The Golden Ratio: The Divine Beauty of Mathematics - NIBM ...

TheGoldenRatioTHEDIVINEBEAUTYOFMATHEMATICS

GaryB.MeisnerFounderofGoldennumber.netandPhiMatrixTM

http://Goldennumber.net

CONTENTS

Introduction

IGOLDENGEOMETRY

IIPHIANDFIBONACCI

IIITHEDIVINEPROPORTION

IVGOLDENARCHITECTURE&DESIGN

VGOLDENLIFE

VIAGOLDENUNIVERSE?

APPENDICES

AppendixA:FurtherDiscussion

AppendixB:GoldenConstructions

Notes&FurtherReading

Acknowledgments

ImageCredits

Index

W

INTRODUCTION

hatmakesasinglenumbersocaptivatingthatithaspersistedin

ourimaginationsformorethantwothousandyears?Souniversalthatitis

foundinthewritingsofanancientGreekmathematician,themusingsofa

revolutionarycosmologicalscientist,thedesignsofatwentieth-century

architect,andtheintrigueofabest-sellingthrillernovel-turned-movie

blockbuster?Sopervasivethatitappearssimultaneouslyinthegreatest

architecturalmonumentoftheancientworld,thepaintingsofhistory’s

mosteminentRenaissanceartist,andtheatomicarrangementofrecently

discoveredquasicrystallineminerals?Andsocontroversialthatit

engendersconfusingandpolarizingclaimsaboutitsappearancesand

applications?

Youmaythink,orhavebeentold,thattheevidencehasalreadybeenpresented,theanswershavealreadybeenfound,andthatthiscaseisclosed.Thegoldenratioisnotanewtopic—muchhasbeenwrittenaboutitsinceancienttimes.

Whatcouldpossiblybenew?Theanswersmaysurpriseyou.Fortunately,technologyandknowledgecontinuetoadvanceatanever-increasingpace,constantlyprovidingnewinformationthatwaspreviouslyunavailable.JustasnewtechnologiesinDNAevidencecanrevealnewtruthsthatcompletelyoverturnapastverdictinacriminalcase,newtechnologyisgivingustheinformationandtoolstoshowthatpastverdictsonthistopicwerealsolackingintheircompletenessandaccuracy.We’reabouttooverturnsomepastconvictions,too—notconvictionsoffelonsheldinaprison,butratherconvictionsofbeliefheldinthemind.Beliefscanbetheirownformofprison,andweoftendon’tknowhowimprisonedourmindsareuntilweseetheworldfromavarietyofdifferentperspectives.

OurnewtoolsforcollectingforensicevidencearetheInternet,newsoftwareapplicationsonmuchfastercomputingtechnology,andagrowingglobalcommunityofpeoplesharinginformation.In1997,theInternetwasusedbyonly11percentofthedevelopedworldandonly2percentglobally.1By2004,mostUSuserswerestillaccessingtheInternetonslowdial-upconnections,2andWikipediahadlessthan5percentofthearticlesthatithadby2017.3IlaunchedGoldenNumber.netin2001,andfollowedin2004withmyPhiMatrixsoftware,whichallowstheanalysisofdigitalimagesinjustseconds.Thereisnowamind-bogglingcollectionofimagestostudy,manyofwhichwerenotreadilyavailableinhigh-resolutionuntilthelastfivetotenyears.ManyoftheinsightsI’llsharewithyouwerecontributedbyusersaroundtheworldwhohadnowayofconnectingwitheachotheruntilveryrecently.So,indeed,someoftheinformationwrittenonthistopicjustadecadeortwoagocannowbeshowntobeincompleteinitsfactsandconclusions.AndIfullysuspectthattechnologiesandinformationavailabletenortwentyyearsfromnowwillbringnewinsightsthatweren’treadilyavailableasIwritethesewordstoday.

Whetheryou’reamathematician,designer,phiaficionado,orphiskeptic,Ihopeyou’llfindsomethingnew,interesting,andinformativeinthisbook,andIhopeitchallengesyoutoseeandapplythisnumberinnewways.Furthermore,Ihopetokindleafireinyouaswejourneyacrosstimeandspace,exploringtheveryunusualanduniquemathematicalpropertiesofthisubiquitousnumber—knownbyvariousmonikersthroughtheages—thathasinspiredsomanyofhistory’sgreatestminds.

http://GoldenNumber.net

Thisc.100CEfragmentfrompapyrifoundatOxyrhynchus,Egypt,showsadiagramfromBookII,Proposition5ofEuclid’sElements.Thefirstreferencetothe“extremeandmeanratio”appearsinthedefinitionsandProposition30ofBookVI.

WHATISPHI?

Let’sbeginthisongoingjourneyofdiscoverywithabasicunderstandingofthisintriguingnumber,gettoknowsomeofthepeoplethroughouthistorywhoselivesittouched,andexplorewhereitappearsandthewaysinwhichithasbeenusedoverthemillennia.Representedinshorthandbythedecimal1.618,phiisanirrationalnumberfollowedbyaninfinitenumberofdigits,andisaccurateenoughforalmostanypracticalpurposeweaskittoserve,takesmuchlesstimetowrite,andsavesaninfinitenumberoftreeswhenprinted.Thefamiliarnumber3.14,whichrelatesacircle’scircumferencetoitsdiameter,isrepresentedbytheGreekletterπ(pi).Similarly,1.618isrepresentedbyanotherGreekletter,Φ(phi),althoughithastakenonotheraliasesindifferenterasofhistory.InmathematicscirclesitissometimesdenotedbytheGreekletterτ(tau).Today,itismostoftencalledthegoldenratio,butithasalsobeenknowninrecenttimesasthe“goldennumber,”“goldenproportion,”“goldenmean,”“goldensection,”and“goldencut.”Furtherbackintime,itwasevendescribedas“divine.”

This“divine,”“golden”numberisuniqueinitsmathematicalpropertiesandfrequentappearancesthroughoutgeometryandnature.Mosteveryonelearnedaboutthenumberpi(π)inschool,butrelativelyfewcurriculaincludephi,wherewe’llusetheuppercaseGreeksymbolΦtodesignate1.618,andthelowercaseϕtodesignateitsreciprocal,1/1.618or0.618.Thisisperhapsinpartbecausegraspingallitsmanifestationscantransportonebeyondanacademicsettingintotherealmofthespiritual.Indeed,Φunveilsanunusuallyfrequentconstantofdesignthatappliestosomanyaspectsoflife,art,andarchitecture,butlet’sbeginwiththepurestandsimplestoffactsaboutΦ,whicharefoundinthefieldofgeometry.

MAKINGTHEGOLDENRATIO“GOLDEN”

Thegoldenratiowasn’t“golden”untilthe1800s.ItisbelievedthatGermanmathematicianMartinOhm(1792–1872)wasthefirstpersontousetheterm“golden”inreferencetoitwhenhepublishedin1835thesecondeditionofthebookDieReineElementar-Mathematik(ThePureElementaryMathematics),famedforcontainingthefirstknownusageofgoldenerschnitt(goldensection)inafootnote.4ThefirstknownuseofthetermgoldenratioinEnglishwasinan1875EncyclopediaBritannicaarticlebyJamesSulleyonaesthetics.Butthetermdidn’tappearinamathematicalcontextuntilScottishmathematicianGeorgeChrystal’s1898bookIntroductiontoAlgebra.5

HistoryrecordstheancientGreekmathematicianEuclidasdescribingitfirst—andperhapsbest—inBookVIofhismathematicstreatiseElements:

“Astraightlineissaidtohavebeencutinextremeandmeanratiowhen,asthewholelineistothegreatersegment,soisthe

greatertotheless.”6

So,where’sthemagicandwonderinthat?Let’sstartwithanexample.IfIaskedyoutodividealine,youcoulddosoinmanyplaces.Ifyoucutitinhalf,you’llcreatethis:

Thewholelineis1.Let’scallitA.Thefirstsegmentis1/2.Let’scallitB.

Thesecondsegmentisalso1/2.Let’scallitC.

Here,theratioofAtoBis2to1,andtheratioofBtoCis1to1.Let’scutthatlineagain,andthistimethinkofitassomethingtobeshared

betweenyou(B)andme(C),suchasabarofchocolate.I’lltakejustone-third,becausethat’sthekindofguyIam:

ThewholelineAisstill1.ThelongersegmentBisnow2/3.TheshortersegmentCisnow1/3.

HeretheratioofAtoBis3to2andtheratioofyourpieceBtomypieceCis2to1.

IfIonlytookaquarterofit,thoseratioswouldbe4to3and3to1.AndifIonlytook10percent,thoseratioswouldbe10to9and9to1.

Aswecutthelineindifferentplaces,wegetavarietyofdifferingratiosforAtoB,andtheynevermatchtheratiosforBtoC…exceptwhenwecutitinthat

one,uniqueplacethatEuclidmarveledovermorethantwothousandyearsago.Atthatsinglepointofequilibrium,wefindthattheratioofAtoBis1.618to1,andtheratioofBtoCisalso1.618to1!

Thisisoneuniqueaspectofthegoldenratio:theratioofthewholesegment(A)tothelargersegment(B)isequaltotheratioofthelargersegment(B)tothesmallersegment(C).Inotherwords:

A/B=B/C

Butphihasmanyuniquemathematicalproperties.Forexample,itistheonlynumberwhosereciprocalisonelessthanitself,as1/1.618=0.618.Statedmoresimplyandelegantly:

1/Φ=Φ–1

As1.6182=2.618,phiisalsotheonlynumberwhosesquareisonemorethanitself;thatis:

Φ2=Φ+1

Totakethenextstepinunderstandingwhyphianditsmathematicalpropertieshaveramificationsbeyondbeinganinterestingexerciseinmathematics,I’dliketointroduceyoutoPhiMatrix,thesoftwareapplicationIdevelopedin2004andre-releasedwithanewversionin2009.Ilearnedobject-orientedprogrammingattheageoffifty-fourjustfortheexpresspurposeofcreatingthisoneprogram,whichisnowusedbythousandsofverytalentedandenthusiasticartists,designers,photographers,andothersinmorethanseventycountriesaroundtheworld.PhiMatrixmakesitveryeasytofindandapplythegoldenratiotoanyimageonyourscreen.Asanexample,considerthelinesegmentwejustdividedaccordingtothegoldenratio,byoverlayingitwitharectangularPhiMatrixgrid(showningreen):

Asyoucansee,thegreendividinglineintersectsthepointrepresentingthegoldenratioonthesegment.Simpleenough,isn’tit?You’llseesimilarrectangularoverlaysusedthroughoutthisbooktovisuallyindicateapplicationofthegoldenratio.

Aswewilldiscover,theappealofthisgoldenproportionextendsfromdesigners,mathematicians,andmysticstodoctors,biologists,andinvestors.Thegoldenproportionispresentinthenaturalworldandisevenintrinsicallyrelatedtoourperceptionsoffacialbeauty.Throughouthistoryithasbeenusedtocreatebeautyinmanygreatworksofclassicartandarchitecture,anditisstillinusetodaytocreatevisualharmoniesingraphicdesign,productdesign,photoandvideocomposition,logos,userinterfaces,andmore.Somebelieveit’sevenfoundintheproportionsofthesolarsystem,aswellasthepriceandtimingmovementsofstockmarketsandtheforeigncurrencyexchange.

ThisportionofLeonardodaVinci’sJohntheBaptist(c.1516)showssomecompellingproportions.CoulddaVincihavebeenintentionallyreflectingthegoldenratiointhispainting?

ACONTROVERSIALNUMBER

Withalltheattentionithasreceived,youwouldthinkthatthisnumberwouldberecognizedasanimportantuniversalconstant—certainlyaswell-knownaspi—butthiscontroversialnumberusuallygetslittlemorethanapassingmentioninthecurriculaofmostacademicinstitutions.Why?

Indeed,manyconfusingandpolarizingclaimshavebeenmadeaboutitsappearanceandapplication.Eventhesmallminorityofpeoplewhoknowofitreallyknowverylittleaboutit.Doesitbelongintherealmofconspiracytheory,orarethesecuriousmindswhodiscernitslatenttreasuresontosomething?I’llletyouinonthemanyclaimsandcounterclaimsandunveiltheevidencelikeagoodmysterynovelorepisodeofCSI.Inthiscase,though,youarethedetective,judge,andjury.Youdecideforyourselfiftheclaimsaretrueorfalseorifthey’regroundedinmathormyth.Intheend,youmaynotknowforcertainifitwasjustaverystrangecoincidenceorevidenceofagranderdesign.

Intrigued?Themoreyouunderstandaboutthemathbehindthegoldenratio,themoreyou’llappreciateitsappearancesinnatureaswellasthearts,andthemoreyou’llbeabletoapplyitincreativeartisticexpressionsthatarevirtuallylimitlessintheirapplication.

Let’sbeginourexplorationofthisverybroad,deep,andfascinatingsubjectbytakingalittlewalkthroughhistory,exploringthelivesofseveralofthediversecastofcharacterswhohaveplayedaroleinthistimelessstory.

SacredGoldenRatioSculpturebyOliverBradyandCarmelClark.Thismagneticsculpture’sdesignisbasedonthe180-degreegoldenspiraldiscussedshownhere.

I

GOLDENGEOMETRY

“Geometryhastwogreattreasures:oneisthetheoremofPythagoras,theotherthedivisionofalineintomeanand

extremeratio.Thefirstwemaycomparetoamassofgold,thesecondwemaycallapreciousjewel.”1

—JohannesKepler

A lthoughtheproportionknownasthegoldenratiohasalwaysexisted

inmathematics,geometry,andnature,exactlywhenitwasfirstdiscovered

andappliedbymankindisunknown.Itisreasonabletoassumethatithas

beendiscoveredandrediscoveredthroughouthistory,whichexplainswhy

itisknownbyseveralnames.There’ssomecompellingevidenceof

awarenessandapplicationofthegoldenratiobytheancient

mathematiciansofBabylonandIndia,butlet’sfirststartwithGreece.

ThisengravingbyJeanDambrun(1741–c.1808)portraysPythagorasasdepictedonaRomancoinfromthethirdcentury.

ThispaintingbyRussianartistFyodorBronnikov(1827–1902)showsthecultofPythagorascelebratingsunrise.

ANCIENTGREECE

Mostofthecontentintoday’sgeometrytextbooksisderivedfromthediscoveriesoftheancientGreeks,andtheearliestreferencestowhatwenowknowasthegoldenratiomayhavecomefromthetimeofPythagoras,amathematicianandphilosopherwholivedfromabout570BCEto495BCE.Itisthoughtthatthefive-pointedstar,orpentagram—inwhichthelengthofeverylinesegmentisinagoldenratiorelationshiptoeveryotherone,asshownbelow—wasthesymbolofhisschool,andthatheandhisfollowerswerethefirsttodiscoversomeoftheuniquepropertiesofthegoldenratio.

ThepentagonatthecenterofthepentagrammakesanappearanceintheworkoftherenownedGreekphilosopherPlato(c.427–347BCE)—specificallyhisc.360BCEdialogueTimaeus,whichdescribesauniversemadeupoffourelements,representedbyfourfundamentalgeometricsolids(nowknownasthePlatonicsolids).Thefifthsolidisrevealedtobethedodecahedron—anassemblageoftwelvepentagonsintendedtorepresenttheshapeoftheuniverse.Inhisdialogue,PlatoalsowroteofameanrelationshipbetweenthreenumbersthatmightbeadirectprecursortoEuclid’s“extremeandmeanratio”:

Thegoldencutofthepentagram.

Theratiosoftheredsegmenttothegreensegment,thegreensegmenttothebluesegment,andthebluesegmenttothepurplesegmentareallequaltophi(Φ).

ThisillustrationofthefivePlatonicsolidsandtheirassociatedelementsappearsinJohannesKepler’sMysteriumCosmographicum(1596).

2“Whenthemeanistothefirsttermasthelasttermistothemean,…theywillallbynecessitycometobethesame,andhavingbecomethesamewithoneanotherwillbeallone.”2

Tothisday,however,itisunclearwhetherthisisadescriptionofmeansingeneral,orwhetherthisisaspecificreferencetothegoldenratio.

Althoughlittleisknownabouthisorigins,EuclidlivedinancientAlexandriaaroundthethirdcenturyBCE,whenPtolemyI(c.367–c.283BCE)ruledovertheHellenistickingdomofEgypt.Comprisedofthirteenbooks,Euclid’sElementscontainsillustrateddefinitions,postulates,propositions,andproofscoveringgeometry,numbertheory,proportions,andincommensurablelines,whicharethosethatcannotbeexpressedasaratioofintegers.Itwasafoundationalworkinthedevelopmentoflogicandmodernscience,andtodayitisregardedasoneofthemostinfluentialtextbookseverwritten.Firstprintedin

1482,itwasoneoftheearliestbooksonmathematicstobeproducedaftertheinventionoftheprintingpressbyGermanblacksmithJohannesGutenberg,anditislikelysecondonlytotheBibleinthenumberofeditionspublished.AbrahamLincolnstudieditintenselytohonehislogicalthinkingskills,andin1922thePulitzer-winningAmericanpoetandplaywrightEdnaSt.VincentMillaypennedapoementitled“EuclidAloneHasLookedonBeautyBare.”

