople, or used to reproduce those lethal goods). And so the studyofHistory as a superorganismhas 2 differentparts:

-Onone sidewecanconsideraperfect,efficient superorganismofHistorywith the lawsof vital topology justbyimitatingperfectefficientsuperorganismsofNature,whicharethemajorityofthem–Historyisinfactanexception–aswesaidasickorganism.

- On the other side, we can study our superorganisms, as they have evolved in time, spotting their degrees ofcorruption and sickness of its 3 physiological systems, the life Earth, Gaia that sustains them; the economic andfinancialsystemthatreproducesitsgoods,andtheinformativeculturalandlegalsystemthatsynchronizesitscitizenscells.Itisthenwhenitwillbecomeevidentwhatwentwrongwith‘human’history,oneofmanylikelysubspeciesintheinfinitefractalplanetsoftheorganicUniversethatlikelywillnotmakeitintothefuture…

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In the next graphwe can see a historic super organism based inwelfare,whealthymemes that allow humans tosurvive,belongingtothe‘idealspecies’thatwouldbeefficient,providetothe100%andsurvive…

Historyisasuper-organismmadeofhumancells,extendedoverageographicalbodyofenergycalledGaia,thevital-spaceorbody,wherehistoric,socialorganismsevolve. Inthatsense,ahumanandasocialorganism,anationoracivilization,haveincommontheelementsofallsuper-organisms:

1. Cellularunits.Thesearecellsinahumanbody,citizensinahumansociety.

Networksofenergyorvitalspace;providedbyGaia, a super-organismmadeof livingbeings, joinedbya commonnetworkofvisual informationandnetworksof lifeenergy,gatheredaroundher ‘riverveins’.SinceGaia isalso thelivingorganismthathoststhesocialorganismsofMankind.

2. Networks=Languagesofinformation.Thisisthenervoussystemthatorganizescellsinabody,ortheverbal/visualinformationthatorganizeshumansocietiesthroughlaws,ethicsandart.

3. Networks that reproduce energy and information. These are sexual systems in individuals; and economicalnetworks that reproduce goods and cultural networks that define howhumans reproduce in societies. This is theblood system in a human body and the economic networks of production & transport in a society that favor‘WHealth’,thatis,goodsthatsatisfytheneedsofthe‘3physiologicalnetworksoflife-existencethatmakeushumanbeings,whose‘properframeofreference’(left)isNOTbasedonfinancialvalues,manipulatedbyfinancialinstitutionsandcompaniesthatsetprices,butby‘biologicaltruevalues.’

SoahumanorganismIsapopulationofDNAcells,relatedbynetworksofnervousinformationandenergy(blood).Andasupœrganismofhistoryisapopulationofcitizens,relatedbylegal,cultural,informativeandeconomic-energynetworks‘predating’overaterritorialgeography,ontheouter‘membrane’ofplanetEarth.

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PARTII.CALCULUSOFDIMOTIONSOFEXISTENCE.THE3AGESOFANALYSIS.

TheunderlyingorderofallstructuresoftheentangledUniversebetweenitsS,Tand∆Componentsoncemoreshowsinthe3agesofcalculus,whichwecantermsasthescalarage,whenthemainquestionwasthatbetweenpartsandwholes(fromGreecethroughNewton),thetemporalagewhenitsmainfocuswasthedescriptionofthe5Dimotionsof physical systems (from Leibniz to Heaviside) and finally the spatial view, when its main focus is, besides thecompletionofthepreviousages,itsusetothedescriptionofmentalspaces(fromGaussthroughRiemannandHilberttoEinsteinandquantumspaces).

Thustoputsomeorderinsuchavastsubject,weshalldoasusualadiachronicanalysisofitsinformativegrowthincomplexityin3ages,barelytouchingtheessentialelementsofeachofthem;fromits:

-IAge:Scalarview, fromtheGreekstoNewtonnsLeibniz.Thebeginningofcalculuswasverbal, logic, intheGreekage, with the discussion of finitesimals (5D infinitesimals with a minimal size), and Universals. This philosophicalanalysis was retaken by Leibniz.Whereas the dualityof derivatives as limits vs. differentials – tangents of change(Newton'svs.Leibniz'sapproach),representsthedualityofaminimalquantainspacetime(Leibniz'sinfinitesimal)orinscale (Newton's limit), hardly explored in philosophy of mathematics, but a key concept in 5D Planes, UniversalConstantsandquantumphysics.

Newtonontheotherhand,apracticalEnglishmanwithlittleinterestforthewhyscametotheconceptthroughthestudyoflimits,ofpowerseries–thescalarview,withoutmuchinterestonwhattheymeant.TheywhyswerecoveredbyYahwehandhisbiblicalstudiestoprovethatGodhadsenthimcometstoteachhimgravitationasthe‘chosenone’afterKepler,whoknewthat ‘Himhadwaited5000yearstofindan intelligence likehis,meKepler, toshowhimhisclockwork’.Aftersomuchevidenttruth,whowerethosehumblebelieverstocontestGod’swisedecisions?Leibnizthoughwasmore interested inmeaningandsohedid findthetrue finitesimal,1/x.Tothequestionofwhocopiedwhotheanswerisobvious,andthefactisnotyetresolvedmerelyshowsthatmathematiciansstilldoNOTunderstandthefoundationsofcalculusinitstrinityusefulfor∆,SandT,the3componentsofreality.Becausetheycamethroughdifferentmethods, Newton found the ∆-scalar power series of finitesimal changes that grow internally in ‘speed ofchange’astheyaccumulatelargerpowerseriesfactors;soeachsummandofthepowerseriescanbetakenasascalareverlargerchangeperunitoftime;whileLeibnizfoundtheSTgeometricanalysisofexternalchange,mostlyusefulforlocomotionvs.thehigherinterestofpowerseriesunderstoodasasummandatatime,forinternalchangeandgrowth.

Both are completelydifferent approaches that serve an essential duality between internal evolutionary ‘biological’growthvs. external physicalmotion,which insteadofopeningaproperphilosophyof calculusbased in thewholerangeofchangesintime(bestservedbyderivatives),spacepopulations(bestservedbyintegrals)andscalargrowth(bestservedbysumsofseries),justbroughtthequintessentialmonologicegocentered,ænthropicman,Mr.Newton,toargue,ashehaddonealsowithBoyle,ontheprimacyof its‘ceterisparibus’discoveryofavastregionofmentalmathematical spaces, suit to study ALL forms of change=time, which truly made ¬Algebra, the queen of allexperimentalsciences,ofwhichphysics,giventhereductionismofitspractitionersisjustasub-discipline.

Soashumindsstill ignorethatall isaboutthetrinityof∆,SandT(powerseries, integralsandderivatives)andtheiregocyistheonly∞truth(Einstein),theyhavenotyetunderstoodwhattheyfounddiscoveryingcalculus.

-II Age,Motion view: Needless to say because power series are yet the less understood, and internal growth andbiological scalar series ignored, the Newtonian approach had less obvious uses than the approach of calculatinglocomotionsintimesequencesandspatialexternalevidentgrowthinintergralforms.SofromLeibniztoHeavisideitsmethodsbecamethefundamentalapplicationstophysicsof locomotion,anditstwoessentialdimotions,Ts,andTT(locomotion and entropy), which became the magic of calculus. While the level of complexity of ∆∫∂ studies ismaintained in strict realist basis, as physicists try to correspond those finitesimals andwholeswith experimentallysoundobservationsoftherealworldatthecloserangeofPlanesinwhichhumansperceive.Whiletheformalismofits

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functionsisbuiltfromLeibniz'sfinitesimal1/nanalysistotheworkofHeavisidewithvectorsand∇functions.Partialderivativesarekeptthenatthe'holographiclevel'of2dimensions(secondderivativeson∆±2).

∆willbethusthegeneralsymbolofthe5thdimensionofmentalwholesorsocialdimensionand∫∂thesymbolofthe4thdimensionofaggregatefinitesimalsorentropicdimension.

-III Age, spatial view: from Riemann and Einstein to the present. The extension of analysis happens to infinitedimensionswiththehelpoftheworkofRiemannandHilbert,appliedbyEinsteinandquantumphysiciststothestudyofPlanesofrealitybeyondourdirectperception(∆≥|3|).

Thisimpliesthatphysicistsaccordingto5Dmetrics,P$txTƒ=Kmustdescribemuchlargerstructuresinspaceextensionand time duration (astrophysics) and vice versa, much faster populous groups of T.œs in the quantum realm; so'functionals' - functions of functions - ad new dimensions of time, and Hilbert quasi-infinite spaces and statisticalmethods of collecting quasi-infinite populations are required in the relentless pursuit of huminds for an all-comprehensive 'mentalmetric'of ablockof time-space,whereall thepotentialhistoriesandworldcyclesof all theentitiestheystudycanbe'mapped'.

Theimpressiveresultsobtainedwiththoseexhaustivemappingsbarewitnessofthemoderncivilisationbasedinthemanipulation wholesale of electronic particles, but the extreme 'compression' of so huge populations in time andspace blurs its 'comprehension' in 'realist' terms, and so the age of 'idealist science', spear-headed by Hilbert'simaginationofpointslinesandcongruencesdetachesmathematicalphysicsandbyextensionanalysisfromreality.

±¡:Thedigitalandexistentialera, isthe lastageofhumindmathematics,whereComputerswillcarrythisconfusingfromtheconceptualperspective,detailedfromthemanipulativepointofview,Analysistoitsquantitativeexhaustion.Butforethicreasons,asa'vitalhumind',weshallnotcommentoradvancetheevolutionofthefuturespeciesthatismakingusobsolete.

Insteadweconsideradifferentversionofcalculusofchange–existentialalgebra.

ThegeneratorequationofAnalysis'ages.

Ifwewere tomakeaGeneratorequation in timeof the 'bodyofanalysis'and itspreandpost-Planesof study,wecouldwritethe3±∆fieldsofobservanceofthescalarUniversethroughmathematicalmirrors:

Γ Analysis: ∆-i: Fractal Mathematics (discontinuous analysis of finitesimals) < Analysis - Integrals and differentialequations(∆º±1:continuous=organicspace):|-youth:ODEs<∑Ø-PDEs<∑∑Functionals≈<∆+i:Polynomials(diminishinginformationonwholes).

The3±∆approachesofmathematicalmirrorstoobservethePlanesofreality isthusclear:Fractalmathsfocusesonthe point of view of the finitesimals, and its growing quantity of information,enlarging the perspective of the@-observerasweprobe,enlargingsmallerPlanesofsmallerfinitesimals;andintheoppositerangepolynomialsobserverlarger Planes with restriction of solutions, asbasically the wholes we observe are symmetric within its internalequations,andtheeasiestsolutionsarethoseofaperfectholographicbidimesionalstructure(whereevenpolynomialscanbereducedtoproductsof2-manifolds).

Now within analysis proper, we find that the complexity or rather 'range' of phenomena studied by each age ofanalysisincreases,fromsinglevariables(ODEs)tomultiplevariables(PDEs)tofunctionsoffunctions(Functionals).

Sothemostbalanced,extendedfieldisthatofdifferentialequationsfocusedonthe∆±1organic(henceneitherlinealnotvortexlikebutbalancedS=T),PLANESofthebeing,wherewefocusonfindingtheprecisefinitesimalthatwecanthenintegrateproperlyguidedbythefunctionofgrowthofthesystem.AndwedistinguishthenODE,whereweprobeasingleSTsymmetryorPDEobviouslythebestmirror,asweextendouranalysistomultipleSandTdimensionsandmultipleS-T-S-TvariationsofthoseSTepmotions;giventhefactthata'chainofdimensions'donotfairwellbeyond

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the 3 's-s-s', distance-area-volume dimensions of space and t-t-t-t deceleration- lineal motion-cyclical motion-accelerationrelatedtimemotionsthatcan'change'agiveneventofspace-time.

SofurtherODEderivativesareonlysignificanttoobservethedifferencesbetweenthedifferentialand/orfractalandpolynomial approaches - this last comparison, well established as an essential method of mathematics, worth tomentioninthisintro.

Aspaceofformal¬Algebrathusisafunctionofspace,whichcanbedisplayedasacontinuoussumofinfinitesimalsacrossaplaneofspace-timeofahigherdimension.

InsuchageographyofDisomorphicspace-timethenumberofdimensionmatterstoobtaindifferentoperationsbutwearejustglidingonthesimplernotionsoftheduality¬Algebra=polynomialsvs.Analysis:integralsofinfinitesimals.

Yet soon the enormous extension of 'events' that happen between the 3 ∆±1 planes of T.œs as forms of entropicdevolutionorinformativeevolutionacross∆±i,convertedanalysisinabulkystiencemuchlargerthanthestudyofanST-singleplaneofgeometry,the2planesoftopologyandthepolynomialsof¬Algebra-whichroughlyspeakingareanapproximationtothemoresubtlemethodsoffindingdimensionalchangeproperofanalysis-evenifhumindsfoundfirst the unfocused polynomials and so we call today Taylor's formulae of multiple derivatives,approximations toPolynomials.

Since Derivatives & integrals often transcend planesrelating wholes and parts, studying change of complexorganicstructuresthroughitsinternalchangesinagesandform.

Polynomials arebetter suited forsimpler systems,Planesof socialherdsanddimensional volumesof space,witha'lineal'socialstructureofsimplegrowth.

So in principle Analysis was a sub-discipline of ¬Algebra. But as always happens,time increases the informativecomplexityof systemsand refines closer toabetter linguistic focuswith finerdetails the first stepsof themind. So¬AlgebrabecamewithAnalysismoreprecise,measuringdimensionalpolynomialsanditsfinitesteps.

Inanycasesuchhugesizeof∆-nalysisisaclearproofthatinmathematicsandphysicsthe∆STelementsofrealityarealsoitsunderlyingstructure.

Assuchsince∆-PlanesarethelessevidentcomponentsoftheUniverse,Analysistooklongtoappear,tillhumansdidnotdiscoveredmicroscopes to see thoseplanesbutwhilemathshasdealtwith the relativismofhuman individualplanes of existence, philosophy has yet to understand Leibniz's dictum upon discovery 'finitesimals', 1/n, mirrorreflections of the (in)finite whole, n: 'every point-monad is a world in itself'.Analysiswasalreadyembedded in theGreekPhilosophicalage, in thedisquisitionaboutUniversalsand Individuals.ThusabriefaccountofAnalysisinits3±1ages,throughitstime-generator:

Ps (youth:Greek age) < St:Maturity (calculus) > T (informative age:Analysis) >∆+1:emergence:Functionals (HilbertSpaces)<∆-1:Huminddeath:DigitalChipthought...

Whereasits3'Planes'are:∆-1:Derivatives>∫∆:integrals>∆+1differentialequations.

ThusAnalysisalsostudiesthescalesofthe5thdimensionanditsevolutionofpartsintowholes.

Derivativevs.integralintimeandspace.

Derivatives&integralscalculateratiosofchangeinthe3elementsofreality,space,time=changeanditsPlanes,∆STThe main error of ‘axiomatic’ analysis is to force continuity, infinitesimals and infinities, without considering thediscontinuouslimitsofderivativesandintegralsofpartsandwholesbetweenplanesofexistence.

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Leibniz vs. Newton already arguedwhat is an infinitesimal part, theunit of derivatives. The answer in a Universe,whichisafractalscalarsystemofstœpsofspace-time,S<T>S,orS=S=S,T=T=T,isaminimalquantainSCALE,TIMEorSPACE,whichbyvirtueof5Dmetric,SxT=K,willbea‘minimalunit’onthatequation(Min.SxMax.T),forthequantaofspace,Min.TxMax.Sforthequantaoffrequency,orminimalcyclicalbitofinformation.

Soforexampletheinverse,ƒ=1/Tofalongdurationintime,willbeitsshortquanta.Theinverseofapopulation,takenasawhole,1,1/nwillbeascalarspacequanta.Soallsystemshaveaninfinitesimal,whichisacut-offlimit,NOTreallyan‘infinitelysmall’(anerrorofthecontinuousdogmaoftheaxiomaticmethod),buta‘finitesimal’.Sowecanobtainthroughaderivativeafinitesimalunitoftime,spaceorscale,aminimalaction,aminimalpoint-volumeoraminimalcellularquanta.

Andforthatreasonwecanapproachfinitesimalswith‘differentials’,whichbecometheminimal‘linealsteps’ofalongcurve. And for that reason we can use ‘affine functions in space’ and lineal approximations, as by definition theshortest pathbetween twopoints is a line, and so in a discontinuousUniverse of Stœps, quantaareminimal linealunits;minimalfractalpointsofapopulationorawhole.

Itisacompleteoverhaulingofthedogmaticattemptstoprovethe‘hypothesisofthecontinuum’,butitisnotmyfaultthathumansaresooff-trackwiththerealityoftheUniverseasitis,notastheiregotriestoimpose.

Theminimalquantaof∆-1spaceandtime.ChainsofDimotionsexpressedaschainsofequations.

DerivativesaretheessentialquantitativeminimalactionabsorbedbyanyTœ,(ab.space-timeorganism).

Integralsthensumaminimalderivativequantainspaceoraminimalactionintimeforanybeinginexistence.

They are best for spatial growth of information as the 3 st-ages orst-ates of the being through itsworld cycle ofexistence, have discontinuities or changes of phase that cannot be integrated. Hence time sequences are betterstudiedwith existential algebra. Further on sequences can be comemore complex, if we consider tridimensionalactionsascombinationsofSandTstates,stt,tst,tss,ssswhichistheoriginamongotherthingsofthe3!=6variationsofspeciesaccordingtohierarchyonitsphysiologicalnetworksstudiedintrinity.

Furtheron,andthiswillconstantlybethelimitofmathematicalanalysisofreality,weshouldstressoncemorethanthe larger more complex actions of gender reproduction and social evolution are qualitative, taking place both inlongertimespansandlongerspatialsurfaces–hencebetterdescribedwithqualitativelogiclanguages.

So existential algebra studies in depth the qualitative connection of the a,e,i,o,u actions between Planes. Thatqualitativeanalysisatthelargerscaleandforsequencesthatimplychangesofstatecannotbeoverlookedandrequiretheapproachofexistentialalgebra.

Andcalculusisitsmathematical,analyticdevelopment(quantitativeunderstandingof1stsecondand3rdderivatives-extracting'1,2,3Dimotions'fromtheinvisiblegravitational,lightspace-timeorfeedingPlanes.

The limit of calculus however is bridgedby the fact that theother families of social operands (±, x÷, xa can reflectbetterthe‘socialre-product-iveactions’.

Soanotherdualitywaytodifferentiatealgebraicoperandsistoconsiderthatclassicpolynomialoperandsmirrorsocialcomplexactions,andcalculusoperandsreflectbetteralsoin‘trinityscales’(3firstderivativesorternaryintegrals)thesimplex actions. Each Operand specializes in one Dimotion (angular sine/cosine in Perception, ± in back and forthlocomotions, x÷ in complementaryand social evolution, log xª in reproduction)andOVERall of themanewPlaneofexistence isaccessedbyanalysis.Sooperandsguide themathematicalequationsthroughavitalprocessof stœps(stopsandsteps)andwillallowus to 'vitalize'equations,aswehavedonewithpointswith 'numericalparts'as theessenceofamathematicalT.œ

¬ÆthussetsalimitednumberoflogicpropositionsthatcanhappenwhenasystemorgroupofT.œsinteractthroughits5Ðimotions,asanpointofview,canpotentiallychangeitsstatebetweenthose5Dimotions,andthelimitsofits

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function of existence, such as the being can only exists without permanent disruption of its 'vital constants',(conservedenergy,angularandlinealmomenta-energyandmembrain).Allsystemscanexistwiththeinfinitecut-offlimitsofspace(membrain)andtime(death),whicharesetaspartofthefractalUniverse.Onlythewholeifpotentialorrealinexistencecanbetalkedoffasafunctionofinfinitybutnotperceived.

So a pointwill start any of the 5Ðimotions andweneed formal symbols to address theÐimotion of any being inexistence,andthestatesofswitchbetweenÐimotions.

DoesthebeingstopbeforeswitchingÐimotion?Ifsoitwouldsimpletoestablishthenforeachsequentialstepsofabeing:

∆¡Ð1,3,2,4,3....andsoonasasimple5letterprocessoftheactionsofabeing(whereas4Ðentropyreferstofeedingsnotdean,onlyinitfinalstatebeing'thatentropy'...Soweknowallsequentialofabeingendsin4Ð.

Canthenwerunasequentialforanyspeciesthroughitslifeasacompletedeterministicsequence?

Thereistherethesequenceofallsequences,theperfectworldcycle=lifesequence?

Questionsthoseforadvancedexistentialalgebra.

RECAP. Actions in timespace are the main finitesimal part of reality, its quantity of time or space. In pentalogicoperandsmirrorasactionsthe5Dvowels(a,e,i,o,u)thatdefinethefivedimotionsofexistence.Incalculustheyfirstextracttheminimaltimespacequantaoftheactionsofthebeing,integratingthemacrossapopulationofspaceoralengthoftime.Thusactionsvitalizetheoperandsofcalculus,relatingthemtoexistentialalgebra.

Thefantasyofthecontinuumsubstitutestherealityofdiscontinuoussum.

The age of calculus represents a great advance over simpler polynomial operands and statistical≈T-probabilisticmethodsofstudying,thepartsandwholesofasystem;anditsS=Tinteractionbetweenitsspatialformandtemporalmotions; as it differentiated by studying the change of each of the previous 5 Dimotional operands of algebra, allmodesofchange=timenaturaltoanysystemoftheUniverse–asthebestmathematicallanguagethatmirrorsthem.

Thisdualityof∆-scalesandS=Tdimotionsrepresentedinalgebrabynumericalsystemsandoperandswasstudiedinabulk manner with polynomials and S-tatistics=T-probabilistic methods that appeared first, using inverse scalaroperandsontheladderof3scalesofnumbers,thesum,theproductanddifferentexponentialsandinverserootsandlogarithms-whichreducedtherangeofexponentialstothe3scalesthatmattertotheUniverse,theLog10orsocialevolutionoftriads into3x3+Inewdecametricscales,theexponentialLnofmaximalgrowthordeath=decayandthelog2or‘powerset’ofallpartsofawhole.

Probability showed in time (frequency events) or space (statistical populations) that events were not perfect, butaccumulatederrorsinitsreproduction,shapinganormaldistributionoftenconvertedintoanentropicpopulationofdisconnectedindividualsformingherdsoreventsthatwitheredawayduetothoseerrorsofreproduction,whentryingtoreproducetheperfectmean;toformanidenticalstatistical‘boson’species,aperfectformthattranscendedscale–a feat of becoming a whole ∆+1 that was only achieved in Dirac’s distributions by immortal point-particles orresonancesthatamplifiedtheperfectinformationof∑finitesimal0’stobecomethewhole∆+1.

ButwhataboutalltherangeofvariationsbetweentheentropicGausscurveandtheDiracperfection?AllthedifferentspecificactionsofspeciesactinginholographicTs,ST,Stdimotionsthatwereneitherperfectresonancesofevolvingform,ordisaggregatedherdsofentropicaleatorymotions?

Thisrequiredtocalculatethefinitesimalformofchange=timeofeachspecificdimotionoftheUniversemirroredbyeachspecificoperand,andaddagivennumberof‘stœps’thatmeasuredtherepetitionofsuchminimaldimotionsofat.œinagivenlengthoftimeorvolumeofspacetogettheoutcomeofanexistentialdimotion.

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Thusbycalculatingeachfinitesimalandthenintegrateitthroughtheentireintervalinspace,timeorscaleinwhichitwasperformed,calculusgetsamoreaccuratedepictionofthewholeevent,nolongeranentropicaleatorychange,butapurposefulaction.Andbecausebothderivativesandintegralscouldbedoneinspace,timeorscale,alltherangeofpossible variations, could be studied, regardless of complexity, from changes of time, to variations in spacerepresentedas0’actions=,δ->0’;happeninginminimaltime(Lagrangians)…

Thedetailandspecificationreachedbythefinitesimalofanspecificdimotion=operandoftime=change;eitherangularperception – sin/cos; social evolution, ±; re=production or locomotion as reproduction of information, x÷; entropicdecayandgrowth,ex,madecalculusthequeenofalloperands,almostamagictool,asitsfoundationswereignored.

Mathematicians unable to understand time=change invented the concept of limit and the continuum to form apedanticscaffoldingofaxiomatictruthswhichhadnothingtodowiththerealityofcalculus.Sowehavetoclarifythatconcept.Asallisperceptionacontinuumismerelyasumofdiscontinuousstœpsinwhichpartofthewholeprocessishiddenbytheselectionofinformationbyamentalspace.Sothefantasyofthe‘continuum’canbereachedwhenthedetailofeachstœpisignored,andweobtainonlythemeasureofarelativeinfinity,∝,ofsuchstepssummonedup,∫,to calculate thewhole change in the long time period, T, or total domain studied. And that is fine, as long as weunderstandthatindetailthecontinuumismadeofasumofdiscontinuousstœpsofchange,wecallfinitesimals.

Inaprocessofcalculusthusthesumis‘smoothed’,eliminatingthe‘stop’states,byreducingtotheminimal0’thesizeor rate of change, so they cannot be seen in detail but can be ‘integrated’ in a long period of time, obtaining ameaningful result for the ∆0 scale of the experimenter, which only went down to ∆-1 to be able to calculatewithaccuracy∑∆-1=∆º-theemergenceofchangeinthesizescaleoftheexperimenter.

Sointhesamewaywecarelittleforthe∆-1<∆østageoftheseminalreproductivefetus–tothepointhumanshavetherightto‘murder’it,aslongasitisnotanemergentchild,theobservercareslittleforthefinitesimal,¡ndifferenttohim.Andyetimprescindibleforthewholetobe=come.

This ‘goal oriented’ view of Nature is common tomost beings; hence calculus became amuch bettermethod thatdiscrete probabilities of change. Since the cumbersome reality of each stop and step of a dimotion that had to beanalyzedineachindividualstœpasithappensinthe∆-1realitycouldbesimplifiedtofacilitatethestudyofchangesthroughalongerperiodoftime.

Socalculuswasenormouslyadvantageoustocalculatelongstretchesoftimeperiodsandspacevolumesatthehumanscalewithminimallossofdetail,providingthatthephilosophyofrealitymadeofstopsandstepswasnotforgotten.Orelsewewouldenter intoan idealizationof realityconfusingthemathematicalmirrorwiththeUniverseas it is. Butthatwaspreciselywhat ithappened.Sincetheessentialerrorofallminds istoconfusetheUniversetheydon’t fullyperceivewiththemindthatreducesit.

So the wrong creationist solution was taken as truth, obscuring our understanding of the principles of reality. Sopunditsofcalculusignoredthenatureoffinitesimalsofchange,anditssmallstopandmotion,S<T>Sbeats.

Sowhenyoustudycalculustheyapologizeforrealityasitis-aseriesofthinstepsandstops,ofsmallfinitesimalstoadd. And consider discontinuity – the real broken space-time an approximation to idealist simplification of acontinuousgraphthatloosesinformationabouteachfinitesimalstœp.Insteadtheartifact,thecontinuousfunctionisconsideredthetruthandrealityamethodtoreachtheidealtruth.

AsthegraphofDescartes,theartifact,becamethenatureofspaceandtime,thesubstancesofwhichweareallmade.Andsuddenlyhumanswerenotlongermadeofcyclicaltimeandfractalspace,brokenbythelimitofamembrain,butGod had drawn a Cartesian graph below as background absolute Newtonian lineal spacetime – something mostscientists still believedue to creationistmathematics,and theegocyparadox that confuses themind languagewithrealityitself;thesimplificationofcalculusthatsmoothedthefinitesimalsbecamereality.

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Factishfinitesimalsarereal,andneverreacha0thatdoesn’texistashorrorvacuumworks,andsomathematiciansshouldteachrealityandthenacknowledgethattheexperimentallanguageofmathematicssimplifiesthatreality;andthatisOkforpracticalreasons.Nottheotherwayaround.

WhenIwasawonderkidbeforehighschoolIhadamathprofessorwhocameonedayworriedaboutthefacthehadtoexplainusthe‘limit’ofh->0andthoughthisstudentswouldn’tunderstand.Ofcoursestudentsneverunderstandadotofit,justmemorizeandsaytheyunderstand.ButIunderstooditwasallwrong.Thusheendedupthrowingmeoutoftheclass(:whenItoldhim,thelimitcanneverreach0.SinceIfnothingexists,thereisnowaytodefineit,hencecalculateit.Nothingmightbeanything.Onlyifsomethingisnolongerzerobutleavesatracewecandefineit.SoItoldhimundefined things should not be the realmof an ‘exact’ science.That blewhim away and having no answer heresortedtoauthority(:SoIwaskickedoutasIwouldn’teveryieldtoauthoritybutreason.Itwasmyfirstrealizationthatwhenhumansdon’tunderstandsomethingorsomethingiswrongbuttheywantitanywaytheyputupadogma,apostulate,apedanticdefinitionor‘self-evidentaxiom’(:

Factisifh->0makesanequationundefined,hmuststopbeforeitloosesthequalityofthewhole–hencehisthelastatom,lastcell,lastfrequency,lasttemperaturevibration.

Ashejustignoredmewiththepretentiousauthoritasofanoldmanwhodoesn’tseeanelephantinthedrawingofahat, lepetitprincehastosearchforhimself.Which iswhat Ihavealwaysdone. Sothequestion iswheretostopafinitesimalportion.Andtheanswerisaswesaid,beforeitbecomesrandom,¡ndifferent.

Forexamplee isdefinedas (1+1/n)n.Yet ifn is∞, theparenthesis is1+0 .0000000…up to infinity.And thoseareundefinedlimitsofreality.

Physicists at least acknowledge physical laws are idealizations of reality and that is Ok (even if they deny non-mathematicalpropertiesthat‘cannotbemeasuredastgeorganicandsentientpropertiesofparticles,theunitsoflife).Sotheiregocyparadoxisabitdifferentfromthatofmathematicianswithitsaxiomatictruths.

StillmoreprofoundmathematiciansdounderstandacontinuousfunctionasoneinwhichS=Thappens,sotheXandYcoordinatesdonotmakeverydifferentchanges,andstœpsof‘present’canbeputoneafteranother.OrasLeonardosaid:“The instantdoesnothavetime;time ismadefromthemovementofthe instant. Inrivers, thewaterthatyoutouchisthelastofwhathaspassed,andthefirstofthatwhichcomes.Sowithtimepresent.Observethelight.Blinkyoureyeandlookatitagain.Thatwhichyouseewasnotthereatfirst,andthatwhichwasthereisnomore.”

Asthesecond,theglimpseofaneyeisthequantaofhumanpresent,thereisalwaysforalltimespacesaquantaweshallcallaninstantoftime-presentorafinitesimalofpopulations.

Andwhen the ratioof change in time, itsquanta, isbalanced to theminimal formof space it change, so inanS=Tgrapheachpoint-position-stopofthemoving-stepfunctionisclose,witha‘tangentangle,s/t’thatcanbesmoothedinaseries…thesumofdiscontinuousstœpsofadimotion,canbemeasuredasacontinuous,largerstœpofspace-timeofalarger∆+1scale.

Thuscontinuityisnotthelimitinwhichh->0,butthelimitinwhichS≈T....andhenceaninstantofpresentchangethatisharmoniousbetweentheSandTcomponentsofthebeingthatdoesn’tchangeinternallybutonlyexternally(asS=Tremainsunchanged)takesplace.

Thishasdeepimplications,asitimpliesthatthesystemiscontinuousbecauseitlasts,anditlastsbecauseits‘actions’tendtozerointernalchange,becomingconservedcyclesofenergyfortheinnerstructureofthebeing.Changesthusarealways ‘returning tobalance’.Andwhenchange isextreme,as inan internalxexternalTT-entropicchange, thesystemcollapses, the ‘tangent’ tendstozeroor infinite intheY(s)orX(t)axisandthe ‘functionends’.Which impliesthatcalculusworksontheSt,STandTsdimotionsof locomotion, informationandenergy=reproduction,NOTonSS-formwithnochangeorTT-absolutechangewithnoform.

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Forexample, the firstdimotion tocalculateaschangewas thespace traversedbya systemwhich indetail is justaseriesofsteps,whichaddtothespacetraversed,λ(s)xƒ(t)=S.Thisgiveusalotofdetailinnumericalapproximationsifwefurtherbreakittoasumofsteps,∑λ(s¡)x∑ƒ(te)=S.

Sothemore informationwewant, themoredetailedthediscontinuitybecomes, tillwecan indeedaddeach ‘fractalstep’withalltheinformation,lengthandtimedurationofeachstep.Butwedon’twantthatmuchinformation,sowecansimplifywith‘statisticalmeans’sinceaswehaveseeninstatistics,thelawofgreatnumbersbringameanforeachstœpandthat isthe justificationofsimplifying intocontinuity–NOTthenonexperimental idealist,mind-generated,creationisthypothesisofthecontinuum.

So it is finetobehumbleandmarvelatthefactwecan‘transcend’the∆-1finitesimalscale, intoparametersofthewholebymaking ¡ndifferent the informationat∆-1 throughstatisticalmethods thataverageeachstep,erasing theuneededinformationoneachstop(whenthemotiontouchesthefloor,orthemoverlooksandgathersinformation).Butwhen trying tounderstandingparadoxes suchas the speedof light constancy, as eachelectronemits light inarelative entangled stop position to the perceiver that measures its speed, hence in a stop distance, (Lorentztransformations) it is good to know that there is no magic on it. And the ‘idealized’ form is the mathematicaltransformation thateliminates thestopstateof theelectron foracontinuousmotionwedoNOTobserve inNature(theelectronisalwaysobservedasastopparticlewhenemittinglight,andmovesinzig-zagasifitwereallthetimecalculatingitstrajectoryinastopposition).

The methods of calculus are awesome and once we realize the cruelty of the Universe that cares nothing for thedifferences between the ¡ndifferent finitesimals of amass-group herded and ruled by the ∆0 larger scale bymass-methods,includinghumansasweshowinthemodelsofhistoryandeconomicsherdedtodaybyfinancierswithcreditratiosestablishedbyanonymousbigdatacomputers,herdedbypoliticianswithequalizinglawsforeach‘socialclass’,herdedbymilitaryassoldiersornumbersofaconcentrationcamp…andpastoverthethoughtthateachatommightbeagalaxyofitsown,anddaretoexplodeitinanacceleratorwithotheratom,whichmightonthemicroscopicscaleprovoke thebiggestgenocideof infinite relativeplanets,andconnectwith theTaoand feel just tobe,∆@st,whichmattersnothingandworshipthe¡ogicofGoÐoGtheinverseDimotionsofexistence,wecanrestinpeace,R.I.P.asdustofspacetimethatdustshallbecome.Andsocalculus…

Themethods then arewell knownandwe cannot butmake a few comments beyond the philosophy of its science,which is themainpurposeof5Dmathematics inthissimplifiedtexts. We justneedtocalculatea finitesimal,whichwasthefirstthingdiscoveredinthe‘1stageofcalculus’andsummonthemuptogetthewhole,whichwasfirstdoneasinreality,withtheexhaustionmethodbytheGreeks.

Yetthefinitesimal0’initselfisimportantasitgiveusinformationabouttherateofchangewithasinglenumber,thetangenttothecurve.Sincewecanapplytocurvatureinspaceitssynonymousintime–speed-whenwerealizethats=Tmeansintermsofcurvesthecurve’stangentrepresentingthe‘speedofchange’S/Tofthefunction.FurtheronasSxT=C,meansthatthesmallerspaceisfasterintimecycles,themorecurvedacycleisthesmalleritbecomesandthefasteritmoves(vortexequation,accelerationprincipleinRelativityoveracurvedspace).Sowealsorealize,thesecondtangentofthecurveistheaccelerationofchange,y”,increasedasthesystemcurvesfurther.

ManymorewondersthenkeptappearingincalculusasweplaywithS=T,SxT=C5Metriclaws,andseethemthroughcalculus;speciallyconsideringthatmostcurvesarethefunctionsofexistenceofaneventbetweenits0’pointsofinitialandfinalconditions.Andsowhatweshalldointhisbriefintroductionto5Dcalculusistohighlightfortheseeminglymost simple equations of calculus the underlying insights they provide on the processes of time change of the 5DImotionsoftheUniverse.

Weshalldo it inahistoric,easier tounderstandnarration,as indeed, the first thinga languagesees is ‘space’,and‘realityasItis’initssimplerterms.Sothefirstageofcalculuswasthatofthesearchforfinitesimalsbothinpraxisandmeaning,wrongly resolved in favorof the conceptofabsolute zeroand limit, thenmathematicianseraseduneeded

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informationonthosefinitesimalstepsandstopsestablishingthemethodoftangentsandcontinoussums,∫,andthenapplied those functions of breaking a whole into parts to re-build it to specification to the different dimotionaloperandsfromthesintotheexponentialofmaximalchange,marvelingthatthelimitofchangeperunitoffrequencytimewasthechangeofthewhole,thatistheequationofdeathinwhichthewholedissolvesintoitspartsinasinglequantaof time, thenegativeexponential so everpervading in studiesof entropicdeath.Of course theyunderstoodnoneofit–theystilldon’t,butthemethodworkedtomirrorthedimotionsofreality,whichtheyneitherunderstoodinan orderly manner. So calculus became magic. And as it got more complex, as the Universe does by repetitions,transformations,scalingintomorecomplex‘packages’ofparts,tothepointthatIcouldsay,asEinsteinputit–Idon’tunderstandrelativitysincemathematiciansgotintoit(:Idon’tunderstandcalculussincemathematiciansgottoit:).Thatiscalculustodayissocomplicatedinitsmorepowerfulanddetailedanalysisthatonlyacomputercancalculateits results. Whichultimately try toanticipate the futureofchange inasynopticmannerby transformingsequentialpatternsofchange intoparallelsimultaneousspatialcomponents(multiplevariables inPDEs),happening inmultiplepointsofviewatthesametime,togatherintoawholeresult,whichisstill impossibleforthemostintelligent, liquidstatesofmultiplechanges(Navierstokesequations)unlessyoutrickitwithafasterdigitalmind,whichwilldothosecalculusfortheslowerhumind,jumpingaswealwaysdopasttheintermediatesequentialstepsfrombeginningtoend.

Theproperconceptforfinitesimals.Reproductiveunitofchange.

Thegreatadvanceofcalculusintheunderstandingof∆STchangesistheconceptofafinitesimalofchange,h,whichinthe symmetrybetween scales, populations in spaceand time frequencieshas the same role: to increasea ‘seminal’unit, the system (or decreate in inverse fashion), becoming theUnit of reproductionof an∆st system, at a point inwhichs=T.

IntheUniversethefundamentalformofchangehappenswhenS=T,thefunctionofpresenttimefindsabalanceandsymmetryinscal,eformandmotionthattriggersthereproductionofthesytemandits3parameters.

Latterwewill study, the simplest case of polynomial reproduction,whereas an X2 has as unit of change, 2x;whichmeans the square grows through both sides reproducing its form, in ‘s=t’ balance, in a manner that the squarepreservesitsform.

Themethodofcalculus.

HowdifferentialequationsshowusthedifferentactionsoftheUniverse?

Thecorrespondencetoestablish isbetweenthe final result, theåction,andthe finitesimalquantas, thesystemhasabsorbedtoperformtheaction,∫∂x,suchas:å=∫∂x,whereasxisaquantaoftimeorspaceusedby∆ø,throughtheaction,åtoperformaneventofacceleration,e-nergyfeeding,information,offspringreproductionoruniversalsocialevolution.

Itisthenwhenwecanestablishhowcalculusoperationsareperformedtoachieveeachtypeofactions.

Firstwenoticethatthespacebetweentheactorandtheobservablequantaisrelative,soeveniftherearemultiple∆-planes between them the actor will treat the quanta as a direct finitesimal, pixel, bit, or bite which it then willintegratewithapolynomialderivativeorsinusoidalfunctionthatreflectsthechangesproduced.

Wewillconsiderinthisintroductorycourseonlyafewofthefinitesimal∫∂actionswherethespacestateisprovidedbytheintegralandthe∂finitesimalactionbythederivative.

Derivativespointouttothemainconsequenceofthesumofthoseactionsinanybeinginexistence,namelythefactthatitssumstendtofavorgrowthofinformationonthebeingandthensignalthe3st-agesand/orst-atesofthebeingthroughitsworldcycleofexistence,whichinitssimplestphysicalequationsistheoriginof...themaximalandminimalpointsofawell-behavedfunction.

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So to establish the action - the final result – we have to isolate the finitesimal quanta/moment of spacetime thesystemhasabsorbedtoperformtheaction,∂x,andintegratethemoverasurfaceofspaceoralengthoftime,suchas:å= ∫∂x,whereasx isamoment/quantaoftimeorspaceused inrepeatedfrequenciesorquantities,∫∂x,by∆ø,through the action, å to perform an event of acceleration, e-nergy feeding, information, offspring reproduction oruniversalsocialevolution.

Wecanthenestablishwhichoperandisbestsuittoperformeachtypeofactions.I.e.theactionofreproduction,mostoftenisexpressedforquantitativesimplephysicalsystemsthroughtheoperationofre=product-ion.

We ascribe each operand to a single dimotion, but they are 'once more' entangled operations, which besides itspreferentialDimotion,doparticipateofalltheothers-rememberlanguagesasmirrorsofrealityhavealsothesameentangledpropertiesofthepentalogic,¬∆@STuniverse,lookingatallitselements.Soweshallnowanalyzetheminmoredepth.

Weestablishdirect relationshipsof operands andactions- taking into account that for eachoperandwemust alsodistinguishthedualitiesof'space-likeintegralofvolumesanditsderivativequanta'and'time-likemomentsofmotionsand its frequencysumtocompleteao-sumworldcycle'.Andtoachievethosebalanced 0’ sums finallyweneedtodefineinverseoperationsforallactions.Sowedepartfromaceterisparibusanalysisandsearchforafinitesimal,andthenwemust study how theymerge and entangle in space and time. This is done generally speaking,with a firstpartialderivativeinspaceortime(PDE)definesthosedimotionsonlyasSorT,whiletheintegralofdoublederivativesputbothprocessestogether,tofindthewholeaction:å(st)=∫∫dsdt

Howeverasallplanesofexistencehavediscontinuitiesbeyonditsminimalquantaandlargerwhole,analysisthroughmultiplePlanesbeyondthoseof∆ø<<∑∆-1entropicdeath,tendtobedistorted.

Stilltheycanbestudiedwithpowerpolynomialsandfurtherapproached(Taylorseries)with∫∂operatorsthatcrossplanesofexistenceforcertainhighlysymmetricactionsacross.

Butagainitisbesttouseexistentialalgebra,asthefundamentallimitofthemathematicallanguageisoneofsyntheticunderstandingoftheorganicvitallawsoftheUniverse,reasonwhytheoriesthatareonlymathematicalinthelargestscalesanddoNOTunderstandthattherearenotequationsthatgotoinfinity,asallhavealimitthatbringsachangeofstate,suchasthebig-bangtheoryoftheuniverse,arefalse.

Reasonwhysystemsdohavebesidesspatialmentalspacesof‘calculus’,along-timerangelanguageoflogicnaturetoexpressthevitalgamesofworldcycles,andthis isthefunctionofexistentialalgebra,westudyfirsttothenconsiderthebasicsofCalculus.

Finallyinthisbriefintroductiontonoticea‘revealing’factoftheinversionbetweenfinitesimalsandintegrals.Astheabsolutearrowoftime-futureissocialevolutionofpartsintowholes,whileafunctionhasonlyaderivative,thatis,allmoleculescanbereducedtoasetofatoms;alllivingbeingstothecell;theoppositeisnottruth:creationofcomplexfuturesismultiple.SoanintegralhasaCvariationalconstantandadifferentialequationmultiplesolutions:thefutureisopen,thepastisonlyone.

RECAP.Analysisstudiesthefinitesimalquantaoftime,spaceandscale;NO∞initssmallness–anerrorofthemindPx.searchingforcontinuity.Atruephilosophyofcalculusthusdealswiththemeaningof‘finitesimals’inspace,timeand scale, as a first ‘seed’ of a ‘clone species’ multiplies, creating the regularities of ‘social numbers’ that make‘analysis’ towork its ‘magic. Integrals in inverse fashion act after ‘calculating’ thisminimal point, often as a ‘linealshorteststep’(differentials),toreachthefinal‘wholevalue’ofthesystem.Thebeautyofthefieldrevealingthenatureofdimotionsanditswideapplications,thuswillrequireanentireIIbookon5Dmathematics,whichshouldr=evolvethediscipline.

Thusonlytheintegralandderivativecanstudyallthosedimotionsofspace-time,hencetheyarethekingandqueenoftheoperatorsof¬Algebra,reasonwhyanalysisissoextended.

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Below, Analysis’ multiple perspectives on the 5 Dimotions= functions of existence & 5 simultaneous structuralelements,¬∆@S≈T,thatconformallsystemsintimeandspace.So3±D¡pointsofview(trinityorpentalogic)findsahighertruth&appliestoalllanguagesmirroringrealityasanalysisdoes:

S-topology:Analysisisusedtostudy(left)structurallytheroleofthe3elementsofatopologicspatialsuperorganism:Its membrane’s curvature and tangential value (line integrals), its vital space (surfaceintegrals) and singularity(derivatives).

∆-Planes: Its inverseoperandsstudy5DPlanes:derivativemeasure thevalueof1of itsinfinitesimal 'cells'. Integralsgiveusitsinternalvolumeofspatialenergy.Whiledoublederivativespeersdown2∆±1planes

@:Weextractinformationonitscentral@-singularity,whichcommandsthelinealmotionofthewholesystem.

Time-cycles:it canmodel thestandingpoints,maximalandminimal,which signal the changesbetweenages,where thederivatives,becomenull,as the 'worldcycleofexistence'changes its'phase'.

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THEDIFFERENTGEOMETRYANDMETHODSDEPENDINGONDIMOTIONS.

BecausecalculusisaboutDimotions,andmathematicsmainlyaboutspace,itfollowsthatcalculusstudiesmostlythe2dimotionswith a larger content of T, TT-entropy and Ts-locomotion.While it has also an important role due to itsunderstandingofS=T,finitesimalchange,inthepresent-reproductivegrowth.

ItishoweveroffarlesserusetostudySSandSt,changesininformation,thoughsinusoidaloperandsgiveusinsightsonit.Itisthenobviousthatforafuture5Dresearcher,ifthereiseveranyonebesidesthiswriter,theobservanceofthestructureof anequationof calculus and its geometrywill give insights about the typeof dimotion it studies; sincetherearesomebasicdifferencesbetweenthem:

Ts-locomotion:Acleardifferencehappensat firstsightbetweenasimpleranalysisof locomotion,whichconcernsasinglepoint-like formthroughspace insequential, lineal time,as there isno internaldimotion,and theT.œcanbetreatedfromthepointofviewofitsmind-wholesingularity(soforexampleamovingrock,regardlessofrotationsinitslinealmotioncanbetreatedasthemotionofitsgravitationalcenter)

-Entropicmotionontheotherhandisadualmotion, internalandexternaltothebeing.Entropicmotions thenareeasier treated ifwe consider the point of explosion or death as a fixed point (which is often the case as entropyhappensafterdeath,whichleavesthewholesystemunchangedinmotion).Thenweshallobservethattheintegralofallthemotionsoftheentropicsystemremainszero,becausethenegativesidesoftheframeofreferencecancelthosedimotions in the positive side, which is essentially the meaning of death, and so the fundamental change of anentropicmotionhappensinthevolumeofspaceofthesystemwhichiswheretheinternaldimotionofthebeingendsup,transformedintoanexternaldimotion.

- Reproductive growth coincides with entropic motion in the factor of expansion in space. However reproductivegrowth is a real growth that fills space, NOT merely expands the distribution of its ∆-1 elementary parts on thebackground ∆-¡ space. So the differences with entropic motion are easy to spot: Reproductive growth does NOTchange the density of form in the vital space it fills. Entropicmotion becomes rarefied in its dwindling density, abubblethatexpandsandthendissolves.Reproductivegrowthisfarslower,unlessithappensinatrulyfriendlydenseinenergyplacentalworldwhere ithappens inageometric2x factorofmaximalgrowth;buteven then itwill seemslowerthanabig-bangifthespeedofdeathisfast.Andasdeathisacollapseinasinglequantaoftime,twoscalesdown,∆º«∆-2,almostallprocessofdeathanddecayexpandfasterinspace.

Whataboutsystemsofmultipletime-changes,‘PDEs’sotospeaknotinitshowbutwhyexistentialprocesses?

Combined reproductiveandentropicmotion.There is themost important casewhenweobserveadual sequentialprocess, inwhich first thedeathof the systemdoesnot seemtochange in space,asgrowth is internal through theradiationofthe‘predator’species.

Thishappens incosmologicalbig-bangs(betadecays,quasarbig-bangs,novasandthehypotheticalfalsecosmicbig-bang, studied on physical papers), when the death of the system is due to the birth of a denser form of matter(strangelets in silly-nilly planets like Earth that do accelerator experiments or star novas, topquark quasars in BCBstars=black holes. A similar processes, whereas death is parallel to the growth of the predator species, inside out(organicdeath).Inallthosecases;initsfirsttimesequence,thesystembecomeslessmotileandoftenshrinksinsize,as it is being carved inside out, and then in the second phase it explodes in a single quanta of time, as the faster,smallerformorherdofformsspreadsonalargerspace.

Inpraxis thenyoucanactaspartialdifferential equationsdo, justperforming two sequential calculusbecauseandthatisthebeautyoftheUniversethatfacilitatesitscomprehensionaswehaverepeatedlystatedinallourparagraphson existential time, at the level of actions, we follow a series of finitesimal steps which seem to be continuous(conceptsclarifiedinthenextparagraphs).SotheduadimotionofanewformfeedinginaT.œsbody,tothenexplodeitanexpand,STx»SSy«TTx+Tsywrittenasasequenceofexistentialalgebra(abodySTxfeedsanewspecies in itsseedform,SSy…thatwillwalkawayTsyastheformcollapsesinentropyTTx);becomesaseriesofpartialderivatives.

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Inthedeepestsensethisistheexistentialwhyofthemethodsofcalculusofpartialderivativesandthekeydifferenceinthe concepts of continuity in classic calculus and ‘stœps’ of discontinuity, in 5D calculus, bridgedwith the commonconceptofasmoothtransitionthrou

gh finitesimalchanges,which in realityarediscrete (thebodycorrupts indiscretestepseven ifas thebacteriagrowexpenentiallyeachstepislarger),butfromahigherpointofview¡ndifferenttothedetailcanbecalculatedaslongasitissmooth.

It is also the reasonwhyboth, reproductive andentropicmotions canbedescribedby e±x functionswhich are themaximal‘rateofchange’(asthederivativeisequaltothefunction,andsince∂x≤x,ismaximal).

Theefunctionhassomultiplemeaningspreciselybecauseofthe‘horrorvacuum’andthirstforexistenceofitsspatialfractalpoints.So itcanalsobeused in its imaginaryform,whichasthenameindicates isrelatedtothecreationof‘mentalSS=§paces’,initsexiform,connectedtothesinusoidalfunctions,inwhichwecanobserve,asinACcurrents,abackandforthmotion=translationinspace,coupledwitharotationalperceptivemotion.Inthoserotationalmotions,thecomplexplaneandexponentialfunctionissousefulbecausewhatperceptionISreallydoingis1)collapsingatthefastestpossibleratetheUniverse intothefinitesimalmind-mappingofthepoint(hencetheexfunction involved); itdoessowithaclearbiasinfavorofthelengthdimensionofthefocusedperception,whilethei-dimensionofheightisgreatlycompressedtofitthesystem(hencetheusefulnessofaframeofreferencewhereY=√x;andfinallyitdoessoina periodic pattern, scanning back and forth the same worldcycles to convert them into mental space; hence therecursiveuseof±sin,cosinefunctionsinvolved.

Letusconsiderthisessentialequivalenceofmathematicsinmoredetail.

Connectionbetweenexponentialsandsinusoidalfunctions:derivativesasanglesofperception.

One very realized role of a derivative as a tangential division of the height in the dimension of information anddistance-linealmotiontotheobserverisameasureoftheangleofthebeing,whichrecedesinspacetimetillreachingthenon-perceptionasarelativefinitesimaloutoftheterritorialmind-worldoftheobserver,whichconnectsdirectlyderivativeswiththe1Dfirstdimotionofperceptiveexistence.Thebeingmightstillbeofcertainsizebutasafractalpointhehasrecededinthemental-spaceoftheworldoftheperceiver.

Thefirst'timespace'numbers:Polygonsasrootofunity

By their very nature, as numbers that probe planes of the fifth dimension, exponentials are closely related to thecomplexplane.Letusconsideronlyonecase,deMoivrenumbers,whichareanycomplexnumberthatgives1whenraisedtosomepositiveintegerpowern:

Annthrootofunity,wherenisapositiveinteger(i.e.n=1, 2, 3, …),isanumberzsatisfyingtheequation:

Theyarecomplexnumbers (including thenumber1, and thenumber –1 ifn is even,whichare complexwitha 0’imaginarypart),andinthiscase,thenthrootsofunityare:

This formula shows that on the complex plane the nth roots of unity are at theverticesofa regularn-sidedpolygon inscribed in theunitcircle,withonevertexat1.Thisgeometricfactaccountsfortheterm"cyclotomic"incyclotomicpolynomial;itisfromtheGreekroots"cyclo"(circle)plus"tomos"(cut,divide).

Euler'sformula,eix=cosx+sinixwhichisvalidforallrealx,canbeusedtoputtheformula for the nth roots of unity into the form: e2πi k/n 0≤k<n. Which is aprimitiventh-root if and only if the fractionk/nis in lowest terms, i.e.thatkandnarecoprime.

We find therefore the first timespace numbers, in the roots of unity. And as suchtheywillbecome'thecreativeprocess'ofdividingthe'whole',1,intocyclical'ticsof

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time'ofincreasinglyfasterfrequency,inaprogression,fork=1,2,…,n−1,whichwillgeneratethefrequenciesofallclocksoftime,tillreachingthecircle,whichcanthenbeconsideredinbidimensionalspacetime,the'Infiniteclock,ofinfinitesimaltimetics'.Dohavethoseinfinitesimalticksa'limit'asallrelativeinfinitesdo?Inphysicsitisbelievedtheminimaltickwillbe10ˆ43orPlanck'stime,whichthereforewouldbecomethelimitof'points'thatformatimeclock.

Anotherfundamentalthemebeingthereasonswhythe'clock'iscounterclockwiseinitsdirection,asitwillalsobeitscomplexrepresentation in4Drelativity theory.Thereasonbeingthat inPlanesof the fifthdimension,aswecreatenewdimensions from the lower planeswithmore entropy, the emergent dimension 'sucks' part of the entropy oflinealspaceofitslowerdimensions,'contracting'itasitrisesonheight.I.e.apicircleismadeof3'curved'diameters(withopenholesbetweenthem),butitdoesnotmeasure3but1inthelengthdimension.

Italsomeansweareaddinganewtimedimension,withanegativeentropicpropertyforthe'dimensionofrealspace',whichthereforecanbewrittenalsowiththenumberofentropy,e,

4thscalarDimotionsofEntropy

A theme them of profound importance and depth of meaning is the relationship between the sine and cosinefunctions thatallowanangularperceptionof thewholeandtheexponential function that reduces thewhole to itsdecaying elements (Euler's formula). We could say then that the whole is 'split' between the entropic negativeexponentialpartthatisdischarged,andthesinusoidal,informativeelementsthatareabsorbedbythemathematicalmirrormapping.

Itisinterestingtonotetheconnection,whichoccursbetweentheexponentialandtrigonometricfunctionswhenweturn to the complex domain, through series, since both functions can be approached by exponential series.If we

replace z by iz,we get: Grouping everywhere the termswithout themultiplieriandthetermswithmultiplieri,wehave:

Euler’sformulassolvedforcoszandsinz,get:

Aswesaidthekeyinsightof5Dinpowerseriesistheunderstandingofthemasaseriesof‘sequentialsteps’inthe∇∆dualscalargrowthanddiminutionscalesofthefifthdimension,whereasthe±summandelementrepresentsastepintime-change,a‘period’ofafrequencyofgrowthanddiminution,butinthecaseoftheuseofanIfactoritcreatesasinusoidal processof a repetitiveworldcycleof perception, short of anopening and closening glimpseon reality, abackandforthmotioninanACcurrent,alifeanddeathcycleinafasttimequantumparticle.Itisanessentialinsighttoresolvethewhysofallthosehowsofmathematicalphysics.

In the complex plane, 1D (sin/cos)combine to represent a full worldcycles, interesting enough through the eix, 4Dexponentialdecay function.This ispossiblebecauseheexponential functionswitchesbetweengrowthandnegativedecrease, as the sine and cosine switch between informative and energetic perception; but the sine function, theinformativeDimotion grows less, as it happens in nature,whereheight and informationhas less energy, and so inparametersofsizeandvolumemattersless.

2newqualitiesmake interesting tocast trigonometric functions in termsof the functionofentropy:weareaddingboth cosine and sine 'on and off' SMH for a value of 1, the total value of aworld cycle, sowe can use frequencyequations(asinelectromagnetism)torepresentthisexponentialworldcycle.Andwesuperposeboth,thefunctionof'space-form'thesineandtime-motion-linealdistance,thecosine,toobserveaharmonicbalanceasthefunctiongoesupanddownbutneverpassesbeyondthevalueofthewhole.

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Thecomplexplaneisrealbecausethecosisrelatedtolinealmotionandthesintoperceptiveheight,whoseactioninstopmodecanbeseenasanegativeslowdownofmotionforacontinuousview(S=-T);asinrelativity(-cT).Butthedeepest level of understanding of those functions and equivalences happens when we carry the worldcycle ofexist¡encetothecomplexplane.

Dualityoncalculus:∆-Newtonv.SóTLeibniz

FinallytonoticetheextraordinaryfactthattheST-cartesiangraphandthecomplexplanecoincideintherootofunity,whichessentiallydividesthebeingintoitsinternalandouterparts.Onlythenthemembraincanassesswithaccuracybothrealitiesinobjectiveterms.Butitwillperceivethemwithdifferent‘volume’ofinformation.

Forthemembraintheexternalworldwillbemeasuredinthecomplexplane,witha lesserdimensionofheightthatwillmaketheworld‘flat’initsperceivedgeometry,astheY(i)planewillbethe√rootoftheX-plane.

Internally though for the o-singularitywhichONLY observes the root of unity circlewith an equivalent height andwidth,this‘verbal,temporalmind’observingthe‘spatialbiasedmembrain’NOTtheuniverse,thathasalreadymadeaselectionofinformation;inthesamemanneryourinternalverbaltemporalthoughtonthe0-1unittemporalsphereor the quantum 0-1 particle on the biased information provided by the harmonic spherics of his electronic eye(rememberallisthesame,allishomologyinfunctionevenifitchangesinform);willthinktheUniverseis‘perfectlyregular’andfavoringitsbiaseddimotionoflength,NOTrealizingoftheequalimportantoftheflatteneddimotionofheight-information.

Essentially all systems have 2 brains, the spatial membrain and the temporal singularity at the center of the 0-1temporal sphere. The spatial membrain already bias reality and as the singularity of time only sees the spatialmembrain itwillactupon it,as its ‘territory’ inwhichtoenact its5dimotionsofexistence,qualifyingrealityas themembrainhasalreadydone.Andthatisfinebecausesingularitiesareselfishself-centeredknotsorelsetheywillbepreys of other self-centered knots. But that makes so difficult objective knowledge as we shall see in the bias ofhuminds. What does then the humind brain observe? The distortion we know exists considering the membrainhomunculusforwhichhandsensations(enzyman’sactions)andmouth(entropicfeedingandsocialcommunication)occupymostofthespacewhilelegs-locomotionregardlessofphysicistsegomatternothing.

Locomotion as reproduction and death. This usefulness of the derivative ofmaximal rate has a deep philosophicalconsequenceofthemanyinsightsaproperunderstandingofthesymmetriesbetweenexistentialalgebraandcalculusmethodsprovidestothe5Dresearcher, ifthere iseveranyonebesidesthiswriter(:repetitivequipofallmytexts :)Considerthepreviousgraph,whichisinfactatrinitysequenceofevents,asthephotonparticle(ifthewaveweretorepresentaLIGHTray),DIESeverycompletewaveandthewaverepresentsitsentireworldcycleofexistence.Butintheprocessitalsotranslatesinspace,andreproducesinthepointofmaximalexistentialmomentum(Max.ST),whichifthegraphwereoneofSTnotitsderivativeofchangewillbeatthepeak,whenthephotoninfactisasthe‘head’-particle state in the top of the dimension of height-information (in static space); but if the wave represents itsderivativeofchange,itwillhappeninthepointinwhichittouchesitsaxis;wherefurtheronthewaveis‘feeding’onthe‘stringoftachyonneutrino’orquantumpotentialthatguidesthewave…Sointhatbriefperiodbetweenbirthanddeath,whatitamountsfinallytoa0STchangeintheexistentialmomentumofthewave,therearethepointsofSS-birth, TT-death, TT-entropic feeding on a lower plane, ST-reproduction, sT-locomotion and finally the cyclicalperceptionthatwillhappenatthemaximalheightinthephotonstate.

Thosearethewhysofexistentialalgebraonameregeometryofmotion:

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Butamathematicalphysicisthaszerointerestinunderstandingthe30yearsoldgraph(henceitspoordigitaldesign)of the5Dimotionsofexistenceofawaveof light;butaphilosopherof scienceentangled to theUniversewill findfascinating that even the smallest formof our light space-timeUniverse has all the properties of 5Dimotional lifeencoded in its mathematical equations; the bridge between those equations and existential algebra being theunderstandingofmathematicsasanexperimentalscienceofvitalspace-time:

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TRILOGICONDERIVATIVESANDINTEGRALS:∆ST:THE3GREATFIELDSOFCALCULUS.Trilogiconcalculus.3±¡ages,scales,andDimotionsmirroredbycalculusoperands.

Aswearemadeof∆STelements,limitedbythereachofasupœrganism,self-centeredandexpressingtheprogramofthe5Dimotionsofexistence in@mind;anysystematicanalysisofanorganismor languagethatmirrors itdepartsfromtheone–thewhole,thenexplainsitsSpaceandtimestates,itsevidentdualityinasingleplane,SóT,thenitstrinity,as∆ST,and finally itspentalogicensembleof¬∆@st.Variationson those themesmightdeliveran ¡mmensenumberofexplanationsofasubjectorspecies. Inthecaseofcalculusthoughas inmostdevelopmentsofasubjectthe best consideration is a ternary analysis of its ∆ST elements in the historic growing complexity natural to theevolutionofabeingthrough3ages.

Ontheotherhandasrealityisentangledin∆±1scales,S-populationsandT-imeDimotionsandages,Calculushasinasynchronous analysis 3 great fields: the study on how systems changes in size and scale through the growth ordiminutionof its ‘finitesimals’, thestudyofgrowthofpopulations inspace,andthestudyofTimedimotions,whichareoftenbased,andthisisthemiracleofNature,inthesameconceptofafintiesimals.

Letusconsideratrilogicexampleonhowanalysis’operandsrepresentthose5Dimotions.

Spatialview:Analysisasatooltoextractquantaofwholesocialpopulations.

The fundamental particle of the Universe is a T.œ. a fractal point or scalar timespace superorganism,which in itssimplest,commonestformhastheshapeofacirclewith3canonicalregions:

@-Mind-Center,measuredbyitsradius,itsaxiallength=motionaroundthe'Territory'oftheorganicsystem.

Amembrainofangularmomentum,orexternalclockthatwemeasureasitscircumference.

Andanareaofvitalenergy,whichcanbemeasuredbythearea.

Sowegetthevalueofthe3elementsofadisk,andexpandingitto3dspheres(graph)wegetavolume,wefinda‘volume’forthevitalenergy,asurfaceofansphereforthemembraneandaperimeterforthewanderingsofthesingularity.

Asitturnsout,thecircle’sareaisπR2,andthecircumferenceis2πR,whichisthederivative.

ThevolumeofasphereisV=/3πR3,andthesurfaceareaisS=πR2,whichisagainthederivative.

Andinversely,theintegralofthecircumferenceisasurfaceandthesurfaceintegralisthevolume.

Theexampleshowsthemainuseofanalysisinstaticspace:todescribethrough3‘∆±1Planes’the3partsofthebeing,whichistheultimatereasonwhyonly2derivativesareofpracticaluseto‘descend’fromthewholedowntwoPlanes,tothefinitesimalquanta,beyondwhichanentirenew‘world’withinthequantaappears,withadifferentcontent,notsuitabletobecalculatedwithinthesameplane.

SoanalysisbecometheessentialtooltounderstandthesocialdimotionsofpartsanditsgrowingPlanesintowholesofahigher∆+1scalarplaneofthefifthdimension, inacorrespondencebetweenanalysisand5DPlanes,motionsandpopulationsofspace.

Temporalview:Analysisasameasureofatemporalmotion.

Yetanalysisismostoftenusedintemporalterms.Thiswasthoughlikelyitsfirstuse(tocalculatevolumesfromareas).Itisinfactusedtostudymotion,changeintime,andweshallarguealsoPlanes;andinthatsense,asweshallrepeatadnauseam,theentangledUniversewhichshowsaclearcorrespondencebetweenthemirrorelementsof3motionsintime, 3 topologies of space and 3 Planes of size, wholes and parts that bring together the 3 x3 (+2mental) = 11Dimensionsofrealityisfullyrealizedinthefactthatanalysisworkstoexplainthe3'ternarysymmetries'.

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Intheexample,wecanconsiderthespheretobethewholesumofparts,whereeachpartisacircumference.Soourplanetisthesumofallits'parallels'withcenterinthepoles.Andthenthevolumeaseachinternalspherecanbeintermsof5Dmetric,$Xð=Khavethesameco-invariantvalue,canbeconsideredthesumofallthoseequal5Dvaluedspheres,soagainwecantalkof∫∫¡-1=circumference->∫¡0=sphere->¡+1=volume.

Whataboutthethird'ternarysymmetry',thatoftime-change?Thisagainisthefundamentalusetodayanalysishas,tostudytherateofchangesofasystem,anditcanbeseeneasilythatthe3elementsofthe't.œ'AREmeasuresoftime-changewhenwe studynot amere locomotion,but the 'change-rate' of 'growth'moreproperof theworldcycleofexistencefrom'seed'(theinternalminimalsphere')toemergentsystem:

Ifyoudescribevolume,V,intermsoftheradius,R,thenincreasingRwillresultinanincreaseinVthat’sproportionalto the surface area. If the surface area is given by S(R), then you’ll find that for a tiny change in the radius, dR,dV=S(R)dRordV/dR=S(R),

Increase involume,dV, is theamountofnew 'cellular layers' theirsystemgrows,andtheamountofcells formthemembrane,whichisthesurfacearea,S(R),timesthethicknessofthegrowth,whereeachunitisalayer,dR.

Thissameargumentcanbeusedtoshowthatthevolumeisthe integralofthesurfacearea(justkeepadding layerafterlayerofatomsorcells).

FinitesimalsinTimevs.space

Space is symmetric; in its directions and they co-exist together. Time is not symmetric and itis experienced as asequentialpatternofsingleTimecycles.SoTimeparametersareshorterinform,spaceisamoreextendedsystem.Oftimeweseeonlyaninstant,ofspaceweintegrateinstants/cyclesoftimeandsumthemasfrequencieswhichallplaythesameworldcycle.

Timethoughoftenisjustthereproductionofanewunitofspace.Thus,timecyclesbecomepopulationsofaspatialherdduetoitsreproductionofa'seed'form.

Space thus is the 'mirror reproductive symmetry' of 'frequencies in time', its tail of memories, by reproduction,expansion,andradiationalongthepathofthesingulartimelineofthewave.

So in broad strokes derivative and integrals cover a wide range of 5D themes: the infinitesimal unitsoftimefrequenciesandcomplexherdsofspacepopulations.

Whereas given the simultaneity properties of space, integrals tend to be used to calculate space populations, andgiven the individual sequential structure of time frequencies, derivatives are most often used to calculate timemotions.

Thusthekeyconceptof5Dmathematicalanalysisisthefinitesimal,whichwasrightlydefinedbyLeibnizas:

∆:1/n;theminimalpartofawhole.

S:Whileinspaceisanindividualunitofasocialpopulation.

T:Whileinlinealtimedurationistheminimalbitofafrequencyƒ=1/t,orquantaoftime.

Thus a finitesimal is adiscreteminimal unit in any scaleof the fifthdimension - h-planckton, cellular units, atomicunits.

And by the equivalence between space-form and time-motion, S=T, as most time actions require a fractalreproductionofform,foreachquantaoftime,weshallsetheexistenceofareproductionofaquantaofspace...

On the other hand its inverse Integral 'integrate' an amount of such units of time, space or scale to obtain asimultaneouswhole,asupœrganism,a'T.œ',∫ds,∫dt,∫∆-1.

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Ofthose3typesofderivativesandintegrals,asfrequencyandtimedurationareinverseparameterscurrentlyusedinallsciences,thelessunderstoodis∫∆-1,whereas∆-1istakentobetheinfinitesimalorminimalquantaofawhole,∆º,(cell,atom,individualinasociety),anditsintegral,aSocial4Ðimotionthatmimicsthecreationofwholes.

Adualderivative,TT,ortSS,will thenextracteitheranentropicunit2scalesbelowtheformoraswefoundintheanalysis of the sphere The Point, NOT ANY point but the Center of mass or charge in a physical system, its mindsingularity. Because derivatives 'extract' the first finitesimal quanta, or fractal point from a function of exist¡ence(T.œ),oftendirectlyasinlogx':1/x,itcanleaddirectlytothevalueofthemind,or'centerpoint'ofthesystem–the‘finitesimalwhole’;anditsinverse,anintegral,whichadsfinitesimalstillreachingthewhole,asinthecaseofavolumeofpopulations,butalsoilluminatesthedissolutionofawholeintoitsintegratingparts.

Whatkindofpointaderivativegivesus,dependsontheconfigurationofthewholeweanalyze.I.e.Inaheatequationthewholelacksacenter,asitisafluxofkineticenergy,soderivativeswillextractanyunit…

In5Danalysisdependingonwhatwestudy'motion',or‘space’or‘scale’upordowntheplanesofthe5thdimensionweshallapplyeitheranintegralcommonestforspatialsumsofpopulationsoraderivative,mostoftenforinstantsoftime,anddoublederivativesforreproductivefunctions.Sincespaceandtimeareinverse,perpendicularfunctions,initsmin.SxMax.T,andMax.SxMin.Tstates,butsymmetricinS=T.Sogoesforthe2differentarrowsofentropy,adissolutiondownwardsandsocialevolutionupwards.

So the ∆ST trinity of integrals and derivatives gives a huge range of possible interpretations for the equations ofmathematicalphysics.Infinitiesthoughdon'texist,asallhasafinitemembraneandafinitedurationintime.Beyondthethirdderivative,asthescalarUniverseisa'ternarygame',thereisnosignificancetothemathematicaloperationsofderivativesandintegrals-astrongproofthat5Distruthasit limitsrealitytoternaryPlanes,topologiesandtimeages.

Soaqualitativeanalysisisrequiredtospecifywhatdimotionweare‘calculating’,withderivativesandintegrals:timemotions,spacepopulationsorreproductivemotions.

5d∫∆-1pentalogiconintegrals

∑s-1=S0: Integrals, on the other hand represent thegrowth of a space population, till it reaches a wholeness in acloseddomain.Sowecando'lineintegrals','surfaceintegrals','volumeintegrals',insimultaneousspace.

Suchintegralsmustbepositiveinitsresults,becauseweareasinthecaseof+v.–numberscalculatinga‘statisticalpopulationinspace’.

(to->t) ∑a = 0: Integrals though arealsorelated to aworld cycle, as the continuous sum of steps in a sequentialduration of time that must therefore have a 0’ final result as all worldcycles when chosen in the apppropiateparameters of ‘energy and information’ end up returning to its origin. Such integrals when properly written mustthereforegiveusa0value.Theclassiccasebeingasinusoidalfunctionofawavewithpositiveandnegativesidesfortheworldcyclethatendsina0value,whenweaddthesurfacesbelowandabovethecurve.

T=S: However when we express those ‘actions=dimotions=stœps’ of the worldcycle with the ‘simpler, first age’formalismof probability;wheras an individual event is a ‘finiteismal’ of time, and the sumof all events a ‘1 value’distribution, if we integrate the probability to get the sum of all events, whole entity as an event, which is byconventionvaluedas‘1’;theresultofsuchintegralmustbe‘renormalized’to1.

Thisisacomplicatedwaytocalculatea0’-worldcyclebutasithasbecometheformalismchoseninquantumphysics,it is constantly carried out to calculate the sum of events of an electron that give birth in space to an statisticalpopulationofallthepotentialpositionsoftheelectroninspace(themselvestakenin∆-1asdensephotonpoints).Astheelectron in trilogiccanbeseenasacloudof∆-1densephotons,asan∆owhole inspace,oras thesumof thesequentialpointsitoccupiesintime,buthumansaremonologic,alotofconfusionisnaturaltoquantumphysics,themoresowiththeadditionoffurthercomplexitywithrenormalizationmethodsandprobabilisticinterpretations.

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T=S: Integralsarealsonecessarytoaddalocomotionoftime,closertotheactionofreproductioninspace,asnatureis 'constantlybuilding integrated wholes by the accumulation of single time actions of reproduction that become'clone'cells-atoms-citizensofanintegratedsupœrganism.

¬:Integrationofanyofthoseactionshoweverneedstobe‘defined’duetotheuncertaintyofinfinities,byconstrains(initialtimeandfinaltime,ora-bintervalofdomaininspace),whichactastheintegrallinemembrain,becomingtheRiemannintegralor‘Cauchy’conditionforittohaveasolution..

As a functionof entropy integrals can also portraythe growthor diminutionof populations in space,withmost ofthosegrowth/decayinversefunctions,representedbye±xor10±xwhicharethestandardconstantsofgrowth.

Theyaremaximalwhenasystemdecreasesandthespaceisdyingwithnoconstrainatmaximalspeedinaquantaoftime–henceusingthemaximalgrowthofe-function.Howeverwhenitgrowssociallyitdoessoslower,mostoftenindecametricscales;sowefindalsodifferentspeedsonthetwotimedimotionofthe5thdimension.