Plato’sAcademyisportrayedinthisfirst-centuryBCERomanmosaicfromPompeii,Italy.

FlemishpainterJustusofGhentdepictedEuclidinhisc.1474series“FamousMen.”

ThisfirstprintededitionofEuclid’sElementsfrom1482showspropositions8–12fromBookIII.

ThisArabictranslationofEulid’sElementswascreatedbyPersianpolymathNasiral-Dinal-Tusi(1201–1294.)

InwhatEinsteinreferredtoasthe“holylittlegeometrybook,”Euclidreferredto“theextremeandmeanratio”anumberoftimes,alongwithconstructions(includingthepentagram)showinghowitisderivedgeometrically.BeginningaquicktourofEuclid’sfundamentalworkonthegoldenratio,wefindthefollowingconstructioninBookVI:3

Proposition30.Tocutagivensegment(AB)inextremeandmeanratio(E).

Here,EuclidasksustoconstructsquareABHCwithsidesequaltoourinitialsegmentAB,andthenconstructrectangleGCFDwithareaequaltothatofABHC,whereGAEDisalsoasquare.WhensegmentAC=1,wefind:

•TheareaofsquareABHC=1•TheareaofrectangleCFEA=1/Φ•TheareaofbothsquareGAEDandrectangleEBHF=1/Φ2

EuclidintroducesthissameconstructioninBookIIbeforeratioshavebeenintroduced,creatingthemidpointEofACandthenusingEBasthearctodeterminelengthsofthesegmentsEFandAFasfollows:

Proposition11.Tocutagivensegment(AB)sothattherectangle(BDKH)containedbythewhole(AB)andoneofthesegments(BH)equalsthesquare(AFGH)ontheremainingsegment(AH).

OtherexamplesinvolvingtheextremeandmeanratioappearinBookXIII,illustratedbelow:

Proposition1.Ifastraightline(AB)iscutinextremeandmeanratio(C),thenthesquare(DLFC)onthegreatersegmentaddedtothehalfofthewhole(CD)isfivetimesthesquare(DPHA)onthehalf(AD).

Proposition2.Ifthesquare(ALFB)onastraightline(AB)isfivetimesthesquare(APHC)onasegmentofit(AC),then,whenthedoubleofthesaidsegment(CD)iscutinextremeandmeanratio(B),thegreatersegment(BC)istheremainingpartoftheoriginalstraightline(AB).

Proposition3.Ifastraightline(AB)iscutinextremeandmeanratio(C),thenthesquare(ABNK)onthesum(BD)ofthelessersegment(BC)andthehalfofthegreatersegment(AC)isfivetimesthesquare(GUFK)onthehalfofthegreatersegment(AC).

Proposition4.Ifastraightline(AB)iscutinextremeandmeanratio(C),thenthesumofthesquaresonthewhole(AB)andonthelessersegment(BC)istriplethesquare(HFSD)onthegreatersegment(AC).

Proposition5.Ifastraightline(AB)iscutinextremeandmeanratio(C),andastraightlineequaltothegreatersegment(AD)isaddedtoit,thenthewholestraightlinehasbeencutinextremeandmeanratio(A),andtheoriginalstraightline(AB)isthegreatersegment.

InProposition6,Euclidintroducestheconceptoftheapotome,whichhedefinesaseach“irrational”segmentthatmakesupa“rational”linethathasbeencutinextremeandmeanratio.JumpingaheadtoPropositions8and9,wediscoverthegoldenpropertiesofthepentagon,followedbythegoldenrelationshipbetweenthesidesofthesix-sidedhexagonandten-sideddecagon.

Proposition8.Ifthestraightlinesofanequilateralandequiangularpentagon(AC,BE)subtendtwoangles,thentheycutoneanotherinextremeandmeanratio(H),andtheirgreatersegments(HE,HC)equalthesidesofthepentagon.

Proposition9.Ifthesideofthehexagon(CD)andthatofthedecagon(BC)inscribedinthesamecircleareaddedtogether,thenthewholestraightline(BD)hasbeencutinextremeandmeanratio(C),anditsgreatersegmentisthesideofthehexagon(CD).

Areyoureadyforthejumpintothree-dimensionalspace?Thislastpropositiondescribesthegoldenratiorelationshipbetweenacubeandadodecahedron:

Proposition17.Toconstructadodecahedronandinscribeitinasphere…andtoprovethatthesideofthedodecahedron(UV)istheirrationalstraightlinecalledapotome.Corollary:Therefore,whenthesideofthecube(NO)iscutinextremeandmeanratio,thegreatersegment(RS)isthesideofthedodecahedron.

Inthelastexample,Euclidshowsthatthesideofthedodecahedron(e.g.,segmentUV)isanapotome—thatis,thegreateroftwoirrationalsegmentsthatmakeuparationallineequivalentinlengthtotheside(e.g.,segmentNO)oftheinscribedcube.Inordertoillustratethisrelationship,thesidesofthecubearebisectedatG,H,K,L,M,N,andO,andthenGK,HL,HM,andNOareconnectedtoformsegmentsrepresentingthewidthofthecube.ThenthesegmentsNP,PO,andHQ—whichrepresenthalfthewidthofthecube—arecutinextremeandmeanratioatpointsR,S,andT.SincesegmentsRUandSVareat

rightanglestothecube,thelengthofsegmentRS,whichisthegreaterapotometotherationallineNO,isequalinlengthtothesegmentUV,whichrepresentsasideoftheequiangularandequilateraldodecahedronUBWCV.

CONSTRUCTINGTHEGOLDENRATIO

Euclidgaveusawonderfulfoundationforunderstandingthemanyappearancesofthegoldenratioingeometry.Butwecanmakethisevensimpler.Let’slookatsomeoftheothersimplegeometricconstructionsthatcanbeusedtocreateagoldenratio,startingwiththeline,andthenproceedingtothethree-sidedtriangle,four-sidedsquare,andfive-sidedpentagon.UnlikeDavidLetterman’s“Top10”Lists,I’mgoingtostartwithonethatisperhapsthemostamazing,byvirtueofitssheersimplicity.(Iliketodescribethisapproachas“incrediblysimple,yetsimplyincredible.”)

THREELINESIfEuclidhadseenthiselegantlittleconstruction,historyprobablywouldhaverecordedhimratherthanArchimedesastheonerunningnakedthroughthestreets,shouting,“Eureka!”

1.Gatherthreesticks(dowels,chopsticks,straws,orwhathaveyou)ofequallength.2.Placethefirstoneinaverticalposition.3.Laythesecondoneagainstthemidpointofthefirst.4.Laythethirdoneagainstthemidpointofthesecond,sothatoneendofeachstickislinedup,asshown.

Figure1.ThegoldenratiocutoflineACispointB.

THREESIDES:TRIANGLEHere’sanothergeometricconstructionthatissimplerthananyofthoseprovidedbyEuclid.

1.Withtheaidofacompass,drawacircle.Theninscribeanequilateraltriangleinsideit.2.Drawalinethroughthemidpointoftwosidesofthetriangle,extendingthelinetotheedgeofthecircle,asshown.

Figure2.ThegoldenratiocutoflineACispointB.

FOURSIDES:SQUAREThisconstructioniscloselyrelatedtoEuclid’spropositionsthatapplyanarctothemidpointofasquare,butwe’redoingtheconstructioninreverse.

1.Withtheaidofacompass,drawacircle.Thendivideitintotwosemicircles.2.Insertasquareinsideonesemicircle,asshown.

Figure3.Inthisconstruction,thegoldenratiocutoflineAC,again,ispointB.

FIVESIDES:PENTAGONThisconstructionisthefirstcontainedinElements,appearingasProposition8ofBookXIII.

1.Withtheaidofacompass,drawacircle.Thencreateapentagonbyinsideitbyconnectingfiveequallyspacedpointsonthecircle.

2.Connecttwooftheverticeswithaline,andthenconnectanothertwoverticeswithanotherline,asshown.

Figure4.ThegoldenratiocutoflineACispointB,wherethetwolinesintersect.

Seehoweasythisis?Goldenratiosjustseemtoappearwithoutmuchplanningoreffort.SeeAppendixBtoexploreothergeometricconstructionsofthegoldenratio.

PYTHAGORASANDKEPLERWALKINTOA…TRIANGLE?

Haveyouheardthejokethatstarts,“PythagorasandKeplerwalkintoabar”?Probablynot,butasyouwilldiscover,thefindingsofthesetwohistoricalmathematicianshelpstoillustrateoneofthegoldenratio’suniqueproperties.Pentagramsaside,Pythagorasisbestknownforhiseponymoustheorem,whichstatesthatarighttrianglewithsidesoflengtha,b,andc(wherecisthehypotenuse),hasthefollowingrelationship:

a2+b2=c2

Asstatedintheintroduction,wealsoknowthatphiistheonlynumberwhosesquareisonemorethanitself:

Φ+1=Φ2

TwothousandyearsafterPythagorasdevisedhisfamoustheorem,GermanmathematicianJohannesKepler(1571–1630)noticedthesimilaritybetweenthesetwoequations.Thisledtohisdiscoveryofauniquetriangle,nowappropriatelyknownastheKeplertriangle,withsidesequalto1,√Φ,andΦ.

This1610portraitofJohannesKeplerbyanunidentifiedpaintercomesfromaBenedictinemonasteryinKremsmünster,Austria.

KeplerobservedanothercharacteristicofthistriangleandwrotetohisformerprofessorMichaelMästlin:

“Ifonalinewhichisdividedinextremeandmeanratiooneconstructsaright-angledtriangle,suchthattherightangleisontheperpendicularputatthesectionpoint,thenthesmallerleg

willequalthelargersegmentofthedividedline.”4

Here,heisreferringtothetwolegsofthetrianglesbelowwithadimensionof1.

Asshown,whenyoudrawalineperpendiculartothehypotenuseoftheKeplertrianglethroughitsrightangle,thesegmentsoneithersideofthelinehaveagoldenrelationship,andtheresultingtwotriangleshaveidenticalproportionstothatoftheoriginalKeplertriangle.

ThePythagorean3-4-5triangleistheonlyrighttrianglewhosesidesareinanarithmeticprogression,inwhicheachsuccessivetermiscreatedbytheadditionofacommondifference:

3+1=44+1=5

Curiously,the√Φ-1-ΦKeplertriangleistheonlyrighttrianglewhosesidesareinageometricprogression,inwhicheachsuccessivetermiscreatedbythemultiplicationofacommonratio.Inthisuniquecase,thatratioisthesquarerootofthegoldenratio:

1×√Φ=√Φ

1×√Φ=√Φ√Φ×√Φ=Φ

CirclingbacktoPythagoras,inthepentagramwefindtwoothertriangleswithgoldenratioproportions—thatis,twotriangleswithaΦto1relationshipbetweenthebaseandsides.

Thepentagram(above)canbedividedintoseveralgoldentriangles(below)andgnomons(below),eachofwhichhasatleastone36-degreeangle.

Theobtusetriangleabove,center,isknownasagoldengnomon.Theacuteisoscelestriangleontherightisknownasagoldentriangle.These,inturn,formthebasisofanimportantmathematicaldiscovery,Penrosetiling(seehere).

THEGOLDENRATIO,ORIGAMI-STYLE

Ifyouknowsomeonewhogetstiedupinknotsbymathorgeometry,trysharingthislastgoldenratioconstructionwithhimorher,becauseitrequiresneither.Allyouneedisastripofpaper.Foldapaperintoasimpleknotandpresstoflatten.(Don’toverthinkit!)This“knot”formsapentagon,withbothvariationsofthegoldentriangle,whosebaseand

sideproportionsaredefinedbythegoldenratio.

HARMONYOFTHESPHERES

BothPythagorasandKeplersawmathematicseverywhere,fromthevibrationsofastringedinstrumenttothemotionoftheplanets.Thoughnooneknowsforsure,itisbelievedthatPythagoraswasthefirsttoidentifytheinverserelationshipbetweenthepitchofamusicalnoteandthelengthofthestringproducingit,andhemayhavegonefurtherinlinkingtheorbitalfrequenciesofdifferentplanetstoinaudiblehums—atheorythathaspersistedthroughtheagesundersuchnamesasmusicauniversalisand“HarmonyoftheSpheres.”

Kepler’sowninterestsrangedintothemystical,andheexploredtheideaoftheuniverseasaharmoniousarrangementofgeometricalformsinhis1596bookMysteriumCosmographicum(CosmographicMystery),aswellashis1619bookHarmonicesMundi(HarmonyoftheWorld).Intheformer,KeplerproposedthattherelativedistancesbetweenthesixplanetsknownatthattimecouldbeunderstoodthroughanestingofthefivePlatonicsolids(seehere),eachenclosedwithinaspherethatrepresentedtheirorbits,withthefinalsphererepresentingtheorbitofSaturn.Thismodelturnedouttobeinaccurate,buthecontinuedinhispursuittoexplainthecosmos,andin1617hepublishedthefirstvolumeofEpitomeAstronomiaeCopernicanae,inwhichheunveiledhismostimportantdiscoveries:thetrueellipticalnatureofplanetaryorbitsandthefirstofhisthreelawsofplanetarymotion.

ThisreproductionofKepler’smodelofthesolarsystemshowsthefivePlatonicsolidsinanestedformation.

EventhoughthehypothesisofnestingPlatonicsolidsinMysteriumCosmographicumdidnotholduptoscrutinyintheend,Kepler’searlymodeloftheuniversewasmathematicallybrilliantinitsownright.Auniquepropertyofthesesolids,whichinclude(below,fromlefttoright)thetetrahedron,cube,octahedron,dodecahedron,andicosahedron,isthateachcanbeconstructedwithidenticalfacesmeetingateachvertex.

TwoofthesefivebeautifulPlatonicsolids,thedodecahedronandicosahedron,aregeometricallybasedonthegoldenratio.Eachoftheirvertexpointscanbedeterminedbyasimpleconstructionusingthreegoldenrectangles(i.e.,rectangleswhoselength-to-widthratioisequaltophi).

Thethreegoldenrectanglesontheleftcanbeassembledintotheinterlockingshapeontheright.Thisinterlockingshapecreatesthebasisforthetwelve-sideddodecahedronandthetwenty-sidedicosahedron.

Inthecaseofthedodecahedron,the12cornersbecomethe12centersofeachofthe12pentagonsthatformthe12pentagonalfaces.

Dodecahedron.

Inthecaseoftheicosahedron,the12cornersbecomethe12pointsofeachofthe20trianglesthatformthe20triangularfaces.

Icosahedron.

Ifwemaptheinterlockinggoldenrectangleconstructioninthree-dimensionalCartesianspace,thecoordinatesofthe12(X,Y,Z)verticesoftheicosahedronwithanedgeoflength2,centeredattheorigin,arerepresentedasfollows:

x-zplane(green,y=0):(±1,0,±Φ)y-zplane(blue,x=0):(0,±Φ,±1)x-yplane(red,z=0):(±Φ,±1,0)

Next,mappingthedodecahedroninthree-dimensionalCartesianspaceprovidesthefollowingcoordinatesforthe20(X,Y,Z)verticesofadodecahedronenclosingacubewithanedgeoflength2,centeredattheorigin:6

orangecube:(±1,±1,±1)y-zplane:(greenx=0):(0,±Φ,±1/Φ)y-zplane:(blue,y=0):(±1/Φ,0,±Φ)y-zplane:(red,z=0):(±Φ,±1/Φ,0)

Givenwhatweknowabouttheproportionsofapentagon,adodecahedronthatenclosesacubewithedgesoflength2shouldhaveedgesoflength2/Φ.

GOLDENTILES

MappingthesurfacesofeachPlatonicsolidintwo-dimensionalspace,asshownhere,areacanbefilledcompletelyandsymmetricallywithtilesofthree,andfour,sides,butwhatabouttilesintheshapeofafive-sidedpentagon?Thelinesofapentagonstar,orpentagram,havebeautiful,goldenratioproportions,butitlongappearedthattheycouldnotbetiledliketriangles,square,andhexagons.EnterEnglishmathematicalphysicistSirRogerPenrose(b.1931).Intheearly1970s,Penrosenoticedthatthetwotriangleswithinthepentagonthathavegoldenproportions(seehereandbelow,topleft)canbeassembledinpairs,formingall-newsymmetricaltilesthatcanbecombinedintodifferentpatterns.Forexample,twoacutegoldentrianglescanbecombinedtoforma“kite”(thegoldpartinfigureb),whiletwoobtusetriangleswithgoldenproportionscanforma“dart”(theredpartinfigureb).ThekiteanddartcanbecombinedtoformarhombuswithsidesoflengthΦ,asshown(figurec).Thetwotrianglescanalsobecombinedtoformdiamond-shapedtiles,asshown(figurec).Althoughpentagonsalonewillnotcompletelyfillatwo-dimensionalspace,these“Penrosetiles,”whichhavegoldenproportions,will(figured).