Recap.Integralsareoverwhelminglythemeasureofchangeinafictiousmentalspaceconstructed.

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1stAGESCALARVIEW:FINITESIMALS.UNIVERSALS≈WHOLES

Universals

Perhapstheclearesthistoricproofofthenatureoffinitesimalsasthepartsofwholesisthefactthathebeginningofcalculus was not related to the study of rates of change in continuous motion but precisely to the relationshipbetweenpartsintowholes.

SoGreeksstudiedinphilosophicaltermstheintegration=growthofasocialsystemfrommicrotomacrocosms,fromindividuals intoUniversals, andmathematically through 'finitesimal'minimalquantaor partsof thewhole, through'series'andexhaustionmethods.

This age extended from the Greeks to Newton, which was the last of the ancients, changing the use of thoseexhaustion methods from spatial series of growth to temporal series of change, but he failed to represent themproperly through the space=time symmetryofY(s)=X(t), in aCartesian frameas Leibnizdid,adding thepropertyof‘continuity’asexplainedbefore,notthelimitinwhichh->0,butthelimitinwhichS≈T....

Platomaintained that exemplifying aproperty is amatter of imperfectly copying anentity he called a form,whichitselfisaperfectorpureinstanceofthepropertyinquestion.Severalthingsareredorbeautiful,forexample,invirtueoftheirresemblingtheidealformoftheRedortheBeautiful.Plato'sformsareabstractortranscendent,occupyingarealmcompletelyoutside spaceand time. They cannot affectorbeaffectedbyanyobjector event in thephysicaluniverse.This is correct, though theerror lies inpositioninguniversalsoutsidespaceand time.Theyare in fact theultimatepropertiesofSE-spatial‘kineticenergy+entropy’andTO-Temporalinformation,which‘emerge’ineachnewscale.

Few philosophers now believe in such a “Platonic heaven,” at least as Plato originally conceived it; the “copying”theoryofexemplificationisgenerallyrejected.Nevertheless,manymodernandcontemporaryphilosophers,includingGottlob Frege, the early Bertrand Russell, Alonzo Church, andGeorge Bealer are properly called “Platonic” realistsbecausetheybelievedinuniversalsthatareabstractortranscendentandthatdonotdependupontheexistenceoftheirinstances.

They are closer to the truth, but they should substitute theword ‘transcendent’ for ‘emergent’ in the parlance ofgeneralsystems.

ForthatmatterGeneralSystems(5DST)reducesthemeaningof‘transcendence’toitsfirstsemanticmeaning:

Vb:Ltranscenderetoclimbacross,transcend,fr.trans-+scanderetoclimb.

vt:toriseaboveorgobeyondthelimits.

Indeed,Universalsarefoundbeyondthelimitsofitsfinitesimals,inthenextn+1scale.

Dimensionalgrowthareafinitesimalsas:reproductionofspatialform

Finitesimalswerefirstfoundinspace,asthemeanstoquantifyasimultaneousareasasthesumof∆-1discontinuous,fractal parts. Let us remember this concept, key philosophical discussion evenwith the greeks - it is the Universecontinuousordiscontinuous,madeofUniversalwholesorindividualparts?

This concept was the earlier idea of Leucipus and Democritus regarding the composition of physical systems; andAnaximander, regarding the composition of life systems,with its ‘homunculus’ concept (weweremade of smallerbeings)

Anaximenes'assumptionthataer iseverlastingly inmotionandhisanalogybetweenthedivineairthatsustainstheuniverse and the human “air,” or soul, that animates people is a clear comparison between a macrocosm and amicrocosm.

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Italsopermithimtomaintainaunitybehinddiversityaswellastoreinforcetheviewofhiscontemporariesthatthereisanoverarchingprincipleregulatingall lifeandbehavior.Soherethere isa firstbridgethatmergesuniversalsandfinitesimals.

And of earlier mystiques, regarding the composition of a superior God, as the subconscious collective of all itsbelievers’minds,fusionina‘bosonic’wayintothesoulofthewhole.

The 3 were right as finitesimals are clone beings with properties that transcend into the Universal, being thehomunculusthe‘futurecell’.

Universalwholesandindividualfinitesimals.

Becausethepraxisofcontinuitywasnotyet‘erasedbyidealismreality’theGreeksacceptedasrealtheirexhaustionmethods,butPythagorismopenedtheroadtoidealism.Sothefirstageofanalysishadagreatdealofphilosophicaldisquisitionsonthenatureofwholesandparts,connectingdirectlywiththegreeklogicargumentsonthenatureofindividualsanduniversals.

Thehistoricaloriginsofanalysiscanbefoundinattemptstocalculatespatialquantitiessuchasthelengthofacurvedlineortheareaenclosedbyacurve.

Asweknow,acurve, isalwayspartofaworldcycle,witha finitenumberof steps,andso theconclusionsof thoseearlier studies can be extended to understand better the space-time worldcycle in a general way: a circle can becalculated as a polynomial number, which becomes nearly undistinguishable, past the 10-20-100th ‘fractal points’stœpsofsocialscalesofnumberallpervadinginNature.

Thisleadtotheexhaustionmethodofcalculatingirrationalnumbers,frompartsintowholes.

o-1:∆-1:1/nfinitesimalscalevs.1-∞:∆+1:wholescale.

Soonlyaquestionofthatsectionisworthtomentionhere,onhowto‘considerPlanes’,whichtendtobedecametric,good!OneofthefewthingsthatworkrightonthehumanmindanddonohavetobeadaptedtotheUniversalmind,fromd•stto∆ûst.

Shallwestudythemdownwards,through‘finitesimaldecimalPlanes’orupwards,throughdecametric,growingones?TheanswerisanessentiallawofAbsoluterelativitythatgoesasfollows:

‘Thestudyofdecametric,§+Planes(10§≈10•10∆≈∆+1)issymmetrictothestudyoftheinverse,decimal∆>∆-1scale’.

Orinitsmostreduced‘formula’:(∞=(1)=0):(∞-1)≈(1-0)

Whereas∞istheperceptionofthewhole‘upwards’inthedomainof1,theminimalquantatotherelative∞ofthe∆+1scale.While1istherelativeinfiniteofasystemobserveddownwards,suchas∆+1(1)iscomposedofanumberof‘finitesimalparts’whoseminimalquantais0.

ItisfromthatconceptfromwhereweacceptasthebestdefinitionofaninfinitesimalthatofLeibniz:N(whole)=1/N(Finitesimal).

Soinabsoluterelativitythe∆-1worldgoesfrom1to0,andthe∆+1equivalentconceptgoesfrom1to∞.Andsonowwecanalsoextractofthe‘infinitorumthoughtreceptacle’Jakeydifferencebetweenbothmathematicaltechniques:

A conceptual analysis upwards has a defined lower point-quanta, 1 and an undefined upper ∞ limit. While adownwardsanalysishasanupperdefinedwholelimit,1andanundefined‘finitesimalminimum,+0).

Finallytonoticethatasall∆-Planeshaverelativefinitesimal+0andrelativeinfinities(see∞|ºtounderstandthelimitsandmeaningofnumbersand itsPlanes),essential toall theoryof calculus is the studyof thedomain inwhich thesystemworks,andthe'holes'orsingularitiesandmembraneswhicharenotpartoftheopenball-system.Sofunctions

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canbedefinedwithcertainsingularitypointsandborders;hencefunctionsneednotbedefinedbysingle formulas.ThiswouldbeunderstoodbyLeibniz-whoelse:)

Unlike Newton, who made little effort to explain and justify fluxions, Leibniz, as an eminent and highly regardedphilosopher,wasinfluentialinpropagatingtheideaoffinitesimals,whichhedescribedasactualnumbers—thatis,lessthan1/ninabsolutevalueforeachpositiveintegernandyetnotequalto0’.

For thosewho insisted in infinities, Berkeleywould reveal those contradictions in thebook 'TheAnalyst'. Therehewroteaboutfluxions:“Theyareneitherfinitequantities,norquantitiesinfinitelysmall,noryetnothing.Maywenotcallthemtheghostsofdepartedquantities?”

Definitionof∆t,∆s,finitesimals:Aquantumoftimeandspace.

Berkeley'scriticismwasnotfullymetuntilthe19thcentury,whenitwasrealizedthat,intheexpressiondy/dx,dxanddyneednot leadan independentexistence.Rather, thisexpressioncouldbedefinedas the limitofordinary ratiosΔy/Δx.

Andhere iswhereweretake it;before the formalageofmathematics,madea 'pretentiously rigorousdefinitionofinfinitesimal limitsandthethelogicianA.Robinsonshowedthenotionof infinitesimaltobe logicallyconsistent,butNOTreal.

Aswebelievemathematicsmustbe real tobe 'consistent' (Gödel’s theorem),wereturn to the finitesimalconcept,±∆y, either as a 'real' increase/decrease of a quantity, with a variation ±∆x of either the surface of space or thedurationintimeofthebeing.

Thus finitesimals depend for each species of the 'quanta' of spaceor 'minimal cell' andquanta of timeorminimalmoment,whichthesystemcanmeasure.

Forman,forexampletimeactionsaremeasuredwithitsminimaltimequantaofasecond,belowwhichitisdifficulttoperceiveanything;ananosecondinthatregardinthehumanplaneofexistenceisNOTworthtomeasure,asnothinghappening inanano-secondwillbeperceivedasmotionorchange.Foranatomhoweverananosecondisaproperfinitesimaltomeasurechanges.

Inspace,mandoesnotperceivesensationsbelowcertainlimits,whichvaryforeachsense,amillimeter,100hertzsofsound,thefrequencyofinfraredwaves;andsoon.

TherewasonlyatthisstageamathematicalapproachtotheconceptbyArchimedes-themethodsofexhaustiontocalculateareasandratios,notablythepiratio.

Themethodofexhaustion…

wasfirstusedbyEudoxus,asageneralizationofthetheoryofproportions.

Eudoxus'ideawastomeasurearbitraryobjectsbydefiningthemascombinationsofmultiplepolygonsorpolyhedral.Inthisway,hecouldcomputevolumesandareasofmanyobjectswiththehelpofafewshapes,suchastrianglesandtriangularprisms,ofknowndimensions.Forexample,byusingstacksofprisms(seefigure),Eudoxuswasabletoprovethatthevolumeofapyramid isone-thirdoftheareaof itsbaseBmultipliedby itsheighth,or inmodernnotationBh/3.

Loosely speaking, the volume of the pyramid is “exhausted” by stacks of prisms as the thickness of the prismsbecomes progressively smaller. More precisely, what Eudoxus proved is that any volume less than Bh/3 may beexceededbyastackofprismsinsidethepyramid,andanyvolumegreaterthanBh/3maybeundercutbyastackofprismscontainingthepyramid.

ThegreatestexponentofthemethodofexhaustionwasArchimedes(c.285–212/211BC).Amonghisdiscoveriesusingexhaustionweretheareaofaparabolicsegment,thevolumeofaparaboloid,thetangenttoaspiral,andaproofthat

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the volume of a sphere is two-thirds the volume of the circumscribing cylinder. His calculation of the area of theparabolic segment (see figure) involved the application of infinite series to geometry. In this case, the infinitegeometricseries:

1+1/4+1/16+1/64+…=4/3

isobtainedbysuccessivelyaddingatrianglewithunitarea, thentriangles that total1/4unitarea, thentrianglesof1/16,andsoforth,untiltheareaisexhausted.Archimedesavoidedactualcontactwithinfinity,however,byshowingthat theseriesobtainedbystoppingaftera finitenumberof termscouldbemadetoexceedanynumber less than4/3.Inmodernterms,4/3isthelimitofthepartialsums.

Hispaper,‘MeasurementoftheCircle’isafragmentofalongerworkinwhichπ(pi),theratioofthecircumferencetothediameterofacircle,isshowntoliebetweenthelimitsof310/71and31/7.

Archimedes'approachtodeterminingπconsistsofinscribingandcircumscribingregularpolygonswithalargenumberof sides. It was followed by everyone until the development of infinite series expansions in India during the 15thcenturyandinEuropeduringthe17thcentury.Thisworkalsocontainsaccurateapproximations(expressedasratiosofintegers)tothesquarerootsof3andseverallargenumbers.

It is then interesting to consider Archimedes' main role on the perception of problems today forgotten after theabsurddogmaticgermanic'foundationsundertheaxiomaticmethod'ofanalysis.

2problemstroubledhimandindeedtheywereveryimportantproblems:thecomparisonsofdifferentpis,(itisthepisquarewith2dimensionsthesamethanthepioftheperimeter)anditspropercalculusbyapproximation.

Approximationsingeometry.

The unit of space is the area and the unit of time the cycle, and so both are bidimensional, and hence thetransformationofone intoanother isnotalwaysperfect,asthere isnotaperfect 'quadrature'.Butasthishappensconstantly a part is lost as 'entropy' in all time-space transformations, or as 'a bit of a circle', that is amotion orparticle,aswheninparticlereactionstherearealways'forces'escaping(neutrinos,gammarrays).Sothismeansthatpiisnotexact,neither√2,thetwokeyconstantsforthesquaring...Yetthatdoesn'tmeanthetransformationhappensallthetime,anditwasthewayinwhichthegameofanalysisstartedwithArchimedes:

The transformationofacircular region intoanapproximately rectangular region. Ingraph∆ST theoryeliminatesallinfinitesimalsproblemsasinfinitiesarelimited,soarethe0s,whichmustberegardedasthe+0minimalquantaofthedomain - the need for furtherinfinities is an error of the mind, the dogmatic truth and the singlespace-time'continuum).Inthatregardpiisnot∞,butitscalculusbecomes'chaotic'beyondalimitof±40decimals,whichisreallyallwhatthehumanmindcanconceivenitslargestfinitesimalanalysis.

Itisthenwhenthe'GreekAge'becomesjustasintheArchimedeancalculusofpibyexhaustionthesameconcept,justwithlessdetail.

Asimplegeometricargumentshowsthatbothprocessesaresimilarwithdifferentdegreesofapproximation:

The idea is to slice the circle likeapie, intoa largenumberof equalpieces,andtoreassemblethepiecestoformanapproximaterectangle(see figure).Thentheareaof the“rectangle” iscloselyapproximatedbyitsheight,whichequalsthecircle'sradius,multipliedbythelengthof one set of curved sides—which together form one-half of thecircle's circumference. As the slices get very thin, the error in theapproximationbecomesverysmall.

ThesimplegraphaboveshowsfromthepointofviewofanS=Tsymmetryifwetakethecircleasanangularmotion,the∆STTrinityofchangethatalwayscanhappeninscale,spaceortime.Inscaleeachminimal‘radius’,isaquantaof

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change.Intimethecirclebecomesanangularmotion,soeachtriangularsectionbecomesarateofchangeperunitoftime, related to the angular speed of the circle. Yet the circle as awave give us also the linealmotion that keepsreproducingquantaafterquantaofchangethewave.

∆STTrinitythenbecomesonceandagaintheleitmotifofchange

Thedualityoffreelines/planesv.closedorder.

Itisinterestingtonoticethatingeneralwhenwegrowinscale,wechangefromfreedomtoorderorviceversa-thatis the fundamental | v.O,past vs. future,part vs.whole, formvs.motion,dualitiesof∆@st changes. Sowhenweintegrateopenlinealtriangles,withitsvertexasthe@-forwardmind≈futurepath,inthecircleitbecomesaninternallocked,social,circularmind-aclosedpointofalargersingularityinacyclicalform.

Theapproximationofsquarespacetocyclicalpoints.Ratiosandir(ratio)nalnumbers,itsfinitesimallimits.

AthemethatwillbesooncastedontermsofnumbertheorywasalsostudiedbyArchimedesbyexhaustionmethods.

Beforetheinventionofthenewmethodsofcalculation,ithadbeenpossibletofindtheareaonlyofpolygons,ofthecircle,ofasectororasegmentofthecircle,andoftwoorthreeotherfigures. Inaddition,Archimedeshadalreadyinventedawaytocalculatetheareaofcurvesbyexhaustion,leavingasounderroraccordingtotheminimalstephetook,whichraisesthequestion,doeshaveacircleafinitesimalminimumstep?ItisthenpiandallotherS>tconstanttransformationsand'ir(ratio)nalnumbers/ratios,limitedbyafinitesimalerror?

The answer is yes!, Normally a decametric limit define the 'valid value of an ir(ratio)nal numbers, which is not anumberinstrictsense(asocialnumber)butaratioofanS/Taction/function.Theexamplesofthetwofundamentalir(ratio)nalswillsuffice:

-piisreallytheratioof3diametersthatformaclosedcurve,whosevaluedependsonthelineal'stepsizes'.

Sopihasaminimalvalueof3,whichisthehexagonwithits6stepsof1/2value(triangulationin6immediatelygivestheresult,asthetriangleistheradius,soarethe6triangularsides:1/2x6=3);whichhappenstobethevalueofpiinextremegravitationalfieldsonrelativity,whichbringsanotherinsight:blackholesdecomposethecircleintoultimatelineal flows of pure 'dark energy' shot through the axis, by converting the curvature of a lightcircle on the eventhorizonina6-pihexagon.Butthisiswellbeyondthescopeofthisintro.

So what is the 'decimal limit' of pi, before it breaks into meaningless (non-effective) decimal Planes, with littleinfluenceonthewhole?

While this ishypothetical Iwould say fordifferent reasonsexplained in thearticleonnumber theory, as it isquiteoftenthecaseitrespondstothegeneral∆≈S≈Tternarysymmetries,socommonintheperfectUniverse.

So pi responds to the symmetry between its spatialminimal, 6 x 1/2=3 hexagonal steps,whichmeansit breaks inthe6th∆-scalingdecimals,3,1415...9.So,3,1416,whichincidentallyisbasicallywhateverybodyusesisthe'realvalue'ofpi,andwhyitisthatvalueisstudiedelsewhere(deducingfromitoneofthemostbeautifulsimpleresultsofget-mathematics,thevalueofdarkenergyinanysystem,oftheUniverse,asthepartnotperceivedthroughtheaperturesofapicycle:π/π-3=96%of'darkness'whichthesingularityofapisystemcannotseeasitsaperturesareonlyπ-3=0.14

Discontinuityandlimitsofmentalspaceandphysicalspaces.

Wegetnowtotheheartofthematter;whichistheparadoxbetweencontinuousmentalspacesanddiscontinuous,fractal spaces, between infinitesimal and infinities vs. finitesimals and relative infinites (∝), between the axiomaticmethodandtheexperimentalmethodthatkeepssurfacingallthese5Dv.4Dpapers.Thespace-timecontinuumisnotsuchwhenwe‘taketheaccordion’ofthe5thDimensionandenlargethewholeUniverseintomultipleplanesofspace-time,whichareconnectedthroughthe‘differentgeometries’oftheconvex,hyperbolicregionsbetweenplanes.

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Thegenerallawsof5DoutlinedinotherpaperswhichweshallpostatAcademia.edusometimeinthe‘future’usetheformalismofexistentialalgebratoliedownallthoselawsdepartingfrom5Dmetric.

Inwhatreferstocalculusitcanbeexpressedintermsofthe‘praxis’ofmathematicalphysicsthatusessystematicallythedifferentialequationvs.thetheoryofidealmathematiciansthatprefertoargueonthe‘passingtothelimit’andsinceCauchyputitinnice‘bullShit=pedantic’talkseemstobeproved.

ManyimportanttruthsofthefractalUniversearededucedpreciselybydenyingpedanticdefinitions,postulatesandaxiomstomakearightwrong.

Now,theotherconstante,whichistheratioofdecayACTIONS,ordeathprocesses(ST<<S), isalongertwo'Planes'downprocess,of self-destructionofa system,unlike thepi, single scalingprocess, S>T. So it breaksat10decimals:2.718281828...459045

Indeed.Now,why 5 and not ten if the Planes are 10¹º? Because 10 Planesare in terms of space-time actions, the'whole' dual game of two directions of time up and down, which happens only inreproductive actions. And thisconnectswiththeS>T<SRhythmsofmotiongo/stop/gobackandforthbetweentwoarrowswhichhappensbothinst-singleplanesand∆±motions.

Theproof?Verysimple.TheexperimentaltruthtellsusthattheUniverseisagameof9-119-11planesofexist¡enceandifwecalculateeas(1+1/10000000000)100000000000wegetthe‘real’ewhichmustbeanumberthatisNOTirrational.Asthatistheexperimentale-numberfortheoverwhelmingquantityofsystemsofNaturemadeof1010’-11parts.Alas,weobtain2.718281828…323131…arationalserieswhoseprofoundmeaningisthatofthefastestprogressionofgrowthordecayofafinitesimalseedintoaperfectwhole(:

Doesthismeanwecannotfind‘largersystems’?Yes,butifyougotanyofthefundamentalconceptsofthe5DscalarUniverse‘runningaroundyourbrain’–realityis∞inthefieldofpureTT-entropictimeflowsandPlanes,butfortheperceiverandany languageofperception thatmake sense,beyond the1011perfect formunitofa larger∆+1 case,perfectionbreaksdown,systemsmalfunction,monstersappearandegetsitsirrationalform.

The reader is leftwitha funnyexercise forwhichhe should receive theFieldsmedalsofmathematic (just joking –thosewho rebel against the axiomaticmethod and its Cantorian Paradises shall not enter the kingdomof nitrolifegaseousheadsbubblingegocywithgo(l)d….ButPIDOEShavea limit.And thismeans theUniversehasnot infinite

Planesofperfectorder– itdoesNOThaveaGodthatcanseethroughall thePlanes,asamindthatordersitsinfinityinspace,scaleandtime…

There is then a limit for existential planes? The 'meaningless' breaking down of e, the 'number ofentropicfunctions'seemstosignalthis.Butitwouldbeanerrortoconsiderthelimitsofe-regularityasit only indicates the LIMIT of entropic death. Death happens and when a system breaks down itsnatural10ˆ±10Planestoitsfinitesimal1/npartsitstopsasthesystemisdead.

The limit thatmatters is the limit of the pi-circle as an Archimedean spiral that lets information enter through its±neverclosingspiraltoperceiveorfeed intheexternalmicro-bitsandbitesoftheUniverse.Andaswecannotfindneither a limit nor a regularity, we could conclude that themost important dimotions of angular perception, andcreationofinnermirrorsoftheouterworldbyapi-spiralhavenolimit.

Whataboutlocomotion?Canweexhaustthelimitofaseriesofsteps?Again,thisismoreevidentlyno,eventhoughtheGreeksthoughtso,inthesocalled...

Theproblemofequivalencesconfusedasidentitiesbetweenlinesandareas.

It is absurd to talk about continuity of a real number, pi, e, and √2, beyond the 10 decimal. This is easily provedbecause those ratios are normally obtained by limits in which certain terms of the infinitesimal are despised, by

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postulatingthefalsitythatthereareinfinitesmallishparts,andsox/∆canbethrowoutwhen∆->∞.Butsincex/∆,thefinitesimalhasalimit,thepretentiousexactitudedoesnothappen.

This in turn leads to questions about the meaning of quantities that become infinitely large or infinitely small—concepts riddledwith logical pitfalls in a simplifiedworld of a single space-time continuum,where on top humansLOVEtoconsider‘identities’ofthemindabsoluteidentitiesinthelargerinformationofthedetailedUniverse,whichareneverso,asd@st≈∆ûst(themind,worldviewismerelysimilartotheUniversalview).

In our exampleexample, a circle of radius r has circumference 2πr and area πr2, where π is the famous constant3.14159…. Establishing these two properties is not entirely straightforward, although an adequate approach wasdevelopedbythegeometersofancientGreece,especiallyEudoxusandArchimedes.Itisharderthanonemightexpecttoshowthatthecircumferenceofacircleisproportionaltoitsradiusandthatitsareaisproportionaltothesquareofitsradius.Thereallydifficultproblem,though,istoshowthattheconstantofproportionalityforthecircumferenceispreciselytwicetheconstantofproportionalityforthearea—thatis,toshowthattheconstantnowcalledπreallyisthesameinbothformulas.

Thisboilsdowntoprovingatheorem(firstprovedbyArchimedes)thatdoesnotmentionπexplicitlyatall:theareaofacircleisthesameasthatofarectangle,oneofwhosesidesisequaltothecircle'sradiusandtheothertohalfthecircle'scircumference.

HoweverinÐSTtheory,those2pisarenotthesame,becausetheybelongtotwodiscontinuous,‘differentspecies’oftopology,theStarea,andtheST-membrane.

Aneasy,immediateproof.Ifwemakethemidentical,thenwecanfindacircle,where:2πr≈πr2.So2r=r2.Hence2=randwegettotheconclusionthatthethinmembraneofanopenballisidenticalinareatotheinternalSTvolumeofthe being, which is ‘conceptually absurd’ (the area intuitively has more surface, as it is bidimensional, the line,infinitelythin).

What’s the problemhere?We cannot in true form, unlesswe deal alwayswith less dogmatic concepts of relativesimilaritieswith‘linesasiftheyweresquares’.Theyaredifferentrealities.Inthefirstequivalence,wecomparealineradiuswithacircleperimeter,inanS>tstructure.

Inthesecondaswecompareπ²,acyclicalareawiththesquareoftheradiuswearealsoingoodfooting.ButwhenwedotheS>STcomparison,weareinaDynamictransformationof∆-Planes,from∆,theworldoflines,to∆+1theworldofsquares(asapolynomialsquareisobviouslyagrowthfromacomplete∆-entitytheline,intoan∆2=∆+1one,thearea).Itisthenwhenwecandosome‘dynamicequivalence’analysis,andtheequivalencehasmeaning,statingthatfora‘perfectcycle’ofrelativeradius2,themembraneabsorptionofbitsanbitesofenergyandinformation,canfully,fill,theinternalarea,makingequivalent,a‘lineandasurface’ integral.Andfinallystatethatall ‘dynamicvorticesofforce’ruledbyNewtonian/Coulombianequationsonthe∆-1and∆+1Planes,arerelativeperfectsystemsofradius2.

Andherewefindthe‘whys’ofthedualitiesofMaxwell’slaws,whichcanbewrittenbothways:

Orinsimplerterms,wearetalkingwhendoingthoseequalitiesofpropertiesthatbecomedynamicandtranscendthestaticmindofmathematicsintotherealityofphysicalsystems.

Finallyaswedefinedrealnumbersasnon-existent (see|∞posts),butapproximations toa±0 infinitesimal, in themeasureofasquare,uncertaintygrowsfurther,π2,thushavethesquare‘error’ofpi.

Allthisofcourseisimportanttoconceptualizereality,inpraxisasweknowwealwaysworkinanuncertaingamewitherrors and deaths. So analysis does work, and all this ‘search for dogmatic proofs’ is just ‘absolute bull$hit’ forabsoluteego-centeredscholarhuminds.

Butontheotherhandthegraphalsoshowsthatbothpis,theoneofthe'surface'andtheoneofthe'perimeter'arenotequal,astherewillbealimitonthenumberof'bidimensionaltriangles'wecancut.

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Asatriangleisindeedthebidimensionalline,thatis:|-$t(one-dimension);∆-$t(2dimension).

Soitisnottheline.

Soastheapproximationwill finda finitesimalquantaor limitofdetail,provethetheorem,thiserror,howevertiny,remainsanerror.Thisminimalquantathusexistinallrelative∆>∆+1measuresofPlanesastheminimaluncertaintyofallmathematicalcalculus,andjustifiesinphysics(∆-1quantumtheory)thattheeisalwaysanuncertaintyofaminimalquanta,whichisprecisely/2;thatish/2π;theminimalquantaofourlightspace-time.

Only in the absolutist imagination of dogmatic axiomaticmathematicians itmade sense to talk of the slices beinginfinitesimallythin,sotheerrorwoulddisappearaltogether,oratleastitwouldbecomeinfinitesimal.

As it happensquantum theoryprovedexperimentally the case tobewrong.Andaswe stress (Lobachevski,Gödel,Einstein)mathematicsmustbeconfrontedwithrealitytorealisewhatis'real'inmaths.

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∆:THENEWTONIANWAY.FINITESIMALSERIES.

In5Dtheconceptofseriesisanimportantone;asitestablishesforeachstœpoftheseriesaquantityofgrowththatconvergestowardsawhole,valuedbyafinitenumber;andthentheseriesisameaningfulmirrorofan∑∆-1=∆º

processofNature.Whentheseriesdivergehoweveritisoflittleinterest,asitisanexponentialgrowththatatbestcansignifyanentropicprocess.Seriesthusarepredecessorsofcalculuswhereeachtermrepresentafinitesimalof

change,andthewholesumofthesequencethe‘wholeworldcycle’intime,or‘volumeinspace’.

Thelimitsofvalueforserieswerealsoinstrumentaltounderstandtheparadoxesofidealmathematics,(Achiles’paradox)showingthatindeedchangerequiresfinitesimal0’saslimitx->0isNEVERabsolutezero;orelseAchileswillnevermeettheTURTLE.Onlyhumanegocyinsearchofmentalsimplifiedabsolutetruths,relativelyfalseexplains2300yearsofdisquisitionsontheobvuoussolutionoftheachilesparadox,whichwillintroducethetheme.Sothemaincommentsonmathematicalseriesareontheconceptsofrelative‘finitesimals’andrelativeimmensities(not

infinities)properof5Dmath.

AquilesParadox.Birthoftheconceptofseriesandlimits.

Inmathematics,aseriesis,roughlyspeaking,adescriptionoftheoperationofaddingmanyquantities,oneaftertheother, to a given starting quantity, in 5D each quantity is a new finitesimal of a series, hence 3 series can bedistinguishedbydimotion:

-Divergentgrowingseriesofidealsocialevolutionandreproductiontillalimitofcarryingcapacity,whichinrealitywillmaketheidealseries‘flatten’itsgrowth.

-Equalseriesofpresentstatesinwhicheachstepsequalthepreviousone,whichreducestosimplesums.

- Convergent series that diminish in size till a finitesimal is reached, that should be perceived inversely from thefinitesimaltothewhole.

Thestudyofseriesisthusamajorpartofcalculusanditsgeneralization,mathematicalanalysis,sinceitisthe‘discretemanner’tocalculateandonemightarguemorereal.Thegreeksstartedtheirstudyinphilosophyandrightlysolvedit(Aristotle)deducingthatabsolute0andinfinitydidnotexist.Modernegocydismantledthosefindingsforthesocalled‘rigorousproofsoftheaxiomaticmentalmethod’whoseaimistoconvinceegocenteredmenthatthe‘simplificationofmindspaces’ thateliminate thedarkholesbetweenpointsandexpand limits to infinitiesandabsolutezerosare‘reality’,notthementalselectionofit.

The paradox of Achiles: in a discontinuous Universe of fractal parts, achiles should never reach the turtle. But ifmotion is reproduction of form, the faster systemmerely 'reproduces' its information faster in adjacent regions ofspace,andmotionbecomes 'rational' -andproves further the reproductivenatureof realityaseven locomotion ISreproduction.

For a long time, the idea that such a potentially infinite summation could produce a finite result was consideredparadoxicalbymathematiciansandphilosophers.

Thisparadoxwasresolvedusingtheconceptofalimitduringthe19thcentury.

Zeno'sparadoxofAchillesandthetortoiseillustratesthiscounterintuitivepropertyofinfinitesums:

Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, thetortoisehasreachedasecondposition;whenhereachesthissecondposition,thetortoiseisatathirdposition,andsoon.

ZenoconcludedthatAchillescouldnever reachthetortoise,andthusthatmovementdoesnotexist.Zenodividedtheraceintoinfinitelymanysub-races,eachrequiringafiniteamountoftime,sothatthetotaltimeforAchillestocatchthetortoiseisgivenbyaseries.

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Theresolutionoftheparadoxisthat,althoughtheserieshasaninfinitenumberofterms, ithasafinitesum,whichgivesthetimenecessaryforAchillestocatchthetortoise.

Thephysicalexplanationoflocomotionthoughdefinesitasareproductionofforofthelowerscale,soitestablishesafinitesimalstœp,equivalenttotheminimal∆-¡quantaofthewave-particledualmotionstates:

Locomotion is a series of stœps that imprint a lower plane with the information of the upper plane: a quantummotioninwavestateandparticle,stopstateofreproductionform(complementarityprinciplewave-particle).

Inmodernterminology,any(ordered)infinitesequence(a1,a2,a3,…)ofterms(thatisnumbers,functions,oranythingthatcanbeadded)definesaseries,whichistheoperationofaddingtheaioneaftertheother.

Toemphasize that therearean infinitenumberof terms, a seriesmaybe calledan infinite series. Sucha series isrepresented(ordenoted)byanexpressionlike:a1+a2+a3+⋯or,usingthesummationsign:

The∞sequenceofadditionsimpliedbyaseriescannotbeeffectivelycarriedoninafiniteamountoftime.

However,ifthesettowhichthetermsandtheirfinitesumsbelonghasanotionoflimit, it issometimespossibletoassignavaluetoaseries,calledthesumoftheseries.Thisvalueisthelimitasntendstoinfinity(ifthelimitexists)ofthe finite sums of the n first terms of the series, which are called the nth partial sums of the series. That is:

What thismeans in 5Dthough is slightly different: because the infinite numberof time-stepswillmakeimpossibletodoanycalculus,alllimitsmusthavein'reality'beyondtheidealizedmirrorofmathematics,alimitofstepsandalimitofsizeofthosesteps.Whichisindeedwhathappensinreality.

Whatthismeansin5Dthoughisslightlydifferent:becausetheinfinitenumberoftime-stepswillmakeimpossibletodoanycalculus,alllimitsmusthavein'reality'beyondtheidealizedmirrorofmathematics,alimitofstepsandalimitofsizeofthosesteps.Whichisindeedwhathappensinreality.

Theturtlehasatime-cycleandasizeofsteps,measurable.Andwhenexplainingthereproductionofmotion,weshallsee that limit is the reproductionon the lowestplaneof lightandparticle forcesof theentire formof thebeing indiscontinuousadjacentspaces.

Inotherworlds,theword'limit'intheformulaeshouldnotbeinfinite.Buta'finiteinfinite',forwhichweshalluseadifferentsymbol:∝

Relativeinfinitiesandfinitesimals

The simplestwhyof the fractal, scalar structureof theUniverse, from theperspectiveof themind: as a linguisticmirrorimageofrealityinasmallerspace,minds'create'fractaldiminishing,infinitePlanes

Thenewsymbol fora 'relative infinity'and its inverse1/, 'finitesimals',becomethenessential to5DAnalysisand itgets rid of all infinite paradoxes from Zeno's to Cantor, further showing the idealized mirror-image nature ofmathematics;asamirrorrecedesapparentlyintoinfinitybutatacertainpointitceasestobeobservableandhenceitdoesNOTexistanymore.

Themeaningofseriestheninrealexistencesbecomesclearasitisanotherwaytodescribeindiscontinuousmanner,whatderivativesonthecontinuousplane(rememberthedualityofdiscretenumberviewvs.Continuousgeometricview),shows:AtravelupanddownthePlanesofthefifthdimension.

Ratesofchange.Thestopandgomotion:stœps.

The discrete, geometric, spatial, static numerical analysis of calculus is the power series, which can be taken asdiscretestœps(stops+steps)inamotiondownthefifthdimensionfromthewholetothe1/npart,whereaswecountalsothestaticform(asweseeonlyinamoviethestaticframe)NOTthestepofmotion.

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ThiswasthentheworkfromArchimedesandearlierGreekstoNewton,whichcaninthatsensebeconsideredthelastoftheancients.

WhileasallS=T,thatisthereisalwaysasymmetrybetweendiscretenumbersandcontinuousmotions,Leibnizwithits geometric interpretation and farmore profound understanding of finitesimals,which he rightly defined as 1/n,representsthefirststepinthefutureofthediscipline,therenovatoranddeepunderstandingof it-whichNewton,whichcanbeconsideredmerelyanautomatonmathematician,specializedbrain,asmostmodernscientistsis-heisindeedthefatherofthewrongviewofscience-understoodnothingofit.

Indeed,Leibniz,theclosestpredecessorofthisblogISthegenius,Newtonthetalent.

FinitesimalchangesarerelatedtothefundamentalbeatoftheUniverse,thestop-form-space-perception,go-motion-time,beatoftheUniverse,whichweshallcallastœp,thediscretewayofmotionoftœsthroughSPace,whichoftenasinmoviesweperceiveincontinuousmodeeliminatingthestopelement:

∆S(top)->∆t->∆S-≥∆(S)t(ep).

MoreovermostofthoseStœpswillhaveeitherinatravelthrough5D,orthroughasingleST,aunitof'expenditureofvitalenergy',transformedinthelength-motionofthelowerscaleinwhichtheimprintingofmotionasreproductionofform,happens(studiedin2Dlocomotion).Soeachstœpbecomesan∆-4unitoflocomotion.

Thusifweconsiderarelativeconstantorfunctionoftheexistence,∆-1:œ,asafinitesimalofitslargerwhole,∆Œ,weobtain2simplefunctions:

œ=∆s/∆t andœ=∆t/∆s as themathematicalmeasureof a 'time stœp'or locomotionand 'volume-density stœp'orfinitesimalquanta.

We shall call the first form a spatial finitesimal or step in space - a quanta of constant speed that moves andreproducesthebeinginspace.

Andifweagainchangethisquanta,withasecond‘derivative’wegetaquantaofitsconstantacceleration.