Asyouexpandthetilingtocovergreaterareas,theratioofthequantityoftheonetypeoftiletotheotheralwaysapproaches1.618,thegoldenratio.Dependingonhowtheyarearranged,thetilingmayexhibitfive-foldrotationalsymmetry.Smallpocketsoffive-foldsymmetry,suchasstarsanddecagons,mayalsooccur.Aswewillseeinchapter5,thissamekindoffive-foldsymmetricalarrangementalsoappearsinnature.

VariousformationsofPenrosetiles.Noticetheproliferationoffive-sidedfigureslikethepentagramandpentagon.

II

PHIANDFIBONACCI

“[Theuniverse]cannotbereaduntilwehavelearntthelanguageandbecomefamiliarwiththecharactersinwhichitis

written.Itiswritteninmathematicallanguage.”1

—GalileoGalilei

T hemathematicalworkoftheGreekswaskeptaliveinninth-century

Baghdad,wherecaliphHarunal-Rashidfoundedagreatlibrarythat

becameknownastheHouseofWisdom.Here,Muslim,Jewish,and

Christianscholarsmettodiscussanddebatesubjectssuchaschemistry

andcartography,andtranslatedancienttextsfromGreeceandIndiainto

Arabic.Manyincredibleadvancesinscienceandmathematicsweremade

duringtheensuingIslamicGoldenAge,whichlasteduntilthethirteenth

century.Forexample,thescholarMuhammadibnMusaal-Khwarizmi(c.

790–c.850)wasamongthefirstmathematiciansintheworldtousezero

asaplaceholder,andhistreatiseHisabal-jabrw’al-muqabala(The

CompendiousBookonCalculationbyCompletionandBalancing)introduced

thewordalgebrafromtheArabical-jabr,whichmeans“completion.”The

wordwasreferringtotheprocessofreducingaquadraticequationby

meansofremovingthenegativeterms,whichgavebirthtothefieldof

algebra.Interestingly,inthesamebookhepresentedaquadraticequation

thatrepresentedalineoflength10dividedinto2segmentswithgolden

ratioproportions.

This1983Sovietstampabovebearsthevisageofal-Khwarizmi,aninfluentialninth-centurymathematicianandtoweringfigureinBaghdad’sHouseofWisdom,depictedatbelow.

THEFIBONACCISEQUENCE

Ahalfcenturyafteral-Khwarizmi,AbuKamilShujaibnAslam(c.850–c.930),anIslamicmathematicianfromEgypt,appliedcomplexalgebratogeometricproblems,solvingthreenon-linearequationsforthreedifferentvariables.Healsopresentedequationsonvariouswaystodividealineoflength10andtoinscribeapentagonwithinasquare.AbuKamilwasthefirstmathematiciantoemployirrationalnumbersassolutionstoquadraticequations,2andhisKitābfīal-jabrwaal-muqābala(BookofAlgebra),whichexpandedontheworkofal-Khwarizmi,wasinfluentialinEuropefollowingitstranslationintoLatininthetwelfthcentury.

Theworkofal-Khwarizmi—particularlyhisdiscussionofHindu-Arabicnumerals—latercaughttheattentionofayoungItalianboyduringavisittoanAlgerianportcitywithhisfather,awealthymerchantfromPisa.Theboy,LeonardoFibonacci(c.1175–c.1250),wouldlaterbecomeoneofhistory’smostfamousmathematiciansafterthepublicationin1202ofhisbookLiberAbaci,whichpromotedtheHindu-ArabicnumberingsystemthroughoutEurope.

Thesepagesfroma1342editionofal-Khwarizmi’sBookofAlgebradisplaygeometricalsolutionstotwoquadraticequations.

Arabastronomersuseanastrolabeandcross-stafftodeterminelatitudeinanobservatoryinConstantinople(present-dayIstanbul,Turkey)duringtheIslamicGoldenAge,whichlastedfromaboutthemid-eighthtomid-thirteenthcentury.

ThispagefromFibonacci’srevolutionary1202workLiberAbaci,whichintroducedHindu-ArabicnumeralstotheWest,showstheassociationbetweenRomannumeralsanddifferentquantities.

InwritingLiberAbaci,FibonaccireliedonmanyArabicsources,includingtheproblemsofAbuKamil.DrawingtheconnectionbetweentwoofAbuKamil’sequationsfordividingalineoflength10andtheresultthatproducesthegoldenratio,Fibonaccigavethelengthsofthesegmentsas√125–5and15–√125,3whichcanalsobewrittenas5(√5–1)and5(3–√5).Theseareboth

expressionsofthetwogoldenratiopointsonalineoflength10.Now,dividebothoftheseexpressionsby10,andyouhavethealgebraicformulasforphi’sinverse(1/Ф,0.61803…)and1–1/Ф(0.38197…).Recallfromherethatphiistheonlynumberinwhichitsreciprocalisonelessthanitself,andderivethealgebraicformulaforphiitselfbyadding1tobothsidesoftheequation:

1Ф=(√5–1)2=Ф–1Ф=(√5+1)/2

Inhisbook,Fibonaccialsowroteasimplenumericalsequencebasedonatheoreticalproblemofgrowthinapopulationofrabbits.Thatsequence—thefoundationforanincrediblemathematicalrelationshipbehindphi—wasknownasearlyasthesixthcenturybyIndianmathematicians,butitwasFibonacciwhopopularizeditintheWest.

Fibonacci’ssequencecanbeexplainedusingthefollowingexample.Supposewehaveanewlybornpairofrabbits,onemaleandonefemale.Supposerabbitsareabletomateattheageofonemonth,soattheendofitssecondmonthafemalecanproduceanotherpairofrabbits.Supposeourrabbitsneverdieandthatthefemalealwaysproducesonenewpair(onemale,onefemale)everymonthfromthesecondmonthonward.ThequestionFibonacciposedwashowmanypairswilltherebeinoneyear?Theansweris144,whichisfoundasthetwelfthnumberinthesequenceofgrowthbelow,correspondingtothetwelfthmonthofnew-bornrabbits.Startingwith0and1,eachnewnumberinthesequenceissimplythesumofthetwobeforeit:

0+1=1

1+1=2

2+1=3

3+2=5

5+3=8

8+5=13

…andsoon,resultinginthefollowingsequence,namedafterFibonacci:

0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,…

YoucanestimatethenthnumberintheFibonaccisequence,usingΦand√5,withtheequation:

f(n)=Φn/√5

Forexample,thetwelfthnumberoftheFibonaccisequencecanbecalculatedthus:

Φ12√5=321.9969…2.236…=144.0014…,whichroundsto144!

IntheFibonaccisequence,theratioofeachsuccessivepairofnumbersconvergesonphi.Tovisualizethisphenomenon,notethateachsuccessivevalueoftheratiogetscloserandclosertophi,asshown:

1/1 = 1.000000

2/1 = 2.000000

3/2 = 1.500000

5/3 = 1.666667

8/5 = 1.600000

13/8 = 1.625000

21/13 = 1.615385

34/21 = 1.619048

=

55/34 = 1.617647

89/55 = 1.618182

144/89 = 1.617978

233/144 = 1.618056

377/233 = 1.618026

610/377 = 1.618037

987/610 = 1.618033

ThismarblestatueofFibonacciwascreatedbyItaliansculptorGiovanniPaganucciin1863.

Atthefortiethnumberinthesequence—102,334,155—theresultingratiomatchesphito15decimalplaces:

1.618033988749895

DespitetheobviousconvergenceofFibonacci’ssequenceonthevalueofphi,theItalianmathematiciandidnotwritespecificallyaboutthegoldenratio.Infact,anotherfourhundredyearselapsedbeforesomeonemadeanexplicitconnectionbetweenthetwo.4ThatpersonwasJohannesKepler(seehere),who,inaletterfrom1609,becamethefirstpersonknowntoclearlymentionthattheratiosofsuccessivenumbersintheFibonaccisequenceapproximatesthegoldenratio.

In1653,FrenchmathematicianBlaisePascal(1623–1662)developedhiseponymoustriangleformation,visuallydescribingthealgebraicexpansionofbinomialcoefficients(i.e.,twopositiveintegersthatformasum).Asshownbelow,startingwithanapexof1,everynumberinthetrianglebelowisthesumofthetwonumbersdiagonallyaboveittotheleftandtheright,andthenumbersondiagonalsofthetriangleaddtotheFibonacciseries.Pascal’strianglehasmanyunusualpropertiesandavarietyofuses,includingthefollowing:

•Horizontalrowsaddtopowersof2(i.e.,1,2,4,8,16,etc.)•Thehorizontalrowsrepresentpowersof11(1,11,121,1331,14641)forthefirstfiverows,inwhichthenumbershaveonlyasingledigit.

•Addinganytwosuccessivenumbersinthediagonal1-3-6-10-15-21-28…resultsinaperfectsquare(1,4,9,16,etc.)

•Whenthefirstnumbertotherightofthe1inanyrowisaprimenumber,allnumbersinthatrowaredivisiblebythatprimenumber.

GIVINGFIBONACCIHISDUECREDIT

WhileKeplerwasthefirsttoconnectFibonaccinumbersandphi,5in1753ScottishmathematicianRobertSimson(1687–1768)wasthefirsttoprovethattheratiosofsuccessivenumbersintheFibonaccisequencedo,indeed,convergeonthegoldenratio.6In1877,thesequenceofwhichFibonacciwroteinhisLiberAbaciwasfinallynamedinhishonorbyFrenchmathematicianEdouardLucas(1842–1891),whodevelopedtherelatedLucassequencedefinedbytheequation:Ln=Ln-1+Ln-2,whereL1=1andL2=3.

Also,Pascal’strianglecanbeusedtofindcombinationsinprobabilityproblems.If,forinstance,youpickany2of5items,thenumberofpossiblecombinationsis10,foundbylookinginthesecondplaceofthefifthrow(notethatyoudonotcountthe1sinthisapplication).

Thiscoloredengravingfrom1822portraysFrenchmathematicianBlaisePascal,developerofPascal’striangle.

THEFIBONACCISPIRALANDOTHERCURIOSITIES

IfyouhavepokedaroundontheInternetonthetopicofFibonaccisequences,there’sagoodchanceyoucameacrossimagesofFibonacciorgoldenspirals.There’salsoagoodchancethatyou’veseensomeofthemasoverlaysoneverythingfromtheParthenontotheMonaLisatoDonaldTrump’shairline.Typically,thespiraliscreatedwithagoldenrectangleatitsfoundation.Dividethegoldenrectangleatitsgoldenratiopointandyou’llbeleftwithasquareandanothersmallergoldenrectangle.Dothesametothesmallergoldenrectangleagainandagaintocreatetheimagebelow:

Nowwedrawaquartercirclearcineachsquaretocreatethegoldenspiral:

AcloselyrelatedspiralistheFibonaccispiral.Here,insteadofcreatingasuccessivepatternofgoldenrectangles,ourbuildingblocksaresquareswhosesidelengthsareequaltothenumbersoftheFibonaccisequence,asshown:

Technicallyspeaking,noneofthesearespirals.They’recalledvolutes.Thedifferenceisalmostimperceptible,butatruegoldenspiralisaunique,equiangular(thatis,logarithmic)spiralthatexpandsataconstantrate.Intheillustrationbelow,thegreenspiralisconstructedwithasuccessionofindependentquarter-circlearcswithineachsquare.Theredspiralisatruelogarithmicspiralthatexpandsbythegoldenratioevery90degrees.Theportionsthatoverlapappearinyellow.Now,you’reoneofthefewwhoknowsthedifferencebetweenthem!

CREATINGA“FIBONACCITRIANGLE”

NothreesuccessivenumbersintheFibonacciseriescanbeusedtocreatearighttriangle;however,everysuccessiveseriesoffourFibonaccinumberscanbeusedtocreatearighttriangle.Todothis,considerthelengthsofthebase(a)andhypotenuse(c)asbeingdeterminedbythesecondandthirdnumbers,andtheremainingsidebeingthesquarerootoftheproductofthefirst(b’)andfourth(b’’)numbers.Thetablesbelowshowhowthisrelationshipworks:

TheFibonacciSeries

b' a c b"

0 1 1 2

1 1 2 3

1 2 3 5

2 3 5 8

2 3 5 8

3 5 8 13

TheFibonacciTriangle

a2 b'×b" a2+b'×b"=c2

1 0 1

1 3 4

1 2 9

2 3 25

3 5 64

Thedimensionsofthistrianglearereflectedinthefifthrowofthetableontheleft.

TherearemanyunusualrelationshipsintheFibonacciseries.Forexample,foranythreenumbersintheseriesf(n–1),f(n),andf(n+1),thefollowing

relationshipexists:

f(n–1)×f(n+1)=f(n)2–(–1)n

3×8=52–15×13=82+18×21=132–1

Here’sanother:EverynthFibonaccinumberisamultipleoff(n),wheref(n)isthenthnumberoftheFibonaccisequence.Given0,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,notethefollowingresults:

•Every4thnumber(e.g.,3,21,144,and987)isamultipleof3,whichisf(4).

•Every5thnumber(e.g.,5,55,610,and6765)isamultipleof5,whichisf(5).

•Every6thnumber(e.g.,8,144,and2584)isamultipleof8,whichisf(6).7

TheFibonaccisequencealsohasapatternthatrepeatsevery24numbers.8Thisrepetitivepatterninvolvesasimpletechniquecallednumericreductioninwhichallthedigitsofanumberareaddedtogetheruntilonlyonedigitremains.Asanexample,thenumericreductionof256is4because2+5+6=13and1+3=4.ApplyingnumericreductiontotheFibonacciseriesproducesaninfiniteseriesof24repeatingdigits:

1,1,2,3,5,8,4,3,7,1,8,9,8,8,7,6,4,1,5,6,2,8,1,9

Ifyoutakethefirst12digits,addthemtothesecond12digits,andthenapplynumericreductiontotheresult,youfindthattheyallhaveavalueof9.

Thiscolorfularrangementofrectanglesrepresentsthefirst160naturalnumbersassumsofFibonaccinumbers.

Asdiscoveredin1774byFrenchmathematicianJosephLouisLagrange,thelastdigitofthenumbersintheFibonaccisequenceformapatternthatrepeatsaftereverysixtiethnumber.Theseare:

0,1,1,2,3,5,8,3,1,4,5,9,4,3,7,0,7,7,4,1,5,6,1,7,8,5,3,8,1,9,0,9,9,8,7,5,2,7,9,6,5,1,6,7,3,0,3,3,6,9,5,4,9,3,2,5,7,2,9,1

Whenthesesixtydigitsarearrangedinacircle,asshownbelow,additionalpatternsemerge:9

JosephLouisLagrange,anotherprominentFrenchmathematicianwhostudiedtheFibonaccisequence,ispicturedinthisengraving.

•Thezerosalignwiththe4cardinalpointsonacompass.•Thefivesalignwiththe8otherpointsofthe12pointsonaclock.•Withtheexceptionofthepairsofzeros,thenumbersdirectlyoppositeeachnumberaddto10.

CALCULATINGPHI

In1567,Kepler’smentorMichaelMaestlin(1550–1631),aGermanastronomerandmathematician,presentedthefirstknownapproximationofthegoldenratio’sreciprocalinalettertohisformerstudent,describingthegoldenratioasadecimalfractionof“about0.6180340.”10

CombiningFibonacci’svaluefor1/Фwiththefactthat1/Ф=1–Ф,wewereabletogenerateanequationforthevalueofФshownhere.Butthere’sanotherwaytoderivethatsamevalueusingbasiclogic.Rememberfromherethatatthegoldenratiocut,illustratedinthediagrambelow,theratioofthewholetothelargersegmentisthesameastheratioofthelargertothesmallersegments,representedbytheequationA/B=B/C.

WealsoknowthatthetwolinesegmentsBandCaddedtogetherareequaltoA,whichisalgebraicallyrepresentedasA=B+C.

Now,ifwecombinetheseequations,weseethat(B+C)/B=B/C.MovingallvariablesononesideoftheequationandmakingC=1,wearriveatthisfamiliarequation:

B2–B–1=0

Becausethisequationisnowintheformax2+bx+c=0,wecanapplythequadraticformula,whichallowsustosolveforxafterplugginginthevaluesfora,b,andc(1,-1,-1):

Therefore,ourtwopossiblesolutionsare(1+√5)2and(1–√5)2.ThepositivesolutiongivestheexactvalueoftheΦ.

Asweknowalready(seehere),theratioofsuccessiveFibonaccinumbersconvergesonphi,butthisisnottheonlyseriesinwhichthatrelationshipexists.

Youcanpickanytwonumberstocreatethesuccessiveratiosandtheresultwillalwaysconvergeonphi.Asanexample,separatethedigitsof1.618into16and18,andthenaddtwonumbersandtaketheratioof18to16,asshownbelow.Ifyouthensumthenexttwonumbersinthesequenceanddeterminetheirratio,andsoon,afamiliarpatternemerges:

16+18=34,andtheirratiois1.125

18+34=52,andtheirratiois1.888889…







Now,let’sreturntotheotheruniquepropertyofphi,describedhere:

Φ2=Φ+1

ThiscanalsobewrittenasΦ2=Φ1+Φ0,leadingtoournextrevelation:Foranynumbern,eachtwosuccessivepowersofphiaddtothenextone,expressedmathematicallyasfollows:

Φn+2=Φn+1+Φn

Anotherlittlecuriosityinvolvesraisingphitoapowerandthenaddingorsubtractingitsreciprocal:

•ForanyevenintegernwefindthatΦn+1/Φnisawholenumber(e.g.,Φ2+1/Φ2=3).

•ForanyoddintegernwefindthatΦn–1/Φnisalsoawholenumber(e.g.,Φ3+1/Φ3=4).

Phicanalsobecalculatedasthelimitofavarietyofiterativeexpressionsof

Phicanalsobecalculatedasthelimitofavarietyofiterativeexpressionsoflimits,includingthese:

Finally,asweobservedwiththeconnectiontothepentagonandpentagram,phihasaspecialrelationshipwiththenumber5.IfwerewritetheexpressionforФ,(1+√5)/2,usingdecimals,wecomeupwiththisequationthatcanbeusedinExcelorcoding“(^isasymbolforexponent,orraisedthepowerof.):11

Ф=.5^.5*.5+.5

Hereisyetanotherequationforphi:

PerhapsKeplerwasontosomethingwhenhedescribedthegoldenratioasa“preciousjewel.”Afterall,thisisthepersonwhosecuriosity,persistence,andinsightledtothediscoveryoftheellipticalnatureoftheplanetaryorbitsaroundtheSun,revolutionizingourunderstandingoftheuniverse.Inthenextchapter,we’llexplorehowthesebeautifulconceptsofgeometryandmathematicsareexpressedinthearts.

III

THEDIVINEPROPORTION

“Withoutmathematicsthereisnoart.”1

—LucaPacioli

“Wherethespiritdoesnotworkwiththehandthereisnoart.”

—LeonardodaVinci

N owwewillexamineavarietyofapplicationsofthegoldenratioin

Renaissanceartandbeyond.Indoingso,westepfromtheworldofthe

absoluteprecisionandverifiableproofsofmathematicsandgeometryinto

themoresubjectiveworldofbeautyandaesthetics.Thusventuringintoa

domainwhereourhearttellsuswhatlogiccannot,we’llalsobestepping

intoaworldofcontroversy,fullofconflictingandpolarizingclaimsthat

leadtomuchmisinformationandconfusionaboutthegoldenratio.Thisis

whereyouplaytheroleofdetective,judge,andjury.DidtheRenaissance

masterstrulyandintentionallyincorporatethegoldenproportioninto

someoftheirmostreveredworks?I’llpresentthebestevidenceavailable,

andyourtaskwillbetoexaminetheevidenceandcometoyourown

conclusions.

FrenchartistHoraceVernet’s1827paintingshowsPopeJuliusIIorderingarchitectDonatoBramanteandRenaissancemastersMichelangeloandRaphaeltobuildthelargestchurchintheworld,St.Peter’sBasilica.

DIVINEPROPORTIONS?TOOLSANDRULESOFENGAGEMENT

Beforewebeginourinvestigationintothepresence,orlackthereof,ofthegoldenratioinsomeofhumanity’sgreatestpaintings,letmeproposethe“toolsofengagement”and“rulesofengagement.”Theanalysisofanyimageorobjectforgoldenratioscanbeundertakenwithsomesimplebutspecializedtools.Physicalobjects,suchasstatues,buildings,andevenhumanfaces,canbemeasuredwithgoldenmean–gaugecalipers.Onetypeofagaugecaliperhastwolegsjoinedattheirgoldenratiopointsothattheoppositeendsareingoldenratioproportion.Anothertypehasacenterlegthatremainsingoldenratioproportionbetweenthetwoouterlegs.

Foranalyzingdigitalimages,thePhiMatrixsoftwareIdevelopedisperfectforinvestigatingandapplyingthegoldenratio.Thesoftwarecanbeusedtofindthegoldenratioofanydimension,horizontallyorvertically,withpixel-levelaccuracy.Itcanalsoshowgoldenratiosofgoldenratios,inwhicheverylineisingoldenratioproportiontotheonesoneithersideofit,asshowninthelastgridbelow:

Withthesetools,youmightbegintonoticeexamplesofgoldenratiosallaroundyou.Sometimes,theseproportionsmayhavebeenintendedbythecreator;atothertimes,theymaybejustacoincidence.Bearingthisinmind,Iproposethefollowingguidelinesforidentifyingthegoldenratioasabasisforcomposition:

•Relevance:Appearancesshouldbebasedonthesubject’smostprominentorrelevantfeatures.•Ubiquity:Appearancesshouldappearinmorethanoneplacetodemonstrateknowledgeandintentratherthancoincidence.

•Accuracy:Appearancesshouldbewithinabout±1%ofthegoldenratio,measuredwithasmuchaccuracyaspossible,andbyusingthehighest-resolutionimagesavailable.

•Simplicity:Appearancesshouldbebasedonthesimplestpossibleapproaches,thosethatmostlikelywouldhavebeenappliedbytheartistordesigner.

Uptothispointinthisbookwe’veseenthatthereisbeautyinmathematics,butastheItalianfriarLucaPacioliastutelyobserved,thereisalsomathematicsinbeauty.Euclid’sElementswasreintroducedtoEuropeviaaLatintranslationaroundtheyear1120,anditbecameoneofthemostwidelycirculatedbooksaftertheinventionoftheprintingpressinthe1450s.Althoughnootherwrittenreferencetothegoldenratioappeareduntilthelate1490s,thereisclearandcompellingevidencethatsomeofthegreatestartistsofthisperiodwereapplyingitinthecompositionsoftheirpaintingsasearlyasthe1440s.Theapplicationofthegoldenratiototheartswaslaterrevealedtobea“secretscience,”and,aswe’llseenext,itseemsthatmanyofthegreatRenaissancemasterswereinonthesecret,includingthelikesofPierodellaFrancesca,LeonardodaVinci,Botticelli,Raphael,andMichelangelo.However,itwasPacioliwhoproducedthefirstcomprehensivestudyofthisspecialnumber,whichhedubbed“thedivineproportion.”

Thispaintingfrom1495showsPacioliinhisFranciscanhabitdrawingamathematicaldiagramwithhislefthanduponanopenbook.Intherightcornerofthetableisadodecahedron.Theyoungmanbehindhimisprobablyastudent—possiblytheGermanartistandpolymathAlbrechtDürer,whowasinhisearlytwentiesandvisitingItalywhenthepaintingwasmade.

DEDIVINAPROPORTIONE

LucaPacioli,wholivedfrom1447to1517,wasamanofvariedinterestsandtalents.HewasaFranciscanfriar,mathematician,andfriendofLeonardodaVinci,withwhomhecollaborated.Knownasthe“FatherofAccountingandBookkeeping,”hewasalsothefirstauthorinEuropetopublishadetailedworkonthedouble-entrysystemofaccounting.Soonafterthepublicationofhissix-hundred-pageSummadearithmetica(SummaryofArithmetic)in1494,hewasinvitedbytheDukeofMilan,LudovicoSforza,totakeupresidence.ThisledtohisfatefulmeetingwithdaVinci,whobecamehispupilinmathematicsasPacioliworkedonDeDivinaProportione.Writtenbetween1496and1498,andpublishedin1509,thisbookconnectedmathematicstoartandarchitecture,exploringthepresenceandusesofphithroughouthistory.HisillustratorwasnoneotherthandaVincihimself,wholivedwithPacioliduringthelate1490s.

ThetitlepagesfromPacioli’sSummadeArithmeticaandDeDivinaProportione,withLudovicoSforza,whopresidedoverthefinalandmostproductivestageoftheMilaneseRenaissance.FamedasapatronofLeonardodaVinciandotherartists,hecommissionedTheLastSupperaround1495andbroughtPaciolianddaVincitogether.

Inhismonumentalthree-volumetreatise,Paciolicapturedthebreadthanddepthofthistopicintheopeningwordsofhisintroductionandstatementofintent:

“Aworknecessaryforalltheclear-sightedandinquiringhumanminds,inwhicheveryonewholovestostudyphilosophy,

perspective,painting,sculpture,architecture,musicandothermathematicaldisciplineswillfindaverydelicate,subtleand

admirableteachingandwilldelightindiversequestionstouchingonaverysecretscience.”2

Bydiscussingmathematicalproportion—especiallythemathematicsofthegoldenratio—anditsapplicationinartandarchitecture,Paciolihopedtoenlightenthegeneralpublicaboutthesecretofharmonicforms.Aswe’veseenalready,somegeometricsolids,suchasdodecahedronsandicosahedrons,haveinherentgoldenratioswithintheirdimensionsandinthespatialpositionsoftheirintersectinglines.However,herevealedotherexamplesofgoldenratioproportionsinthedimensionsofGreco-RomanstructuresandRenaissancepaintings.WeevenfindthegoldenratiointheletterGofhisbeautifularchitecturalscriptletters!

UntilPacioli’stime,phiwasknownasthe“extremeandmeanratio”describedbyEuclid.Althoughlongrecognizedforitsuniquenessandbeauty,itwasPacioliwhofirstdubbed1.618as“divine.”Thetheologicalimplications,coupledwithdaVinci’spreciserenderingsofthree-dimensionalskeletonicsolids,popularizedthestudyofphiandgeometryamongartists,philosophers,andmore.

ItalianartistRaffaelloSanzioMorghencreatedthisengravingofamiddle-agedLeonardodaVinciin1817.

ThiswoodcutofthebeautifulgateofSolomon’sTempleinJerusalem,whichappearsinthe1509editionofDeDivinaProportione,containsgoldenproportions.

DaVincidrewallofPaciolil’soriginalpolyhedronsinhisbook,includingthedodecahedron(above)andtheArchimedeantruncatedicosahedron(below).

Pacioli’sGdisplayscleargoldenratioproportions.

PIERODELLAFRANCESCA

ThethirdvolumeofPacioli’sDeDivinaProportionewasanItaliantranslationofPierodellaFrancesca’sShortBookon(the)FiveRegularSolids,whichwaswritteninLatin.Whileknowninhisowntimemostlyasamathematicianandgeometer,PierodellaFrancesca(1415–1492)isnowprimarilyrecognizedforhisworksasanartist.

PierowroteDeProspectivaPingendi(OnPerspectiveforPainting)laterinhiscareer,buthisunderstandingandappreciationofperspectiveandproportionisevidentinhisearlierworks.Intheveryfirstofhisextantpaintings,TheBaptismofChrist(c.1448-1450),weseethatPierohasChristperfectlypositionedbetweenthetwogoldenratiosformedbythesidesofthecanvas,andalsobetweenthetwotrees.TheFlagellationofChrist(seehere)wasprobablypaintedbetween1455and

1460,anditisrecognizedforitscomplexcompositiononapanelofonly23by32inches(58by81cm).BritisharthistorianKennethClarkcalledit“thegreatestsmallpaintingintheworld.”3UsingmyPhiMatrixsoftware,it’seasytoseethatPierocarefullyappliedthegoldenratiointheroomtotheleft.TherewefindChristatthegoldenratioofthewidthoftheroom,whethermeasuredwherethefloortileschangeorfromthecolumnsatitsentry.Thearchitecturalfeaturesofthebuildingsalsoshowalignmentwiththegoldenratiogridlines(green).

AnotherpaintingthatdisplaysgoldenproportionsisPolyptychoftheMisericordia,(seehere)completedbetweentheyearsof1445and1462.HereweseethecrownedMadonnastandingwitharmsoutstretched.Atthegoldenratioofherheight,thereisasashtiedaroundherwaist.Thewidthofthesashatherwaistisingoldenratioproportiontothelengthbetweenheroutstretchedhands.

Examiningthepaintingevenmoreclosely,wecanseethatPieroappliedthegoldenratiotwicemore—oncehorizontallyintheoff-centeredknotofthebelt,andagainverticallyinthelengthsofropehangingfromtheknot.

Thus,sixtyyearsbeforeDeDivinaProportionewaspublishedbyLucaPacioli,wefindevidencethatRenaissancepaintersimplementedthegoldenratioasameansofcreatingvisualharmonywithinpaintings.Furthermore,inreligiousartthegoldenratiomayhavebeenusedbytheartiststoincorporateanelementoftheeternalorthedivineintotheirworks.

BaptismofChrist,c.1449.

MadonnadellaMisericordia(OurLadyofMercy),1445-1562.

TheFlagellation,c.1457.

GoldenproportionsalsoaboundinthispaintingoftheburialofChrist,whichappearsdirectlybelowtheMadonnainPierodellaFrancesa’sPolyptychoftheMisericordia.

LEONARDODAVINCI

Ahalfmillenniumafterhisdeathin1519,westillcelebrateLeonardodaVinciforhisbrilliantinsightsasaninventorandscientist.Butthispolymathicgeniuswasalegendinhisowntimeaswell,beingdescribedbyhiscontemporariesasa“divine”painter.AstheillustratorofPacioli’sDeDivinaProportioneandasacentralfigureintheplotofDanBrown’s2003bestsellerTheDaVinciCode,Leonardohaslongbeenassociatedwiththegoldenratio.However,aswewillsee,daVinci’sassociationwiththegoldenratiorunsmuchlongeranddeeperthanmanyofusrealize.

WhilestillayoungmanunderthetutelageoftheFlorentineartistandsculptorknownasVerrocchio(“trueeye”),daVincipaintedAnnunciation—asceneshowingtheannouncementtoMaryVirginbytheangelGabrielthatshewouldbecomethemotherofChrist—whichdisplayssomeinterestingproportions.Paintedaround1472–1475,itisthoughttobehisearliestsurvivingwork.

Asshown,thegoldenratioappearstobethebasisforthedimensionsofthewallsandentrywayofthecourtyard,aswellotherkeyelementsofthecomposition.Theornamentalcarvingsatthebottomofthetablearepositionedatthegoldenratiosofitswidth,andMary’necklineisatthegoldenratiofromhersashtothetopofherhead.Furthermore,abasicgoldengridrevealsthatthepaintingcanbedividedintothreeverticalsections,withthetwooutersectionshavingaphi-basedrelationshiptothemiddleone.

AnnunciationbyLeonardodaVincic.1472–1475.

THEMONALISA

DaVinci’smostfamouspaintingisLaJoconde,ortheMonaLisa.Applicationofthedivineproportiontothispaintingisalsothemostsubjecttointerpretationanddebate.UnlikeTheLastSupperandAnnunciation,theMonaLisahasfewstraightlines,orarchitecturalelements,touseasreferencepointsinmakingthisdetermination.Searchtheinternetfor“MonaLisagoldenratio,”andyou’llfindsomeverycreativeinterpretationsofgoldenratiosintheMonaLisa,withgoldenspiraloverlaysofvaryingpositions,orientations,andsizes.Thiscanseemveryarbitraryandinconsistent,andtheycannotallberight.It’sunlikelythatLeonardoeverusedthegoldenspiralthatisnowsocloselyassociatedwith

thegoldenratio,sincesuchlogarithmicspiralswerefirstdescribedmorethanonehundredyearslaterbymathematicianRenéDescartes(1596–1650).AlthoughitmaybedifficulttoknowdaVinci’soriginalintentinhiscomposition,thesimplestandmostobjectiveapproachistooverlaygoldenratiolinesbasedontheheightandwidthofthecanvas,andthefewavailablereferencepointsofherhead,neckline,andhands.Herewefindthatherlefteyeispreciselycenteredinthepainting,andherhairisroughlyboundedbygoldenratiolinesfromthepainting’scentertothesidesofthecanvas.Wealsofindpossiblegoldenratioproportionsbetweenthetopofherheadandherarmatherchinandneckline.DidtheRenaissancemasterintentionallydividehiscompositionasshown?Itseemsvery

plausible,butwewillprobablyneverknowforcertain.

MonaLisa,theworld’smostfamouspainting,isonpermanentdisplayattheLouvreMuseuminParis,France.