Andweshallcallthesecondfunction,atimefinitesimal,achangeinthedensityofinformationorcyclicalspeedofthebeingasasecondchangeinrelationtoitspositioninspace.

Classicconceptsofmathematicsappliedto5Dinseries.ImmenseGeometricseries

Thissaidsomeclarificationsareneededinclassicseriestheorymostlyrelatedtothefactthat0isnotinfinitesimalbut0’and∝islimited.LetusdenotebySnthesumofthefirstntermsoftheseries;wewillcallitthenthpartialsum.Asaresultweobtainasequenceofnumbers:

andwemayspeakofavariablequantitySn,wheren=1,2,···.

Theseriesissaidtobeconvergentif,asn→∝,thevariableSnapproachesadefinitefinitelimit.Soinsteadofinfinitynisanimmensenumber.

Thislimitiscalledthesumoftheseries,andinthiscasewewriteLimn->∝Sn=Sw

WhereSwisthe‘populationorworldcyclevalueoftheseries’–itstotalinspaceortime.Itfollowsthatofinterestareseries that converges to 0’ sums, as they will be worldcycles in time, or to finite values, as they will represent acarryingcapacityofthewholeasapopulationinspace.

Butif,asn→∝,thelimitSndoesnotexist,thentheseriesissaidtobedivergentandinthiscasethereisnosenseinspeakingofitssum.Theseriesisaninflationarycaseofthemathematicalmirror.

But thanx God things are not so simple, because as we have seen there are two digital mirrors of worldcycles ofexistence,the0’-1unitcircle(palingeneticworldcycle)andthe1-∝,whichdifferinthe‘certaintyofoneofitsterms’–

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inthe0’1,thewhole1iscertainthefinitesimal0’uncertain,inthe1-∝,the1,fintiesimaliscertainandthe∝relativeuncertain.Asbotharemirrorsofeachother,wecanconsiderthecertaintyofoneofthetwolimitstocalculatehowfarnintheSnseriesoftheotherlimitreaches(rememberin5Dndoesnottendtoinfinity).Sowecanmakeusefulsomeinfinityseriesbycalculatingits¬entropicn->∝value.

Whilewecandiscoverthatinfiniteseriesinthe0’-1spherearenot.

Asa simpleexample (weshallalwaysusesimpleexamples inallour textsandstiences,aswewant toeducate the‘pro’inaphilosophyofstiencecommontoallplanesofspace-time,forhimorfuture5Dresearchersifeverthereisonebesidesthiswritertocompletethework:),letusconsidertheseries:

whosetermsformageometricprogressionwithcommonratiox.Thesumofthefirstntermsisequalto:if|x|<1thissumhasalimit:1/1-x=1+x+x2…

If | x | > 1, thenobviously the limit is∝,whichhas no value in classicmathematics as theseriesdiverges,butitdoesin5Das∝willbeanumber,normallyofthetrinity->decametricscale.

Adifferentsituationholdsforx=1,astheseriesbecomesthenadefinitionofthenaturalnumbers,suchasSngivesusthevalueofthennaturalnumber,andsoitexpresseshownaturalnumbersareborninsequentialtime.

Finally,ifx=–1thevaluesbecome,1,-1,1,-1,whichareinversevaluesforadimotion,representingthereforeinitspartialsumsthattakethevalues+1and0alternately,aworldcycleofexistenceinrepetitivepairs(0,1).

Theexample illustratesourcase for5D0’,∝ realistvalues for the ‘∆-1’and∆+1 limitsofaT.œdomain:weobtainmoreinformationinsuchacase,asallthecasesoftheseriesDOhaveameaning,whileinclassicmathematicsonlyfor|x|<1theseriesismeaningful;alltheothervaluesaredivergentSn->∞

Thesedifferencescanbebreachedwiththenexttheoreticalaxiomofclassicseries:

To each series there corresponds a definite sequence of values of its partial sums S1, S2, S3, ··· such that theconvergenceoftheseriesdependsonthefactthatthesumsapproachalimit,butalsoon5Dseriestheinversethatthere isa limittothenumberofsums;that is theconcept limit isNOTonlyappliedtothewholeSn(x)butthepartsN(S);which is thepentalogic justification, ifweweretodevelopherethemoreadvanced ‘concepts’ofmultiple timelogic (that is, therearearrowsof time, from∆-1 to∆1, fromSSóTTandStóTsandSóT) so for everything it isaworthyexercisetostudytheinverse,forA->B,B->A.

Itisthenpossibletodefineconversely,anarbitrarysequenceofnumbersS1,S2,S3,···whichcorrespondstoaseriespartial sumsofwhichwillbethenumbersofthesequence.

Thusthetheoryofvariablesrangingoverasequencemaybereducedtothetheoryofthecorrespondingseries,andconversely.Yeteachofthesetheorieshas independentsignificance in5D.Thepreviousseries is relevantbecause itsignifiesthecommonestprocessof‘erasing’ofprevioustermsinatimesequence;hencetheseriesinrealitytendsnottobethevalueofthesum,butthestepsoftime,as‘previousgenerations’dieaway;andthisindeedisthecaseforthemost famousseriesof themall, theFibonacci series,whichmimicsbestprocessesof reproduction in time,andsimilarmorecomplexconceptsasthe‘logcurve’

It is then when in 5D series we can prove the ‘natural tendency of all worldcycles’ towards zero, as If the seriesconverges,thenitsgeneraltermapproacheszerowithincreasingn,since:

Moreover,thedivergence=uselessnessofageometricprogressionwithcommonratiox>1followsimmediatelyfromthefactitsgeneraltermdoesnotapproachzero.Sowemightsaythatallmemorilessseriesrepresentaworldcycleofexistence,whichapproachestoazerosum,asmore‘timequanta’happen.

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Anothersimilarcriteriatofindthenifaseriesisusefulcanbeobtainednotfromthesimple‘memorialtimesumormemorilesssubstaction’asthepreviousmethods,butthroughthenextlevelofdimotionoperands,thex,÷.

It is thesocalledD’Alambertmethod:Letussupposethat,asnapproaches immensity, theratio (Un+1)/Unhasalimitq.Thenforq<1thesequencewillcertainlyconverge,whileforq>1itwilldiverge.Butforq=1thequestionofitsconvergenceremainsopen.

Thus the useful series are those that converge either in its sum as a whole in space towards a number or in itsdifferencebetweentermstowardsa0’sumintimeandoftenhaveareflectiononNature.WealreadymentionedtheFibonacciseries;wecanconsideranotherFinitesimalseriesthatconverge,exampleof:

Geometricseries.

Graphicalillustrationofthepointsofviewofa∝-finitegeometricseries.Before understanding calculusmathematicians were concernedwith‘relative’infinitesimalseries.

Sincesimilarparadoxesoccurinthemanipulationofinfiniteseries,suchas:1/2+1/4+1/8+⋯Thisparticularserieshasitsvaluepreciselyat1,thewhole,whichistheconceptualmeaningof∝ Immensity,aworldwitha limitingmembrain.Butthewaytodefineitisactuallytheinversetowhichthepowerseries

is written. That is to consider the 0’ undefined finitesimal where the series starts and then as ithappensinanyprocessofreproduction,considerita2xseriesthatwillgivebirthtothe1=∝relativewhole after the finitesimal 0’=1 becomes that whole, illustrating the relativity of the concepts offinitesimalandinfinite.Itisthusnotaconstantpartitionofasystem,butratheraconstantgrowth.

Toseewhythisshouldbeso,considerthepartialsumsformedbystoppingafterafinitenumberofterms.Themoreterms,thecloserthepartialsumisto1.Itcanbemadeascloseto1asdesiredbyincludingenoughterms.YetoncewearrivetotheMinimalquantaofthephysicalrealitywedescribe(cell,atom,individual,etc.)thereisNOneedtogobeyondexceptinerrorsofthemind.

Thusaseriescanbeconsideredbothascalar'searchforitsfinitesimalpart'butalsotheinversegrowthofaseedtothewhole.Becausefromthehumanwholeperspectivethe¡ndifferentelementisthefinitesimalwetendtowriteitbackwardsintime,butfortheUniversemostserieshappenfromthefinitesimaltothewhole.Yetbothcasestheyarealways'limited'bythesizeofthe'finitesimal'.

Ageometricseriesisaserieswithaconstantratiobetweensuccessiveterms.Sotheseries½+1/4+1/8…isgeometric,becauseeachsuccessivetermcanbeobtainedbymultiplyingtheprevioustermby1/2.

Eachofthepurplesquareshas1/4oftheareaofthenextlargersquare(1/2×1/2=1/4,1/4×1/4=1/16,etc.).Thesumoftheareasofthepurplesquaresisonethirdoftheareaofthelargesquare.

Wecanthenconsidertobeaseriesthatdiminishestill it reachesthe 'finitesimal'1/npartof thewhole.And itcaneasilybecastedasapolynomial;sincethetermsofageometricseriesformageometricprogression,meaningthattheratioofsuccessivetermsintheseriesisconstant.Thisrelationshipallowsfortherepresentationofageometricseriesusingonlytwoterms,randa.Thetermristhecommonratio,andaisthefirsttermoftheseries.

Intheexamplewemaysimplywrite:

a+ar+ar2+ar3…a=1/2andr=1/2

Thebehaviorofthetermsdependsonthecommonratior:

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If r isbetween−1and+1, the termsof theseriesbecomesmallerandsmaller,approaching 0’ in the limitand theseriesconvergestoasum.Inthecaseabove,whererisonehalf,theserieshasthesumone.

Ifrisgreaterthanoneorlessthanminusonethetermsoftheseriesbecomelargerandlargerinmagnitude.Thesumofthetermsalsogetslargerandlarger,andtheserieshasnosum.(Theseriesdiverges.)

Ifrisequaltoone,allofthetermsoftheseriesarethesame.Theseriesdiverges.

Ifrisminusonethetermstaketwovaluesalternately(e.g.2,−2,2,−2,2,...).Thesumofthetermsoscillatesbetweentwovalues(e.g.2,0,2,0,2,...).Thisisadifferenttypeofdivergenceandagaintheserieshasnosum.

ForexampleinGrandi'sseries:1−1+1−1+···.

Geometricseriesareamongthesimplestexamplesofimmenseserieswithfinitesums,althoughnotallofthemhavethisproperty.

Historically,geometricseriesplayedanimportantroleintheearlydevelopmentofcalculus,andtheycontinuetobecentralinthestudyofconvergenceofseries.

Geometric series are used throughoutmathematics, and they have important applications in all sciences, as all ofthem are obviously scalar in its form, and respond to any of the 3 possible behaviors of systems, 'convergentinformation',divergententropyandrepetitive=reproductiveoscillation.Andfinallytheyarestablein itstimemotiongivenbytheconstantratiobetweenitsgeometricterms.

Thuswecansayageometricseriesisagoodmirrorofabalanced∆STrepetitive‘present’eventandassuchreal.

Ofthemanymirrorcorrespondencesbetweenseriesand5Dwewantnowtostresstherelationshipbetweenthepartandthewhole,aselementstheternarystructureofanyT.œwithitssingularity,thatcanbeconsideredthea,initialterm, the FINITESIMAL above all other finitesimals, the king of the hill so to speak, its membrane and the spacebetweenthem.

Thisrelationshipistrulyenlighteningofthesymmetrybetweenthe3regionsinspaceofabeing,andits3regionsinscale.Whereasthecentralfinitesimal@-mindisthefinitesimalofthelowerplane,theexternalmembranethe'largerterm'arnoftheseries,andvitalenergywithinthem,theintermediatetermsoftheserieswhichareirrelevant.

Soasthesingularity@=a,oftheseriesexpandsthroughthevitalenergyelementsingrowing'circles'toreachthefinal'membrane'arn,magicallythoseirrelevantvitalspacecellswilldisappearinthefinalcalculusofthevalueoftheseries.

Furtheron,thosesumswillbelimitedbyn,whichISTHEvalueoftheNUMBEROF'Planes'withinthevitalenergy(concentriccircles)requiredtoarrivetothesurfaceofit.

Soacanalsobeviewedastherelative'radius'ofthesingularitymind,whichgivesconceptualbirthtotheformulaoftheangularmomentumoftheseries,wherermv,signifiesr=sumofsingularityradius(imaginetheinnerregionofthesystemasanArchimedeanspiral)mthevitalenergymass,andvthemembrane.

All this is expressed in terms of discrete numbers - not geometric continuous motion - by the classicformula:

Forr≠1,thesumofthefirstntermsofageometricseriesis:

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Aswesee,the@singularityvalueand its final term,arnaretheONLYvaluesthatmatter,withall the intermediateterms 'absorbed' in the dynamic relationship betweenmembrane and singularity by them. If s, is the value of theseries for the singularity, without the membrane, rs is the value of the system for the membrane, without thesingularity.Asthevitalenergywithinhasboththesingularityandthemembraneasits'Klein'limitsofanon-euclideansphere,whichtheyneverreach.Andsowerestfromthe 'Singularity',Splus ITS perceptionofthevitalenergy,themembrane,'rs',anditsfeeding(negativevalue)ofthevitalenergy,S-rS,tosearchfortheSolutionofthepowerserieswhichisnotthemembraneviewbutthesingularityview,s:

And so the solution as always is that of the mind view (in any discrete, numerical self-centeredanalysis)s=a(valueofthesingularity)multipliedbytheparenthesis.

Thenwecaneasilyseethesymmetryofthattopologicalexplanationoftheseries,withitsscalartranslationasatraveldownascalefromthewholetothefinitesimals.Sinceaswedifferentiatethoseseriestoconvergeandmakesense,becausewearetravelingdownthescaletothefinitesimals,r,asngoestoImmensity,mustbelessthanonefortheseriestoconverge.Thesumthenbecomes:

…theleft-handsidebeingageometricserieswithcommonratior.

Thebeauty and simplicityof the formula showsbyOccam's razorprinciple indeed its 'essential nature' in termsoftime-spacelaws.

Itisquiteinterestingthentounderstandintermsofthe5Dimotionsando-1=1-,time-spacedualsphere(essentialforquantumphysics) thevariationsof thepower series.As theywork for theo-1 sphere, inwhich the series travelsascaleofthefifthdimension:from1∆downto∆-1vs. itsentropicdivergentexpansionwhenr is largerthan±1,asittravels in the1- sphere,which shouldhavea solution,whenwedefinea relative infiniteas thevalueof thewholeperceivedfromthefinitesimalpointofview,whichmeansarelativeinfinite.Thenwemakeatravelupwardsfromthe∆-1finitesimalor∆-beingtothe∆+1world.

So those series represent the1Dand4-5Dimotions,while the3rd reproductivedimotionhappenswhen r=1, as thereproductive sum that creates terms of a reproductivewave,which in a lineal sum of stepswill represent the 2Dlocomotionof thebeing. Finally if r is -1 the series forms a 'steady state' 0’ sumworld cycle, anoscillationof twovalues.

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So the key concept of a proper 5D scalar interpretation of series (this analysis on the simplest of all series for 5Dadvanced theorywould obviously expand to power series Taylor series etc, but we leave this work for the futurepouringofmynotebooksorincaseIlikelydieearlier,forfutureresearchers)istheconceptoffinitesimalsandrelativeinfinites,

Thelimitofasequence

InthatregardweamendtheworkoftheGermanmathematicianKarlWeierstrassanditsformaldefinitionofthelimitofasequenceasfollows:

Considerasequence(an)ofrealnumbers,bywhichismeantaninfinitelist:a0,a1,a2,….

It is said that an converges to (or approaches) the limit a as n tends to Immensity, if the followingmathematicalstatementholdstrue:Foreveryε>0,thereexistsawholenumberNsuchthat|an−a|<εforalln>N.Intuitively,thisstatementsaysthat,foranychosendegreeofapproximation(ε),thereissomepoint inthesequence(N)suchthat,from that point onward (n>N), every number in the sequence (an) approximates a within an error less than thechosenamount (|an−a|<ε). Stated less formally,whennbecomes largeenough,an canbemadeasclose toa asdesired.

Forexample,thesequenceinwhichan=1/(n+1),thatis,thesequence:1,1/2,1/3,1/4,1/5,…,goesonforever.

Everynumberinthesequenceisgreaterthan0’,but,thefartheralongthesequencegoes,thecloserthenumbersgetto0’.Forexample,alltermsfromthe10thonwardarelessthanorequalto0.1,alltermsfromthe100thonwardareless than or equal to 0.01, and so on. Terms smaller than 0.000000001, for instance, are found from the1,000,000,000th term onward. In Weierstrass's terminology, this sequence converges to its limit 0 as n tends toImmensity. The difference |an−0| can bemade smaller than any ε by choosingn sufficiently large. In fact,n>1/εsuffices.So,inWeierstrass'sformaldefinition,Nistakentobethesmallestinteger>1/ε

This examplebrings out several key features ofWeierstrass's idea. First, it does not involve anymystical notionofinfinitesimals;allquantitiesinvolvedareordinaryrealnumbers.Second,itisprecise;ifasequencepossessesalimit,then there isexactlyone realnumber that satisfies theWeierstrassdefinition.Finally,although thenumbers in thesequencetendtothelimit0,theyneednotactuallyreachthatvalue.

Nowthisn>1/εisexactlywhatLeibnizwithoutsomuchpedanticformalismconsideredthefinitesimal,whatwecallthequantaofan∆-1scaleandwhatphysicistscall in itsstudyofdifferentPlanes,theminimal 'error-quanta'/2π,k-entropy,or'Planckmass'(Blackholeofacomptonwavelengthvolume,orminimalquantaofgravitational∆+1Planes).

Inthegraph,1/±10²isthelimitconsideredthefinitesimalofthisparticular'graphperception'.Andalsotheerrorofourmeasure,asifweaddanother1/±10²,theseriesbecomesawhole.

Thusmostparadoxesofmathematicsarisefromnotunderstandingthosesimpleconcepts,aswellasthemeaningof'inversenegativenumbers'.

Forexampleaninfiniteserieswhicharelesswell-behavedaretheseries:1−1+1−1+1−1+⋯Ifthetermsaregroupedoneway:(1−1)+(1−1)+(1−1)+�,thenthesumappearstobe:0+0+0+�=0.

Butifthetermsaregroupeddifferently,1+(−1+1)+(−1+1)+(−1+1)+�thesumis1+0+0+0+�=1.

It would be foolish to conclude that 0=1. Instead, the conclusion is that theserieshas adue value, and so it iscreativeoscillatoryserieswithatimedynamicthatcannotbemerelysaid,nottohaveasolution,buthas2.

Ithasthereforeaninternaldualstructure,whichinmodern¬Algebraisthegroup:

'a':1-1=0.Andsoifweacceptthatinternal∆-1unitfortheseriesgroupingandits'realvalueis:

a+a+....=0+0+0...=0.

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Sowecanwriteitintermsofthegeneratoras:

∑$t(+1)<≈>∑ðƒ(-1),whichdefinesgenericallyafeed-back'worldcycle'whosesumis0’.

Inclassicmathsofasinglespace-timecontinuum,thedifferencebetweenbothseriesisclearfromtheirpartialsums.Thepartialsumsof1/2+1/4...getcloserandclosertoasinglefixedvalue—namely,1.Thepartialsumsofa+,withoutitsinternal∆-1(a)structure,alternatebetween0and1,sotheseriesneversettlesdown.

Aseriesthatdoessettledowntosomedefinitevalue,asmoreandmoretermsareadded,issaidtoconverge,andthevaluetowhich itconverges isknownasthe limitof thepartialsums;allotherseriesaresaidtodiverge.But in∆STmanydivergingseriesbecomewhenconsideredalsoitsinternalstructure,convergentandwell-behaved.

Actually,withoutevenexperimentalevidence, thereexist subtleproblemswithsuch 'infinite' construction. Itmightjustifiably be argued that if the slices are infinitesimally thin, then eachhas 0’ area; hence, joining them togetherproduces a rectanglewith 0’ total area since 0+0+0+�=0. Indeed, the very idea of an infinitesimal quantity isparadoxicalbecausetheonlynumberthatissmallerthaneverypositivenumberis0itself.

Thesameproblemshowsupinmanydifferentguises.Whencalculatingthelengthofthecircumferenceofacircle,itisattractive to think of the circle as a regular polygon with infinitely many straight sides, each infinitesimally long.(Indeed,acircleisthelimitingcaseforaregularpolygonasthenumberofitssidesincreases.)Butwhilethispicturemakessenseforsomepurposes—illustratingthatthecircumferenceisproportionaltotheradius—forothersitmakesnosenseatall.Forexample,the“sides”oftheinfinitelymany-sidedpolygonmusthavelength0,whichimpliesthatthecircumferenceis0+0+0+�=0,clearlynonsense.

Sobyreductioadabsurdum,thelimitsofinfinitesimalsarealwaysan∆-1quanta.THISofcoursealsoresolvesalltheCantor'snonsenseofdifferentinfinitiesanditsparadoxes.Itisjust'math-fiction'andworthlesstostudy.

In5Dmaths,liestheexhaustionmethoddoeslimitthepartstofinitesimals,asarealistmethod,whichimpliesnaturealso limits its divisions. This conceptwould be lost in the 3rd formal age, alsowith the ‘lineal bias’ introduced onDedekind’sconceptofarealnumberNOTasaproportion/ratio,betweenquantitativeparametersofthe‘parts’ofawhole,orthe‘actions’ofasystemanditsSt<ST>Tsparameters,whichiswhatitis,butasan‘abstractcut’inalinealsequentialorderof‘abstractnumbers’.

In the classic STi balanced age, both the limitsmethodand finitesimalmethodof Leibniz considered infinitesimals,finitesimalsthatiswitha‘cut-offlimit’andrealnature.

Thoselimitsareminimal‘steps’ofanyscale(intime-motion),orminimalparts(inspace-forms).

Furthercomments

Asusualwecannotbeexhaustiveinanythemeof5Dbutjustgivea‘feeling’ofthedisciplineandhowitcorrectstheerrorsoftheaxiomaticmentalmethodofjustificationofhumind’smathematicalspace,asreality.

Seriesinthatsensearealsoconnectedtotheconceptsofdifferentinfinities,socherishedinmodernalgebra(Cantor’scardinal infinitiesandall that jazz)…Theirparadoxesdisappear thoughwhen thenumberofelementsof the seriesreduces to∝, so thenwe can always compare. In fact it is the classic method to define a series as divergent orconvergent.

Ifwearegiventwoseries:

with positive terms such that for all values of n, beginningwith a certain one, we have theinequality:

thentheconvergenceofthesecondseriesimpliestheconvergenceofthefirst,andthedivergenceofthefirstimpliesthedivergenceofthesecond.Considerthesimplestcaseoftheharmonicseries:

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which itmight seem to converge to K, as the numbers diminish. But if wecompareitwiththetermsofaserieswherethesumoftheunderlinedterms

ineachcase isequal to½butthe last termof thosepartialsumsSncoincideswiththesametermoftheharmonicseries(S4=S4,S8=S8,etc.):

It is clear that the sumSnof the2nd series (S2) approaches Immensitywithincreasingn,(∑1/2)andconsequentlytheharmonicseriesdivergesto‘potentialinfinity’evenifitremainssmallerthanS2

Can then we make the harmonic series converge towards a constant, as it seems a ‘natural series’ of growingfinitesimals(whenaddedinverselyfrom1/max.n).Yeswedo,whenwerisetoapoweritsterms:

Againthiscanbeprovedcomparingwithaseriesthatconvergesto1,thewhole:

withpositivetermsconvergestounityasitssum.SinceitspartialsumsSn,areequalto:

Ontheotherhand,thegeneraltermofthisseriessatisfiestheinequality:

fromwhichitfollowsthattheseries:

converges.

Againwedofininthepreviousseries1/n-1–1/n,whichwrittenbackwardsstartinginthefinitesimal,formsanaturalprogressionfrom1/nto1,withthememorialerasingofthepreviousterm,asimplenaturalformtogrowto1;whichdohavemanyalternativepaths/series.

Universalconstantsasseries.

Itisthenobviousthatallthe‘numbers’weconsidered‘ratios’offundamental‘dimotions’ofreality;thatis,Universalconstantscanbewrittenaspowerseries,whichshowstheirsymmetryin∆¡-1>∆,essentialtounderstadtheconstantentanglementbetweenscale,spaceandtimeintheUniverse.

Aswestudythemindifferentpartsofthosetextsoncalculusandalgebra,wereferthereadertothem.

Such series converge, themore sowhenwe transfer them to the complex plane,which due to its ±1 ‘I’ variation,convertslinealprocessesintocyclicalones,mirroringbetterallfunctionsrelatedtotime.

Series arealsoa justification forpolynomialsbeyond the simplest spatial viewof them in3 stepsofdimensionsofspace(point,line,volume)ormotionsoftime(distance,motion,acceleration):

Polynomialsasdivergentorconvergentscalarseries.

In mathematics, a power series (in one variable) is an infinite series of the form

where an represents the coefficient of the nth term and c is aconstant.an is independentofxandmaybeexpressedasa functionofn (e.g.,an=1/n!).Powerseriesareuseful inanalysissincetheyariseasTaylorseriesofinfinitelydifferentiablefunctions.

Inmanysituationsc(thecenteroftheseries)isequalto0’,forinstancewhenconsideringaMaclaurinseries.Insuchcases,thepowerseriestakesthesimplerform

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Any polynomial can be easily expressed as a power series around any center c,althoughmostofthecoefficientswillbe0’sinceapowerserieshasinfinitelymanytermsbydefinition.Forinstance,thepolynomialf(x)=x²+2x+3canbewrittenasapowerseriesaroundthecenterc=0as

Or around any other center c One can view power series asbeing like "polynomials of infinite degree," although power

series are not polynomials. These power series are also examples of Taylor series,which are the key dimotions ofscalarmotion(1/1-x),entropy(exponential)and1Dimotion(Sin).

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IIAGEOFCALCULUS:THEOPERANDS∫∂ANDITS‘SENTENCES’:PDES&ODES

WestudythesecondageofAlgebra, theageofcalculuswith its reflectionof the5DimotionsofTimespace,as thebestdisciplinetostudythelawsoftimes=changes,whicharethefundamentallawsofaUniversemadeof‘timespacedimotions’,inwhichspatialmentalspacesareaMayaofthesenses.

Theconceptsofmathematicalanalysis,suchasthederivativeortheintegral,astheypresentedthemselvestoNewtonand his contemporaries, had not yet completely “broken away” from their physical and geometric origins, such asvelocityandarea.Infact,theywerehalfmathematical incharacterandhalfphysical.Theconditionsexistingatthattime were not yet suitable for producing a purely mathematical definition of these concepts. Consequently, theinvestigator could handle them correctly in complicated situations only if he remained in close contact with thepracticalaspectsofhisproblemevenduringtheintermediate(mathematical)stagesofhisargument.

Newtonwasguidedatallstagesbyaphysicalwayoflookingattheproblem.ButtheinvestigationsofLeibnizdonothave such an immediate connectionwith physics, a fact that in the absence of clear-cutmathematical definitionssometimesledhimtomistakenconclusions.Ontheotherhand,themostcharacteristicfeatureofthecreativeactivityof Leibniz was his striving for generality, his efforts to find the most general methods for the problems ofmathematical analysis; and its depth of philosophical understanding of finitesimals and wholes, shown also in hissuperior symbolism. The evolution of the concepts of mathematical analysis (derivative, integral, and so forth)continued,particularly in theworkofCauchy,which idealizedtheconceptofa limitandused itas thebasis forhisdefinitionsofcontinuity,derivative,differential,andintegral.Thesedefinitionsarewidelyusedinpresent-dayanalysisandmustbecorrectedbacktotheGreekandLeibniz’sscalarview.

Regardingpracticalapplicationsuch idealismrequiresthe limitationofthetruefinitesimalsofchangeroftheactualworld, solvedwith theexpedientmethodof usingdifferentials. Thismeans that at every stepof ourmathematicalargument the results obtained will contain certain errors, which may accumulate as the number of steps in theargumentincreases.Butmathematicalidealismdeniesit.

Still no other discipline of science was so close to understand time=change as calculus was even if its whys werehidden in the ‘magic’ of its techniques, as it subconsciously applied the pentalogic of different derivatives for eachdifferentfunctionrepresentingeachdistinctdimotionofspace-time.Andthenoncethosederivativesofchangeswereapplied,thewholefunctionofexistencecouldbeintegratedasawhole(ODEs).Whilemultipleprocessesoffinitesimalcalculation of different variables, S=T equivalences between curvature and motion, etc. could be applied to theresolution of complex events between multiple T.œs performing different dimotions through PDEs and calculus ofvariations.

So not only pentalogic special analysis of change could be performed with derivatives and integrals of differentoperands, but very complex ‘sentences’ of sequential changesmodeledwith the 3 fundamental complex syntacticequationsofcalculus,ODEs,PDEsandcalculusofvariations,withintheentropiclimitsofanA-Bdefiniteintegralorthefirstandfinalmomentonatimepath,orthemaximalandminimalorthepointsinwhichthefunctionbecamezerocuttingtherealline,orinitsfirstorsecondderivative.

So each equation of mathematical physics studying the motion of physical systems hid a sequence of ‘existentialalgebra’forthephysicalpartsofthesimultaneousensembledofvariablesstudied.

Weshallconsider inthisbrief introductiontothegoldenageofcalculus,apentalogicanalysisofderivativesand itsinverse integrals, consideringhowcalculus representsa second layerof complexityover theoperands it can furtheranalyzeextractingitsminimalquantaofchange,and/orintegratingtheminnew‘dimensions’.

Tostudythenthecomplexcombinationsofcalculus(multiplederivativesandintegralsontimeandspace)-ODEs,PDEsand variationalmethods, and finally consider the fundamental simpler equations ofmathematical physics, growth,reproductionanddecay,studiedwiththem…

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THECONNECTIONBETWEENINTEGRALSANDDERIVATIVESIN∆ST:FINITESIMALSASLIMITS:5DAPPROACH

Those subtle philosophical considerations could only be done, once Leibniz established the tangent as the propermeasureofchange,whereY(S)andX(t),couldbeconsideredsimilar;S≈T.

Then once the condition of present balance is reached; calculusworks on praxis inmost caseswith the value of adifferential,whichistheequivalenttothefinitesimalminimaltimequantaofreality.

In other words, calculus consists on ‘calculating’ for a spatial present state, its quantity of time-change through aperiod of existence, which for worldcycles will be zero as we shall add, as explained in the worldcycle, both thepositivegrowthandnegativedecreasesofthey’sinusoidalfunctions.

Thosefunctionsthenwillbelimitedbythepointsofbirthandextinction.

While inversely intheintegrationofexponentialgrowthordecay,wewillreachrelative∝growthandthefunctionwillbe‘cut’byalogofmaximalgrowth.

Thefinitesimalsofphysicalscales.H,t,cc.

In the S=T homology amind stopsmotion into form, hence converts T into S. And this happens in an asymmetricmanneraccording to the choiceof upperor lower scalesof 5D:Huminds arebuilt, to see the largerwhole as slowspaceandthesmallerpartsas time,becauseof5Dmetrics (smaller fasterclocks, slower largerwholes).Andsowedistinguish finitesimals of time in lower scales (angularmomentum); and finitesimals of space associated to largerscales(c2);withanintermediatekfinitesimalofBoltzmanninthethermodynamicscale.

Inpraxistheexistenceofminimalquantawouldonlybecomeevidentwiththediscovery inmathematicalphysicsoftheminimal quantaof energy,which cannotbe cancelled, as it becomes the ‘minimal’ amount that givesorigin tovirtual particles (Heisenberg’s residual h/2). Yet again this minimal Planckton that cannot be eliminated WASinterpretedwith theweirdmentalpointof view that itwasan ‘uncertaintyofhumind’smeasure.Abizarrewayofcomplicatingreality;whichamountstosaythatthefirstcellisanuncertainmeasureoflife,thefirstatomanuncertainmeasureofmatter–andsothePlanckton,minimalquantaofenergybecametheoriginofoneofthemostarrogantdeludedinterpretationsofrealityhumind’simaginationhasdeviced;eversinceanassbreedersawabushburningandthought it was G. Bush talking to him – Copenhagen interpretation of quantum physics in terms ofmathematicalcreationismstillgoingstrong,alongMosaiccreationismforthethroneofhumind’sphilosophyoftheUniverse.Backtoreality,inanysystemweshallfindaminimalquantathatdoesnotgoaway.Inphysicalsystemsisthe'Planckton'(H-Planckconstant),thefirstquantaofangularperception,thefirstdimotionalspin,oflightspace-time,ourminimalpartinthe∆-3scaleoftimespace–asscaleIStheabsoluterealityandappearsalwaysassociatedtoaminimalspaceor time parameter, in which a scale starts to exist. The same finitesimal would be found in the Thermodynamicmolecular scale as absolute zero cannot be reached, and in the scale cosmological scale as cc, a quanta =area ofentropiclightmotionexists.

It is then essential to understand in depth 0’s finitesimals for the workings of the Universe to fully grasp therelationshipbetweentheelementsofPlanesofspacetime–itsfractalpointsandlinesthatadthemintimeorspaceandanalysis;inbothdirections-downasderivativesandupasnewdimensionscalledintegrals.

FinitesimalQuanta,asthelimitofpopulationsinspaceandtheminimalaction=dimotionintime.

In the idealized viewofmathematics this amounts to a sumof smallish areas either taken as finitesimals of space(Riemannintegral)orfinitesimalsoftime(Lebesgueintegral),withdifferentapplicationstocalculateitssumthroughanintervalofeventsorpopulations.Itisthenonlyneededtobeawarethatwhatmathematicianscallalimitofh->0IS,a limit that stops when h=0’; that is when h finds its real finitesimal value on the ‘units of measure’ we use tocalculate.ThisfinitesimalmightbeassmallasanatomwhichfromthehumanscaleintermsoftheAvogadronumberreducesto10-24partsbutregardlessofitstinysizetheconceptwewanttostressisthatforalimittoexisthmuststopat0’.

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Themethod of integrals (or rather its reality) becomes then a clear proof of the discontinuity of scalar spacetimesystems,whicharemadeofsteps,sincetheprocedureof integrationcalculates thevalueof anY(S)parameter foreachmin.X(T)=1/tfinitesimal‘stœpoftimequanta’measuredbyitsfrequencyintime.

It is thenNOT real topretend that this1/t=minimal timeeventmust reachabsolute zero.As Leonardonoticed, aninstant (zero)hasno time.Soasapointwithnopartshasnospace,anddoesnotexist.Timeexistsbecause it isaminimal interval of motion, separated often by a stop state of space, and a dimotion of perceptive information,conformingadiscontinuousseriesofstœps,S=T,thebasicbeatofreality,which iswhatweintegratebymovingupandrightXxYtofindanareaandthensummonthemup.

So to solve an integral we proceed as follows. We divide the interval [a, b] into n parts, notnecessarilyequal.WedenotethelengthofthefirstpartbyΔx1,ofthesecondbyΔx2,andsoforthupto the final part Δxn. In each segment we choose points ξ1, ξ2, ···, ξn and set up the sum:

Letussupposethatacurveabovethex-axisformsthegraphofthefunctiony=f(x).WeattempttofindtheareaSofthesegmentboundedbytheliney=f(x),bythex-axisandbythestraightlinesdrawnthroughthepointsx=aandx=bparalleltothey-axis.

ThemagnitudeSnisobviouslyequaltothesumoftheareasoftherectanglesshadedinfigure:

Thefinerwemakethesubdivisionofthesegment[a,b],thecloserSnwillbetotheareaS.Ifwecarryoutasequenceof such constructions, dividing the interval [a, b] into successively smaller and smaller parts, then the sums SnwillapproachS.

Thepossibilityofdividing[a,b]intounequalpartsmakesitnecessaryforustodefinewhatwemeanby“successivelysmaller”subdivisions.WeassumenotonlythatnincreasesbeyondallboundsbutalsothatthelengthofthegreatestΔxi in the nth subdivision approaches 0’. Thus the calculation of the desired area has in thisway been reduced tofindingthelimit:

Wenotethatwhenwefirstsetuptheproblem,wehadonlyanempirical ideaofwhatwemeanbytheareaofourcurvilinearfigure,butwehadnoprecisedefinition.Butnowwehaveobtainedanexactdefinitionoftheconceptofarea.Itisthelimit:

Wehavenowanintuitivenotionofarea,onthebasisofwhichwecancalculatetheareanumerically.

Wehaveassumedthat:ƒ(x)≥0’.Iff(x)changessign,theninfigure,thelimitwillgiveusthe¬Algebraicsumoftheareasofthesegmentslyingbetweenthecurvey=f(x)andthex-axis,wherethesegmentsabovethex-axisaretakenwithaplussignandthosebelowwithaminussign.

Definiteintegral.Theentropiclimitsofadomain.

TheneedtocalculatetheintegralSumlimitarisesinmanyotherproblemsinwhichanewdimensionisreachedbythesumoffinitesimalpaths.Forexample,supposethatapoint ismovingalongastraight linewithvariablevelocityv=f(t).Howarewetodeterminethedistancescoveredbythepointinthetimefromt=atot=b?