PerhapsoneofthebestillustrationsoftheuseofthegoldenratioisindaVinci’sTheLastSupper,whichhepaintedbetween1494and1498.Variousdesignandarchitecturalfeaturesshowveryprecisegoldenratiorelationships.Forexample,examiningthespacebetweenthetabletopandtheceiling,thetopofJesus’sheadappearsatthemidpoint,whilethetopsofthewindowsareatthegoldenratio.Thewidthoftheshieldsisthegoldenratioofthewidthofthecirculararcs,andthestripeswithinthecentershieldareatgoldenratiopointsofitswidth.SomebelievethateventhepositionsofthedisciplesaroundthetablewereplacedindivineproportionstoJesus.

TheLastSupper,1494-1498.

AnotherofdaVinci’smostfamousworksisadrawingcreatedaround1490,theofficialtitleofwhichisLeProporzionidelCorpoUmanoSecondoVitruvio(TheProportionsoftheHumanBodyAccordingtoVitruvius).Asindicated,itisbasedontheidealhumanproportionsasconceivedbytheancientRomanarchitectandmilitaryengineerVitruvius(c.75–c.15BCE).InBookIIIofhistreatiseDeArchitectura,Vitruviusdescribedthehumanfigureasbeingtheprincipalsourceofproportioninarchitecture,withtheidealbodybeingeightheadshigh:

“Thenavelisnaturallyplacedinthecenterofthehumanbody,and,ifinamanlyingwithhisfaceupward,andhishandsand

feetextended,fromhisnavelasthecenter,acirclebedescribed,itwilltouchhisfingersandtoes.Itisnotalonebyacirclethatthehumanbodyisthuscircumscribed,asmaybe

seenbyplacingitwithinasquare.Formeasuringfromthefeettothecrownofthehead,andthenacrossthearmsfullyextended,wefindthelattermeasureequal

totheformer;sothatlinesatrightanglestoeachother,enclosingthefigure,willformasquare.”4

Vitruviusmeasuredtheentirehumanbodyinintegerfractionsoftheheightofaman,asshownbythegridlinesoverlayingdaVinci’sillustration.

Thisillustrationshowstheheightdividedintoquartersandfifths,whilethehorizontalextensionisdividedintoeighthsandtenths.Asyoucansee,thegridlinesalignverticallyatthecollarbone,nipples,genitals,andknees.Horizontallytheyalignwiththewrists,elbows,andshoulders.

However,theVitruvianManalsohassomedimensionsthatsuggestagoldenratiorelationship.Inthedistancefromthetopoftheforeheadtothebottomofthefoot,thefollowingareallatgoldenratiopoints:

VitruvianMan,c.1490.

•thenavel(whichismostoftenassociatedwiththegoldenratioofthetotalheight).

•thepectoralnipples.•thecollarbone.

Inthedistancefromtheelbowtothefingertips,thebaseofthehandbeginsat

Inthedistancefromtheelbowtothefingertips,thebaseofthehandbeginsatthegoldenratiopoint.

In2011,thediscoveryofalostpaintingbyLeonardodaVinciwasannouncedtotheworld.Thispainting,entitledSalvatorMundi(ChristasSavioroftheWorld),hadbeenintheartcollectionofKingCharlesIofEnglandin1649.In1763itwassoldatauctionandthenlostformanyyears.RobertSimon,anarthistorianandprivateartdealer,ledtheefforttorecoverthelostpainting,whichwaslaterrestoredtoitsformerglorybyDianneDwyerModestini.ManyuniquequalitiesofthispaintingledexpertstoconfirmthatitisindeedanoriginalworkofLeonardodaVinci—oneofonlyfifteennowinexistence.In2017,thepaintingwassoldinaChristie’sauctionforarecord-shattering$450milliontoSaudiPrinceBaderbinAbdullahbinMohammedbinFarhanal-Saud,fordisplayinthethenrecentlyopenedbranchoftheLouvreinAbuDhabi.5

Portraitsgenerallyhavefewerdistinctlinesthanpaintingsoflandscapesandarchitecture,buttherearesomeveryinterestingfeaturesintheoverallcompositionofthispaintingthatexhibitgoldenratioproportions.Thedimensionsofitskeyelementsareingoldenratioproportiontooneanother,forexample.Startingwithagoldenrectanglebasedontheheightofthehead,wethenfind:

•thedimensionsofthehandarebasedonagoldenratioofitswidth.•thedimensionsoftheorbarebasedonagoldenratioofitsheight.•thedimensionsofembroideredemblemarebasedonthegoldenratioofitsheightandwidth.

Furtheranalysisrevealsgoldenproportionshorizontallyintheoutsideoftheeyesrelativetothewidthofthecanvas,thewidthofthecenteremblemtothewidthoftheneckline,thewidthofthejewelstotheemblems,andthepositionsofthefingerstothehand.Goldenproportionsappearverticallyintheheightoftheheadtotheneckline(aswiththeMonaLisa),theheightofthejewelstotheemblems,thepositionsofthefingerstothehand,andthepositionsofthereflectionsontheglassorb.

SalvatorMundi,c.1500,themostexpensivepaintingeversold.

WecannotknowwithcertaintywhereLeonardointentionallyappliedthedivineproportioninthispainting’scomposition.Wejustknowthathehaduseditextensivelybefore,thatthispaintingofChristwasbegunwithinafewyearsofhiscollaborationwithPaciolionDeDivinaProportione.AsLeonardooncesaid:

“Therearethreeclassesofpeople:thosewhosee,thosewhoseewhentheyareshown,thosewhodonotsee.”6

AlthoughLucaPacioliwasLeonardodaVinci’smentorformathematics,perhapsPacioli’sappreciationoftheuniqueaestheticsofthegoldenratiothatinspiredhimtowriteDeDivinaProportionecamefromdaVinciandFrancesca,bothofwhomusedthegoldenratiointheirworksmanyyearsbeforeitswriting.

TheseinferiorversionsofSalvatorMundibyItalianpainterMicheleColtellini(above)andBohemianetcherWenceslausHollar(below)helpedtoalertarthistorianstotheexistenceofLeonardo’sversion.

SANDROBOTTICELLI

TheBirthofVenus,paintedbySandroBotticellibetween1482and1485,isoneofthemostfamouspiecesoffifteenth-centuryItalianart.ItisbasedonOvid’sMetamorphoses,aclassicofLatinliterature,andportraysVenus,thegoddessoflove,betweenherhandmaid,theHoraofSpring,andZephyros,whosebreathcreatestheblowingwind.

Here,too,wefindevidenceofknowledgeandapplicationofthegoldenratiowellbeforethewritingofDeDivinaProportione.Thefirstclueisfoundinthedimensionsofthecanvasitself,whichis67.9×109.6inches(172.5×278.5cm).7Theratioofthewidthtotheheightisthus1.6168,avarianceofonly0.08percentfromthegoldenratioof1.618.Toputinperspective,forthecanvastohavebeenanexactgoldenratio,theheightofthecanvaswouldneedtobereducedbylessthanonetwentiethofaninch!Thewidthofthepaintingat109.6inches(278cm)seemssomewhatarbitrary.Thatis,untilonerealizesthattheunitsofmeasurewerenotstandardizedinthisera.Forexample,theSpanishfoot,orpie,oftheMiddleAgeswas10.96moderninches(27.8cm),whichcouldindicatethatthedimensionswerenothaphazardatall,butrathercarefullyplannedtobeexactly10“feet”wide.Eitherwayyoulookatit,it’squitereasonabletoconcludethatBotticelli’sintentherewastobeginthisgreatworkofartwiththeperfectionofthegoldenratio.

TheBirthofVenus,c.1485.

Botticelli’spatronLorenzo“IlMagnifico”deMediciisportrayedinthisdetailofTheProcessionoftheYoungestKing(1459–1461)byBenozzoGozzoli.

Interestingly,theBirthofVenusisthefirstworkeverpaintedonacanvasinTuscany.Arevolutionarywork,itwascreatedbyBotticelliasaweddingpresentforamemberofhispatronfamily,thepoliticallyandfinanciallypowerfulMedicifamily.NuditywasrarelyportrayedinthiseraofChristian-inspiredart,anditsintendeddisplayabovethemaritalbedaddedarathershockingundertoneofsensualityanddesire.Thepaintingwassocontroversialthatitremainedbehindcloseddoorsforanotherfiftyyears.

Severalkeyelementsofthepaintingarealsopreciselypositionedatgoldenratiopoints:

ratiopoints:

•TheverticalgoldenratiolinefromtheleftsidetotherightsidefallsexactlyatthepointatwhichHora’sthumbandfingeraretouching,asthoughsheisgraspingthegoldenratioproportionembodiedinthepainting,perhapsevenreachingforsomethingdivine.

•Theverticalgoldenratiolinefromtherightsidetotheleftsidefallsatthepointwherethelandonthehorizonmeetsthesea.

•Thehorizontalgoldenratiolinefromthetoptothebottomcrossesexactlyatthetopoftheseashell.

•Thehorizontalgoldenratiolinefromthebottomtothetopcrossesatthehorizonline,mostperfectlyontheleftsideofthepainting,andpassesdirectlythroughVenus’navel.

Inaddition,thesubjectVenushashernavelatthegoldenratiopointoftheheightofherbody,whethermeasuredfromthetopofherhairtothebottomofherlowerfoot,fromherhairlineatthetopofherforeheadtothebottomofherupperfoot,orfromthemiddleofthefeettothetopofherheadatthebackpartinherhair.

BotticellialsocreatedanumberofpaintingsoftheAnnunciationbetween1485and1490.Thisevent,whichclearlycapturesthemeetingofthedivinewiththemortal,isanexcellentopportunitytoapplythedivineproportion.Notethatthegoldenratiogridlinesarebasedsimplyonheightandwidthofthecanvasinallbutonecase,sonocreativeinterpretationofplacementisrequired.

Botticelliincludedthisself-portraitinhisc.1475paintingTheAdorationoftheMagi.

TheCestelloAnnunciation,1489.

ThisversionofAnnunciationbyBotticelliisheldatthePushkinMuseumofFineArtsinMoscow,Russia.

AmodernpanoramaofFlorence,thebirthplaceoftheRenaissance.

Annunciation,c.1488–1490,fromBotticelli’sAltarpieceofSaintMark.

RAPHAEL

RaffaelloSanziodaUrbino,popularlyknownasRaphael,wasanItalianpainterandarchitectoftheHighRenaissancewholivedfrom1483to1520.Heisrecognizedasoneofthethreegreatmasters,alongsideMichelangeloandLeonardodaVinci,ofthatperiod.OneofhismostfamousworksisTheSchoolofAthens,afrescointheApostolicPalaceintheVatican.ItcapturesthespiritoftheRenaissanceandisreveredashismasterpiece.Thisworkwasbegunin1509,theyearthatPacioli’sDeDivinaProportionewaspublished,andfinishedtwoyearslater.

Ifthere’sanyquestionwhetherRaphaelusedthegoldenratiointhispainting’scomposition,itcanbeeliminatedwithagooddegreeofconfidencebythegoldenrectanglethatwasplacedfrontandcenterinthepainting.It’sasthoughRaphaelmadeasmallbutundeniablestatementtoanswerthequestionbeforeitwasasked.Thissmallrectangleisabout18by11.1inches(46×28cm)andisaratherunusualfeature.Perhapsitonceborethetitleorsomedescriptionofthepainting?Wemayneverknow.

Self-portrait,byRaphael,c.1504-1506.

Nootherratiowouldaccomplishthesameresultinthiscomposition.Thepaintinghasthousandsofintricatelines,sosomemightsaythatfindinggoldenratioswithinitwouldbeasimpleexerciseinpatternrecognition,whethertheywereintendedornot.Therearetwowaystoovercomesuchanobjection:

1.SettheLineRatiooptioninPhiMatrixsoftwareprogramtoanyotherratiosandseeifyougetthesameabundanceandconsistencyofresultsthanwith

andseeifyougetthesameabundanceandconsistencyofresultsthanwiththeratiosettothegoldenratio.

2.Focusonthemajorelementsofthecompositionalone.Forexample,notethatsimplegoldenratiosofthewidthandheightofthepaintingdefinethepositionoftheclosestarch,thetopofstairs,andthetopofthefarthestarch.

Othergoldenratiosdefineotherkeyelementsofthecomposition,asshown.Raphael’sintricateapplicationofthegoldenratioisobviousaswellasbrilliant.ToappreciatethedetailanddepthofRaphael’splanningandapplicationofthedimensionalproportionsinthispainting,takealookattheimageopposite:

•Eachrectanglebeginsattheleftsideoftheleftcolumninthepainting.Thispointrepresentsthefirstarchitecturalreferencepointoftheactualschoolbuildingasviewedthroughthearchedportalofthefresco.

•Eachrectangleextendstoaprominentcompositionfeatureontherightsideofthepainting.

•Eachdividinglineillustratesagoldenratioformedwithinanotherprominentfeatureofthecomposition.

TheSchoolofAthens,byRaphael,1509-1511.

MICHELANGELO

ThepaintingsoftheothergreatmasteroftheHighRenaissance,Michelangelo(bornMichelangelodiLodovicoBuonarrotiSimoniin1475),provideyetanotherbrilliantexampleofthegoldenratio’sprominenceinRenaissanceart.AnalysisoftheSistineChapelhasrevealedmorethantwodozenexamplesofgoldenratiodimensionsinmajorelementsofthecomposition.

PerhapsthemoststunningexampleappearsatthepointatwhichAdam’sfingeristouchedbythefingerofGodinMichelangelo’siconicpaintingTheCreationofAdam.Thisisfoundatthegoldenratioofboththeirhorizontalandverticaldimensions.

MichelangelobyItalianartistDanieledaVolterra,c.1544.

AviewofMichelangelo’sfinishedSistineChapelceiling,whichwascompletedbetween1508and1512.

TheCreationofAdam.

MichelangelorepeatedthisthemeofthecharacterstouchingthegoldenratiopointinotherpaintingsoftheSistineChapel.Thegridlinesinthephotooppositeshowthegoldenratiooftheheightand/orwidthofeachpainting.Insomecases,thehandsarepositionedasifgraspingthisgoldenproportion,whichcanbeviewedasavisualmetaphorofthehumandesiretograsptheDivine.

TheFallandExpulsionfromtheGardenofEden.

TheCreationofEve.

TheSeparationoftheEarthfromtheWaters.

ThelastoftheseriesofninebiblicalnarrationpaintingsonthecenterceilingoftheSistineChapelisofNoah’sdisgrace.Thepaintingitselfiswithin2percentofgoldenrectangleproportions.Init,thefingersoftwoofNoah’ssonspointdirectlytothegoldenratiolinesfromthepainting’ssides.It’sdoneasiftoshowtheviewerexactlywheretheyare,andthatMichelangelohadindeedappliedthedivineproportion.

TheDrunkennessofNoah.

AmodernviewofVaticanCity,homeoftheRomanCatholicChurch,withSt.Peter’sBasilicaatitscenter.

ThisSistineChapellunettebearsthenamesofSalmon,Boaz,andObed,whoarementionedintheOldTestament’sBookofRuth.Inthisfresco,RuthnursesbabyObed.

IfthereremainsanydoubtthatMichelangelousedthedivineproportioninhisepicpaintings,looktothetabletslistingtheancestorsofJesusonthesidewallsoftheSistineChapel.Theheighttowidthofthenameplatesformagoldenrectangle,withinapixelortwo.Theaverageheighttowidthratioofallthepaintingsis1.62,accuratetowithin1/1000thofthegoldenratioof1.618.

Michelangelo’smagnificentcollectionofpaintingswascreatedbetween1508and1512forPopeJuliusIIandsuccessorPopesoftheRomanCatholicChurch.Giventheirreligioussignificance,itreallyshouldbenosurprisethatMichelangelousedthedivineproportionextensivelytobringbothmathematicalandvisualharmonytothebiblicalaccountsofscripture.Inretrospect,itwouldbemuchmoreofasurpriseifheandtheothermastersoftheRenaissancehadnot.

E

IV

GOLDENARCHITECTURE&DESIGN

“Somesaytheyseepoetryinmypaintings;Iseeonlyscience.”1

—GeorgesSeurat

E verythingthatyouseeorhearcanbedescribedmathematicallyand

geometrically.Thereismathematicsintheorthogonallinesofacityscape

convergingonavanishingpointonthehorizon.Itisseeninthe256values

ofred,green,andblueofeachpixelonyourscreenmonitorthatproduce

the16,777,216uniquecolorcombinations2thatdefineeveryimage.

Everybeautifulmomentofeverysongcanbeexpressedasa

mathematicallydefinedcombinationoffrequenciesandamplitudes.

Aswe’veseen,fascinationwiththemanyuniqueaspectsofthegolden

ratiobyartistsandphilosophershasinspireditsuseinthearts.Whenand

wherethatfirsthappened,wedonotknow,butthereisevidencethatthe

ancientEgyptiansrecognizedthattherewassomethingspecialaboutthis

proportion.

Thefivethousand-year-oldGreatPyramidstoweroverthedesertontheoutskirtsofGiza,Egypt’sthird-largestcity.

PHI,PI,ANDTHEPYRAMIDSOFGIZA

ThepyramidcomplexatGiza,about10miles(16km)southofmodern-dayCairoand5miles(8km)westoftheNileRiver,hashadatoweringpresenceinthecollectivehumanpsycheformorethanfourthousandyears.Threemassive,pyramidal,mortuarytemplesdominatethelandscapeandcommemoratethreepharaohsofEgypt’sprosperousfourthdynasty:Khufu;hisson,Khafre;andhisgrandson,Menkaure.ThefamousGreatSphinx,whichbearsKhafre’svisage,reclinessome546yards(500m)eastofKhafre’spyramid.Eveninouradvancedtechnologicalage,archaeologistsmarvelattheincredibletechnologyandmanpowerthatmusthavebeenrequiredtohaulthousandsof2-ton(1.8mt)limestoneblocksintosuchpreciseandimmenseformations.

THEGREATPYRAMIDTheGreatPyramidofGiza—alsoknownasthePyramidofKhufuorthePyramidofCheops—istheoldestoftheSevenWondersoftheAncientWorld.Itisalsotheonlyonethatremainslargelyintact.ThereisongoingdebateastothegeometricprinciplesusedinthedesignoftheGreatPyramid.Thoughttobebuiltaround2560BCE,itsonceplanar,smooth,outershellisgone,andallthatremainsisthecraggyinnercore,soitisdifficulttoknowtheoriginaldimensionswithabsolutecertainty.Luckily,however,theoutershellremainsattheapex,helpingarcheologistsestablishacloseestimate.

There’slittletodisputeastowhetherthedimensionsoftheGreatPyramidreflectpiandthegoldenratiowithahighdegreeofaccuracy.TheonlydisputeconcernswhethertheancientEgyptiansactuallyknewoftheseconstantsandintentionallyappliedtheminthedesign.SohowmighttheGreatPyramidhaveembodiedeitherorbothoftheseconcepts?Thereareseveralpossibilitiesbasedonvariousmeasurementsandobservationsthatwewillexplore.

ThisillustrationshowsnomadicBedouinsrestingneartheGreatPyramidofGizaduringthelatenineteenthcentury.

1.ApyramidbasedonФvariesbyonly0.07percentfromtheGreatPyramid’sestimateddimensions.

Asmentionedshownhere,phiistheonlynumberwiththemathematicalpropertyofitssquarebeingonemorethanitself,allowingJohannesKeplertoderivehiseponymoustrianglebyconnectingthispropertywiththePythagoreantheorem.UsingtheKeplerrighttrianglewithsides√Φ,1,andΦtodescribetherelationshipbetweenapyramid’sheightandthelengthofitsfoursidesallowsustocreateapyramidwithabasewidthof2andaheightof√Φ,whichisapproximately1.272indecimalnotation.Theratiooftheheighttothebasewidthofthispyramid,then,isapproximately0.636.

TheGreatPyramidofGizahasanestimatedoriginalheightof480.94feet(146.59m)andabasewidthof755.68feet(230.33m),3whichalsocreatesaheighttobasewidthratioof0.636!ThisresultindicatesthattheGreatPyramiddoesindeedrepresentanexampleofaKeplertriangle,atleasttowithinthreesignificantdecimalplacesofaccuracy.Ifthebaseisexactly755.68feet(230.33m),thenaperfectgoldenratiowouldyieldaheightof480.62feet(146.49m),whichvariesfromtheestimatedactualdimensionsoftheGreatPyramidbyamere3.85inches(0.10m),or0.067percent.Thiswouldbeanincrediblecoincidenceifthedesignhadnothingatalltodowiththegoldenratio.

ApyramidbasedonaKeplertrianglewouldhaveotherinterestingproperties.Forexample,thesurfaceareaofthefoursideswouldbeagoldenratioofthesurfaceareaofthebase:

•Theareaofthetriangularsidesoneachfaceisequaltohalfofthebaselength(2)multipliedbytheirheight(Φ),whichyieldsΦ.

•Thesurfaceareaofthebaseis2×2,whichequals4.•Thus,theratioofthesurfaceareaofthefoursides(4Φ)tothesurfaceareaofthebase(4)isΦ.

2.Apyramidbasedonπvariesbyonly0.03percentfromtheGreatPyramid’sestimateddimensions.

In1838H.C.AgnewproposedanotherinterestinghypothesisinAletterfromAlexandriaontheevidenceofthepracticalapplicationofthequadratureofthecircle,intheconfigurationofthegreatpyramidsofGizeh:4WhatiftheEgyptianscalculatedtheheightofthepyramidbasedontheradiusofacirclewiththesamecircumferenceandareaasthepyramid’sbase?Imagineacirclewithacircumferenceof8,whichmatchesthelengthoftheperimeterofthispyramidwithitsbasewidthof2.Ifyoucalculatetheradiusofthiscirclebydividingthecircumferenceby2π,youobtainthevalueof4/π,orapproximately1.273—lessthanone-tenthofapercentdifferentthanthevalueof1.272computedaboveusingKepler’striangle.Multiplyingthe755.68-foot(230.33-m)basewidthofthepyramidbyhalfthisvalueyieldsaheightof481.08feet(146.63m)—adifferenceinheightbetweenthetwomethodsofonly5.5inches(0.14m),andadifferenceofonly1.7inches(0.04m)fromthepyramid’sestimatedheight.

Thisdiagramshowstherelationshipbetweenapyramidwithabaselengthof2andtheradiusofacircularbasewiththesameperimeteroflength8.

3.ApyramidbasedonareasisidenticalingeometrytoonebasedonФ.5

InadditiontotherelationshipsoftheGreatPyramid’sdimensionstoФandπ,it’salsopossiblethatthepyramidwasconstructedusingacompletelydifferentapproachthatcoincidentallyproducedthephirelationship.ThewritingsoftheGreekhistorianHerodotusmakeavagueandoften-debatedreferencetoarelationshipbetweentheheightofapyramidandtheareaofoneofitsfaces,expressedasfollows:

AreaoftheFace=AreaoftheSquareformedbytheHeight(h)

(2r×s)/2=h2

Also,fromthePythagoreanTheoremweknowthatr2+h2=s2,whichmeansh2=s2–r2

Therefore,r×s=s2–r2

Whenr=1,wefindthats=s2–1.RecallfromherethatФistheonlynumberwhosesquareisonemorethanitself,andФisthereforetheonlypositivesolutiontothisequationwhenwesolvefors.Inconclusion,wefindthatiftheheightareatosideareawerethebasisforthedimensionsoftheGreatPyramid,itwouldbeinaperfectphirelationship,whetherornotthatrelationshipwasintendedbyitsdesigners.

4.ApyramidbasedontheancientEgyptiansekedvariesby0.01percentfromtheGreatPyramid’sestimateddimensions.

There’sagoodpossibilitythattheGreatPyramidwasbuiltusingtheseked,ameasurementtechniquethatdescribestheinclinationofapyramidintermsofancientEgyptianroyalcubits,asaratiooftherun(i.e.,halfofthebasewidth)totherise(i.e.,height).ThesekedconceptappearsinexcavatedEgyptianpapyri,includingthefamousRhindMathematicalPapyrusdatingtoaround1550BCE,buttheroyalcubitunitofmeasuredatesasfarbackasthethirdmillenniumBCE6,priortotheGreatPyramid’sconstruction.Theroyalcubitisequivalentinmeasureto20.7inches(52.5cm)or7palms,eachofwhichismadeupoffourdigits.ModernsurveysoftheGreatPyramidsuggestasekedslopeof5.5—thatis,arunof51/2palms(i.e.,5palms,2digits)overariseof1cubit(i.e.,7palms).7Sincetherunisonlyhalfofthebaselength,theheighttobaseratiobasedonthismeasurementtechniqueis7/11,or.63636.Ifwemultiplythemostaccurateandup-to-datebasewidthof755.68feet(230.33m)bythisratio,weproduceanestimatedheightof480.87feet(146.57m)—anincredible0.6inches(0.016m)lessthantheactualestimatedheightoftheGreatPyramid.

ThisportionofthePalermoStonerecountstheNilefloodlevelsduringthereignofKingNynetjer(d.2845BCE),measuredincubits,palms,anddigits.

Wereallydon’tknowwithcertaintyhowthepyramidwasdesigned,andknowledgeofthespecificgeometricrelationshipsandconceptscouldhaveexistedandthenbeenlost.WedoknowtheEgyptiansbuiltthepyramidswithamazingprecisionandleftlittletochance,asevidencedbytheiralignmenttowithin1/20thofadegreefromtruenorth.Thebuildersmayhavechosenapproachesthatproducedalmostidenticalgeometricrelationshipstothoseofpyramidsbasedonphiandpi.

IftheancientEgyptians’knowledgeandapplicationofthegoldenratiowerelimitedtothisoneexceptionallyaccurateappearanceintheGreatPyramid,itcouldstillbearguedthatitwasduetochance.However,wenowhaveadditionalevidencethatsuggeststhatthegoldenratioalsoappearsinthepositionsandrelativesizesofthepyramidsattheGizasite.Theserecentfindingsmakeforamuchmorecompellingcase.

AnaerialviewoftheGizanecropolis.

COMPARINGKHUFU,KHAFRE,ANDMENKAUREConsiderthepyramidcomplexasawhole.Usingsatellitemappingimages,ifyoucreatearectanglewithaperimeterthatoutlinesthebasesofthetwolargestpyramidsatthesite,KhufuandKhafre,you’lldiscoverthattheeasternedgeofKhafre’sbaseiscloselyalignedwiththegoldencut,movingwestwardfromtheeasternedgeoftheperimeterrectangletothewesternedge.You’llalsofindasimilarratiocomparingthedistancebetweenKhufu’snorthernedgeandKhafre’snorthernedgetothedistancebetweenthenorthernandsouthernedgesofKhafre’sbase.

TheserelationshipsareconfirmedbythedistancesattheGizacomplex,ascalculatedbyarcheologistGlenDash.8Forexample,thewidthoftheperimeterrectanglethatenclosesthetwolargerpyramidsisapproximately1,825.5feet

(556.4m),whereastheheightofthissamerectangleis1,894.4feet(577.4m).Meanwhile,thebasewidthofKhafreis707feet(215.3m).9Ifwesubtractthisnumberfrombothrectanglelengths,wediscoveradistanceof1,119.1feet(341.1m)betweentheeasternedgesofKhufuandKhafre’sbases,andadistanceof1,188.3feet(362.2m)betweenthenorthernedgesofKhufuandKhafre’sbases.Dividingtheperimeterlengthsbythesedistancesgivesratiosof1.631and1.594.Theaverageoftheseratiosis1.613,whichismightycloseto1.618.

AnotheranalysisoftheGizacomplexsitebyChrisTedder10providesanevensimplerandmoreelegantrelationshipbetweenthelocationsoftheapexesofKhufu,Khafre,andMenkaure.Therelationshipinvolvestwogoldenrectangles(oneinportraitorientation,oneinlandscapeorientation)whosecornersalignwiththeapexofeachpyramid,asshownonthefollowingpage.

Again,relyingonDash’sveryprecisemeasurementsattheGizasite,theeast-to-westdistancebetweenKhufuandKhafre’sapexesis1,095.5feet(333.9m),andtheeast-to-westdistancebetweenKhafreandMenkaure’sapexesis785.76feet(239.5m).Fromnorthtosouth,thedistancesbetweenapexesare1,162.4feet(354.3m)and1,265.4feet(385.7m),respectively.Thisallowsustoconstructtworectangleswithdimensionsof1,881.2×1,162.4feet(573.4×354.3m,showninblue)and1,265.4×785.76feet(385.7×239.5m,showninred).Thelargerrectanglehasperfectgoldenproportions,whereasthesecondhasproportionswithin0.08ofphi.

TheTeddergridshowsthetwogoldenrectanglesformedinthedistancesbetweentheapexesofMenkaure(left)andKhafre(center),andKhufu(right).Note:Thetopofthediagramfaceswest.

Insum,hereiswhatthegeometricrelationshipsbetweenthemostup-to-dateGizasitemeasurementsshowusaboutthethreemainpyramids:

•TheaverageratioofthedistancebetweentheeasternandnorthernedgesofKhufuandKhafre’sbasestothewidthofKhafre’sbaseisapproximately1.618.

•Theratiooftheeast-westdistancebetweenKhufuandMenkaure’sapexestothenorth-southdistancebetweenKhufuandKhafre’sapexesis1.618.

•TherighttriangleformedbyKhufu’sheight,thelengthofitsfourslantedfaces(hypotenuse),andhorizontaldistancebetweenitsapexandtheperimeterofitsbase(width)producesahypotenusetowidthratioof1.618—identicaltothatoftheKeplertriangle.

THEPYRAMIDSOFQUEENS

THEPYRAMIDSOFQUEENSInthesite’sEastField,nexttotheKhufuPyramid,arethreesmallerpyramids,thoughttocontainthetombsofKhufu’smother,QueenHetepheresI,hiswife,QueenMerititesI,andhisotherwife(orpossiblyhisdaughter),Henutsen.11Asshownbelow,thelengthoftherectanglearoundallthreepyramidstothelengthoftherectanglearoundthebaseofMeritites’sandHenutsen’spyramidscanberepresentedbyФ.

JustsouthoftheMenakaurePyramidarethethreePyramidsofQueens.Althoughirregularlyshaped,thedistancebetweenthecornersoftheirbasesonthesouth-facingsidesrevealtheverysamegoldenratiorelationshipthatappearedinthethreepyramidsnexttoKhufu.

Whenseenfromsatellite,thepositionsofthethreepyramidsadjacenttotheKhufuPyramidappeartoreflectthegoldenratiointheirrelativepositions.

ThesouthernedgesofthesatellitepositionsofthePyramidsofQueensalsoappeartoreflectthegoldenratio.

THEGREATSPHINXThere’sonemoremajormonumentattheGizasitethatI’dberemisstoexclude

There’sonemoremajormonumentattheGizasitethatI’dberemisstoexcludefromthisanalysis:TheGreatSphinx.Again,usingsatellitetopologicalimagesandexaminingtherelationshipbetweenthefulllengthofthemonumentoneachsideandthelengthfromthefrontpawsandbackpawsoneachside,wediscoveryetanotherexampleofgoldenproportions!

Thereisstillmuchtounderstandaboutthehistory,mathematics,design,andpurposeofthebuildingofthepyramids,butonethingisprettyclear:ManykeyfeaturesoftheGizasiteappeartocloselyembodythegeometryofthegoldenratio.Ihopethesediscoveriesandanalysesprovideotherswithanincentiveformoreresearch.Fornow,thequestionpersists:Whydidtheancientbuilderschoosethisparticularconfigurationforthegreatpyramids?Becauseitappearedmorebeautifulandmorealignedwithnature?Ifnot,whydoesthegoldenratioappeartobesoprevalentinthemostnotablesurvivingmonumentsoftheancientworld?

This1904paintingbyRussianartistV.F.Ulyanov(1878–1940)portraysthemajesticSphinxduringtwilight.

Whenviewedfromsatellite,bothsidesoftheGreatSphinxappeartoreflectgoldenproportions.

PHIDIASANDTHEPARTHENON

TheancientGreeksculptor,painter,andarchitectPhidias,wholivedfromabout480to430BCE,deservesspecialrecognitioninourstory,asheinspiredtheuseoftheGreekletterФtodesignatethenumber1.618.Althoughnoneofhisoriginalworksremain,numerouscopiesexist.AmonghisgreatachievementswasthestatueofZeusintheTempleofZeusatOlympia,oneoftheSevenWondersoftheAncientWorld.HealsocreatedthestatuesoftheParthenon,includingthatofthegoddessAthena,andthere’sevidencethatheappliedthegoldenratiointhesedesigns.WhilehisstatuesofZeusandAthenadidnotsurvive,hislegacylivesoninancientGreekcanonandtheenduringstructuresontheAthenianAcropolis,perchedabovethecityasamonumenttoclassicalGreece.

TheParthenoninAthens,builtbytheancientGreeksbetween447and438BCE,isregardedbymanyasaprimeexampleofarchitecturethatmakesuseofthegoldenratio.Ofcoursetherearethosewhodisagree,pointingoutthatmorethanacenturyelapsedbetweenthecompletionoftheParthenonandthefirstdocumentationofthegoldenratioinEuclid’sElements.

Thisnineteenth-centurydrawingportraysPhidias’smassivegoldandivorysculptureofZeusinOlympia.Oneoftheworld’soriginalSevenWonders,itstood39feet(12m)high13andwasadornedwithpaintingsandpreciousstones.