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Letusassumethatthefunctionf(t)iscontinuousinthesenseaforementioned(S≈T;h->0’);thatis,insmallintervalsoftimethevelocitychangesonlyslightly.Wedividetheinterval[a,b]intonparts,oflengthΔt1,Δt2,···,Δtn.TocalculateanapproximatevalueforthedistancecoveredineachintervalΔti,wewillsupposethatthevelocityinthisperiodoftimeisconstant,equalthroughouttoitsactualvalueatsomeintermediatepointξ1.Thewholedistancecoveredwill

thenbeexpressedapproximatelybythesum:

andtheexactvalueofthedistancescoveredinthetimefromatob,willbethelimitofsuchsumsforfinerandfinersubdivisions;thatis,itwillbethelimit:

Whereasthelimit∆twillbetheminimalstepinspaceandfrequencyoftimeofasingleevent.

SinceasthewordsaysareallimitiseachrelativeS(0’)=T(0’)ofchange.

Idealmathematicianstreatthosefinitesimalsundertheobsessionwithperfectmeasureasrealzerosbecausetheywilldischarge them to obtain a finite solution in ∆+1.We have discussed the falsehood in philosophical terms of suchapproach.But and this is the paradoxicalmarvel of theUniverse, as for the ∆+1 system all ∆-1 finitesimalsmatternothing,andare‘expendable’,asacitizenisexpendableforthestateorarmythat‘doesn’tcountcorpses’.

Weshalltrytogivesomeideaoftheseconcepts.Forthispurposeweconsiderthefollowingexample.

Wewishtocalculatetheareaboundedbytheparabolawithequationy=x2,bythex-axisandbythestraightlinex=1.Elementarymathematicswillnotfurnishuswithameansforsolvingthisproblem.Buthereishowwemayproceed.Wedividetheinterval[0,1]alongthex-axisintonequalpartsatthepoints:

andoneachofthesepartsconstructtherectanglewhoseleftsideextendsuptotheparabola.Asaresultweobtainthesystemofshadedrectangles,thesumSnofwhoseareasisgivenby:

LetusexpressSninthefollowingform: =3+¡n

Thequantity¡n,whichdependsonn,possessesaremarkableproperty:Ifnisincreasedbeyondallbounds,thenαnapproaches0’,the∆-1finitesimal.Thispropertymayalsobeexpressedasfollows:Ifwearegivenanarbitrarypositivenumber,∍, ,inclassiccalculus,thenitispossibletochooseanintegerNsufficientlylargethatforallngreaterthanNthenumber¡nwillbeequaltothegiven∍inabsolutevalue.

Whereas ∍ is the ‘real physical value’ of the finitesimal that ‘exists'; where our 5D calculus stops. While in theaxiomaticmethode canbechosenatwill as small as the idealistmathematicianwishes,which isNOTwhat realityshows. Ewill be h in quantumphysics, itwill be the residual k temperature and theminimal c2 part of amass orminimalunitofvacuumspacetimeinthegalaxy;itwillbeacellinanorganism,anatominamatterstate.Orelsewhatwillbeourcountingmadeof?Entelechies?

Thissaid,as∍issosmall,forthelarger∆+1an¡ndifferentitisOKtodischargetheresidualfinitesimalstoobtainanapproximate result. Since absolute precision for ∆+1, the planeof the observer is not needed andwedischarge in

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specificmethodologicalcalculustheremindersmallishsumofsome0’stocalculatearesultthattobemoreaccurateshouldbeadjustedwitha‘±’symboloran≈notan=identityone.

Soweobtainedtheareabelowtheparabolaas1/3rdbydischarginga finitesimaltermofthesum.This ‘discharged’quantityhoweverisreal–itisthetollwepaybutnevergetstozero.Inthemeasureofafractalcoast,thecoastgrowsin size and precisionwith smaller steps, butwemust stop the steps at a certain scale or elsewewould spend aninfinitetimemeasuringit.Informationthusbecomesidealizedtomakepossibleinfinitetimetomeasureit.

Formalsymbolsof5Dcalculusfromexistentialalgebra:-¡n,∍, ,finitesimal,¡ndifferent,¡nfinitesimal

Itisthennecessaryasalwaysin5Dtoslightlychangethesymbolsof5Dcalculus,adaptedtoexistentialalgebra–northat Ithinkhumindswilleverupgradeitschipsbutoncewearegone,fasterthanyouthink, likelymorerigorousAIrobotswilladoptit.

Thefinitesimalisthendefinedas-¡n,theminimalamountwedischarge,thatissubtract(reasonwhyithasanegativesymbol),fromtherealresulttoobtaintheidealizedmathematicalmirroroftheeventorspatialpopulation,asfor∆+1itisindistinguishable,whichwecall¡ndifferent(asimplerworld,withanethicalcomponent,because-¡nisrealandforitselfitmatters,evenifforthe∆+1worldisexpendable).Itcanalsobecalledafinitesimal(preferred)or-¡n≈finitesimalifyoulike.Thesymbol-¡alsomeansitistheunitof∆-¡,theminimalquantityofthewhole;andfinallythesymbols,,∍,,meanthey‘exist’,theare‘real’,0’doeshaveparts,itisanon-euclideanpointofitsown.

Itwouldbeeasytogivemanyexamplesofpracticalproblemsleadingtothecalculationofsuchanimperfectlimit.Wewill discuss some of them later, but for the moment the examples already given will sufficiently indicate theimportanceofthisideathatadaptsclassiccalculustoreality.AsnoweachoftheclassicfinitesimalsusedincalculusareasLeibnizputit‘aworldinitsown’,anditsstudyinreal,philosophicaltermsrevealsimportantpropertiesofthestructure of space-time and its modes of change. The following list of finitesimals can be then assessed in itspropertiesunderthoseconditions:

It is clear that xn, yn, and zn are -¡n≈finitesimals, the first of them approaching 0’ through decreasing values, thesecondthroughincreasingnegativevalues,whilethethirdtakesonvalueswhichoscillatearound0’.Further,un→1,whileυn does not have a limit at all, since with increasing n it does not approach any constant number butcontinually oscillates, taking on the values 1 and −1 – so the ¡nfinitesimal is the whole changing its direction ofexistence.Allof themwillbe subject to further scrutinywhen/ifmy finitesimal lifetimeallowsme topublisha fewpapersonmathematicalphysics,astheyappear inmultiplephysicalequations.Themost importantof them,being,Xn, the fundamental finitesimal, which we shall often comment on (the minimal part of an N whole, thedimensionlessangle/curvatureofamotion)anditscloselyconnected-1/n2,foracceleratedmotions.

Finallytonoticethatintheexample,-¡nmakesthevolumeslightlysmallerthan1/3rdas-1/2n>1/6n2;whichinreversefashion ifweconsiderthe innerregionofthecurvedparabola,makes itslightly largerthan2/3rdswhich isageneralruleforthe internalvolumeofcurvedsurfaces,alwaysslightly largerthanthepolygonal, lineal formitencloses. I.e.thehexagonhasa3perimeter,initsinscribedπ=3,14…circle;whichagainhasimportantconsequencesforthereal,vitalstructureoforgans,asthesmallerpartsare‘lineal’,andfitinthelargercurvedenclosures,leavingasafetyspaceto itswalls,whichoftenhas apertureswhen constructed asmost circles arewith 3diameters, equal toπ-3/π=4%,whichistheidealamountofouterrealityperceivedthroughthoseaperturesleftbythe3diametersthatconstructtheporousmembrain(percentageofenergyandmatterintheUniverseweobserve).

Finitesimalstreatedwithotheroperands.

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Ifwe shall study howoperands are treatedwith calculus it is customary to consider the reverse action of treatingfinitesimalswiththepolynomialoperands,andasnow‘finitesimals’arerealfractalpointwithparts, it isobviousallthelaws,propertiesandoperandsofpolynomialsworkwith‘finitesimals’,theso-calledlawsofoperationswithlimitsofclassicmathematics.

It is then not necessary (but possible to prove with the axiomatic ideal method, consistent in itself beyond thelimitationsweinclude,thatifthevariablesxnandynapproachfinitelimits,thentheirsum,difference,product,andquotientalsoapproachlimitswhicharecorrespondinglyequaltothesum,difference,product,andquotientoftheselimits.Thisfactmaybeexpressedas:

The only case that deserves further analysis, which is when a quotient of two finitesimal 0’s of different value isconsidered.Hereitisimpossibletostateinadvancewhethertheratioxn/ynwillapproachalimit,andifitdoes,whatthatlimitwillbe,sincetheanswertothisquestiondependsentirelyonthecharacteroftheapproachofxnandynto0’; thatmight result ineithera0’oran∝, thusprovingad lateral the falsehoodofCantorianequal infinities,asN2regardlessofCantor’smusingsISlargerthanNbuthasparadoxicallyLESSinformationasasetofnumbers(becausein5DlargerSpatialformshaveparadoxicallylessinformation,storedinthefastertimecyclesofsmallerbeings,SxT=C).Now, ifwe justcareforthe ‘size’thenthepreviousexamples,Xn=1/n&Yn=1/n2meansthat,Yn/Xn=1/n->0’ &Xn/Yn=n->∝

WhileXn/Zn:(-1)n/n->(-1)ndoesnotapproachanylimitbecauseZndoesnot;andinquotientsthedominantelementisthedenominatorwhichtendstobethepredatorthatimposesitspropertiestothewhole.

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DIFFERENTDIMENSIONALMOTIONSOFSPACE-TIMECHANGE

Wethenneedtoconsiderinhowmanydimensionsfinitesimalchangeanditsaggregatedaccountintoacontinuous∆+1parameterofthewholechangeoftheeventcanbeobserved,andhowitcanbediversifiedintoTime,scalarorspatialchange.Letusthenconsidertwoexampleofdualdimensional,holographicchange,inTs-speedandSt-volume,which were the first 2 themessolved historically, to see how calculus methods can be used equally forquanta=frequency=stepsoftime,orquanta=populations=finitesimalsofspace:byvirtueofS=T:

There aremany different parameters of change in space and time in human sciences, due to the lack of clear-cutunifyingconceptsofspaceandtime.

But in all we need to find a 'finitesimal quantity', either in time or in space, to measure 'changes of speeds andfrequencies of time motion for each spatial step', ∆s/∆t , or changes on volumes of space and populations ofsimultaneousspace-beings.Thedifferencebetweenbothanalysisisoneof‘persistenceofchange’or‘simultaneityofchange’studiedinspace,vs.sequentialtimechanges.Inspatialanalysisweoftencalculatea‘whole’domaininwhichpopulationshaveagradientofchangeinitsparameters,eveniftheyco-existtogether.

Intime,this‘gradient’ofchangeor‘acceleration’iscalculatedat‘eachinstantoftime’,forasinglepoint,andthusitschange,haslesser‘dimensionalityinspace’.

So the studyof change in space tends tohavemore ‘volumeanddimensionality’and ‘simultaneity’,as the studyofpuretimechanges(locomotion,entropy)isanalyzedwiththebeingreducedtoatimepoint,orevenloosingitsspatialsimultaneitythroughentropicprocesses.

Addinganewdimensionof'width-energy-population-intensity-density-pressure.'

Let us put an example and resolve it in terms of space-quanta (method of limits) which is the first ‘basicunderstanding’ofcalculusintermsofitsfinitesimalunits:

Quantaofspace.

Aspatialuseofthelimitconceptcalculatesnotatimebutaspacevolume,forebearofdifferentialcalculus:

Example2.Areservoirwithasquarebaseofsideaandverticalwallsofheighthisfulltothetopwithwater(figure1).Withwhatforceisthewateractingononeofthewallsofthereservoir?

Wedividethesurfaceofthewallintonhorizontalstripsofheighth/n.Thepressureexertedat eachpoint of the vessel is equal, by awell-known law, to theweight of the columnofwater lyingabove it. Soat the loweredgeofeachof the strips thepressure,expressed in

suitableunits,willbeequalrespectivelyto:

WeobtainanapproximateexpressionforthedesiredforceP, ifweassumethatthepressure isconstantovereach

strip.ThustheapproximatevalueofPisequalto:

Tofindthetruevalueoftheforce,wedividethesideintonarrowerandnarrowerstrips, increasingnwithout limit.Withincreasingnthemagnitude1/nintheaboveformulawillbecomesmallerandsmallerandinthelimitweobtaintheexactformula:

P=ah²/2

Leibniz rightly considered 1/n the 'finitesimal unit',whereaswe consider 1 thewhole, andn, itsminimal fraction,usually1ofits1010elements(1/1010):thestandardvalueoffinitesimalunits.

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Intheexampleagainthefinitesimallimitisextremelysmall.Howmuch?Weshouldconsiderstatisticalmechanicstofind it is thesizeofmoleculesofwater,which formbidimensional layersof liquid toshapethe3Dvolume,andareabout1020timessmallerthanthewholeintermsofAvogrado’sMols.

TheerrorεissosmallastobeP=(ah²/2)x1.00000000000000000001(1+1/n)

Andthisisageneralruleinmostcases:thefinitesimalerrorisassmallas1/n,wherenisthequantaofthescale.Sowhenwedo∆+1calculationsasinmostcasesitisirrelevant.Buttheoreticallyitisimportantandinfactitwillgiveusa'realist'conceptfortheuncertaintyprincipleofHeisenberg.

Hence unnoticeable,truly ¡n=finitesimal, still important to understand the idealization ofmathematical rules.As itmeanstheoreticallythatthecorrectconceptisadifferentialequation,wherethefinitesimalis‘real’.

The ideaof themethodof limits is thus simple, accurate and amounts to the following. In order todetermine theexact value of a certain magnitude, we first determine not the magnitude itself but some approximation to it.However,wemakenotoneapproximationbutawholeseriesofthem,eachmoreaccuratethanthelast.Thenfromexamination of this chain of approximations, that is from examination of the process of approximation itself, weuniquelydeterminetheexactvalueofthemagnitude.byignoringthefinitesimalerror.

Thesamepracticalproblemcanberesolvedwiththedifferentialusedasanapproximatevalueforthe increment inthefunction.Forexample,supposewehavetheproblemofdeterminingthevolumeofthewallsofasimilarclosedcubical boxwhose interior dimensions are 10 × 10 × 10 cm and the thickness of whosewalls is 0.05 cm. If greataccuracyisnotrequired,wemayargueasfollows.ThevolumeofallthewallsoftheboxrepresentstheincrementΔyof the function y = x3 for x = 10 and Δx = 0.1. So we find approximately:

Speedandacceleration:2DTT

Weusedasimplespatialcaseofagradientwithaclearequation,P=ah2/2tocompareitwithonecaseoftimechangeinwhichthegradientalsocausedbygravitational‘weight’isnotconstrainedbyawall,hencetheforceisreleasedtobecomeatimedimotions.Notsurprisinglyinsuchacase,asitwasestablishedexperimentallybyGalileo,thedistances covered in the time t by a body falling freely in a vacuum is expressed in terms of TT-acceleration by a similarformula:s=gt²/2

WhereasgisaconstantthatmeasuretheaccelerationonEarth,equalto9.81m/sec².

Whatisthevelocityofthefallingbodyateachpointinitspath?

LetthebodybepassingthroughthepointAatthetimetandconsiderwhathappensintheshortintervaloftimeoflengthΔt;thatis, inthetimefromttot+Δt.ThedistancecoveredwillbeincreasedbyacertainincrementΔs.Theoriginaldistanceiss1=gt²/2.

Fromtheincreaseddistancewefindtheincrement:

Thisrepresentsthedistancecoveredinthetimefromttot+Δt.Tofindtheaveragevelocityoverthesectionofthe

pathΔs,wedivideΔsbyΔt:

LettingΔtapproach0’,weobtainanaveragevelocitywhichapproachesascloseasweliketothetruevelocityatthepoint A. On the other hand, we see that the second summand on the right-hand side of the equation becomes

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vanishinglysmallwithdecreasingΔt,sothattheaverageυavapproachesthevaluegt,afactwhichitisconvenientto

writeasfollows:

Whilebothformulaearenevercomparedinclassicphysics,itisworthtonoticethattheyaremimeticbyvirtueofS=T,sonowthespatialgradient,‘h-eight’becomesthetemporalgradient,t-ime;theintensityofthegradient,givingforPbythea-massofliquid,isnowgivenbytheg-forceofthemassoftheEarth;andtheoutcomeisaninversedynamicparameteroftimechangePressurevs.astaticparameterofspatiallength,distance.

PressureisanenergySTdensity.Soapressuregradientisanenergydensitygradient,accelerationisaTT-doubletimemotion.Andyetasbothareultimately‘holographicST,TT’dualfunctionsitsequationsin‘scalarterms’arethesame,akeyconceptforallthesimilarequationswewillfindregardlessofthe‘holographicdimotion’weintegrateorderviate.

THEUPPERLIMIT:¡MMENSITIES:∝

Therelative∝ .+¡mmensity.

It is then customary in calculus to teach the inverse concept of an infinitemagnitude, whichwe also reduce to arelativeImmensity,∝;asinfinitiesloosemeaningandbecomeentropic,uncertaininthebordersof∆,orbeyondthedomainofexistenceintimeandspaceofthefunctionwedealwith.

Following the rules of ¡nglish that change slightly the wording of 5D we thus substitute the word infinity for+¡mmensity.Immensityasweallknowisnotinfinite,butitisimmensurablewhichisthemeaningthatmattershere,as such ‘largesse’ becomes the +¡wholeworld for the -¡nfinitesimal that finds it ‘infinite’, because it no longer canmeasureit.

Wedoestalkofan¡mmenselylargemagnitude,whichisdefinedasavariablexn(n=1,2,∝),withthepropertythatafterchoiceofan¡mmenselargepositivenumberM,thelimitofmeasureforthe-¡n=finitesimal it isnotpossibletofindanumberN>Mthatwecan‘countinreality’,withinthelimitsoftimeandspaceofour¡n≈finitesimalexistence,suchthatforalln>N,

M+¡ thus become the +¡mmense value,∝, that limits the world of X. Such a magnitude xn is said to approach+immensity.Ifitispositive(negative)fromsomevalueon,thisfactisexpressedthus:xn→+∝(xn→−∝).

Forexample,forn=1,2,∝;limlog1/n=-∝,limn2=+∝;limgtan(π/2+1n)=-∝

Itiseasytoseethenthatifamagnitude+¡Mis¡mmense,then-¡n=1/Misimmenselysmall,andconversely.

Something that as, 5Dmathematics is experimental,moving into the realm of ‘reality’, not an ‘idealmathematicalentelechy’ must have an experimental consequence on the study of ‘real immense creatures’. And indeed, weimmediatelynoticethatthelargestspeciesfeedon¡ndifferent¡n≈finitesimalsthatareparadoxically¡mmenselysmall.I.e.thelargestmammals,Whales,feedonthesmallestanimals,KrillsandPlanckton;thelargestcosmologicalbodies,gravitationalblackholes,feedinthesmallestquanta,‘gravitons≈neutrinos?’,etc.

Whichjustifiesthe∆±¡structureofnestedUniverses,wherethelargestbeingsaremadeofthesmallestparts.

This also shows thatwhenwe talkof an ¡ndifferent ¡n≈finitesimal,wearemostly referring toaquantaofentropicfeeding,whichisindifferenttothebeing.I.e.wereduceevenfurtherourfoodtoits¡ndifferentaminoacidsthatwewillthenreformtoourspecificinformation.

Thetopologicalviewissimple;thelimitsofthedomainofafunctionfor itsfinitesimalparts,arethemembrainandsingularitytheycannotreach,asthewholeforitsinnerpartsisanopenballtransitedbythefinitesimal.

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It is thus essential in calculus the question of the 'boundary conditions', in which the membrane determines thevolumewhichisintegratedastheSpace-timearea,surroundedbythebeingthatismeaningfultoitsterritory.

The area integrated on a function has then 2 S=T meanings. Either is a measure of its vital energy between thesingularityat0’point and itsmembrane, the limitsof thedomain in space (albeit elongated in the linealCartesianframe)orifweintegrateamotionbetweenitsinitialconditionwhereitreceivesitsmomentumanditsfinalstop.

Whenapointmovesbackandforthwithinitsworldasitperformsrepetitivedimotionsofexistencewecanintegratethepath toextract informationof itsmotion. For the larger∆+1world, the finitesimal integratedalong thepath isoftenapointofenergyorinformationsharedbetweenthemembrainandsingularityastheinitialpointandboundaryconditionofitsworldseenasatopologicalopenball,wherethemembrainisits‘birth-seed’stateandthesingularityitsfinalpointofdeath-entropy.Itisthusthequantathatthemembrain‘sendstothesingularity’forittoperceiveorfeed,whichmovesinapathofminimalaction=consumptionofenergybetweenboth.Andfinallythefinitesimalmightrepresent,an'ex-foliated'unitofangularmomentum,askinlayerofvitalspacesubtractingpiecebypiece,finitesimalbyfinitesimal,fortheT.œtocommunicateintheouterworld

Thelimitiscalledthedefiniteintegralofthefunctionf(x)takenovertheinterval[a,b],anditisdenotedby:

Theexpressionf(x)dxiscalledtheintegrand,aandbarethelimitsofintegration;aisthelowerlimit,bistheupperlimit.Andveryoftentheyaretheinitialandfinalpoint,the0’singularityandmembrainthatcancelaworldcycle.Andsoin5DaandbareNOTpartoftheintegral,asthefinitesimalifitisatimepathdoesnotexistonthose2limits,wemight consider to belong to ∆±1, or if it is a volume, represent themembrain and singularity of an open ball, of a‘differentsubstance’,nottobeintegrated.

RECAP.We change following the transformation of sciences into slightly different stiences, the concept of infinitefroma relative infinite,,andan infinitesimal fora finitesimal.The firstbeing thewholeofan ¡-planeof reality, theseconditsminimalpart.

Theconnectionbetweendifferentialandintegralcalculus.

Theproblemconsideredthenreducestocalculationofthedefiniteintegral:

AnotherexampleIStheproblemoffindingtheareaboundedbytheparabolay=x².

Heretheproblemreducestocalculationoftheintegral:

Wewereabletocalculateboththese integralsdirectly,becausewehavesimpleformulas forthesumofthefirstnnaturalnumbersandforthesumoftheirsquares.Butforanarbitraryfunctionf(x),wearefarfrombeingabletoaddupthesum(thatis,toexpresstheresultinasimpleformula)ifthepointsξi,andtheincrementsΔxiaregiventosuitsomeparticularproblem.Moreover,evenwhensuchasummationispossible,thereisnogeneralmethodforcarryingitout;variousmethods,eachofaquitespecialcharacter,mustbeusedinthevariouscases.

Soweareconfrontedbytheproblemoffindingageneralmethodforthecalculationofdefiniteintegrals.Historicallythisquestioninterestedmathematiciansforalongperiodoftime,sincethereweremanypracticalaspectsinvolvedinageneralmethodforfindingtheareaofcurvilinearfigures,thevolumeofbodiesboundedbyacurvedsurface,andsoforth.

WehavealreadynotedthatArchimedeswasabletocalculatetheareaofasegmentandofcertainotherfigures.Thenumberofspecialproblemsthatcouldbesolved,involvingareas,volumes,centersofgravityofsolids,andsoforth,graduallyincreased,butprogressinfindingageneralmethodwasatfirstextremelyslow.Thegeneralmethodcouldnotbediscovereduntilsufficienttheoreticalandcomputationalmaterialhadbeenaccumulatedthroughthedemandsofpracticallife.

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TheworkofgatheringandgeneralizingthismaterialproceededverygraduallyuntiltheendoftheMiddleAges;anditssubsequentenergeticdevelopmentwasadirectconsequenceoftherapidgrowthintheproductivepowersofEuroperesulting from the breakup of the former (feudal) methods of manufacturing and the creation of new ones(capitalistic).

Theaccumulationoffactsconnectedwithdefiniteintegralsproceededalongsideofthecorrespondinginvestigationsof problems related to thederivativeof a function. The reader already knows from that this immensepreparatorylaborwascrownedwithsuccessinthe17thcenturybytheworkofNewtonandLeibnitz.ItisinthissensethatNewtonandLeibnitzarethecreatorsofthedifferentialandintegralcalculus.

One of the fundamental contributions of Newton and Leibnitz consists of the fact that they finally cleared up theprofound connection between differential and integral calculus, which provides us, in particular, with a generalmethodofcalculatingdefiniteintegralsforanextremelywideclassoffunctions.

Toexplainthisconnection,weturntoanexamplefrommechanics.

Wesupposethatamaterialpointismovingalongastraightlinewithvelocityv=f(t),wheretisthetime.Wealreadyknowthatthedistanceacoveredbyourpointinthetimebetweent=t1andt=t2isgivenbythedefiniteintegral:

Nowletusassumethatthelawofmotionofthepointisknowntous;thatis,weknowthefunctions=F(t)expressingthedependenceonthetimetofthedistancescalculatedfromsomeinitialpointAonthestraightline.Thedistanceσcoveredintheintervaloftime[t1,t2]isobviouslyequaltothedifference:σ=F(t2)-F(t1)

Inthiswayweareledbyphysicalconsiderationstotheequality:

whichexpressestheconnectionbetweenthelawofmotionofourpointanditsvelocity.

FromamathematicalpointofviewthefunctionF(t),maybedefinedasafunctionwhosederivativeforallvaluesoftinthegivenintervalisequaltof(t),thatis:

F'(t)=ƒ(t).Suchafunctioniscalledaprimitiveforf(t).

Wemustkeepinmindthatifthefunctionf(t)hasatleastoneprimitive,thenalongwiththisoneitwillhaveaninfinitenumberofothers; for ifF(t) isaprimitivefor f(t), thenF(t)+C,whereC isanarbitraryconstant, isalsoaprimitive.Moreover,inthiswayweexhaustthewholesetofprimitivesforf(t),sinceifF1(t)andF2(t)areprimitivesforthesamefunctionf(t),thentheirdifferenceϕ(t)=F1(t)−F2(t)hasaderivativeϕ(t)thatisequalto0’ateverypointinagivenintervalsothatϕ(t)isaconstant.*

Fromaphysicalpointofviewthevariousvaluesof theconstantCdetermine lawsofmotionwhichdiffer fromoneanotheronlyinthefactthattheycorrespondtoallpossiblechoicesfortheinitialpointofthemotion.

Wearethusledtotheresultthatforanextremelywideclassoffunctionsf(x),includingallcaseswherethefunctionf(x) may be considered as the velocity of a point at the time x, we have the following equality:

whereF(x)isanarbitraryprimitiveforf(x).

This equality is the famous formula ofNewton and Leibnitz,which reduces the problemof calculating the definiteintegralofafunctiontofindingaprimitiveforthefunctionandinthiswayformsalinkbetweenthedifferentialandtheintegralcalculus.

Manyparticularproblemsthatwerestudiedbythegreatestmathematiciansareautomaticallysolvedbythisformula,statingthatthedefiniteintegralofthefunction.f(x)ontheinterval[a,b]isequaltothedifferencebetweenthevalues

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of any primitive at the left and right ends of the interval.‡ It is customary to write the difference (30) thus:

Example1.Theequality:(x³/3)'=x²showsthatthefunctionx³/3isaprimitiveforthefunctionx2.Thus,bytheformulaofNewtonandLeibnitz:

Example 2. Let c and cʹ be two electric charges, on a straight line at distance r from each other. The attraction Fbetweenthemisdirectedalongthisstraightlineandisequalto:

F=a/r²(a=kccʹ,wherekisaconstant).TheworkWdonebythisforce,whenthechargecremainsfixedbutcʹmovesalongtheinterval[R1,R2],maybecalculatedbydividingtheinterval[R1,R2]intopartsΔri.

Oneachofthesepartswemayconsidertheforcetobeapproximatelyconstant,sothattheworkdoneoneachpartisequal to: . Making the parts smaller and smaller, we see that the work W is equal to the integral:

Thevalueofthisintegralcanbecalculatedatonce,ifwerecallthat:

Sothat:

Inparticular,theworkdonebyaforceFasthechargecʹ,initiallyatadistanceR1,fromc,movesouttoImmensity,is

equalto:

From the arguments given above for the formula of Newton and Leibnitz, it is clear that this formula givesmathematicalexpressiontoanactualtieexistingintheobjectiveworld.Itisabeautifulandimportantexampleofhowmathematicsgivesexpressiontoobjectivelaws.

Weshouldremarkthatinhismathematicalinvestigations,Newtonalwaystookaphysicalpointofview.Hisworkonthe foundations of differential and integral calculus cannot be separated from his work on the foundations ofmechanics.

Thedefiniteintegral

Returningnowtothedefiniteintegral,letusconsideraquestionoffundamentalimportance.Forwhatfunctionsf(x),defined on the interval [a, b], is it possible to guarantee the existence of the definite integral: Namely anumbertowhichthesum:

TendsaslimitasmaxΔxi,→0?Itmustbekeptinviewthatthisnumberistobethesameforallsubdivisionsoftheinterval[a,b]andallchoicesofthepointsξi.

Functionsforwhichthedefiniteintegral,namelythelimit(29),existsaresaidtobeintegrableontheinterval[a,b].Investigationscarriedoutinthelastcenturyshowthatallcontinuousfunctionsareintegrable.

Buttherearealsodiscontinuousfunctionswhichareintegrable.Amongthem,forexample,arethosefunctionswhichareboundedandeitherincreasingordecreasingontheinterval[a,b].

Thefunctionthatisequalto0’attherationalpointsin[a,b]andequaltounityattheirrationalpoints,mayserveasanexampleofanonintegrablefunction,sinceforanarbitrarysubdivisiontheintegralsumsn,willbeequalto0’orunity,dependingonwhetherwechoosethepointsξi,asrationalnumbersorirrational.

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Letusnote that inmanycases the formulaofNewtonandLeibnitzprovidesananswer to thepracticalquestionofcalculatingadefiniteintegral.Butherearisestheproblemoffindingaprimitiveforagivenfunction;thatis,offindingafunctionthathasthegivenfunctionforitsderivative.Wenowproceedtodiscussthisproblem.Letusnotebythewaythattheproblemoffindingaprimitivehasgreatimportanceinotherbranchesofmathematicsalso,particularlyinthesolutionofdifferentialequations.

Aswestatedbeforeintegralsaremostlyusefulwhenwearestudyinga'defined'fullS<ST>Tsystemwithamembraneorcontourclosingthesurface.Asintegralsaremoreconcernedwith'space'and'derivatives'withtime.Andfurtheron,thosewhichintegratespace-timesystems,ordoubleandtripleintegrals.

CalculusofALLtypeofvitalspaces,enclosedbytimefunctions,witha‘scalar’pointofview,parameterthatmeasured what the point of view extracted in symbiosis with the membrane, from the vital space itenclosed.Alas,thisquantityabsorbedandab=usedbythepointofview,onthevitalspacewouldbecalled‘Energy’, thevitalspace ‘field’, themembrane ‘frequency’, the finitesimal ‘quantaorUniversalconstant’,

andthescalarpointofview‘activemagnitude.

Thefundamentallanguageofphysicsaredifferentialequations,whichallowtomeasurethecontentofvitalspaceofasystem.Therichnessandvarietiesof‘worldspecies’willdefinemanyvariationsonthetheme.Sometimestherewillnotbeacentralpointofview,andwetalkofaliquidstate’,wherevolumeswillnothavea‘gradient’,but‘Pressure’,thecontrollingparameterofthetimemembranewillbeequal,orrelatedtothegradientoftheeternalworldp.o.v.oftheEarth(gravitationalfield).

Thenweshall integratealong3parameters,thedensitythatdefinesthe liquid,theheightthatdefinesthegradientand the volume enclosed. Liquids, due to the simplicity of lacking an internal POV, would be the first physicalapplicationofLeibniz’s findingsbyhis students, theBernoulli family.Nextaviolinplayerwould find thedifferentialequationofwaves–theessentialequationofthemembranesofpresenttimeofallsystems.The3rdtypeofequations,thoseofthecentralpointofview,willhavetowaitamathematician,Poisson–latterrefinedbyEinsteininhisGeneralRelativity.

This is theerrorofNewton.All cyclesare finite,as theyclose into themselves.Allworldcyclesof lifeanddeatharefiniteastheyendastheybeguninthedissolutionofdeath.Allentropicmotionsstop.Alltimevorticesoncetheyhaveabsorbedalltheentropyoftheirterritorybecomewrinkled,anddie.Newtondied,his‘timeduration’didnotextendtoinfinity.

But thosemindsmeasure from their self-centeredpointof view,only apart of theUniverse, and the rest remainsobscure.Soallofthemdisplaytheparadoxoftheego,astheyconfusethewholeUniversewiththeirworld,andseethemselves larger than all what they don’t perceive. Hence as Descartes wittingly warned the reader in his firstsentences‘everyhumanbeingthinksheisgiftedwithintelligence.

Themeanvaluetheorem.

A differential expresses the approximate value of the increment of the function in terms of the increment of theindependentvariableandofthederivativeattheinitialpoint.Sofortheincrementfromx=atox=b,wehave:

ƒ(b)-ƒ(a)≈ƒ'(a)(b-a).

It is possible to obtain an exact equation of this sort if we replace the derivative fʹ(a) at the initial point by thederivativeat some intermediatepoint, suitably chosen in the interval (a,b).Moreprecisely: If y= f(x) is a functionwhichisdifferentiableontheinterval,thenthereexistsapointξ,strictlywithinthisinterval,suchthatthefollowingexactequalityholds:

ƒ(b)-ƒ(a)=ƒ'(ξ)(b-a)

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The geometric interpretation of this “mean-value theorem” (also called Lagrange’s formula or the finite differenceformula) isextraordinarilysimple.LetA,Bbethepointsonthegraphofthefunctionf(x)whichcorrespondtox=aandx=b,andletusjoinAandBbythechordAB.

Now letusmovethestraight lineAB,keeping itconstantlyparallel to itself,upordown.At themomentwhenthisstraightlinecutsthegraphforthelasttime,itwillbetangenttothegraphatacertainpointC.Atthispoint(letthecorrespondingabscissabex=ξ),thetangentlinewillformthesameangleofinclinationαasthechordAB.Butforthechordwehave:

tanα=ƒ(b)-ƒ(a)/b-a.OntheotherhandatthepointC:tanα=ƒ'(ξ):

Thisequationisthemean-valuetheorem,whichhasthepeculiarfeaturethatthepointξappearinginitisunknowntous;weknowonlythatitliessomewhereintheinterval(a,b).

Itsinterpretationin∆stisthatƒ'(ξ)correspondstothevalueofafinitesimallyingbetweenboth.

FIRST,thefactthat 'membranesmustdeterminethebeginningandendpointofanyfunctionfor ittobemeaningfulandsolvable.Andindeed,onlybecauseweknowwhenitstartsandendsthedomain,wearesuretofindameanpoint.

If we consider then a T.œmean value theorem, where ƒ(b) > ƒ(a) if we are deriving in space, where f(b)=Max. Srepresents theparameter of the membrane, ƒ(a) will represent the singularity and so we shall find in between afinitesimal part of the vital energy of the T.œ with a mean value within that of Max. S (membrane) and Min. S(singularity). And viceversa, if we are deriving in search of the minimal quanta of time, ƒ(a) > ƒ (b), where ƒ(a)representsthetimespeedofthesingularityandƒ(b)thetimespeedofthemembrane.Andthemeanvaluewillbethatoftheinfinitesimal.

But in spite of this indeterminacy, the formula has great theoretical significance and is part of the proof ofmanytheoremsinanalysis.

Theimmediatepracticalimportanceofthisformulaisalsoverygreat,sinceitenablesustoestimatetheincreaseinafunctionwhenweknowthelimitsbetweenwhichitsderivativecanvary.Forexample:

|sinb-sina|=|cosξ|(b-a)≤b-a.

Herea,bandξareangles,expressedinradianmeasure;ξissomevaluebetweenaandb;ξitselfisunknown,butweknowthat|cosξ|≤1

Another immediate expression of the theorem which allowto derive a general method for calculating the limits

andapproximationsofpolynomialswithderivativesis:

For arbitrary functionsϕ(x) andψ(x) differentiable in the interval [a, b], provided only thatψʹ(x) ≠ 0 in (a, b), theequation,holdswhereξissomepointintheinterval(a,b).

Fromthemeanvaluetheoremitisalsoclearthenthatafunctionwhosederivativeiseverywhereequalto0’mustbeaconstant;atnopartoftheintervalcanitreceiveanincrementdifferentfrom0’.Analogously,itiseasytoprovethata functionwhose derivative is everywhere positivemust everywhere increase, and if its derivative is negative, thefunctionmustdecrease.

Andsothe'classicfunctionofmean-valuetheorem'allowustointroduceanessentialelementof∫∂whichwillopenupthe∆stcalculusofworldcyclesofexistence,thestandingpointsofafunction.

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Maxima and minimum. The 3 standing points of aworld cycle.Themean value sets for the region between thelimitingpointsofthecurve-whichmustbetakeninhigherstep-timespaceastwosectionsofabi-podalsphericalline,partofthemembraneofa3Dform,givesusthenavalueforthevitalenergytobeexpressedwithascalar.Andthe

initial and final point of the segment become themaximal andminimal of the function in F(f)=xvalues.

It is thenbetween those two limitsaquestionof findpointsof thevital energyamong them thesingularityMax.Sxt,tohaveawell-definedTOEinitsmembrane(maximalminimalvalues)volumeofenergy,meanvalueandMaximalpointoftheSingularity.

Wecanthendissectthesphere inantipodalpointsrelatedtothe identityneutralnumbero-1thesphereoftimeprobabilitiesthatthelargestwholemaximisesinitsantipodalpoints.Ifweconsidertheantipodalpointstheemergentandfinaldeathpoint,whichimperfectmotionsstillcloseto0’-

sums,themaximalmiddlepointwillbethesingularity,Max.SxMaxt.

Dimensionalintegration.Dimensionsofformthatbecomemotionsandviceversa.

Now the key to fully grasp the enormous variety of integral and derivative results obtained in all sciences, is tounderstandthatallspaceformscanbetreatedasinstantsintime,oreventsofmotion,andallmotionsintimecanbeseenasfixedpresentmomentsinspace.

Theseseriesofcombinationsoftimeandspace,S>T>S>T,whichleavesatraceofstepsandfrequenciesanditswholeintegration,whichemergesasan∆+1newscaleofrealityisatthecoreofallfractal,reproductiveprocessesofreality.

Forexample thes-Tduality isat thecoreof theGalileanparadoxof relativity (epursemuoveepurnomuove),ofEinstein'srelativity,ofZeno'sparadox.

Sowecanconsidermotionintimeasreproductionofforminadjacenttopologiesofdiscontinuousspace.

Wecanconsiderthestopandgomotionsof films,picturebypicture, integratingthose 'spatialpictures' intoatime'motionpicture'.

Weconsiderthewave-particleparadox,aswavesmovebyreproductionofformandparticlecollapsebyintegrationofthatforminspaceintoatime-particle.

Inthosecasesintegrationhappensbecauseasystemthatmovesintime,reproducesinspace.Andviceversa,stepsinspacebecomeamemoryoftime.

Nowitisimportantalsotostudycasebycaseanddistinguishproperlywhatarewetrulyseeingpopulationinspaceoreventsintime,aswecanandoftenithappensthathumansconfuseinquantumphysicswheremotionissofastthattimecyclesappearasformsofspace.Weshallthenunveilmanyerrors,whereaparticleintimeisseeingasaforceinspace(confusionofelectroweak,transformativeforceasaspatialforce,andsoon).

Allsystemscanbeintegrated,aspopulationsinspacetocreatesynchronoussuperorganismsandasworldcyclesintime,creatingexistentialcyclesoflifeanddeath.Thepopulationintegralwillbehoweverpositiveandtheintegralintimewillbe0’.

Since. systemsof populations in spacedohave volume. Yet thewholemotion in time, canbe integrated as closedpathsof time,orconservativemotions thatare 0’ sums,andthisallowsus toresolvewhat is time integrationandspaceintegration.

Considertofullygraspthis,thereproductionofawave,whichconstantlyreproducesitsformasitadvancesinspace,andcannotbelocalised(Heisenberguncertainty)becauseitisapresentwaveoftime,aslightmovesNOTintheleastspacebuttheleasttime.Now,consideraseminalwave-you,whichreproducesintime,butbecomesaherdofcellsthatintegratedemergesintoalargerscale.Inbothcasesthefinalresultisinspaceandsoitispositive.

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Soeachcasemustbestudiedtoconcludeweareeitherobservingatimeeventoraspatialorganism.

InthatregardthemostimportantandhencefirstviewoftheRashomonEffecton∫∂is:this

∫≈∂aretime=spacebeats/stepsinanyd²

Itfollowsthenthatwecanescapethememorialcreation,stepbystepofthespatialform,assomethingwhichforusisnolongerneeded,whenweareinterestedonlyinintegratingthespace,andforthatreasontheintegralworkmerelyasanintegralofavolume,asurface-whosecreationintimehasalreadyhappened.

Butwestillhavetofindaquantaofthat'creation'nowamere'populationinspace'.

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WHYDIFFERENTIALS=FINITESIMALSNOTINFINITESIMAL0sARETHEREALTHING

FinitesimalsasDifferentialofafunction.Thestœpsofmotionandformthatshapetheflowsoftimespace.

The essential connection between ∆ST actions and dimotions performed by a larger whole and algebra occurs incalculus,whereasthewholeistheintegralandtheminimalindistinguishablepartits(in)finitesimal.Againherethekeyconceptisoneofperception.SoinfactwetalkNOTofaninfinitesimallimitbutmoreonthethoughtofLeibnizofan1/nquanta,sosmallthatthewholedoesn’tdistinguishandsoitcanbedischarged.Themesthoseweshallanalyzeindepthwhenstudyingcalculus.

Thisdeepthoughtfact-thatsmallstepsare'lineal'andlongeronesarecurvedandultimately0’sumclosedpathsisthe justificationfor theuseofdifferentials insteadofderivatives inmostapplicationsofcalculustoreality.Becausewhilethelimittoinfinitydoesnotexist,thereisafundamentalparadoxbetweenlinealapproximationsandopenfreesteps in thesmall realmandcurvedclosedworlds in theupperrealm,whichmakesevery ‘finitesimalbeing’ to feelhappyandunbounded,tracing‘zig-zag’stopandgomotions,wherethemotionisalwaysalinealstep(trytowalkincurvedfashion)…butduringtheformalobservationinstop-modeofthenextsteptorun,the∆+1enclosurewholewilldeviateyourabsolutemotionintoacyclicalstep.Andsowhilesmallstepsaredifferentials,thesumofthem,withthelength-motion, height-perception, represented by the X and Y components of the tangent gives us finally a closedcurve,oronewhichwillhavelimitsofitsvalidity–mathematicaldomain–imposedbythehigher∆+1world.

Thuswhenweobserverealityinanyscaleatthemaximaldetailofitsstœpsitappearsexactlyasthetwosidesofthetangentofaderivative,ifweconsiderthe‘absoluteframeofcoordinates’,whereXisthemeasureofstepsofmotionin lineal continuous time,andY thecoordinate of formandperception,whichwealreadydefined forevolution inbiology,vitaltopologyin¬EGeometryandwave-particlestatesinquantumphysics,inotherpapers,turnouttobeinmathematicstheidealformof‘ST,space-time,formandmotionstates.

Soweseeelectronsinastopandgomotion,deviatingitspathateachstageeveniffinallywilltraceitsnaturalflowoftime,andweseeBrownianmotioninparticlesthattrytogostraightbutareconstantlydeviatedbythelargerworld.

AndtheEarthlooksflatbutElcanoreturnedwenttothewestandreturnedthroughtheeast,andwethinkweshallliveforeverwhenweareyounglookingatthefuturebut when we are old we only see the past; and so in time, space and scale theparadoxes of curved order and small freedoms carry the exist¡ential momentumthrough itsworldline that always becomes aworldcycle for thosewho gobeyondthe‘shallow’4Dcontinuumintothesuddenstopsanddiscontinuitiesofmovingon,toassessthenewdirectiontotakeinfrontofinsurmountablelargerwalls.

Differentialsinessenceare'lineal'ratesofchangeinsmall'intervals'ofanyfunctionthatiscurved,andwhoseexact,ideal,non-linealrateofchangeinalongstretchisdifficult to calculate. And in reality is used everywhere instead of the idealderivative.Andthejustificationin5Distheconceptofafinitesimalminimalquanta,

andthefractalnatureofpointsandstœps,theminimalquantaofchange.Thatischangeisneverinfinitesimal,butachangeimpliesaminimal¡-1unitofthebeing,eitheritsfrequencysteporreproductivecell,etc.Sothat'quanta'ofchange,whichisbettermeasurebythe'diameter'or'height'orlengthofthesphericalortallorflatform(cell,atom,individual)isadifferential.

Themathsofit,arewellknowntoanystudent.Asitissoessentialto5D‘experimentalmathematics’weshallbringithereforfurthercomments.

Let us then consider a function S = ƒ(t) that has a derivative. The increment of this function: ∆s = ƒ (t+∆t) -ƒ(t)correspondingtotheincrementΔt,hasthepropertythattheratioΔs/Δt,asΔt→0,approachesafinitelimit,equaltothederivative:∆s/∆t->ƒ'(t)

Thisfactmaybewrittenasanequality:∆s/∆t->ƒ'(t)+a

BDisdy,alinealfinitesimalofchange

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wherethevalueofadependsonΔtinsuchawaythatasΔt→0,aalsoapproaches0’;sincein∆sttheminimalstepofanyentityalwayshasalinealform.

Thustheincrementofafunctionmayberepresentedintheform:

∆s=ƒ'(t)∆t+a∆twherea→0,ifΔt→0.

ThefirstsummandontherightsideofthisequalitydependsonΔtinaverysimpleway,namelyitisproportionaltoΔt.Itiscalledthedifferentialofthefunction,atthepointtncorrespondingtothegivenincrementΔt,andisdenotedby:ds=ƒ'(t)∆t

Thesecondsummandhasthecharacteristicpropertythat,asΔt→0,itapproaches0’morerapidlythanΔt,asaresultofthepresenceofthefactora.

ItisthereforesaidtobeafinitesimalofhigherorderthanΔtand,incasefʹ(t)≠0,itisalsoofhigherorderthanthefirstsummand.

BythiswemeanthatforsufficientlysmallΔtthesecondsummandissmallinitselfanditsratiotoΔtisalsoarbitrarilysmall.

Practicalstienceonlyneedstomeasureadifferentialeither inspacedy=BD+BCor in time,asa fractionof theunitworldcycle,ƒ(x)=cos²x+sin²x=1whichbecomesaminimallinealst-eporaction,ƒ(t)=Sstep.

Ingraph,decompositionofΔSintotwosummands:thefirst(theprincipalpart)dependslinearlyonΔTandthesecondisnegligible forsmallΔS.ThesegmentBC=ΔS,whereBC=BD+DC,BD= tanβ · ΔT = fʹ(t) Δt = dS,andDC isaninfinitesimalofhigherorderthanΔt.

Forsymmetryinthenotationitiscustomarytodenotetheincrementoftheindependentvariablebydx,inourcasedttocallitalsoadifferential.Withthisnotationthedifferentialofthefunctionis:ds=ƒ'(t)dt

The derivative is the ratio, fʹ(t) = ds/dt of the differential of the function, normally a 'whole spatial view' to thedifferential of the independent variable, normally a temporal step or minimal change-motion in time.The differential of a function originated historically in the concept of an “indivisible”, similar to our concept of afinitesimal and somuchmoreappropriate for ∆st than the abstraction of aninfinitesimalwith ∆t->0, since time isdiscrete and there is always a minimal step of change, or reproductive step in a motion of reproduction ofinformation.

Differentialsofcalculusarepracticalinfinitesimalsanditsknowledgeforanyfunctionactsasan∂stlimit.

Ontheotherhand,thereisforanygroupthatwecantakeasvitalspace-time,findsusamiddlepoint.

Rightly then the indivisible, and later the differential of a function, were represented as actual infinitesimals, assomethinginthenatureofanextremelysmallconstantmagnitude,whichhoweverwasnot0’.

According to thisdefinition thedifferential isa finitemagnitude,measurable in space,foreach incrementΔtand isproportionaltoΔt.Theotherfundamentalpropertyofthedifferentialisthatitcanonlyberecognizedinmotion,sotospeak:ifweconsideranincrementΔtwhichisapproachingitsfinitesimallimitthenthedifferencebetweendsandΔswillbearbitrarilysmallevenincomparisonwithΔt-till itbecomes 0’.Theerrorof interpretation inclassiccalculusbeingthatitisthedifferencewhatapproaches0asfinallythefunctionwillbealsolineal,not∆t,whichwillbecomea'quanta'-asquantumphysicistswouldlatterdiscover.

Asthisistherealmodel,thesubstitutionofthedifferentialinplaceofsmallincrementsofthefunctionformsthebasisofmostoftheREALapplicationsofthenow-called'finitesimalanalysis'tothestudyofnature.

The¬Algebraic/graphicduality.

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OnviewofourdeeperdeparturefromtheultimateessenceofAnalysis,whichistostudystepsofspace-time. Thatistoput¬AlgebraicS=Tsymmetries inmotion; the¬Algebraicvs.graphic interpretationsof calculus responds toyetanothersymmetryofspatialvs.temporalmethods,consideredonourpostsof@nalyticgeometryand¬Algebra.

Itdoesshowmoreclearlywhatwemeanbythose'steps'asbasicallythe'tangent'ofthecurveisinmostcasesaspace-timestepexpressedbythegeneralfunction:X(s)=ƒ(t)

Obviouslyas sand tare illdefined, itwasonlyunderstood for lineal space-distanceand time-motion. Andsothe'geometrical'abstractconceptremains,voidofallexperimentalmeaning...asa...tangent...itwas...

Spatial:geometricview.

Weareledtoinvestigateapreciselyanalogouslimitbyanotherproblem,thistimeageometricone,namelytheproblemofdrawingatangenttoanarbitraryplanecurve.

Let thecurveCbethegraphofa functiony= f(x),and letAbethepointonthecurveCwithabscissa x0 (figure 10). Which straight line shall we call the tangent to C at the point A? Inelementary geometry this question does not arise. The only curve studied there, namely thecircumferenceofacircle,allowsustodefinethetangentasastraight linewhichhasonlyonepointincommonwiththecurve.

Todefinethetangent,letusconsideronthecurveC(figureup)anotherpointAʹ,distinctfromA,withabscissax0+h.LetusdrawthesecantAAʹanddenotetheanglewhichitformswiththex-

axisbyα.WenowallowthepointAʹtoapproachAalongthecurveC.IfthesecantAAʹcorrespondinglyapproachesalimitingposition,thenthestraightlineTwhichhasthislimitingpositioniscalledthetangentatthepointA.EvidentlytheangleαformedbythestraightlineTwiththex-axis,mustbeequaltothelimitingvalueofthevariableangleβ.

ThevalueoftanβiseasilydeterminedfromthetriangleABAʹ(figureup):

It is thenclear thathis the frequencyquantaof time,or ifweare inverselyusingthe∫∂methodtomeasurespacepopulations,theminimalunit.AndsotheultimateconcepthereisthathNEVERgoesto0.Andtheclearproofisthatifitwerearrivingto0’,x/h=∞.

SoinfinitesimalsdoNOTexist,anditonlybearswitnessoftheintuitiveintelligenceofLeibnizthathesomuchinsistedon a quantity for h=1/n... (and the lack of it of 7.5 billion infinitesimals of Humanity, our collective organism,whichmemorisethish->othatsomuchabstractpaingavemewhenakid-oneofthoseerrorsIannotatedmentallywiththeabsurdconceptofanon-Epointwithnobreath,orelsehowyoufitmanyparallels,ofthelimitofc-speed,howEinsteinprovedthatexperimentally?,andother'errors'that∆stdoessolveinallsciences).

Butforothercurvessuchadefinitionwillclearlynotcorrespondtoourintuitivepictureof“tangency.”

Thus,ofthetwostraightlinesLandMinfigurebelow,thefirstisobviouslynottangenttothecurvedrawnthere(asinusoidalcurve),althoughithasonlyonepointincommonwithit;whilethesecondstraightlinehasmanypointsincommonwiththecurve,andyetitistangenttothecurveateachofthesepoints.

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Andyetsuchacurveisultimatelythecurveofawave,andweknowwavesaredifferentiable.SothetangentISNOTtheultimatemeaningofthe∫∂functions- time/spacebeatsare.Thequestionthen iswhatkindofstbeatshallwedifferentiateinsuchatransversalwave?

Adifferentdimension,normallyaswavesarethe2nddimensionofenergy,asintheintensityofanelectricflow...amixtureofapopulationandamotion;or'momentum'(thederivativeofenergy)...

Andsothenextstageintotheproperunderstandingof∫∂operationsiswhat'kindofdimensionalspace-timechange-steps'wearemeasuring.

∆view:Theinversionofthefinitesimalcalculusof∆-1istheintegralcalculusof5D.

Thetransitionto∆nalysis:newoperations

Themathematicalmethodoflimitswasevolvedastheresultofthepersistentlaborofmanygenerationsonproblemsthatcouldnotbesolvedbythesimplemethodsofarithmetic,¬Algebra,andelementarygeometry.

TheinversepropertiesofSpaceproblemsandtemporalproblems.

Whatweretheproblemswhosesolutionledtothefundamentalconceptsofanalysis,andwhatwerethemethodsofsolutionthatweresetupfortheseproblems?Letusexaminesomeofthem.

Themathematiciansof the17thcenturygraduallydiscovered thata largenumberofproblemsarising fromvariouskindsofmotionwithconsequentdependenceofcertainvariablesonothers,andalsofromgeometricproblemswhichhadnotyieldedtoformermethods,couldbereducedtotwoSTtypes:

Temporalexamplesofproblemsofthefirsttypeare:findthevelocityatanytimeofagivennonuniformmotion(ormoregenerally,findtherateofchangeofagivenmagnitude),anddrawatangenttoagivencurve.Theseproblemsledtoabranchofanalysisthatreceivedthename“differentialcalculus.”

Spatialexamples:Thesimplestexamplesofthesecondtypeofproblemare:findtheareaofacurvilinearfigure(theproblemofquadrature),orthedistancetraversedinanonuniformmotion,ormoregenerallythetotaleffectoftheactionofacontinuouslychangingmagnitude(comparethesecondofourtwoexamples).Thisgroupofproblemsledtoanotherbranchofanalysis,the“integralcalculus.”

Thus2S=Tproblemsaresingledout:thetemporalproblemoftangentsandthespatialproblemofquadratures.

NowthereaderwouldobservethatunliketheageofArithmeticsand¬Algebra,whichstaysinthesame'locus/form';hereweobserve a keyproperty of analysis: thetransformationof a temporal cyclical question, into a lineal spatialsolution.

I.e. the solution of acceleration/speed by a lineal tangent, throughanapproximation; and the calculusof a cyclical,spatialareabytheadditionofsquares.Andthedeepphilosophicaltruthbehindit,whichonlyKeplerseemedtohaverealizedatthetime:

'Alllinesareapproximationsorpartsofalargerworldcycle'

Andsowecanconsider intermsofmodernfractalmathematics,that 'the infinitesimal isthefractalunit,quantaorstep'ofthelargerworldcycle,andasageneralrule:

'Allphysicalprocessesarepartofaconservative0’-sumworldcycle'.

Whichexplainsultimatelytheconservationofenergyandmotion,asmotionsbecomeultimatelyworldcycles,eitherclosedpathsinasingleplane,orworldcyclesbalancedthrough∆±1planes.

SuchisthesimpledualÐSTjustificationofAnalysis,asalwaysbasedin∆…finitesimalsandSt…theinversepropertiesof∫∂.

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Are there other operands of mathematics beyond those of calculus and hence other dimensional motions andcomplexmindsbeyondhumind’smaximalunderstandingofthegameofexist¡encereflectedinthosetexts?

Yes…Wehavenotreallyanalyzedthemostimportantofalloperands,thesymbolofequality,whichmathematicalpraxisusessomerrilyevenoflogiciansandrigurousmathematicianshaverightlystudiedthefactthatequalityasanidentityistherarestofalloccurrences.Evensomoreinaworldinwhichfractalpointsholdaninvisibleinnermind-

worldwhichcanonlybeequalizedwiththerestrictedviewofEuclid’spointswithnobreath.

Differentials-anySt-DimensionalSteps

Thedisquisitionofwhich'minimalistfinitesimal'allowustodifferentiateanS≈T¬Algebraicsymmetry,bringsustothe'praxis'ofcalculustechniquesthatovercomeby'approximations'thequestforthefinitesimalquantainspaceortime,susceptibleofcalculusmanipulation,whichgavebirthtothepraxisoffindingdifferentials,whicharetheminimalF(Y)quanta to work with and obtain accurate results (hence normally an spatial finitesimal of change under a timedependentfunction).Thiswastheoriginofthecalculusofdifferentials.

Asalwaysinpraxis,theconceptisbasedinthedualitybetweenhumindsthatmeasurewithfixedrulers,linealsteps,overacyclical,movingUniverse.SoMindsmeasureAristotelian,shortlines,inalong,curvedUniverse.

So thequestioncomes towhichminimalist lineal stepofamind isworthy tomakeaccuratecalculusof those longcurvedUniversalpaths.

Itisthenobviousthatthederivativeofalinealmotionhasmoresubtleelementsthatitssimplest¬Algebraicform,thex÷ linealoperationof 'reproductivespeed'andso theconceptofadifferential tomeasure thedifferencebetweensteady state lineal reproduction and the variations observed by a curve appeared as the strongest tool ofapproximationofbothtypeoffunctions.

Aswehaveconsideredthatmostdifferentialequationswillbeoftheform:F(s)≈g(t),wheresandtareanyofthe5DimensionsofSpace($,S,§,∫,•)or5Dimensionsoftime(t,T,ð,∂,O),whosechangerespecttoeachother,wearebound to study...showing how a spatial whole is dependent on the change and form of a world cycle, we shallconsidergenerallythaty->sandx->t...

Theresultofthischangewillbeamuchmoregenericconceptofspeedofchangeinanyofthedimensionsofentropy,motion, iteration, informationor formthatdefinestheUniverse, lettingus introduce its3 fundamentalparameters,S/t=speed, t/s=densityandsx t=momentum/force/energy... inanaturalwaywith itsmultipledifferentmeanings,Ðisomorphictoeachother-aswerepeatthesandtofthegeneral

Spacefinitesimalsvs.Timefinitesimals

Wemust'differentiate'whendifferentiating(:

-Spacefinitesimals,whicharetheminimalquantityofaclosedenergycycleorsimultaneousformofspace,easiertounderstand, as they are 'quanta'with an apparent 'static form',which can be 'added', if they are a linealwave ofmotion-reproduction,alongthepath;orcanbeintegrated(addedthroughdifferentareasandvolumes),togiveusa3Dreality.

-Timefinitesimals,whicharetheminimalperiodforanyactionofthebeingandwilltraceahistoryofsynchronicitiesas the actions find regular clocks,which interactbetween them to allow the being to performALL their 5Dactionsneededtosurvive.Sowewalk(A(a)),buttheneatenergy(Å(e)),andwedonotdothemoftentogether.Actionshavedifferentperiodicities, forEACHspecies thatperform5actions.So to 'calculate'all thoseperiodicities inasingleall-encompassingfunctionwehavetodevelopa5Dvariablesystemofequations.

-Spacetimefinitesimals.ButmoreinterestingisthefactthatNatureworkssimultaneouslyintegratingpopulationsinspace andsynchronising their actions in time. So we observe also space-time finitesimals where the

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synchronicityconsists in summoning theactionsofmultiplequanta thatperform in the samemoment the same 'D-motion',whichis'reinforced'becomingaresonantaction.

Andforthecalculusofthosespace-timefinitesimalsthebestwaytogoaroundisby'gatheringthesumof∆-1quanta'intoa'larger∆ºquanta'treatedasanew'1'addingupitsforce.Evenifmostofthemarejustcomplexensemblesofthe simplest actions of many cellular parts - steady statemotions, reproduction of new dimensions and vortex ofcurvatureandinformationabsorption.

Allfunctionsofanalysisthuscanbeconsideredoperationsonactionsofspace-time.

Groups of Finitesimals and their synchronous actions thus meet at ∆º in the mirror of mathematical operations,through the localisation of a 'theoretical' tangent≈ infinitesimal of the nano-scale (∂s/∂t proper) or an 'observable'differential,alargerfinitesimal,whichistherealelement,asanyfinitesimalisafractalmicropointsthathaveafractalvolume,expressedinthedifferential.

Thenwegatherthem,intimeorspaceandstudytheir'inverse'actioninspaceortime.

Sothefirstdistinctionwemustdoisbetweenfinitesimalsexpressedasfunctionsoftimefrequenciesandfinitesimalsexpressedasareasofspace.Andtheactionsdescribedonthem.Inpracticethoughmostfinitesimalsarespatialpartswhosefrequencyofactionisdescribedbytheƒ(x)=tfunction.

The3partsofT.œ.

Everyeventandformmustbeanalysedternary,andsohappenswithintegralsandderivatives,whichoftenrepresentintegralsofspace-timequantabelongingtothevitalenergyofthesystem,constrainedintimeorspaceby the singularity andoutermembrane.Ormight beof timequanta. So how canwedifferentiatethem?

Thusweestablishacorrespondencebetween5Dimotions,TT,St,TsSTandSSanditsintegrals,suchas:

TT:Bothelementsaretimelike.SoinTT-dimotionsweintegrateafrequencyofTimedimotion,thefinitesimalovera

timeduration:

SS:Bothelementsarespace-like.Soweintegrateaquantaofpopulationoveravolumeofspace:

Ts:Ifwecallenergy,Ts,amotionwithalittleform,wearethusconsideringmorecomplexequationstointegrate,andwe can use the obvious example of physical systems, in which energy becomes the integration of existentialmomentum,mv,acombinedSTparameterofspaceandtime,alongapathoftime,givingusE=1/2mv2.Soingeneral

anyTsformwillbeintegratedintheform:

St:Inverselyforchangesintheinformationofsystems,whichareperceivedasspatialformswithalittlebitofmotion,

theintegrationwillbeoftheform:

ST:Andfinallyforsystemsthatexperienceareproductiveprocessinspace-time,asingleintegrationwillnotsuffice,sowewillrequireadoubleortripleintegralofthesystemeitherinspaceortime,requiringamoreprofoundanalysisforeachcase:

€

∂tTo

T1

∫

€

∂sSo

S1

∫

€

∂stTo

T1

∫

€

∂stSo

S1

∫

€

st∂s∂tS,T∫∫

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∫S∂t=∆ebecomestheintegraloftheinnerspatialquantaoftheopenball,surroundedbythemembraneoftemporalcycles, which conserves its Energy and by the sum of all T.œs that of the Universe. Its calculus, after finding a'continuousderivative',surroundedbythemembraneisthenanintegral:∫Spxðƒ=Ke.

And inversely. Ifweconsidera singlequantaof spaceora single frequencyof time,amomentof linealorangularmomentum,theresultisaderivative.

SoAnalysisbecomesthefundamentalmethodtostudytravelsupwardsanddownwardsofthe5thdimension.

In general if we call a spatial quanta a unit of linealmomentum of each scale and a time cycle a unit of angularmomentum,themetricmerelymeanstheprincipleofconservationoflinealandangularmomentum.

Thusanalysis studies theprocesswhichallowsbymultiplicationof 'socialnumbers' ,eitherpopulations in spaceorfrequenciesoftime,asystemto'growinsize';whichistheultimatemeaningoftravellingthroughthe5thdimension.For example,when awave increases its frequency, it increases the quantity of time cycles of the system.When awheelspeedsupitsincreasesthespeedofitsclocks.Andviceversa,whenasystemincreasesitsquanta,growinginmass,orincreasingitsentropy(degreesofmotioninmolecularspace),italsogrowsthroughthe5thdimension.

Andtheintegrationalongspaceandtime,ofthosegrowths, iswhatwecallthetotalEnergyandinformationofthesystem

Itiswhatphysicistscalltheintegralofmomentumortotal'Energyandinformationofthesystem'

So we shall only bring about here some examples of analysis concerned with the definitions of the fundamentalparametersofthefractalUniverse,thatistheconservationprinciplesandbalancesofsystemswhichcanberesumedin2fundamentallaws:

Continuityoffunctions

All this understoodwe can then return to the inflationary nature of languages, which in the case ofmathematicsmeans thatwithout amirror reflection in reality, it tries to introduce false concepts of infinity and continuitywithpedanticaxiomaticmethods,originoftheconceptofabsolutecontinuityofafunction;whenthetrueconceptisthe'stopandstep'natureofmotions,anddark,nonperceivedregionsbetweencontinuouspoints,orfinitesimals.Soitisirrelevantifthefinitesimalisanaturalnumbertotalkofdiscontinuity,asthesystemwillhavecontiguousfinitesimalsof1numbersize.Wethentalkofmeasuremorethancontinuityanderrorsofmeasure,fromanupperƼmind"

Intuitively,afunctionf(t)approachesa limitLastapproachesavaluep if,whateversizeerrorcanbetolerated,f(t)differsfromLbylessthanthetolerableerrorforalltsufficientlyclosetop.

Justasforlimitsofsequences,theformalizationoftheseideasisachievedbyassigningsymbolsto“tolerableerror”(ε)andto“sufficientlyclose”(δ).Thenthedefinitionbecomes:Afunctionf(t)approachesalimitLastapproachesavaluepifforallε>0thereexistsδ>0suchthat|f(t)−L|<εwhenever|t−p|<δ.(Notecarefullythatfirstthesizeofthetolerableerrormustbedecidedupon;onlythencanitbedeterminedwhatitmeanstobe“sufficientlyclose.”)

Butwhatexactlyismeantbyphrasessuchas“error,”“preparedtotolerate,”and“sufficientlyclose”?

Again it is the relative ¡-1quantaof thesystemstudied.The 'error'ofmeasurewill thenbecomeESSENTIAL to theexplanationof theUncertaintyprincipleofHeisenberg,which indeedcanbeobtained from theoryofmeasureanderror,bypuremathematicalmethods.

Soinidealmathematics,havingdefinedthenotionoflimitinthiscontext,withnolimittotheinfinitesimalsizeoftheerror,itisstraightforwardtodefinecontinuityofafunction.Continuousfunctionspreservelimits;thatis,afunctionfiscontinuousatapointpifthelimitoff(t)astapproachespisequaltof(p).Andfiscontinuousifitiscontinuousateverypforwhichf(p)isdefined.Intuitively,continuitymeansthatsmallchangesintproducesmallchangesinf(t)—therearenosuddenjumps.

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Butasthatsmallchangewillalwaysbeindetailanε-quanta,ingreatdetailtherearequantumjumps.Infact,asthereisalwaysanε-quanta,inanyprocessinspaceortime,informandmotion(aswehaveshownwhenconsideringthenatureofmotionasreproductionof form inadjacent spaces) therewillalwaysbeaquantum jump forallmotions.Andmotionwillbethereproductionofforminquantumjumpsofε,nature.

1/n:the(in)finitesimal(in)finite

Withtheconventionthatƒ(x)isnormallyafunctionoftimefrequencies,ƒ(t),ofmotionsoftime,whosesynthoniesofsynchronicityinspaceareexpressedbyan¬Algebraicequation,webringthefollowingunderstanding:

Infinitesimalquantainanyscaleisthedeparturepointtobuildanyfunction,assuchitmusthaveaminimalsize,andƒ'(t)isnormallyagoodmeasure.

Theinfinitesimalstudyasperceivedfromthefinitepointofviewistheviewoffractals,whenindetailandobservingtheclosedworldcyclesthatseparateandmakeeachinfinitesimalawhole.

Aderivativeisthefinitesimalofthefunctionobserved,andsowhenwegoevenfurtherandstudyasenlargedintoourscalarviewtinmaximalinformationweareinthefractalviewofreality.

Soasweexpandourviewthefractalviewbecomesmorereal,tillfinallytheenclosuresobserved∆-1becomefractalandwerecogniseitsself-similarities:∆-1≤∆º.

Foreachderivativethusafunctionshowsits1/ninfinitesimal(notnecessarilythisfunction,whichisthederivativeofthelogarithm).

It follows that functionswhich growginormouslyhave a 'quantaof time' reproducedand so itsminimal derivativefinitesimalisthefunctionitself,eª.

Inthenextgraphweseeinverseequationsofexponentialsandlogarithms.

Exponentialsexpressbetterdecaythanexponentialgrowth,withtheexponent"negative".

Mathematics is a reflection of nature. A small mirror of its ∆º±iStructure and so we need for exponential growth that Natureprovidesunlimitedenergyforgrowth,whichhappensonlyinthe0’-1generationaldimensionofthebeing,orinitsinversedecay/inits4Dentropyage.ofdeath.

OntheotherhandthelimitoflogarithmicgrowthmapsoutbetterinlogisticcurvesrealgrowthbeingagoodfunctiontoexpressƤcales.

So numbers reflect those processes in their inverseexponential/logarithmmathematicalgraphsandnumericalseries.

ST:Asthethreecoordinatesystems,self-centredintoan∆ºpov,whichreflectseachofthethree'topologiesofspace-time'(Cylindrical:lineal,polar:cyclicalandcartesian:Hyperbolic);whiletheinfinitesimalo-1scale,andtheinfinite1-∞scaledividedbythe'1'∆ºrelativeelement,representperfectlythe∆-scalarnatureofsuperorganisms.

∆º±1:Furtheron,wecan'reduce'eachrelativeImmensitytothose3Planes,andrepresentalltimespacephenomenawith the different familiesof numbers that close ¬Algebra (entropic, positive numbers, informative, negativenumbers,presentspace-time,complexbidimensionalnumbers,s/t ir-ratio-nalnumbers,etc.),mathematicsbecomesessentiallythemorerealistlanguagetorepresentthescalar,organic,ternaryUniverse.

The0’-1scale isequivalenttothe1-∞scaleforthe lower∆-1Universe,where1=∆º,thewholeand1-∞isthe∆+1eternalworld.

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Andthisisthesymmetrytograsptheconsequencesoftheo-1-∞fundamentalgraphofthefifthdimension.Letusseehowwithasimpleexample:

Themirror symmetriesbetween the0’-1universe and the1-∞are interesting as they set twodifferent 'limits', anupperuncertainboundforthe1-∞universe,inwhichthe1-world,∆ºexists,andaloweruncertainboundforthe0’-1Universe,wherethe1doesnotseethelimitofitslowerbound.Arethoseunboundedlimitstrulyinfinite?

Thisisthenextquestionwherewecanconsiderthehomologyofboththemicroscopicandmacroscopicworlds.

Of course the axiomatic method 'believes' in infinity - we deal with the absurdities of Cantorian transinfinities inarticlesonnumbers.Butasweconsidermaths,afterLobachevski,GodelandEinstein,anexperimentalscience;wearemore interested in the homologies of ∆±1. For one thing. While 0 can be approached by infinite infinitesimal'decimals', so it seems it can never be reached, weknow since the 'violet catastrophe' that the infinitesimal is a'quanta',a'minimum',a'limit'.AndsowereturntoLeibniz'srightfulconceptofan1/nminimalpartof'n',thewhole'1'.

Thisimpliesbysymmetrythatontheupperbound,theworld-universeinwhichthe1isinscribedwillhavealsoalimit,adiscontinuitywith∆+2,whichsetsupallinfinitiesintheupperboundalsoasfinitequanta,'wholesofwholes'.

So the 'rest' of infinities,must be regardedwithin the rest of 'theoryof information languages' and its inflationarynature,inflationaryinformation.Whatisthenthe'practicallimit'formostinfinitiesandinfinitesimals?InÐST,thestandardlimitistheperfectgameof3x3+0(±1)elements,wheretheo-mindtriplesasitis an ∆-1 'god of its infinitesimals it rules subconsciously, as you brain rules you cells', ∆º,consciousnessofthewholeand∆+1infinitesimalofthelargerworld.

Ano-1 timemirroredquantumworldofprobabilitiesofexistence,as indistinguishable infinitesimalsthroughthesurfacelimitofitsstatisticaldescriptioninthethermodynamicscaleofatomicbeingsend

inthe1unitofourhumancellularspace,wherethermodynamicconsiderationsarereducedtotemperaturegradienttowardsthehomeostaticmassbasedforcesofourhumanlevelofexistence,Ƽ.

SoweconsiderasusualtheKaleiodoscopic,multiplefunctionofanalysis,andthemultiplemeaningsofitsinverse,∆±1operations,derivativesandintegrals;sinceasusualthepotencyof∆stisonthesearchofwhys,notonthediscoveryofnewequations,whichhumansalwaysexhaustbytheMonkeyMethodoftrialsanderrors,sweatandtranspirationmorethentheinspirationofpurelogicthought...

Physicalequationsindifferentialform,ageneraloverviewofitsmainspecies.History

Differentialequationsfirstcame intoexistencewiththe inventionofcalculusbyNewtonandLeibniz.Newton listedthreekindsofdifferentialequations:thoseinvolvingtwoderivativesoneofspaceandtime(orfluxions)andonlyoneundifferentiatedquantity (spaceor timeparameter); those involving 2 derivatives and twoquantities of space andtime;andthoseinvolvingmorethantwoderivatives.

Itsanalysisthuswasrightinthespotashereferredchangestochangeinspaceortime,thus∫∂withST-eps-afactlatterforgottenandtodaythoroughlymissedwiththe'view'oftimeasasingledimensionofspace(1D-linealmotionconfusedwith4D-entropyinphilosophyofscience)

Itisstillagoodclassificationofpartialdifferentialequationsas'time-like'(∂x,∂²x,∂³x),orspacelike(∂²y,∂y,∂³y)orspace-time like (∂x∂y,∂y∂x)asthemainvariationsthatrepresent,T,TT,TTT;S,SS,SSS,ST,TSsteps,whicharethemain5D,4Dand1,2,3DchangesoftheUniverse.

And it speaks of the enormous range of real phenomena ∫∂ functions can describe as the essential operands ofmathematicalphysicsandany∆stphenomena.

Whatallowallthose∆stphenomenatoentertheworldofquantitativemathematicswasthediscoveryofapendulumclocktomeasuretimeinlinealfashionandatelescopetomeasurespace.Bothgavebirthtothe2ndageofscience,

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themathematical/scientificmethod,addedtotheexperimentalAristotelianmethod,whichnowtheisomorphicÐSTageofstiencecompletes.

In1609appearedthe“Newastronomy”ofKepler,containinghisfirstandsecondlawsforthemotionoftheplanetsaroundthesun.