THEΦCONNECTION

Itwasn’tuntiltheearlytwentiethcenturythattheGreekletterphi(Φ)wasfirstusedtodesignatethegoldenratio.Onpage420ofhis1914mathematicsreferenceTheCurvesofLife,SirTheodoreAndreaCookcreditedAmericanmathematicianMarkBarrwithintroducingthe

symbolinreferenceto1.618“partlybecauseithasafamiliarsoundtothosewhowrestleconstantlywithpiandpartlybecauseitisthefirstletterofthenameofPhidias,inwhosesculpturethisproportionisseentoprevailwhenthedistancesbetweenthesalientpointsaremeasured.”12However,somescholarssuggestthattheassociationhadmoretodowithFibonacci,sincephiistheGreekequivalenttotheletterF.

This1887statuebyParisiansculptorAiméMilletshowsPhidiasandaminiaturereplicantofhisfamoussculptureAthenaParthenos,whichoncestoodwithintheParthenon.

TheruinsoftheParthenonareperchedontherockyAcropolis,highabovethemodernAtheniancityscape.

ThereareseveralchallengesindeterminingdefinitivelywhethertheParthenon’sarchitectsintentionallyincorporated1.618intoitsconstruction:

•TheParthenonembodiesavarietyofnumbersandproportionswithitsforty-sixperimetercolumnsandthirty-nineinteriorcolumnsspacedatvaryingdistancesfromoneanother.

•TheParthenonisnowpartiallycollapsed,makingitsoriginalfeaturesandheightdimensionsubjecttosomeconjecture.

Twoofthemostfamiliarappearancesofthegoldenratioarepresentinthedimensionsoftheshortersideofthestructure.TheimagebelowshowstheParthenonwithasuperimposedgoldenrectangleandembeddedgoldenspiral.However,thisassumptionrequiresthealignmentofthegoldenrectanglewiththebottomofthesecondstepofthestructureandwiththeestimatedpositionofthetriangularpediment’soriginalapex.Withthisalignment,thetopofthecolumnsandbaseoftherooflineareinaclosegoldenratioproportiontotheheightoftheParthenon.This,however,isnotthemostcompellingevidencethattheancientGreeksusedthegoldenratiointentionallyinthedesignofthisiconic

theancientGreeksusedthegoldenratiointentionallyinthedesignofthisiconictemple.

ViewintotheHeydayofGreece(1836)byGermanpainterAugustAhlborndepictstheconstructionoftheParthenon.

Phidias(center)showsoffthefriezeoftheParthenontohisfriendsinthis1868paintingbyDutch-BritishartistSirLawrenceAlma-Tadema.

Applyingthegridlinestothebuilding’sentablaturerevealsotherinterestingproportions.Zoominginontheentablaturefriezeandembeddedmetope-triglyphpattern,wediscoverthatthehorizontaldividinglineoftheentablatureisataprecisegoldenratioofitsheight.Wealsofindbeautifulgoldenrectanglesenclosingthemetopes,withyetanothergoldenrelationshipbetweenthewidthofthetriglyphsandthatofthemetopes.

Goldenproportionsareapparentintheremainingstructureofthenearly2,500-year-oldParthenon.

Finally,let’sexaminethefloorplanoftheParthenon,whichshowseightcolumnssupportingtheshortersideandseventeencolumnssupportingthelongerside.Justinsidetheperimeteroneachoftheshortersidesaresixcolumns,followedbyentrancestotwointeriorchambers.Myanalysisrevealsthefollowing:

•Thewallthatseparatestheeastandwestinteriorchambersiscloselyalignedtothegoldencutoftherectanglealignedwiththecentersoftheeastandwestperimetercolumns.

westperimetercolumns.•ThecentersofthefourcolumnsinthewesterninteriorchamberandthebaseoftheStatueofAthenaarepositionedatthetwogoldencutsofthedistancebetweenthecentersofthenorthandsouthperimetercolumns.

•Theentrancestobothinteriorchambersarepositionedatthegoldencutsofthedistancebetweenthenorthandsouthwallsofeachinteriorchapter.

MorethanfourhundredyearsfollowingtheParthenon’sconstruction,RomanmilitaryengineerVitruvius(seehere)proposedwhatheconsideredtheperfectRomanhousefloorplaninhisfamousbookDeArchitectura(c.20BCE).Consideringthequantityofgoldenproportionsthroughout,itseemslikelythathewasawareoftheuseofthegoldenratiobytheancientGreeksintheirartandarchitecture.

AcloseupofartistGodfriedSemper’scoloredreproductionoftheParthenon’sfriezeshowsthegoldenrelationshipbetweenthemetopeandtriglyphmoreclearly.

ThefloorplanoftheParthenondisplaysgoldendimensions.

ThisillustrationofanidealGreekhomefloorplanfromVitruvius’sDeArchitecturaisfullofgoldenrectanglesandotherphi-baseddimensions.

GOLDENCATHEDRALS

Theconstructionofawe-inspiringcathedralsasChristianityspreadthroughoutEuropewasanoutwardexpressionofaninnerreverenceforGod’sgloryandafocalpointforcommunitylifeforcenturiestofollow.ItwasalsoanoutletforthecreativeenergyofmedievalEuropeansociety.Themassivefinancial,technical,artistic,andphysicalresourcesrequiredmadeeachacommunityeffortthatwasapproachedwithgreatambitionandenthusiasm.Constructionoftentookoveracentury,inspiringgenerationstobeapartofsomethinglargerthanoneself.

ThenorthrosewindowofNotre-DamecathedralinParis,France,beautifullyexhibitstheDivineproportion.

Manyphi-basedproportionsarefoundinthewesternfaçadeofNotreDame.

OneofthefinestcathedralsinexistencewasbeguninParisin1163underthedirectionofBishopMauricedeSully.Hediedin1196,andfinally,in1225,constructionofthewesternfaçadewascomplete.Anothercenturypassedbeforetheentirecathedralhadbeenbuilt,andnearlyeighthundredyearslater,itisoneofParis’stoptouristattractions:theNotre-DameCathedral.Interestingly,thewesternfaçade,aswellasthenorth-facingGothicstained-glassrosewindow,reflectthegoldenratiointheirdimensions.

NotlongafterconstructionwasunderwayonNotre-DameinParis,anothercathedralwascommissionedinChartres,about50miles(80km)southwestofParis.Thisonewascompleteby1220,andlikeNotre-Dame,goldenproportions

Paris.Thisonewascompleteby1220,andlikeNotre-Dame,goldenproportionsarefoundthroughoutthestructure.Infact,thegoldenratioseemstoreappearinvariouscathedralsthroughoutEurope.

AboveandBelow:ThewindowsofthewesternfaçadeofNotre-DamedeChartresCathedralalsoexhibitthegoldenratio,asshowninthesedetailsfromadetailed1867architecturaldrawingofthecathedral.

Thestunningnorth-facingtranseptrosewindowofChatresCathedral,constructedc.1235.

Aviewofthecathedral’ssouthernfaçadetoweringovertheChartrescityscape.

ThiscoloredmonographofaportionofasouthtranseptwindowleaveslittledoubtastowhetherphiwasworkedintotheChartresCathedral’sdesign.

ThefloorplanoftheStiftskirche(“CollegiateChurch”),constructedmostlyoverathree-hundred-yearspanbetween1240and1547inStuttgart,Germany,appearstohavesomephi-baseddimensionsbasedonthislate-nineteenth-centurydrawing.

ThewesternfaçadeoftheGothic-RomanesqueLimburgCathedralinHesse,Germany.

In1296constructionbeganonanotheroftheworld’smostfamousandidentifiablearchitecturalwonders:Florence,Italy’sCattedralediSantaMariadelFiore(CathedralofSaintMaryoftheFlower).TuscanarchitectArnolfodiCambiocameupwiththewinningdesign,whichincludedthreewidenavesandanoctagonaldome.Afterhisdeath,workresumedunderaseriesofarchitects,

includingFrancescoTalenti,whoexpandedthelengthofthenavestomakethecathedralEurope’slargestinthe1350s.Healsocompletedthealmost300-foot-(91m)-tallcampanilenearthebasilica’smainentrancein1359.14

Thefamousdomewasoneofthelaststructurestobebuilt.In1418thepowerfulMedicifamilyannouncedacontestforthedome’sdesign,andmastergoldsmithFilippoBrunelleschireceivedthecommission.In1436thedomewasfinallycomplete.Atechnologicalmarvel,itbegan171feet(52m)abovethefloorofthebuilding,spanning144feet(44m)andrisingto375feet(114.5m)intotal,withthecrowninglanternincluded.15Thespanofthedomewastoolargeandtoohighforwoodensupports,soBrunelleschihadtodeviseingeniousconstructiontechniques—nottomentionmorethanfourmillionbricks—toaccomplishhistask,butalltheeffortpaidoffinwhatisstillthelargestbrickdomeintheworld.Ifthatisn’tenough,thismagnificentstructurealsoembodiesgoldenproportions!

Asidefrombeingamasterpieceofengineering,Bruneschelli’sfamousoctagonaldomeappearstoreflectgoldenproportions.

Thegoldenratiocanbeseeninmanyelementsofthecathedral’sfinalfloorplan.

ManyarchitecturalelementsofFlorence’sgiganticCathedralofSaintMaryoftheFlowerhavephi-baseddimensions.

THETAJMAHAL

Almost4,000miles(6,437km)fromGreeceandalmosttwomillenniaintothefuture,weencountertheTajMahal.TheMughalemperorShahJahancommissionedthemonumenttohousethetombofhisfavoritewife,MumtazMahal,followingherdeathduringchildbirthin1631.Withintwelveyearsthebeautifulmausoleumwasmostlycomplete,withotherphasesoftheprojectcontinuingforanothertenyears.

MumtazMahal(bornArjumandBanuBegum,1593–1631)andherhusband,ShahJahan(1592–1666),areshownintheseminiatureportraitsfromUdaipu,India,whicharepaintedoncamelbonewithinlaidsemipreciousstones.

LocatedinAgrainnorthernIndia,theTajMahalisconsideredoneofthefinestexamplesofarchitecturethatexiststoday.PersianarchitectUstadAhmadLahoridirecteditsconstruction,employingaroundtwentythousandartisansintheeffort.Evidenceofthegoldenratioasafoundationalaspectofitsdesignisobservedinthewidthofthecenterarchinrelationtothewidthofthebuilding.

Thegoldenratiocanalsobeseeninthewidthandpositionofthearchedwindowsatthecenteroftherectangularframearoundthecentralarch.Other

windowsatthecenteroftherectangularframearoundthecentralarch.Othergoldenproportionsappearthroughout,includingtherelationshipbetweentheheightandwidthofthecentralstructureandthoseofthetowersoneitherside.

ThemonumentalivoryandmarbleTajMahalmausoleumhasobviousphi-basedproportions.

SEURATANDTHEGOLDENRATIO

FrenchpainterGeorgesSeurat(1859–1891)iswellknownforhisinitiationoftheNeo-Impressionistmovementinthelatenineteenthcentury.Hissignaturepointillistmethodofpaintingisexemplifiedinhisbestknownwork,ASundayAfternoonontheIslandofLaGrandeJatte,paintedbetween1884and1886.However,fewareawarethatSeuratappearedtoincorporatethegoldenratiointomanyofhisworks.AccordingtoRomanianpolymathMatilaGhyika,whowroteongeometryinartandnature,

Seurat“attackedeverycanvaswiththegoldenratio.”16Averyinterestingclaim,butisittrue?SomescholarsinsistthatGhyika’sassertioniswithoutmerit,butlet’sexaminetheevidence.IexaminedSeurat’sfullcatalogofpaintingsandfoundaboutone-quarterofthemtobe

paintedongoldenrectanglecanvasesorpanels,inbothportraitandlandscapeorientations.However,that’snottheonly“coincidence.”Furtherexaminationshowsthattheproportionsandspacingofmanykeyelementsinaboutone-thirdofthesepaintingsalsoreflectsgoldenproportions.

PhotographicportraitofSeurat(1859-1891),whoseartworkblendedimpressionismwithmathematicalprecision,1888.

BathersinAsnières,1884.

VariousSeuratpaintingsongoldenrectanglewoodpanelsareshownbelow.

TheLighthouseatHonfleur,1886.

StudyforASundayonLaGrandeJatte,1884.

PeasantwithaHoe,c.1882.

TheNavvies,c.1883.

WomanwithUmbrella(1884)isoneofseveralportraitpaintingswithgoldenratiocompositiononacanvasofneargoldenproportions.

BridgeofCourbevoie,1886–1887.

TheChannelofGravelines,PetitFortPhilippe,1890.

InSeurat’s1888paintingTheSeineatLaGrandeJattebelow,forexample,thereappeartobeanumberofclearandpreciseapplicationsofthegoldenratio.Theseincludethefollowing:

•Thesailboatisverticallyandperfectlypositionedwithinthebasicgoldengridoverlayingtheentirepainting.

painting.•Thewidthoftheshorelineatthebottomofthepaintingtransitionstowateratthegoldencut.•ThebuildingenclosedbyagoldenrectangleontheshoreoftheSeineisdividedatthegoldencut.•Theheightandwidthofthesmallsailaregoldenratiosoftheheightandwidthofthelargesail.•Therowerispositionedatthegoldencutbetweenthebottomofthesailboatandthebottomofthepainting.

ThisandotherexamplesshownhereindicatethatSeuratmaynothaveattackedeverycanvaswiththegoldenratio,butheseemstohaveapplieditliberallyinhiswork.

LECORBUSIER’SMODULORDESIGNS

ThearchitectureiconknownasLeCorbusierwasbornCharles-ÉdouardJeanneretinSwitzerlandin1887.HewasthesonofanartisanwhodecoratedwatchesandtookfrequenthikesintheJuraMountainswheretheylived.Jeanneretdevelopedaloveofnatureaswellasthedecorativearts,teachinghimselfthebasicsofarchitectureandphilosophybyreadingthroughthebooksofhislocallibrary.Inhisearlytwenties,followingthetrendofotherartistsoftheera,headoptedthepseudonymLeCorbusier.Yearslater,inhisfifties,hedevelopedasystemofdesignbasedonthegoldenratioandthehumanbodycalledtheModulor.Thissystem,whichsoughttounitethemetricandimperialsystemsofmeasurement,wasintendedasauniversalstandardofmeasurethatengineers,architects,anddesignerscouldusetoproduceformsthatwerebothpracticalandbeautiful.Herepresentedthis“rangeofharmoniousmeasurements”withtheabstractformofa6-foot-tall(1.83-m)manwitharaisedarmthatwasbentinalignmentwiththetopofhishead,whichwasconvenientlypositionedatthegoldenratiocutbetweenhisnavelandthetopofhisraisedarm.AustralianarchitectureprofessorMichaelJ.Ostwalddescribesitas:

“ForLeCorbusier,whatindustryneededwasasystemofproportionalmeasurementthatwouldreconciletheneedsofthehumanbodywiththebeautyinherentintheGoldenSection.Ifsuchasystemcouldbedevised,whichcouldsimultaneouslyrendertheGoldenSectionproportionaltotheheightofahuman,thenthiswouldformanidealbasisforuniversal

standardization.”17

InLeCorbusier’sattempttousethemathematicalproportionsofthehumanbodytoimproveboththeappearanceandfunctionofarchitecture,hefollowedinthefootstepsofVitruvius,DaVinci,Pacioli,andtheRenaissancemasterswhousedthestudyofmathematicsandnaturetoimbuetheirartisticmasterpieceswithadivinequality.

Afteritsformulationinthemid-1940s,Corbusierappliedthenewsystemtoseveralbuildings,including:

•theworldheadquartersoftheUnitedNationsinNewYork,NY(completed

•theworldheadquartersoftheUnitedNationsinNewYork,NY(completedin1952).

•severalmodernisthousingdevelopmentsthroughoutEurope,beginningwiththeCitéradieuse(RadiantCity)inMarseille,France(completedin1953).

•ConventSainteMariedelaTourettenearLyon,France(completedin1961).

AnengravingofLeCorbusier’sModulorappearsontheexterioroftheCorbusierhaus.

VariousarchitecturalfeaturesinLeCorbusier’s1958Unitéd’HabitationofBerlin(nowknownasCorbusierhaus)residentialprojectreflectthegoldenratio,includingthewindows,floorheights,andbalconywidths.

LeCorbusier’sModulorsystemformsthebasisoftheUNSecretariatbuilding’sdesign,asshown.