In1609tooGalileodirectedhisrecentlyconstructedtelescope,thoughstillsmallandimperfect,towardthenightsky;the first glance in a telescopewas enough todestroy the ideal celestial spheresofAristotle and thedogmaof theperfect formof celestial bodies. The surfaceof themoonwas seen to be coveredwithmountains andpittedwithcraters. Venusdisplayedphases like theMoon, Jupiterwas surroundedby four satellites andprovided aminiaturevisualmodel of the solar system. TheMilkyWay fell apart into separate stars, and for the first timemen felt thestaggeringly immensedistanceof the stars.Noother scientificdiscoveryhasevermade suchan impressionon thecivilisedworld.

ItalsokilledamethodequallyvalidofthoughtrepresentedbytheGreeksandLeonardo:theidealisedunderstandingofthecanonicalperfectÐSTgameofexistence,ofwhichwewereall impureplatonicforms,bondtodissolveunliketheperfectgameofthe∞Universe,whichisimmortal.

Manneverwentbackbecausealas!whatreallymatteredwasballistics,mechanisms,power.Idealismdiedaway:

Thefurtherdevelopmentofnavigation,andconsequentlyofastronomy,andalsothenewdevelopmentoftechnologyandmechanicsnecessitatedthestudyofmanynewmathematicalproblems.Thenoveltyoftheseproblemsconsistedchieflyinthefactthattheyrequiredmathematicalstudyofthelawsofmotioninabroadsenseoftheword.AndnowwehadmachinestomeasureitbetterthantheartisticSp-eye-T=wordsofthehumanspace-timemind.

Pointsofconstrain,balanceandlimitsofintegrals.

Any equation with a real, determined solution must be a complete T.œ. Hence it will have limits either in space(membraneandsingularityoftheopenball),orintime,initialandfinalconditions,bridgedbyanactioninthe'leasttime'possible.

Thisisthekey∆stlawthatappliestothesearchforsolutionsinbothODEandPDEs.

Maximiseitsðƒ/Sp,densityofinformation/mass,itsSp/ðƒdensityofenergyandhence,reachabalanceatðƒ=Sp

Thissimplesetofequations:max.ðƒxSp->Tƒ=Sp:maxTƒ/SpandMax.Sp/TƒarethereforethefluctuationpointsofsystemsthatconstantlymovebetweenthetwoextremesofinformationandspatialstatesacrossapreferredpointofbalanceSp=Tƒasthisisthemax.SpxTƒplace.

Thusintegrals,LagrangiansandHamiltoniansarevariationsofthosethemes.Themotionofsprings;thelawof leasttimeetc.allarevibrationsalongapointofbalance,Tƒ=Sp,and2maximalinverselimits.

Thedifferenttime-spacebeats.

This of course must be done because reality is bidimensional and a dimension of space goes accompanied by adimensionoftime,generatingasinthepreviousgraphs,themotions=changes,S≈T≈S≈Tthatshapereality.

Anditisthejustificationonwhydifferentialequationsthatmakesystemsdependantofsuchpairofvariableshappen.

But then it followswe shall be able to apply pentalogic and find a use for the pair ∫∂ as expression of an inversebeatingforeachpairofdimensionsofspace-time.

Anddecomposebothspace-timeformsandtime-spaceeventsinS>T<Sbeats.

Andintheprocessofdoingso,learnfurtherinsightsaboutthesymmetriesbetweenspaceandtime.

PRODUCTANDINVERSEDIVISIONINTEGRALS

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Themostabundantofalloperand, themergingproduct requires thereforeamorecomplex rule thanadirect sum,whichactsby'superposition'ofEQUALBEINGS.

It isalsosusceptibletobeoperatedbycalculusand 'derivatives'asnowwe involveforthefirst time,both,ascalarlevel,sincemultiplicationtendstohappeninthelowerscaleofthebeinganddifferentstatesoftimeandspace.Sowenolongeroperateasinadditions,withthesametypeofT.œsinthesameplane.

Themostabundantofalloperand, themergingproduct requires itsownrulewhich interestingenoughshowshowindeedproductisamergingoperation,asthederivativeofaproductoffunctionsmergesfirsteachfunctionwiththechangerateoftheother,andthenoncebotharemerged,superposesthembyaddition:

TheProductRuleusedtofindthederivativeofaproductoftwofunctions,isthusmorecomplexthanthesumeventhough it also keeps as in polynomials the distributive property - which shows once again that the product is a'democraticmerging'thatcangobothways.

Soh'(x)=[ƒ(x)xg(x)]'=ƒ(x)•g'(x)+ƒ'(x)•g(x).

The rule, interesting enough shows how indeed product is amerging operation, as the derivative of a product offunctionsmergesfirsteachfunctionwiththechangerateoftheother,andthenoncebotharemerged,superposesthembyaddition.

Inthatsenseitkeepswiththe'rule'ofmergingatthelower'planelevel'ofitsinfinitesimalparts,inthiscase,takinginsteadof thespatialelementsofXandY, its 'temporal'quantaof change, f(x)' andg(x)',MERGINGthemwith theotherwholes,beforea'superposition'=additioncanbeeffected.

IntheproductrulethusDerivativesactininversefashiontopowerlaws,searchingfortheinfinitesimal.

Whilepowerwholes(integrals)searchthewholeness,andasweknowthetwodirectionsofspace-timearedifferentincurvature,quantityofinformationandentropicmotions.

Here we shall bring a little explained fact - derivativesact in the inverse fashion to power laws, searching theinfinitesimal,whilepowerwholes(integrals)searchthewholeness,andasweknowthetwodirectionsofspace-timearedifferentincurvature,quantityofinformationandentropicmotions.

SoanexternaloperationthatreducesawholewhichisNOTintegratedassuchbutalinealproductoftwowholes,ƒ(x)and g(x), a COUPLE, ismixing the infinitesimals of one,with the otherwhole before herding them; in a process of'geneticmixing'ofthepartsofthefirstsharedwiththesecondwholeandthepartsofthesecondsharedwiththefirstwhole.

This law of Existential ®lgebra simplified admaximal as usual inmathematicalmirrors surprisingly enough is theoriginalsoofgenetic'reproduction',whichoccursattwolevels,mixingthe'parts'-thegenesofthewhole-inbothdirectionstorisethenthemixingtotheƼleveloftheGandFgendercouple.

Thenwhatwillcomeoutofthatgeneticmultiplicationisitsdivisionintotwoequalparts,showinghowtheinteractionof inverseoperandsdoesnot cancel realitybutmerely completesadimotionmovingahead theeternal timespaceuniverse.

Soifapowerfollowedbyalogarithmbringstheinfinitesimalseedintoawholeherd,themultiplicationfollowedbyadivisionofthereproducednewlayerofmixed'axons,genes'orparts,bringsthereplicationofidenticalforms.

Whilethesimplestdefinitionofadivisionisasusualinhumindsanentropicdestructivefeedingaction,thecomplexviewfromtheperspectiveof information isageneticmitosis.Andbotharereflected in thederivativeofadivision,whichisimpossiblefortwoequalfunctions(resultingin0constant)andviceversacangiveusanyconstantvalueinitsintegral-soitdoesnotgiveusanyinformation.

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While inmost cases is NOT a positive communicative act but a perpendicular negative reducing game,where theDOMINANTelementisthe'predator'largerdenominatorthatcutsthefunction,multiplyingitsinfinitesimalf'(x)parts,towhichitwilldeducethelesserpartsabsorbedbythef(x)functionfromit,andthencutitatthe'lower'levelofitspotentialelements(G(x)²):

So thenumerator, thevictim, sharedby thedenominator thepredator so tospeakisfirstabsorbedinitsƒ'(x)parts,g(x)ƒ'(x),subtractingtheg'(x)partsthatthe prey has absorbed in the 'fight', ƒ(x) g'(x), and then shared by the parts,g(x)²ofthewholeasentropicfeeding.

Sowecanconsiderthederivativeofadivisivefunctionasan'idealized'expressionoftheprocessofkillingandfeedingof a system, whereas the predator absorbs the infinitesimal parts of the other being, and feeds its cellular, i-1elementswithit.WhichobviouslyisNOTacommutativeprocess.

Ofcourse,we lovetobringvital interpretationtoabstractmath,butasweapplysuchrulestoparticularcases, theinterpretationsvarybutinallcaseswillbeabletobeinterpretedintermsofsub-equationsofthefractalgenerator.

Whatmightbenoticeinanycaseisthatunlikeinourrather'abstract'dimensionalexplanationoftherulesofpowerlaws,hereweareabletobringrealvitalanalysisof thoseroles in termsevenofbiologicalprocesses,showinghowmuchmoresophisticated is the∫∂operands, thekingof thehillofmathematicalmirrorsonrealst-epmotionsandactions,reasonwhyitsuseissowidespread.

Sothefundamentallawofoperandstovitalizethemisthis:

Bypentalogicalldifferentialoperandscanbecomeanactioninoneofthe5ddimensionalvowels(a,e,i,o,u)thatdefinethefivedimensionsofexistence,asvitalquanta-actionsofthebeing.

Thisisthelogicconceptthattrulyvitalizestheoperandsof¬Algebra.

Sothosepropertiestellusnewthingsaboutthemeaningof∫∂.

Finallythechainrulewhichistrulytheonethatenclosesallothersisusedinthecaseofafunctionofafunction,or

compositefunctionwrites:

And this truly an organic rule, as we are not derivating on 'parts' loosely connected by ± and x÷ herds and linealdimensionalgrowth,butthe'function'isafunctionofafunction-afunctional,asall∆+1ismadeof∆ºwhicharealsofunctionsofxofractalpoints.

Sothisisthemostusefulofallthoserulestomirrorbetterreality.Andweseehowthederivative,thechangeprocessdeepsinatthetwolevels,atthe∆º=g(xo)level,whichbecomesg'(xo)andatthewholelevel,whichbecomesƒ'[g(xo)],whichtelluswecan indeedgodeeperwith∫∂betweenorganicPlanes,which iswhatweshall learn inmoredepthwhenconsiderpartialderivativesandsecondderivativesandmultipleintegrals.

Wearegettingsotospeakintotheinfinitesimalofthepartsofawholefromits∆+2perspective,andthisruleenclosesallothers,becauseitbreaksintothemultiplicationofitsparts-dwindlingtrulyascaledown,andseparatingthewholeandthepartsderivatedintoloosepartsandfinitesimalsnowmultiplied.

Andwhatwillthepartsdowhentheyseetheirpreviousfinitesimalsnowcampingbythemselvesbut'atsight'togetthemto'produce'anoperative'action'(a,e,i,o,uactionsareALLsubjecttothepreviousoperands),ONthem.

And what will come of that multiplication. Normally it will capture them all again and then normally will notre=produceonthem(oneoftheoperandsactionswhicharepossibleunderpentalogic)butdivideandfeedonthemthelastoperationtotreat:

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And its inverse,which isNOTapositive communicative act butoften aperpendicular negative reducing gamealsoconsequentlydiffers.

In that sense themost important ad on that ∆stwill bring to the use of differentials in existential ¬ælgebra, is itstemporaluseasthe'minimalactionintime',ofabeing,afarmoreexpandednotionthattheactionofphysics(whichhoweverwillberelatedtothelinealactionsofmotionon1D).

Finally in the next stage of ¬Algebra, when @nalytic geometry allowed a more clear representation of thosepolynomialsinmoredetail,asusualthroughits3AGESofevolutionandthroughitsPLANESofcomplexityandthroughits pentalogic 'Rashomon effect'; that is, how analysis operates independently to extract information from the 5DImotionsofabeing.

S=T:ANALYSISONSPACE

A2ndconsiderationonthepentalogicshouldbeonanalysisofSPACEandtrans-form-ationsbetweenspace=formandmotion=timestates.SOFIRSTweshallrememberwhatspaceismadeof-namely¬Epoints:

Dimensionsandanalysisarepossiblebecausepointshavevolume.

AUniverseoffractalspacespoint-particleshaveaninnervolumeofinformationasNon-Euclideanpointswhichgaugeinformationinthestillnessofamindsyntax,languagemirroroftheUniverse.Aspointshavevolumes,linesarewavesandplanestopologicalnetworks,whichensembleinternarya(nti)symmetriestoformthetopologicalsuperorganismsof reality across 3 time ages, 3 topological forms and 3 Planes. It is this physical T.œ which we shall study inmathematicalphysics,explainingthemeaningin5Dofthemainmathematicallawsofphysics,whichareenhancedbytheunderstandingofanenhancedgeometryand logicoftime,bornofthefractalcyclicalstructureofboth,apriorielementsofrealitythatthelanguageofmathematicsanditsoperandssoaccuratelymirror.

Thefundamentaltruthderivedfromthissimpleanalysisofderivativesisprofound.Firstitconnectsthemimmediatelywiththepuregeometricnatureofdimensions,which innon-Euclideangeometry(graph)arerelevant inasmuchastheyrepresentmotionsintimebutalsodimensionsinspace.

Inthatregard,itisimportanttounderstandthatinthefractalUniverseadimensionhas'alwaysinnerbreath'asthepoints growwhenwe see them closer.. So it is very simple to consider a single dimensional being, simply as one,whosepreferentialX-dimensionismuchlargerthantheothers,butstilltheotherexistastheparticle-pointindetailisbig:

1Dbeing:X>>Y≈Z,forexampleastring,alinealmomentum...

AndthenatwodimensionalbeingonewhosetwoDarelargerthanZ:

2DBeing:X≈Y>>Z;forexampleagraphenesheet;aplanewave.

Whereasa3Dimotionbeinghasvolume,motiononthe3,forexampleasphericalbeing,anentropicexplosion.

Aderivativethenmerely'annihilates'onedimensionoronemotioninspaceortime-wehaveheretosplitdimotions,ashumansdo,even if it isnottheproperunitoftheUniverse,which isalwaysbidimensional. I.e.even inamotionthereisaparticlethatmoves,soyouhaveapoint-dimensionfortheparticleandoneforthemotionintime...

SoindeedanalysisISthemainmathematical instrumenttostudythe5DUniverseanditsternarymirrorsymmetriesbetween Planes, topologies andmodes of time-change. Andwe can consider a general formulae for analysis, as aspecificversionofthefractalgenerator:

∂(Bodywave of vital energy) = Membrane; ∂Membrane = Singularity path and its inverse, better known as lineintegrals,surfaceandvolumeintegrals.

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Because analysis ismainly used inmathematical physics, in praxis, the previous relationship is connected to the 3elementsofaphysicalsystem:

Field(entropic,locomotionsource)<wave(reproductivebody)>Particle.

Sowemakedoublederivativestoobtainthefield(Laplacians),andsinglederivativestorelateparticlesandwaves-'one-dimensionalspecies.'(Fourierseries).Andthosearetheallpervadinganalyticalfunctionsofthe3partsofthebeing:

Sphericalharmonicsandelectronorbitalsarethesame,becauseourlightspace-timeinparticlestatearephotonsthatformtheelectronicnebulae.Sobotharehomologic.

The result are spherical harmonics, a set of functions used to represent functions on bidimensionalmembranes-surfaceofthesphere-thehigherdimensionalhomologyofFourierseries-periodic,single

variablefunctionsonthecircle.

SphericalharmonicsarethustheeigenfunctionsoftheangularpartoftheLaplacian,representingsolutionstopartialdifferential equations inwhich the Laplacian appears. Since the Laplacian appears frequently in physical equations(e.g.theheatequation,Schrödingerequation,waveequation,Poissonequation,andLaplaceequation)ubiquitousingravity,electromagnetism/radiation,andquantummechanics.

The orbitals of the hydrogen atom in quantum mechanics in fact are totally undistinguishable from sphericalharmonics, showing indeed thatwe are all topologic beings, andmathematical functions for the simplest forms ofspacetimeastheelectronis-adensefunctionof'lightspacetimeparticle-points.

Theintimateconnectionbetweenthe3elementsofthebeingisperfectlyexplainedbythedual∫∂functions.

Inthatregard,variationsoverthesamethemerespondtotheternarystructureofallT.œs:

Inthegraph,whenderivingandintegrating,mostoperationsrefertoa'limited'system,inwhichfirstweextractthefinitesimalpart-element,andthenweintegrateittoobtainawhole;somostlikelythesystemdescribedwithdepartfromatime-changing-variablequanta,andintegrateittoobtaina'staticwhole-spatialview'.

Butvariationsonthesamethemehappenbythenaturalsymmetryofspaceandtimestates.

Sowecanalsostartwithaquantaofspaceintegratedovertimetogetaspatialareaorvolume.

Whatweshallalwaysneedtofind'singlesolutions'istheparametersthatdescribeintimeorspacethe3elementsoftheT.œ:Soweshallstartwithinitialorfinalconditions(definiteintegrals),anddefinemostlyinspaceasawhole,theenclosure or membrane that the limits the domain of the function (which might include as a different limit thesingularity).

All in all the analytical approachwill try to achieve a quantitative description of the unit/variable of 'change', the'finitesimalquantaof space - interval, area, volume'or the 'stepsof time' (frequency), and then integrate itoverasuperorganismofspaceoranintervaloftime,wewishtostudy,oftenbecauseitformsawholeora0’sumworldcycle.

GalileanParadox.linealvs.Cyclicalview.

Inthatregard,theS=Tsymmetrywilloncemorebecomeessentialtothetechnicalapparatusofanalysisasithasdoneinallothersubdisciplines.

Ofthemthe3key 'dualities'betweenlinealperception inshortandcyclicalperception in large, isthekeytoobtainsolutions,asthemindofmeasureislinealmadeofsmallstepsthatapproximatelargercyclicalwholes.Itisinessencethemethodofdifferentialequations,wherethedifferentialdy=ƒ'(x)∆x+α∆x,approachestoalinealderivative,ƒ'(x)∆xinshortincreases,andsowecangetawaywiththesmallerelementthatcurvesinlongerdistancesthesolution.

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FinallythethirdGalileanparadoxbetweencontinuityanddiscontinuityisalsoattheheartofanalysis(andmostformsofdualknowledge).Analysishasacceptedasadogmathecontinuityoftherealnumberandsoitconsiderscontinuitya necessary condition for differentiability but we disagree in a discontinuous Universe, continuity has a loosedefinition(asneithertheaxiomaticmethodistheproofofmathematicalstatementsbutexperiencealsomatters).Socontinuityisdefinedbyasimplerrule:thatthetermα∆xofthediscontinuitybetweenthelinealandcyclicalviewofaninfinitesimalderivativedoesindeeddiminishfasterthecloserwearetothepoint'a'inwhichthedifferentialequationis defined. In brief, continuity means no big jumps and big changes in the direction of a function and the T.œ itreflects.

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PENTALOGICONDERIVATIVES

Pentalogicoffinitesimalchange.

Leibnizdefined the finitesimalas1/x,whereX is thewhole. In termsof topological curvatureandmotion,1/xalsodefines in Leibniz'swork the curvature in space (as theoscular curve, 1/rof a cycle),whichby virtueof S=T IS theminimalunitofcyclicaltimeofawhole).Yetinlinealterms,curvaturewhichisdimensionlesscanalsobeseenasthe‘angle’of change in thedirectionofamotion,eachquantaof timeor stepof space. Soanangle changes in lineal,discrete quanta (1st Dimotion of change). It changes in curvature in a continuous longer measure. It becomes anacceleration,inS=Tmotion.ThenafinitesimalofcurvaturebecomesthefundamentalconceptthatcarriesderivativesofmotionintogeometryinthegravitationaltheoryofEinstein,notyetunderstoodconceptuallyinRelativitytheoryastheS=Trelativityprincipleon5Dphilosophyisignored.

Furtheron,afinitesimalrepresentsan∆-1unitofchangeinscale;andfinally,the‘ultimatefinitesimal’,isthe@-mindsingularityofthesystem.

So finitesimalscanreprsentall formsofchangesof∆@stasminimalscalarparts,minimalstepsofmotion;discreteangular changes, curvature of an acceleratedmotion seen as form; andmind singularity…And its limit is absolutechange or entropic change,which cannot be calculated because the function breaks and there is no derivative, orratherthederivativeisthewholebeing,whochangessomuchthatitisnomore…

Theactionsitdescribes.

The minimal unit for any T.Œ are its a,e,i,o,u actions of existence, its accelerations energy feedings, informationprocessing, offspring reproduction anduniversal evolution. So the immediatequestion aboutmathematicalmirrorsanditsoperations iswhatactionsreflect.Wehavetreatedthethemeextensively inthe¬Algebraicpost,concludingthatbeingmathematicsamostlyspatial,socialmorethanorganiclanguage,itsoperationsareperfecttomirrorsimplesystems of huge social numbers=herds; and as such to describe the simpler accelerations=motions, which arereproductions between two continuous Planes of the fifth dimension; informative processes, where the quantaperceivearetrulyfinitesimal∆-ielementspeggedtogetherintothemirrorimagesofthesingularityandsowetalkofmotions,simplereproductionsandvorticesofinformation,andtime>spaceprocessesofdecelerationofmotionintoform,asthekeyactionsreflectedbymathematicaloperations.

Itfollowsthatwhenwestudythemorecomplexsystemsandactionsofreality,reproductionandsocialevolutionofnetworks into organisms, mathematics will provide limited information, and miss properties for which illogicalbiologicalandverballanguagesarebetter.

Anditfollowsthatphysicalandchemicalsystemsarethebesttobedescribedwithmathematicalequations,eitherin¬Algebraic terms or analytic terms, which fusion together when we try to describe the most numerous, simplersystemsofparticlesandatoms(simplerbecausebycastinguponthemonlymathematicalmirrorsweare limitedtoobtainmathematicalproperties).

Letusnowconsiderthe∫∂operationsforthedifferentdimotionsofrealityonthemainfunctionswithfundamentalrolesin∆standitsderivativesbydividingthemin3great∆st'groups':

@:∫∂ofidentityelements-formsthatdonotchangeexpressedwiththeconceptofanidentitynumber,as0istheidentity of sum and 1 of product. But they also have a clearmeaning as the interval 0’-1 of the generation 'seed'dimensionfrom∆-1to∆º.

And indeed, the surprising result that ∫o dx = C does indeed suggest that the 0’-point is a fractal point that 'hasvolume',orelsehowintegratingthenothingnessofexistenceshallwegeta'constant'whichisasocialnumber?Butifwedostartfromao-1unitits'integral'sumwillgiveusareproductivegroup,or'socialnumber'.

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Andifweintegratethefull'1being',weshallgetanewdimension,thevariableplustheconstant,whichsuggestalsoalittleunderstoodprocessrelatedtotheoperationsofderivativesandintegrals,theswitchingcausedbyoperationsonmotionsofsets(ourdefinitionofanalysis),whichchangeaspatialstateintoatimestateandviceversa.Sothespatial1-form-wholebecomesatime-variableX,whilethevariableXbecomesaspatialderivativeconstant.

SinceconstantnumberdoesNOTchange.Soatimevariablegivesusthespatialidentitynumber.

Finally, thedeepestthoughtonthoseseeminglywellknownoperationsregardsthesubtledifferencebetweenbothoperations:thederivativelocalisesasingle'finitesimalsolution',orminimal∆-1pastpartofthesystem...

But the inverse, 'integral' or 'future 5th Dimensional arrow' of social wholes opens up the possibility of multipleconstant solutions to add to the variables, as the futureis open to subtle variations (∫) but the past is fixed bytheinfinitesimalidentitynumber(∂).

Ofcourseifweinsteadconsidertheintegralnotintimebutasafixedspatialpath,thisconceptoffuturevanishesandwegetadeterminedsinglesolutiontotheintegralwheretheconstantisjustthestartingpoint.

Otherwayofseeingitthoughistoconsidertheidentityelement@,theconstantmindthatdoesNOTchange.

Letusnowstudytheseeminglysimple ‘equationsofchange’ forthebasic functions.Yes,ofcourse, thescholarwillfindnointerestonthem.Whatcanwereallyfindofsomethingso‘obvious’as∂x2=2x.

Butwe shall remind again the reader of 2 ofmy favoritemasters’ quotes, ‘genius targetswhat nobody sees’ and‘simplicityisgenius’(Leonardo).Sincethebeautyof5Dspacetimeconsistonseeingrelationshipsthatnobodycarestowonderabout,whicharefoundinthesimplestrealitiesthatholdfundamentallawsoftime=change.

∫∂ofPOLYNOMIALGROWTH:POLYNOMIALACTIONSvs.DERIVATIVES

ApolynomialcanbeunderstoodasaregularT.œ,withgreatersymmetrybetweenpartsandwholes,expressedasaquantitativesumofitsparts,xa.Thatisinspaceitrepresentsa‘line’,‘square’,or‘cubicform’.

And so as the fundamental form of change happenswhen S=T, in the form of reproduction, the first insight of 5Dcalculusontheoutcomeofpolynomialchangeisthis:

Whenwechangeasquare,X2changedoesNOThappenonlyasitwouldseemnaturalatanXrate–thatisthefrontallineofthesystem,butata2Xrate,in2ofthe4sidesofthesquare.SochangeisNOTamotioninasingledirectionthatwouldreproducethesquarebyanXrate,butithappensin2sidesofthesquare.

Andasaconsequenceitpreservestheformofthesquare.

We thus realizeby this simpleanalysisof∂X2=2X that theMAIN formofCHANGEwhichultimatelyappears inmostrealitiesis‘reproductivechange’,NOTmerelocomotion(ultimatelyaformofreproduction)…

Againwhenweconsiderthecube,X3,therateofchangeis3X2;sochangenowhappensin3sidesofthecube,anditisalsoareproductivegrowth,thatPRESERVEStheformofthecube.Whileinaline,∂nX=n,changehappensinoneofthesidesofthelinethatPRESERVEStheformoftheline,reproducingit,somethingthatcanbeperceivedasthemotionoftheforeheadthatadvancestheline.

Inverselywecanconsiderthattogeneratethroughanintegraloftheunitofchange,thefinitesimalofapolynomialvolumefrom1to4Dimensionsadsfrom3sidesofthecube,twosidesofthesquareand1sideoftheline(asweknowfromalgebra,aquinticdoesnotexistintermsofradicals;becausetheUniverseismadeof4Dimotionsplusdissolvingdeath=entropy.Soit isnotworthtobotherformorecomplexones;whilethefourthdimotionofsocialevolutionor

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motion of a cubemakes the system grow in scales of the fifth dimension; so it is a different case to the 3 basicpolynomialsthathappeninasinglescale).

Nowthismightalsoseemsimplebutwerealizethatifwearecreatingawholecubefrom3faces,theyarebecomingintertwined,webbed together as they penetrate each other to form the cube. So they act in fact as ‘networks’ ofpointswith ‘dark spaces’ throughwhich the other two sides penetrate. And the result is notmerely a cube but anetworkof3different‘orthogonal’,networks,whichistheidealizedconceptofaphysiologicaltrinity.Thesamecanbe saidof theprocess of creationof a square from two ‘orthogonal’ dimotionsof time-space. But, and this is alsointeresting tomeditate in depth, the line isNOTwebbing from the 2 extremes butmerely reproducing and hencemovingasawave,whichisNOTconnectedwiththe‘steps’leftbehind.Thusthelineconstantlymovesbecauseitdoesnot ‘entangle’withthememorial tailof itspreviousstœpsofreproduction.Whilethesquareandthecubebecomeentangledendingitsreproductionwhentheareaisfilledwithmemorialpersistence.

Weconcludeanotherstartlingconsequenceofallthis:

-Onlylinealinertiaiseternallocomotionbecausethelinedoesnot‘last’andhencecanbedisplacedadeternal,butaholographicortrinitydimotionbecomesareproductiveactionthatstops in its finalcompletionofthehigherwholeform,onceitisintegrated.

Thenitmightasaquartic,developanewreproductivemotion,whereasthecubeisapointofalargerscale.

So the derivative give us the ‘units of reproduction’ of the polynomial in space, its minimal growth parts and theintegralreproducesawholebywebbinganumberofparts,thatentangletoformthe∫Xwhole.

Trilogiconpolynomialderivativesforspaceandtime.

Onceall this is resolvedwecanconsiderhowthetrilogicofcalculuswidensthepossibilitiesofasimplepolynomialderivative,consideringthesimplestcaseofitsapplicationtospaceandtime.

Exponentialderivativesandintegral

The first result already considered are the polynomial 'reduced' dimension bymeans of searching its infinitesimal,whichhoweverisforsimplepolynomialsquitelarger,comparedtoadirectxˆn-1reduction.

Furtheron,thelogarithmISclearlythe5Dsocialscalingoperationanditsderivativeisindeedtheabsolutefinitesimal,1/n.

Andinverselythemaximalgrowthisitsinverse,theabsolutedecayofe¯x.

It isworthtotalkof those3co-relatedresults fromthephilosophicalpov: themaximalexpansionofanevent isanabsolutefuturetopast,∆+1<<∆-1entropicdeathexpressedbytheexponential:

Theminimalprocessofgrowth(Log)isaninfinitesimal,themaximalprocessofdecay(e¯ª)isequivalenttothewhole,ina singlequantaof time.Westate in thegeneral law thatdeathhappens inasinglequantaof time, inwhich theentirenetworkthatpeggedtogetherthebeing,disappeared.

1D(singularity)+2D(Holographicprinciple)=3D(vitalenergy).

Inpracticethismeansthe'synchronicityintimeoftheclocksofthe3partsofthebeing'andthesuperpositionofthesolutionsthatbelongtoeachofthe3elementsofanyT.œ

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TT.EXPONENTIALDECAYS.

Finallythelogarithmicfunctionandexponentialfunctiontheratioofchange(derivative)diminishesfromtheabsolutemaximal,eª,which is itsownderivative,totheabsoluteminimal1/athe logderivativewhich isthedefinitionofaninfinitesimalpart(Leibniz),tillitpeaks,convertingan∆-1firstunitintoan∆ºwholeinthepeakofanexistentialworldcycle that then will start an inverse function of decay with -1/x diminution and a final fast collapse in the 3rdage<<deathmomentateˆ-aspeed.

Finitesimalactions.

Thelogarithm'sderivativethusadsasratioofchangeonlyaninfinitesimal,soittendsalwaystoabalancedstaticform(y=c).

Thequantityasystemabsorbstocreateanactionisgenerallydefinedasa'finitesimal',notinfinitesimal.Infinitedoesnotexist ina single continuum,but throughmultiplediscontinuitiesasall systems in timeand spaceare limited inspaceandtime,bothinasinglemembrane,andinwithinthePlanesofthe5thdimension(asinformationandenergydoesn'tfluxbetweenthosePlaneswithoutlossofentropy).

Afinitesimalisthequantityofenergy,motion,informationetc.usedbyaT.œforanactionofspace-timeINanyofthe5Dimensionsofthebeing,'putinmotion'tothataim.

4D:Thusentropyhasanegativeexponentialwhichshowtherhythmofdecayofthesystem.Andinthiscasethereisno need for 'a logarithmic' limit, since for the predator the death body is 'unlimited ¡-1 energy', though once the'relativeinfinite'numberofits¡-1partsareabsorbedthe'e-function'willhaveacutoff.

Howthefunctioncombineswithotherfunctionsshowsthenhowthesuperpositionandmerging-productprocessescombine as Dimotions with the reproductive growth and decay processes, whose results are intuitive: theproduct=mergingoftwopowersofreproductivegrowth,whenits'base=toe'isidentical'superpose=add'thatgrowth.

So the combination of ± exponentials and logarithm curves are also the best way to graph as a bell curve theworldcycleofexistenceinlinealterms.

4thdimension:Entropy:S∂polynomialdeathdimensionofdecay.

Polynomials do not evolve reality towards an impossible infinite growth. THEY ARE the inverse decay process; ofexponentialextinction,eˆ-x.

Wholes are physiological networks, which we analyse mathematically in its parts, mostly performing a motion ofspace-time,anactionthatexchangesmost likelybitsandbitesof timeandspace.Sothe logarithm isanoperationthat reflects the processes of minimal transmission and gathering of information and energy, of bidimensionalholographicquanta...reasonwhyitissopervadingintheconceptsofentropyandinformation.

5thdimension:∫T...

This isunderstoodbetter observing that the inverse function does in factmodel growth in the differentmodels ofbiologyandphysics,limitedbyacarryingcapacitystraightflatline:

The logarithmic functionhas as derivative an infinitesimal, 1/x,whichmakes it interesting as itmodels better thecurveofgrowthfromoto1intheemergentfastexplosive∆-1seedstate,whiletheinverseeˆ-xmodelthedecaydeathprocess.

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Integralsandderivativeswhichhaveamuchslowergrowth,thanpolynomialsontheotherhanddomodelmuchbetterastheyintegratethe'indivisible'finitesimalquantaofasystem,itsorganicgrowthand'wholeness'integratedinspace.

Thusintegralsdomoveupasocialgrowthinnew∆+15Dplanes.Anditsgraphsareacurvedgeometry,whichtakeseachlinealstep(differential)upwards,butasitcreatesanewwhole,partofitsenergygrowthsinksandcurvestogivebirthtothemind-singularity@,thewholenessthatwarpsthewhole,andconvertsthatenergyintostill,shrunkmind-mappingsofinformation,oftenwithinthe3Dparticle-head.

Wewillretaketheanalysisofthemorecomplexst-epson3,4and5D,sincemostofthecomplexprocessrelatedtothe3rddimension,asamixtureofSandT innerPlanes,willrequireamorecomplexdoubleortriplederivativeandintegrals - only the 4D decay entropic explosion can be satisfied as the decay of the single ∆-1 finitesimal with asinglevariable.

TT.EntropyEquations

OneExample.Thelawofdecayofradiumsaysthattherateofdecayisproportionaltotheinitialamountofradiumpresent. Supposeweknow thata certain time t= t0wehadR0gramsof radium.Wewant toknow theamountofradiumpresentatanysubsequenttimet.

LetR(t)betheamountofundecayedradiumattimet.Therateofdecayisgivenbythevalueof–(dR/dt).SincethisisproportionaltoR,wehave:

-dR/dt=kR where k is a constant.In order to solve our problem, it is necessary to determine a function from thedifferential equation. For this purpose we note that the function inverse to R(t) satisfies the equation: -dt/dR=1/kR,sincedt/dR=(1/dR)/dt.Fromtheintegralcalculusitisknownthatequationissatisfiedbyanyfunctionoftheform:T=-1/klnR+C.

whereCisanarbitraryconstant.FromthisrelationwedetermineRasafunctionoft.Wehave:

Fromthewholesetofsolutionsweselectonewhichfort=t0hasthevalueR0.ThissolutionisobtainedbysettingC1=R0ekt0.

Fromthemathematicalpointofview,equation(3)isthestatementofaverysimplelawforthechangewithtimeofthefunctionR;itsaysthattherateofdecrease–(dR/dt)ofthefunctionisproportionaltothevalueofthefunctionRitself.Suchalawfortherateofchangeofafunctionissatisfiednotonlybythephenomenaofradioactivedecaybutalsobymanyotherphysicalphenomena.

Wefindexactlythesamelawfortherateofchangeofafunction,forexample,inthestudyofthecoolingofabody,where the rate of decrease in the amount of heat in the body is proportional to the difference between thetemperatureof thebodyandthetemperatureof thesurroundingmedium,andthesame lawoccurs inmanyotherphysicalprocesses.Thustherangeofapplicationofthoseequationsisvastlywiderthantheparticularproblemoftheradioactivedecayfromwhichweobtainedtheequation.

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1D:PERCEPTIVEACTIONS:Derivativesasanglesofperception.

The 3rd typeof functions are concernednotwith∆±1past tofuture to past d=evolutions butwith present sinusoidalwaverepetitions of the same time-cycle, hence change is cyclicalrepetitive, and so those functions are very useful for the 3rdreproductivedimotioninspace,butalsoforatimedimotionorcycle:

Bothfunctionsthusareclearly inversenotonly inΓstbutalso inthe∆±1Planes-beingthenegativesymboloneofconventionsregardingthechosen±directionofthecyclical,sinusoidalmotion.

Herethoughtheinterestresidesincomparingbothtypeofpresentvs.∆past-futurefunctions:thepresentderivativeisselfrepetitive,aswereturntothesinafter4quadrantderivatives;andideedwereturntothepresentconsideringalsothegenerationalcycle,after4agesoflife.Sowecanmodelasinusoidalfunctionasaworldcycleofexistenceinits4quadrants.

SS:SinusoidalfunctionsofanglesandsinsandcosinesrelatetotheSS-trigonometricperceptivefunctionoforganisms.

growingdistortionanda‘blind’spotforitsinverse5thdimotionofexistence(whoseperformanceisthedenialofitsself).

St:ThussinusoidalfunctionsarealsogoodtomeasureSt-motionsdominantininformation.I.e.waveforms.

One very realized role of a derivative as a tangential division of the height in the dimension of information anddistance-linealmotiontotheobserverisameasureoftheangleofthebeing,whichrecedesinspacetimetillreachingthenon-perceptionasarelativefinitesimaloutoftheterritorialmind-worldoftheobserver,whichconnectsdirectlyderivativeswiththe1Dfirstdimotionofperceptiveexistence.Thebeingmightstillbeofcertainsizebutasafractalpointhehasrecededinthemental-spaceoftheworldoftheperceiver.

WhileasallS=T,thatisthereisalwaysasymmetrybetweendiscretenumbersandcontinuousmotions,Leibnizwithits geometric interpretation and farmore profound understanding of finitesimals,which he rightly defined as 1/n,representsthefirststepinthefutureofthediscipline,therenovatoranddeepunderstandingof it-whichNewton,whichcanbeconsideredmerelyanautomatonmathematician,specializedbrain,asmostmodernscientistsis-heisindeedthefatherofthewrongviewofscience-understoodnothingofit.

Indeed,Leibniz,theclosestpredecessorofthisblogISthegenius,Newtonthetalent.

TheÐimotionofcyclicalperceptionisexpressedbythenegative¡number,withitscyclicalrulesofsummation,whichimpliesthesumoftheangleofperceptionoftheself-centerednumberinitsargument.

Thus next then in the entangled representation of reality through those 'basic operand' comes then the duality ofaddition and subtraction, and its attached physical meanings of superposition and fusion of 'parts' into 'wholenumbers',oritsentropicinverseoperandsofnegativesubtraction.

ItstartstobethenobviousthatalloperandshaveitsinversefunctiontomaintainthebalanceoftheUniverse.

Additionby superposition in ever tighter spacesof similar clone species, is the simple¬Algebraic expressionof thesocialdimotions,bothinitspositive5Ðandnegativeentropic4Ðwhoseadditionofdecaying¡-1T.œsissofastthatitcanbeexpressedasanegativeexponentialgrowth,whichinthismannerwouldcompletethe3'Planes'ofaddition:+,x,xª...

Moreoveradditioncanhappeninsequentialtimeoradjacentspace,forminggrowingprobabilitiesorpopulations.Soasthesimplestmodeofoperandsextendsitsdiversificationthroughspaceortimeitwillmeandifferentthings.Ifweconsiderthehappeningofaneventorfullworldcycle1,probabilitieswillrepresentpartsofthewholeevent. Ifwe

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project it intospace itwillbeapopulationof similarevent,entering the regionofmaximal frequency.Bothwillbemathematicallyprojectedasabell curve.S=T.Same function for theadditionofeventsandpopulations, in timeorspace.

ThefirstmarveloftheUniverseisthesimplicityofitsoriginalprinciples,madecomplexbythedifferentiationacrossthesymmetriesofscale,topologyortime.Indeed,somethingsosimpleasthesumandinversesubtractionISstillthemost importantoperandsof theUniverse,whichgivesusnewnumbers,socialgatheringsof identicalbeings,whichherd together intoparallel flowsadoptingmost likelyabidimensional ST superpositionon laminar states thatkeepaddingthe3rddimensionofthebeing.LikethesimplestfirstmasterpiecesofBach,thearchitectonicalUniverseisasimpleprinciplebeforeorganicismtwistsitsform,inwhichbeingswhichareequalcometogether.

Superpositionofbidimensionalholographicfields issoimportantthatthewholeofquantumphysics isbasedinthissuperposition principle. The sum thus is still themaster of operands. But for sums to happen, the beingsmust beexternallyidentical,tobeperceivedaspartsofaquantifiedmass,eachofthemthesamevalue.AdditionthusistheultimateproofofthesocialnatureoftheUniverse.

infinitesimal'curved'exponentialchangesthathappenbetweentwoplanes,wherelinearityislost.

Wenoticeimmediatelythat4changesturnasinxcycleintoitself,as3-¡,3agesandanentropicdeathclosesa0’-sumworldcycleofanyspeciesbacktoitsinitialpoint.,startinginthiscasewiththesimplestcyclicalclock-motions,whichastheydoNOTmoveinspace,andrepeatitsformintime,areinfactnotoperatedby∫∂measuresofchange.

Nextcomes fromthebottomof that list, the functionsofperception, sinandcosangles;andtheresulthavesome'metaphysical'meanings.Indeed,therateofchangeofourinformativeanglemeasure(thesine),becomesthecos,the

rateofchangeofourmotion,orinotherwords,weSWITCHfromsin-stopstatestocos-movingstates,instœps.Wegofromstop-sintostep-cos;buttheinversedoesn'thold.Thatisifwegofrommotioncostostopsin,thiswillbeperceivedfromtheperspectiveofcos-motionasa'negative'reductionmotion-sine.

1D:cyclicalclocks,angularmomentum

Inthegraph,inthesimplestphysicalsystems1Dismerelytheangularmomentumofitscyclical clocks of time,maximised in the membrane that encloses the system. Strictlyspeakingitdoesnotchangebutbecomesthe'presentfunction'ofarepetitivefrequencyclockwithoutaderivativeofchangeasthetime-spacestepsseemnottovary.Whenweintroduceatorque,changehappens,called'acceleration',theseconddimensionoftime

motioninphysics,whichweshalllatterstudywhenanalysingin5DwiththeGalileanPx.Newton'slaws.Herewejustshallbrieflyexplainwhy in lineal time,ashumansonlyuse t tomeasurechange, the1D is the invariantoneand itsderivativeis0’.

Whatabout'higher'morecomplex,cyclical,andscalarDimensions?Theansweristhataswechangetheformofthedimensions,wehavetochangetheoperandsweuse;andspecificallywhenwestudytheDimensionsofchange,whichistheonedifferential/integralequationsquantify,thoseequationsmustadaptnottheotherwayaroundasmirrorsofrealitytotheformofthedimensionsofspace-timetheydescribe.

Soas1Disasteadystaterotarymotion,strictlyspeakingitdoesNOTchangeinspace-timelocomotion(whichiswhathumanswithitslinealsingletimeexpressinderivatives).Hencebasicallythederivativeofthoseangularmomentumsis0’.Itisconserved.

Letusrecallbrieflythoseclassicdefinitionsandmaths:

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Angularmomentumisavectorthatrepresentstheproductofabody'srotationalinertiaandrotationalvelocityabouta particular axis. In the simple case of revolution of a particle in a circle about a center of rotation, the particleremaining always in the same plane and having always the same distance from the center, we discard the vectornatureofangularmomentum,andtreatitasascalarproportionaltomomentofinertia,Iandangularspeed,ω:

L=Iω:Angularmomentum=momentofinertia×angularvelocity,anditstimederivativeis

dL/dt=dI/dtω+Idω/dtis0’,anddL/dt=0+Idω/dt,whichreducestodL/dt=Iα.

Therefore, angular momentum is constant, dL/dt=0when no torque is applied. And this is the essence of itsconservationlaw,aspecificcaseoftheconservationofthe5Dimensionsofspace-timeoftheUniverse:

'Inaclosedsystem,notorquecanbeexertedonanymatterwithouttheexertiononsomeothermatterofanequaland opposite torque. Hence, angularmomentum can be exchanged between objects in a closed system, but totalangularmomentumbeforeandafteranexchangeremainsconstant'.

But when a torque is applied in a single present plane, ormuchmore relevant to our inquire: when a system issubmittedto theorganisingordisorganisingentropic forceofahigheror lowerplaneofexistence,andaccelerationexists,avortexoftime-spacehappensandweenterintothesocialdimensionsofevolution-the5thDimensionofthemind.

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ODES=TIME-LIKEVSPDES.=SPACE-LIKEEQUATIONS:MATHEMATICALPHYSICS.

Adifferentialequationisamathematicalequationthatrelatessomefunctionwithitsderivatives.Letusapplysimpletrilogictoitsmeaning.

∆:Adifferentialequationbasicallygiveusthedualityofitsfinitesimallocalstatesanditsglobalwholeorganicformsbutatthesimplestlevel,mostoftenwithasinglefieldstate,ornetwork.Thusitisagoodrepresentationofthepartsandwholesof5Dbutformorecomplexsystems,discrete,organicanalysisfaresbetter.

T:ODEs

Inapplications,thefunctionsusuallyrepresentphysicalquantities,thederivativesrepresenttheirratesofchangeofagivendimotion, suchas the firstderivative is its ‘time speed’and the2nd its ‘timeacceleration’,whichare the twoderivativesthatgiveusallthemaximalandminimalstandingpointsofthe‘functionofexistence’ofthechangeintimewestudy.

Oncewefindtherateofchange,orfinitesimalofthedimotionofthespaceparameter;whichoverwhelminglybelongtothosetwomeaningfulones,we integratethroughan intervalofexistenceofaworldcycleandthusthesystemissolved.

Philosophically then the interestof5Dcalculus is toanalyze thegeometryof thesolution thatwill indicateuswhattypeofSS«Ts<ST>St»TTorganoreventofafunctionofexistencewearestudying.

And if the solution is possiblewhichwill give us even deeper insights on the structure of existential algebrawhoseillogicismirroredbythelawsofmathematicalalgebra.

Whenweworkwithasingleparameterweare‘obviously’workingonchangeinlinealtime.

This is the concept behind any ODE. Which is therefore clearly a time-like equation, as it probes mostly a singledimotionofthefunctionofexistenceofasystemanditsrateofchange.

S:PDEs.

Ontheotherhand,aPDEworkonseveralparameterswhichtendtobethe3coordinateshumansestablishforspace;henceitistheanalysisofamotioninspaceasasimultaneouswhole–inthesimplestformsasaherdinmotionwithcontinuity equations. Hencemore often concernedwith locomotions of the herd or system through space. And itschangesofpositioninspace.

ST:TruePDEs.Itisessentialtounderstand5DCalculusthefollowingconcept.Whileamerelocomotioninspacemightappear as a PDEbecauseof the use of ‘vectors in x, y, z’ coordinates that artificiallymultiply the variables for thepositioninspace(whichisasingleparameter,SS,intermsof5D)ofagivenpoint-T.œ-function,thereisaclearcaseinwhichtruePDEsappearofenormousimportanceformathematicalphysics,andanyanalysisofrealityintermsofthe‘fractalgenerator’ofSS»St≤ST≥Ts«TTsupœrganisms:

Thosearefunctionsinwhichthe3presentelementsofasupœrganism,Ts≤ST≥Stappearinthesameequation.WecallthemtodistinguishthemfromPDEsofcoordinatesofposition,‘TruePDEs’.

Andwecandistinguishroughlytwotypesofsuchfunctions:

- Spatial function in which the 3 dimensions of change are NOT coordinates external to a point (locomotion) butinternaltothepoint(areaandvolume):Forexample,theareaofarectangleisafunctionS=xy,of itsbasexanditsheighty.Thevolumeofarectangularparallelepipedisafunctionv=xyz,ofitsthreedimensions.Becausethe3classicdimensions are in vital topology related to its function, width≈reproduction≈ST, length≈locomotion≈sT,height≈information≈St, even though science normally uses themmerely for fixed volumes, they do have in certaincasesof ‘hydrodynamics’ and ‘cellular growth’, or social growthofpopulationsapotential use for adynamic study

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whenproperlyconsideredin‘mentalspaces’,wherethe3dimotionsofchangeareanalyzedasqualitativedimensions(themesofthoseofanadvanced5Dalgebracoursewewillnotattemptforalongtime).

-TImespacefunctionproper,classiconphysics,whenconsideringformulaerelatedtoenergy(ST)expressedbyits2independent space-time parameters, E=H ƒ= K T =M V, in its multiple parameters and variables. The well-knownformula p V = RT expresses the dependence of the volume v of a definite amount of gas on the pressure p andabsolutetemperatureT.

And sowewill findwith those basic type of differential equationswe can calculatemost Fractal generators in itsdynamicchangeinmoststiences.

∆¡:Differentialvs.∆<¡:simplecalculus.

The difference between the simpler calculus of integrals and derivatives vs. differential equations thus is one ofgrowth of complexity, from ‘words’ into ‘sentences’ of time dimotion; which means further extension in time, inspatial population and in the probing through scales, with those derivatives of motion or o information (Fouriertransforms,etc.)

Because suchST-analysisof superorganismsorT, TT, analysisof timeeventsare the fabricofwhich reality is built,even if analysis concentrates in simpler systems and herds, differential equations play a prominent role in manydisciplinesincluding5Dstience,concernedwiththeDimotionsofspacetime.

Differentialequationscanbedivided intoseveral types.Apart fromdescribing thepropertiesof theequation itself,theseclassesofdifferentialequationscanhelpinformthechoiceofapproachtoasolution.

Physical equations are related to the 3 elements of al the existential entities of the Universe. It must then beunderstoodthatwithinthegeneral ƒ(x)≈f(t)andy=S isomorphismbetweenmathematicalequationsandST-eps (notalwaystHecaseassymmetricstepscanrepeatitselfwiththesameparametersinSSSandTTTderivativesaswehaveseen in our intro toODE), partial differential equations,will be combinationsof analysis of systems in its 'primary'differentialfinitesimalsofspaceandtimethenaggregatedinmorecomplexStSYSTEMS,givingasanenormousrangeofpossiblePDEstudies,whichweshall strive toorderaccording to theconcept thatthere isageometricsymmetrybetween¬Algebra(s≈tsymmetries)andgeometry(S-wholessumoft-dynamicpoints)andanalysis(st-eps).

Soitisagoodguidanceforall¬AlgebraequationstomakeacommentofitssignificanceinthevitalternarygeometryofaT.œorcomplexeventbetweenT.œsacrossdifferentplanes,∆§studiedwiththoseequations.

Calculusandphysics:PartialDifferentialequationsas∆@st-equations.

Physicaleventsandprocessesoccurringinaspace-timesystemalwaysconsistofthechanges,duringthepassageofitsfinitetime,ofcertainphysicalmagnitudesrelatedtoitspointsofvitalspace.

This simple definition of space-time processes is at the heart of the whole differential calculus, which with slightchangesofinterpretationapplytoallGST.

AnyofthoseSTprocessescanbedescribedbyfunctionswithfourST,independentvariables,S(x,y),and(z,ƒ),wherex,yarethecoordinatesofapointofthespace,and,andzandƒoftime.

So ideally in a world in which humans had not distorted bidimensional time cycles, the way we work aroundmathematicalequationswouldbeslightlychanged.Aswearenotreinventingthehumanmindof7billionpeople-wearenotthatarrogant,we justwill feelhappytryingtoexplaina fewof thoseprocessesofbidimensionalspaceandtimehere.

Inthestudyofthephenomenaofnature,partialdifferentialequationsareencounteredjustasoftenasordinaryones.Asarulethishappensincaseswhereaneventisdescribedbyafunctionofseveralvariables.Fromthestudyofnaturetherearosethatclassofpartialdifferentialequationsthatisatthepresenttimethemostthoroughlyinvestigatedand

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probably themost important in thegeneral structureofhumanknowledge,namely theequationsofmathematicalphysics.

∆STsymmetries.

Eachpartialdifferentialequationrepresentsadifferentfinitesimalofscale,timeandspace,inthesecondlevel

Letusfirstconsideroscillations inanykindofmedium. Insuchoscillationseverypointofthemedium,occupying inequilibriumtheposition(x,y,z),willattimetbedisplacedalongavectoru(x,y,z,t),dependingontheinitialpositionofthepoint(x,y,z)andonthetimet.Inthiscasetheprocessinquestionwillbedescribedbyavectorfield.Butitiseasy to see that knowledge of this vector field, namely the field of displacements of points of themedium, is notsufficientinitselfforafulldescriptionoftheoscillation.Itisalsonecessarytoknow,forexample,thedensityρ(x,y,z,t) at eachpointof themedium, the temperatureT(x, y, z, t), and the internal stress, i.e., the forcesexertedonanarbitrarilychosenvolumeofthebodybytheentireremainingpartofit.

Thesequantitiescanbedescribedbyfunctionswithfourindependentvariables,x,y,z,andt,wherex,y,andzarethecoordinatesofapointofthespace,ndandtisthetime.

Physicalquantitiesmaybeofdifferentkinds.

∆:Somearecompletelycharacterizedbytheirnumericalvalues,e.g.,temperature,density,andthelike,andarecalledscalars.

S=T:Othershavedirectionandarethereforevectorquantities:velocity,acceleration,thestrengthofanelectricfield,etc. Vector quantities may be expressed not only by the length of the vector and its direction but also by its“components” ifwedecompose it intothesumofthreemutuallyperpendicularvectors, forexampleparalleltothecoordinateaxes.

Inmathematicalphysicsascalarquantityorascalarfieldispresentedbyonefunctionoffourindependentvariables,whereasavectorquantitydefinedonthewholespaceor,asitiscalled,avectorfieldisdescribedbythreefunctionsofthesevariables.Wecanwritesuchaquantityeitherintheform:

U (x,y,z,t)where the bold face type indicates the u is a vector, or in the form of three functions:Ux (x,y,z,t),Uy(x,y,z,t),Uz(x,y,z,t)

whereux,uy,anduzdenotetheprojectionsofthevectoronthecoordinateaxes.

Inadditiontovectorandscalarquantities, stillmorecomplicatedentitiesoccur inphysics, forexample thestateofstressofabodyatagivenpoint.Suchquantitiesarecalledtensors;afterafixedchoiceofcoordinateaxes,theymaybecharacterizedeverywherebyasetoffunctionsofthesamefourindependentvariables.

Inthismanner,thedescriptionofwidelydifferentkindsofphysicalphenomenaisusuallygivenbymeansofseveralfunctionsofseveralvariables.Ofcourse,suchadescriptioncannotbeabsolutelyexact.

Forexample,whenwedescribethedensityofamediumbymeansofonefunctionofourindependentvariables,weignorethefactthatatagivenpointwecannothaveanydensitywhatsoever.Thebodiesweareinvestigatinghaveamolecular structure, and the molecules are not contiguous but occur at finite distances from one another. Thedistances between molecules are for the most part considerably larger than the dimensions of the moleculesthemselves.Thusthedensity inquestion is theratioof themasscontained insomesmall,butnotextremelysmall,volume to this volume itself. The density at a point we usually think of as the limit of such ratios for decreasingvolumes.Astillgreatersimplificationandidealizationis introducedintheconceptofthetemperatureofamedium.The heat in a body is due to the randommotion of itsmolecules. The energy of themolecules differs, but if weconsideravolumecontaininga largecollectionofmolecules, thentheaverageenergyof their randommotionswilldefinewhatiscalledtemperature.

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Similarly,whenwespeakof thepressureofagasora liquidonthewallofacontainer,weshouldnot thinkof thepressureasthoughaparticleoftheliquidorgaswereactuallypressingagainstthewallofthecontainer.Infact,theseparticles, in their randommotion,hit thewallof the containerandbounceoff it. Sowhatwedescribeaspressureagainstthewallisactuallymadeupofaverylargenumberofimpulsesreceivedbyasectionofthewallthatissmallfromaneverydaypointofviewbutextremelylargeincomparisonwiththedistancesbetweenthemoleculesoftheliquidorgas.Itwouldbeeasytogivedozensofexamplesofasimilarnature.Themajorityofthequantitiesstudiedinphysicshaveexactlythesamecharacter.Mathematicalphysicsdealswithidealizedquantities,abstractingthemfromthe concrete properties of the corresponding physical entities and considering only the average values of thesequantities.

Suchanidealizationmayappearsomewhatcoarsebut,aswewillsee,itisveryuseful,sinceitenablesustomakeanexcellent analysis ofmany complicatedmatters, inwhichwe consider only the essential elements and omit thosefeatureswhicharesecondaryfromourpointofview.

I.e.thePoissonandLaplaceequationsareallovertheplace,astheyrepresenttheidealformofmostefficient‘sinks’thatmakean∆-1herdofT.œsfallintoa‘door’toalarger/smallernestedTŒscaleofthefifthdimension(chargesandmasses sink); the Laplace equation,∇2 ƒ=0, represents theway inwhich a herd of ∆-1 fractal points can ‘fill in’ asphericalmembrain,which is the fundamentalmode inwhichmembrains thathavemotioneven ifwe see themasfixedforms(afixedformultimatelyisaπSsurfaceinwhicheachpointthatappearsstaticmerelyisaπd,self-turningdiameter=point).Soitisallpervadinginnature.Infactelectrons,whichareherdsof‘ultradenselightphotons’trappedinthe‘eventhorizon’oftheblackhole-protononthequantumscale,canbemodeledjustasthesphericalharmonicsofits∆-1‘field’ofphotonicpoints;andsoonandsoon.

Soabeautifulwayto‘readphysics’asIusedtodoafewdecadesagowhenmymentalskillswereatitsheight,andmymemoryintact,wasjustto‘seeas∆STfunctions=formsindimotion,theclassicequationsofphysics.Thiswillbethefutureofphysics toentangle theabstractionof thosePDEsandODEsequations,numerated in the following list, intermsofwhattheymeanforthevitallife-deathdimotionalcyclesofitsT.œs

Simpleexamples.

Differential equations are very common in science and engineering, aswell as inmany other fields of quantitativestudy, because what can be directly observed and measured for systems undergoing changes are their rates ofchange.Thesolutionofadifferentialequationis,ingeneral,anequationexpressingthefunctionaldependenceofonevariableupononeormoreothers;itordinarilycontainsconstanttermsthatarenotpresentintheoriginaldifferentialequation.Anotherwayof saying this is that the solutionofadifferentialequationproducesa function that canbeusedtopredictthebehaviouroftheoriginalsystem,atleastwithincertainconstraints.

Differential equations are classified into several broad categories, and these are in turn further divided intomanysubcategories. Themost important categories are ordinary differential equations and partial differential equations.Whenthefunctioninvolvedintheequationdependsononlyasinglevariable,itsderivativesareordinaryderivativesandthedifferentialequationisclassedasanordinarydifferentialequation.Ontheotherhand,ifthefunctiondependsonseveralindependentvariables,sothatitsderivativesarepartialderivatives,thedifferentialequationisclassedasapartial differential equation. The following are examples of ordinary differential equations:

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Inthese,ystandsforthefunction,andeithertorxistheindependentvariable.Thesymbolskandmareusedheretostandforspecificconstants.

Whicheverthetypemaybe,adifferentialequationissaidtobeofthenthorderifitinvolvesaderivativeofthenthorderbutnoderivativeofanorderhigherthanthis.

Theequation:

isanexampleofapartialdifferentialequationofthesecondorder.Thetheoriesofordinaryandpartialdifferentialequationsaremarkedlydifferent,and for thisreasonthetwocategoriesaretreatedseparately.

Instead of a single differential equation, the object of study may be asimultaneoussystemofsuchequations.Theformulationofthelawsofdynamicsfrequentlyleadstosuchsystems.Inmany cases, a single differential equation of the nth order is advantageously replaceable by a system of nsimultaneousequations,eachofwhichisofthefirstorder,sothattechniquesfromlinear¬Algebracanbeapplied.

Anordinarydifferentialequationinwhich,forexample,thefunctionandtheindependentvariablearedenotedbyyandxisineffectanimplicitsummaryoftheessentialcharacteristicsofyasafunctionofx.

Thesecharacteristicswouldpresumablybemoreaccessibletoanalysisifanexplicitformulaforycouldbeproduced.Such a formula, or at least an equation in x and y (involving no derivatives) that is deducible from the differentialequation,iscalledasolutionofthedifferentialequation.Theprocessofdeducingasolutionfromtheequationbytheapplicationsof¬Algebraandcalculusiscalledsolvingorintegratingtheequation.

It shouldbenoted,however, that thedifferentialequations thatcanbeexplicitly solved formbuta smallminority.Thus, most functions must be studied by indirect methods. Even its existence must be proved when there is nopossibility of producing it for inspection. In practice, methods from numerical analysis, involving computers, areemployedtoobtainusefulapproximatesolutions.

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PDES

Abstract.Eachstœpofamethodofsolutionisgroundedinarealpropertyofthe5D∆STsymmetriesandconservationlawsoftheUniverse,whicharethe3Galileanparadoxesbetween∆+1curvedclosedworldcycles,sumoflinealsteps,whichgivesbirthtothemostusedmethodof linealapproximations;theequivalencebetweenSpaceandtime, inallStœps of dimotions,which gives birth to themethod of separation of variables on differential equations andmorebroadlyallowstomovearoundrelativespaceandtimeparametersinequationsjoinedbyanoperandof‘equivalence’(≈ not =). And the 2 conservation laws of the Universe, conservation of those ‘beats’ of existence, S=T in relativepresent,eternalbalance,justifyingtheequivalenceoperands.Andconservationofthe‘volumeofspace-time’ofeachplaneoftheUniverse,byvirtueofthe5DmetricequationSxT=C,whichjustifiesthesolutionofdifferentialequationsbyseparationsof scalesandharmonizes thosescalesallowingconstantbutbalancedtransfersof largerbitesenergyexchangedbysmallerbitsofinformation,St¡=1=Ts¡.

Asinbothcases,becauseS=Tandand∆-1=∆0,wecanseparateparametersandmakethemequivalenttoacommonconstant

TheSimplestEquationsofMathematicalPhysics.Solutionsbasedin5Dlaws.

Theobjectofmathematicalphysicsistostudytherelationsexistingamongtheseidealizedelements,theserelationsbeingdescribedbysetsoffunctionsofseveralindependentvariables.

The elementary connections and relations among physical quantities are expressed by the laws ofmechanics andphysics.Althoughtheserelationsareextremelyvariedincharacter,theygiverisetomorecomplicatedones,whicharederivedfromthembymathematicalargumentandareevenmorevaried.Thelawsofmechanicsandphysicsmaybewritteninmathematicallanguageintheformofpartialdifferentialequations,orperhapsintegralequations,relatingunknown functions to one another. To understand what is meant here, let us consider some examples of theequationsofmathematicalphysics.

Apartialdifferentialequation(PDE)isadifferentialequationthatcontainsunknownmultivariablefunctionsandtheirpartialderivatives.(Thisisincontrasttoordinarydifferentialequations,whichdealwithfunctionsofasinglevariableand their derivatives.) PDEsareused to formulateproblems involving functionsof several variables, andareeithersolvedbyhand,orusedtocreatearelevantcomputermodel.

PDEscanbeusedtodescribeawidevarietyofphenomenasuchassound,heat,electrostatics,electrodynamics,fluidflow,elasticity,orquantummechanics. These seeminglydistinctphysicalphenomenacanbe formalised similarly interms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partialdifferential equations often model multidimensional systems. PDEs find their generalisation in stochastic partialdifferentialequations.

Bothordinaryandpartialdifferentialequationsarebroadlyclassifiedaslinearandnonlinear;suchaslineararesmallstepsofanon-lineallarger∆STperiod,hencewecanapproximateallnon-linealsystemsbylinealequations.

Adifferentialequation is linear if theunknown functionand itsderivativesappear to thepower1 (productsof theunknownfunctionand itsderivativesarenotallowed)v.nonlinearofhigherpowers.Thecharacteristicpropertyoflinearequationsisthattheirsolutionsformanaffinesubspaceofanappropriatefunctionspace,whichresultsinmuchmore developed theory of linear differential equations. Homogeneous linear differential equations are a furthersubclass for which the space of solutions is a linear subspace i.e. the sum of any set of solutions or multiples ofsolutionsisalsoasolution.Thecoefficientsoftheunknownfunctionanditsderivativesinalineardifferentialequationareallowedtobe(known)functionsoftheindependentvariableorvariables;ifthesecoefficientsareconstantsthenonespeaksofaconstantcoefficientlineardifferentialequation.

Thereareveryfewmethodsofsolvingnonlineardifferentialequationsexactly;thosethatareknowntypicallydependontheequationhavingparticularsymmetries.Nonlineardifferentialequationscanexhibitverycomplicatedbehavior

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overextendedtimeintervals,characteristicofchaos.Eventhefundamentalquestionsofexistence,uniqueness,andextendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary valueproblemsfornonlinearPDEsarehardproblemsandtheirresolutioninspecialcasesisconsideredtobeasignificantadvance in the mathematical theory (cf. Navier–Stokes existence and smoothness). However, if the differentialequation is a correctly formulated representation of ameaningful physical process, then one expects it to have asolution.

Lineardifferentialequations frequentlyappearasapproximationstononlinearequations.Theseapproximationsareonly valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to thenonlinearpendulumequationthatisvalidforsmallamplitudeoscillations.

Apartialdifferentialequation(PDE)isanequationinvolvingfunctionsandtheirpartialderivatives;forexample,

thewaveequation

Somepartialdifferentialequationscanbesolvedexactly

Ingeneral,partialdifferentialequationsaremuchmoredifficulttosolveanalyticallythanareordinarydifferentialequations.Theyaremostlysolvedusingthefundamentalillogiclawsof5Dmetrics,evenifhumindsdon’tknowtheyare

usingthem;thatis(:

Anintegraltransform,(findingthe∆-wholeorexactsolution)aseparationofvariables(usingtheS=T=Cmetric)linealprocedures(usingthe∑|¡-1=O¡+1lawofsmalllinealstepsthatadtoTimecycle;or--whenallelsefails(whichit

frequentlydoes)--numericalmethodssuchasfinitedifferences(usingthefactthatcontinuityisreallasumofdiscretestœps).

Letusmakesomecommentsonthosemethodsof5Dcalculus(:

ThemethodsofsolutionsofDifferentialequationscombined5Dcalculus.S=T.

Aswesaidsomewhere,allresumesinthefunctioofpresentexistence,S=T.

This is themethodofsolutionofPDEs,toreducethemtotwoODEs,suchasS=T,henceS=c,T=candwesolvetwoequations.Simple,isn’t?SimplicityisgeniussaidL1.L3agrees(:

Consider the most famous equation of quantum physics, the Schrodinger equation, describing nonrelativisticquantumphenomena:

H2/2m∇2Ψ+V(S)Ψ=−ih∂Ψ/∂t

wherem is themassofasubatomicparticle,h isPlanck’sconstant (dividedby2π),V is thepotentialenergyof theparticle,and|Ψ(s,t)|2istheprobabilitydensityoffindingtheparticleatsattimet.

ThoseEquationshavepartialderivativeswithrespecttotime.AsafirststeptowardsolvingthesePDEs,let’sseparatethetimevariable.WewilldenotethefunctionsinallfourequationsbythegenericsymbolΨ(s,t).

Theseparationofvariablesstartswithseparatingthesandtdependenceintofactors:

Ψ(s,t)≡S(s)T(t).

This factorizationpermitsus toseparate the twooperationsof spacedifferentiationandtimedifferentiation.Asanillustration,weseparatethetimeandspacedependencefortheSchrodingerequation.SubstitutingforΨ,wegeth2/2m∇2(ST)+V(s)(ST)=−ih∂∂t(ST),

DividingbothsidesbySTyields:

−1/Sh2/2m∇2S+V(s)=−i1/ThdT/dt

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Nowcomesthecrucialstepintheprocesscentralargumentoftheseparationofvariables.

TheLHSofEquation(isafunctionofSpacepositionalone,andtheRHSisafunctionoftimealone.Sincerandtareindependentvariables,theonlywaythattheequationcanholdisforbothsidestobeconstant,sayα:

−1/Sh2/2m∇2S+V(s)=α

−I1/ThdT/dt=α

Wehavereducedtheoriginaltime-dependentSchrodingerequation,aPDE,toanODEinvolvingonlytime,andaPDEinvolving only the space position variables. Most problems of elementary mathematical physics have the sameproperty,i.e.,S=T,whichbyvirtueof5DmetricsistheConstantstateofpresent,henceS->αT->=α

Allthisphysicistsdobutdon’tknowwhyitworksforsomanyequation(:

The time-dependent PDEs of mathematical physics can be reduced to 2 ODEs in the time variable and the spacevariablebyvirtueofS=T=Constant,theequationofpresent.

Thusasusualourinterestisnotinrepeatingthewellknown‘pro’methodsofstudyofPDEsandODEsbutthewhyswecanlearnfromitsmergingwith5Daswealreadynoticedinourintroduction.

AsgeometrydefinesforminspaceandbyvirtueoftheS=Tequivalencemotionintime,thegeometryofdifferentialequationsanditssolutionsilluminatesingreatdealthefunctionorpartofanorganicwholethediferentialequationstudies. And greatly reduces the number of ‘real differential equations’we find in nature, helping also to considerwhichsolutionsarevalidandwhichrestrictionsandlimitsmustbeimposed.

Secondorder.

Partialdifferentialequationsofsecond-orderareamenabletoanalyticalsolutioniftheycanbewrritenasalineal

equation.SuchPDEsareoftheform:

Andoncemoreaccordingto5Dtheycanbeclassifiedinto3topologicalvariations,whichmiraculouslyhuminds,withoutknowingvitaltopologyhavecalledexactly…believeitornotifyouareanexperton5D(thatisifyouareme,Iandmyself:)aselliptic,hyperbolic,orparabolic.Whereas,ahyperbolicequationisonedominatedbytheSTbody-wavetopology;thatiswhere∂s∂tislargerthan∂s∂sand∂t∂t;ellipticwhen∂s∂s+∂t∂tisbiggerthanSS+TTandparabolic

whenbothareequal.

Inthelanguageofclassicmath,wesaidthatLinearsecond-orderPDEsarethenclassifiedaccordingtotheproperties

ofthematrix

aselliptic,hyperbolic,orparabolic.

IfZisapositivedefinitematrix,i.e., ,thePDEissaidtobeelliptic.Laplace'sequationandPoisson'sequationareexamples.Boundaryconditionsareusedtogivetheconstraint on∂Ω,where

holdsinΩ.

Ifdet(Z)<0,thePDEissaidtobehyperbolic.Thewaveequationisanexampleofahyperbolicpartialdifferentialequation.Initial-boundaryconditionsareusedtogive

where holdsinΩ

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Ifdet(Z)=0,thePDEissaidtobeparabolic.Theheatconductionequationequationandotherdiffusionequationsareexamples.Initial-boundaryconditionsareusedtogive

where holdsinΩ

Separationofvariablesbyscales.

Finallythe3rdfundamentalmethodbesideslinealapproximationsandseparationofS=Tvalues,takesintoaccounttheconservationofvolumeofspace-timebetweenscale,butalsothefactthatsmallerscaleshavefastertimesspeedsandsmallerparts,perceivedasrelativeinformationcomparedtothebitesofenergyofalargerscale,sotheequivalenceisbetweena‘pool’ofbitsofinformationofasmaller∆-1scaleanditsequivalentvolumeofbitesofenergyofalarger∆0plane.ThisequivalencebeingessentialtoeverythingintheUniverseasitallowsthecouplingoflanguagesandenergy,makingforexamplethephysicaleconomyequivalenttothefinancialeconomy,thelanguageequivalenttotheaction;themathematicsmirror the reality, but as information is smaller andmore abundant, this equivalence tends to beinflationary,themesthosethatappear inalltheplanesofspace-timeandtheinteractionofmindsandlanguagesofinformation(SS,St)withEnergybitesandentropicboosts(TT,Ts).

Letusthenshowthecase,withthepreviousexampleofSchrodinger’sequationfortheorbitofanelectroninanatom.

∂2(ρψ)/∂ρ2=–(ε+2/ρ)ρψ

Here,εistheelectronenergy,ρistheradialcoordinate,andψistheelectronwaveamplitude.

Atsmalldistancesfromthenucleus,ψmayoscillaterapidly,butat largedistances,ψmustdecreaseexponentiallywithr.Thisisbecause,atlargedistances,aboundelectron’spotentialenergyisgreaterthanitstotalenergy.Withanegativekineticenergy,wavenumberkbecomesimaginaryandthenormaloscillatoryexp{ikr}termbecomesexp{–Kr},whereK=k/i.

Weseparatethelarge-scaleexponentialfromthesmall-scaleoscillationsbymakingthissubstitution:

ρψ=g(ρ)exp{–βρ}

Here,β isanarbitraryconstant,andg(ρ) istheunknownsmall-scalefunctionofdistance.Aftera lotofmath,ourdifferentialequationbecomes:

∂2g/∂ρ2–2β∂g/∂ρ+(β2+ε+2/ρ)g=0

Believeitornot,thisisprogress.Let’schooseβ2=–ε,reducingourequationto:

∂2g/∂ρ2–2β∂g/∂ρ+2g/ρ=0

Ifnotfortheρ inthedenominatorofthethirdterm,thiswouldbeasimpleequation.ButwecansolvethiswithaTaylorseries.

Let:g(ρ)=Σkakρk

This techniquewillwork if thecoefficientsakapproachzero for largek.Putting theTaylor series intoourequationyields:

0=Σn{(n+1)nan+1–2βnan+2an}ρn–1

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Here the sum is from n=1 to n=+∞. The above equation is valid for all values ofρ. This can only be true if thecoefficientofeachpowerofρiszero.Thisisanimportantruleforpolynomialsthatiswellworthremembering.Afterrearranging,weobtain:

foralln>0:an+1=an{2(βn–1)/n(n+1)}

Withanychoiceofa1,wecanrecursivelycalculatea2,a3,…intermsofβ,whichisrelatedtotheelectron’senergy.Fortheelectrontobeboundtothenucleus,βmustequalthenafinitesimal,1/nforsomeintegern;whichmakessenseofwhy electrons in atomshavequantizedenergieswhichare its finitesimalparts ( ultradense lightphotonstrappedinthepotentialwelloftheatom).

ThisisthebasisofthePeriodicTable,chemistry,biology,solidstatephysics,digitalelectronics,andeverythingelseweknowaboutatoms;andtheinsightof5Distoconsiderthosequantizedenergiesthefinitesimalbitesofelectrons–itsdensebosonsoflight.

SothequestionofPartialDifferentialequations‘reduces’tothestudyofODEs,whichweshallconsidernowindetail;tothenplungeintotheexperimentalpraxisofthemaindifferentialequationsoftheUniverse,whichwillreflectthebasictopologiesofSt<ST>Tsand∆±¡lawsofexistentialalgebra,regardinglocomotion(Ts)andtransfersofenergyandinformationbetweenplanesandphysiologicalnetworks(St¡-1=Ts¡)

ThuswiththisbriefintroductiontoCalculusweconcludethispaperonNon-Aristotelian,Non-Euclideanmathematics.

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