LetustakealookatoneprominentexampleofLeCorbusier’sModulordesignapproachandhisutilizationofgoldenratios.In1947theBrazilianarchitectOscarNiemeyerandLeCorbusierjoinedforcestodesigntheUNheadquartersinNewYorkCity,a505-foot(154-m)towercalledtheSecretariatbuilding.18Atthetime,LeCorbusierwasintheprocessofdevelopinghisModulordesignsystem,andNiemeyer,anothergiantintheworldofmodernarchitecture,washighlyinfluencedbythisSwiss-bornartist,designer,andurbanplanner.AsarchitectRichardPadovandescribedinhisbookProportion:Science,Philosophy,Architecture:

“LeCorbusierplacedsystemsofharmonyandproportionatthecenterofhisdesignphilosophy,andhisfaithinthe

mathematicalorderoftheuniversewascloselyboundtothegoldensectionandtheFibonacciseries,whichhedescribedas‘rhythmsapparenttotheeyeandclearintheirrelationswithone

another.Andtheserhythmsareattheveryrootofhumanactivities.TheyresoundinManbyanorganicinevitability,thesamefineinevitabilitywhichcausesthetracingoutoftheGoldenSectionbychildren,oldmen,savages,andthe

learned.’”19

FortheUNproject,LeCorbusierconceivedatall,centralbuildingthatwouldhousealloftheSecretariatoffices.Knownasproject23A,themainbuildingdimensionsconsistedofthreestackedgoldenrectangles.Niemeyer’sproject32,ontheotherhand,featuredatall,slightlywidercentralbuildingwiththedimensionsofthegoldenrectangle.ThefinaldesigncombinedelementsfrombothNiemeyer’sandLeCorbusier’sschemesbutusedthreestackedgoldenrectanglesasthebasisofthedesign.

Atfirstglance,thefournoticeablebandsonthefaçadeofthebuildingmakeitseemlikemostofthethirty-ninefloorsaredividedequallyintothreerectangles,butcloserinspectionrevealsthattheirdimensionsdifferslightly.Thefirstrectangleisonlyninefloorstall,whereasthesecondandthirdrectanglesareelevenandtenfloorstall,respectively.Also,whilethebuildingwidthisstableat287feet(87m),thebuildingheightrangesfrom505to550feet(154to168m),20dependingontheelevationasonemovesfromstreetlevelatthebuilding’sfronttotheshoreatitsrear.

LeCorbusier’sUNSecretariatbuildingoverlooksNewYorkCity’sEastRiver.

Ifthebuildingwereaperfectgoldenrectangle,asNiemeyerhadproposed,thebuildingwouldrisetoonly464feet(141m),whichiswithin0.5percentofitsoccupiedheight.However,abuildingpreciselycomposedofthreestackedgoldenrectangleswithalengthof287feet(87m)wouldresultinabuildingthatis532(162m)feettall.Theaverageheightofthebuildingis,infact,527.5feet(160.7m)–0.9percentlessthanaperfectstackofthreegoldenrectangles.That’sasmalldiscrepancy,butinadditiontotheunevenelevationofthelandbetweenthestreetandtheriver,wheretheUNbuildingstands,thereareafewexplanationsforthis:

1.Thegoldenratioisanirrationalnumberthatcannotbeexpressedinintegers,whereasarchitectsarefacedwithanumberofrealworldconstraintsthatarebasedonintegers,suchasthenumberoffloorsandwindows.

2.Thestandarddimensionsofconstructionmaterials,suchasdrywallandbuildingframingcomponents,aresubjecttovariousbuildingstandards.

3.Therearetheengineeringconstraintsrequiredtoconstructa500-foot(152-m)-tallskyscraperthattakeprecedentoveritspureartisticdesignelements.

Regardless,ifyouapplygridlinestothebuildingbasedonLeCorbusier’sModulorsystem,whichinvolvesmultiplyingtheheightofeachdimensionby1.618,averyrevealingpatternemerges,asshownopposite.Also,ifyouapplygoldengridlines,you’llfindthatseveralkeyelevationsinthebuildinghaveaphirelationship.Bothapproachesdemonstratethepresenceofthegoldenratiointheoveralldesign.

ThisattentiontodetailintheconsistentapplicationofdesignprincipleswelcomesvisitorsastheyentertheUNbuilding,too.Thefrontentrancedisplaysgoldenproportionsinthefollowingways:

•Columnsoneithersideofthefrontentranceareplacedatthegoldencutofthedistancefromthemidpointoftheentrancetotheedgeoftheentrance.

•Thetransparententrancestotheleftandrightofthecenterentranceareaaregoldenrectangles.

•Thedoorsontheleftandrightsideofthecenterentrancearegoldenrectangles.

•Therectanglesformedbythecentralfloor-to-ceilingwindowsandtheentrancesoneithersidehavegoldenproportions.

Thewindowswithinthehorizontalbandsonthefaceofthebuildingarealsoacollectionofgoldenrectangles,andthebandsaredividedattwogoldencutstoframethewindowsattheircenters!

AsillustratedbytheintricatenestingofgoldenratiosinhisModulordesigntemplate,LeCorbusier’spassionandvisionforthegoldenratiowasfarmoresophisticatedthansimplydesigningabuildingintheshapeofagoldenrectangle.Nodetailwasoverlookedinthebuilding’sdesign,andtheintricatebeautyofhiscreationonlygetseasierandeasiertoappreciateasthesegoldenproportionsarerevealed.AsexemplifiedintheworksofLeonardo,Michelangelo,Raphael,andothersthatfollowed,thisisthe“verydelicate,subtleandadmirableteaching”and“verysecretscience”ofwhichPacioliwrote.Theapplicationofthegoldenratiotogreatmasterpiecesofartanddesigninordertocreatevisualharmonypersistsinourmodernworld.

Colorgridlinesdefinethevariousphi-basedrelationshipsinthedimensionsoftheUNSecretariatbuilding.

PHOTOGRAPHYCROPPINGANDCOMPOSITION:THERULEOFTHIRDS

Ifyou’vedabbledmuchinphotographyorexploredthecompositiongridsthatareavailableinyoursmartphoneordigitalcamera,youwilllikelyhavecomeacrosstheruleofthirds.Datingbacktoatleastthelateeighteenthcentury,whenJohnThomasSmithproposeditasabasisforpaintingcompositioninhisbookRemarksonRuralScenery(1797),thistoolisbasedondividinganimageintothirdsverticallyandhorizontallytocreatenineequally-sizedsections.Importantcompositionalelements,suchashorizonsorpeople,arethenplacedalongtheselinesorneartheirintersectingpoints.Thisisbelievedbymostartistsandphotographerstocreatemuchmoreinterestandvisualappealthansimplycenteringthesubjectinthemiddleofthepicture.Althoughtheruleofthirdsiseasytocomprehendandcreate,itjustprovidesarough

approximationofthegoldenproportionsutilizedinmanymasterpiecesofartanddesignthroughouthistory.Theruleofthirdshasdividingpointsat1/3and2/3(0.333and0.667),whereasthegoldenratiogridhasdividingpointsat1/Φ2and1/Φ(0.382and0.618).Variationsonthisbasicgoldenratiogridprovidegoldenratiosofgoldenratios,aswellasotherexpressionsofphiincludingthegoldenspiralandgoldendiagonals.Toillustratethedifferencebetweenthetwomethods,takealookattheimagesbelow.The

compositionoftheimageontheleftisbasedontheruleofthirds,whereasthecompositionof

Toillustratethedifferencebetweenthetwomethods,takealookattheimagesbelow.Thecompositionoftheimageontheleftisbasedontheruleofthirds,whereasthecompositionoftheimageontherightisbasedonthegoldenratio.Theruleofthirds,whileundoubtedlyuseful,canbesomewhatlimitingtoartisticexpression.

Bycontrast,thegoldenratiogridallowsyoutocreativelyresizeandpositionthegridtocreatemultiplevariationsofthecroparea,applyinggoldenproportionsagainandagainwithinasinglecomposition.ThisisthesametechniqueofvisualharmonyusedbyLeonardodaVinci,GeorgesSeurat,LeCorbusier,andothermastersofartanddesignduringthelastfivehundredyears.

RuleofThirdsgrid.

DiagonalPhiMatrixgrid.

Phigrid.

SymmetricalPhiMatrixgrid.

Aboveisanexampleofimagecroppingbasedonphi.

Aboveisanexampleofimagecroppingbasedontheruleofthirds.

LOGOANDPRODUCTDESIGN

Inadditiontoitsuseinpainting,architecture,andgraphicdesign,thegoldenratioalsoappearsinnumerousproductdesigns.Insomecases,itenhancesperformanceoftheproduct.Forexample,manystringinstrumentsdisplaygoldenratioproportions.Asanillustration,theworld-famousStradivariusviolins—developedintheseventeenthandeighteenthcenturiesbytheItalianStradivarifamily—seemtoexhibitgoldenproportions.Knownfortheirsuperiormaterials,construction,andsoundquality,todaythesesought-afterviolinscanfetchmillionsofdollarsatauction.

Inothercases,thegoldenratioaddsstyleandaestheticappeal.Corporationsinvestmillionsofdollarsonbrandingandlogodesign,knowingthattheymustcapturetheheartsandmindsofasmanypotentialcustomersaspossibleinaninstant.Theyaresoprotectiveoftheirpowerful,iconicsymbolsthatIcannotvisuallypresentinthesepagesalltheexamplesofgoldenproportionswhichappearinlogodesign.However,Icantellyouwheretolook.

Googlecapturedtheattentionofthedesignworldin2015withtheirannouncementofamajorredesignoftheirlogo,fonts,andotherbrandingsymbolsandicons,butonethingtheysmartlyretainedandenhancedwastheuseofphiindeterminingthedimensionsandspacingoftheletters.Forexample,oncloseinspectionit’sclearthattheratiooftheheightoftheuppercaseGandlowercaseLtotheheightoftheotherlowercaseletters(exceptingthelittletailoftheG)equalsФ.TheratioofthewidthofthecapitalGtothatofthelowercaseGisalsogolden,asisthepositionofthesearchfieldinrelationtothetopofthelogoandthebottomofthesearchbuttonsontheGooglesearchhomepage,whichalsohappenstobethemostvisitedwebsiteintheworld.Eventhelittlemicrophoneiconattherightofthesearchbarreflectsgoldenproportions,andyetmoregoldenrelationshipsabound.Itappearsasifthegoldenratiowasusedforjustabouteverydecisiononpositionandproportionforthismultinationaltechgiant.Ifyoudidn’tknowbetter,youmightwonderifLucaPacioliandLeonardodaVincithemselveshadbeenleadingthedesignproject!

GoldenproportionsaboundintheLadyBluntStradivariusviolin,constructedin1721byAntonioStradivari.In2011itfetchedarecord-setting$15.9millionatauction.

Thesediagramsshowthevariouswaysinwhichthelogosofsomeoftheworld’sbiggestcorporationsreflectthegoldenratio.

Googleiscertainlynotthefirsttousethegoldenratiointheirbranding.MeasurethethreeovalsthatconstitutetheToyotalogo,andyou’llfindthatthewidthofthesmall,narrowovalinthecenterisdefinedbytwogoldencutsofthewidthofthelargestoval.Theinneredgeofthemiddleovalontopispositionedatthegoldencutofthelogo’soverallheight.EventhecrossbaroftheAandtheforkinYarepositionedatgoldencuts.

Otherexamplesaboundinlogosofsomeoftherichestandmosthighlyregardedcompaniesworldwide.ThepositionofthehorizontalbarintheNissanlogoisdefinedbythetwogoldenratiocutsofthelogo’sheight.TheconcentriccirclesthatmakeuptheyellowandgreenBP“flowermandala”logoareingoldenproportiontooneanother.TheNationalGeographiclogoissimplyagoldenrectangle.

Amongcartoonandvideoanimators,useofthegoldenratioincharacterandscenedesignmaybemorethanjustanoccasionaltipofthehat.OneformerDisneyanimatorsharedwithmethatalthoughthegoldenratioisneverdiscussedamongdesigners—mostareverysecretiveabouttheirprocess—hehasbeensystematicallyapplyingitinhisartisticcreations.TheDisneylogoitselfisa

stylizedversionofWaltDisney’ssignature,whichappearstoveneratethegoldenratio.ThedesignoftheDusesthegoldenratioatleastthreetimesintheproportionsandpositionsofitsscrolledarcandverticalstroke,butevenmoreobviously,the“dot”abovetheIstronglyresemblesthesymbolrepresentingphi.Also,theYisnotlikeanyYyou’veeverseenbefore,butitdoesresembleascriptlowercasephisymbol.

Oneoftheworld’spremierluxurycarbrands,AstonMartin,hasappliedthesameconcepttothedesignofitscars.DescribingtheRapideS,DB9,andV8Vantagemodels,AstonMartinmadesuretoemphasizethecentralityofthegoldenratioinmanyaspectsoftheirdesignintheiradvertisingcampaign,toutingthecars’balance,perfection,elegance,harmony,purity,andsimplicity.21

ColorgridlinesdefinethevariousgoldenproportionsinthedesignofAstonMartin’sDB9coupe.

Giventheeleganceofitsdesign,it’snottoosurprisingthatgoldenproportionsareincorporatedintothedesignoftheStarTreksUSSEnterprise.Inthe1960s,seriescreatorGeneRoddenberryturnedtoMattJefferies,anaviationandmechanicalartist,withhisrequestto“designaspaceshipunlikeanyother,withnofins,rocketexhausttrails,powerfulandcapableofexceedingthespeedoflightwithacrewofseveralhundredonafive-yearmissiontoexploreunknowngalaxiesinouterspace.”22Jefferiesstartedwithablankpage,amarker,andaverypragmaticdesignethic,producingashipwithverydistinctphi-basedproportions.

Hisdesigndocumentsrevealedthathespecifiedthedimensionsonhisdesignstothe1/10,000thofaninch.Thiswasclearlybeyondtheaccuracyrequiredfortheconstructionofthesmall-scalemodelsusedontheStarTrekTVsetandindicatesthathewasworkingwithamathematicalprecisionbasedongeometricformulasandproportions.Manyothergoldenratioscanbefoundin

Jefferies’designoftheEnterprise,inthefrontandsideviewsaswellasinthefinedetail.Jefferiesclearlyunderstoodtheconceptofapplyingthegoldenratiotojustabouteverydesigndecisiononproportionandposition.

GoldendimensionsalsoaboundinMattJeffries’sdesignofStarTrek’sUSSEnterprise.

Youmaybeamazedtofindthatthegoldenratiohasbeenrightthereinfrontofyouallthetime,gentlynudgingyoutobuyaproductoruseaservice.AccordingtoDarrinCrescenzi,formerDesignDirectorofInnovationatInterbrandNewYork,andoneofFastCompanyMagazine’s“MostCreativePeopleinBusiness,”

“Thevisually-inclined—artists,architects,anddesigners,historicallykeenobserversanddocumentariansofbothnatureandthehumanconditionandwhowecanthankformuchof

whatweknowabouttheworld—haveforagesincorporatedthisratiointotheirworkduetoitsintrinsicallyalluringbalance

betweensymmetryandasymmetry.”23

PHIANDFASHION

Thevisualallureofthegoldenratiohas,ofcourse,notgoneunnoticedintheworldoffashiondesignandstyling.In2003,fashiondesignerSusanDell,wifeofcomputerbillionaireMichaelDell,embracedthegoldenratioconceptwithherintroductionof“ThePhiCollection,”ahigh-fashionlineofclothinginwhichsheincorporatedthespecialnumberintothemeasurementsandfeaturesofmanyofherdesigns.In2007,styleconsultantsandidenticaltwinsistersRuthandSaraLevycreatedTheFashionCode®,whichappliesthegoldenratiotoeachwoman’suniquebodymeasurementstoproviderecommendationsontheproportionsinclothingthat

willresultinthebestlook.Intheimagesatright,agoldenrectangleframingthewoman’sbodyfromheadtotoeidentifiesthemostappealinghemlinelocation.Asecondgoldencutwithinthelargersegmentoftheoriginalthenpinpointsthelocationofthenecklineorthenaturalwaist,whichshouldbeemphasizedwithabeltorfittedgarmentforthemostpleasingsilhouette.Theoutfitonthelowerleftisanexampleofwhathappenswhenphiisnotused.Herjacketis

tooshortandhertanktopistoolong,makingheroutiftlooklessappealing.

AssuggestedbyCrescenzi,thegoldenratiooffersmuchmorethanjustamorenaturalalternativetotheruleofthirds.Infact,itisamathematicallyuniquesystemofratioswithinratiosthatcankeepanentirecompositioninvisualharmony,andeventhoughit’sjustoneofmanytoolsthatgooddesignersusetoachievegreatcomposition,nodesignershouldbewithoutknowledgeofitsconceptsandapplication.“Thegoldenratioisintendedtobeinvisible,a

compositionalorganizingprinciplethatisfeltratherthanunderstood,”saysCrescenzi.Furthermore,headded:

“Itistheuniquevisualtensionbetweencomfortingsymmetryandcompellingasymmetry,anditsthoughtfulapplicationcan

bringbeautyandharmonyandintriguetoallmannerofdesignedthings.”24

Thepossiblevariationsinapplicationsofthegoldenratiotoanydesignarelimitedonlybyourcreativity,whichonlymeansthereisnolimitatall.

You’venowdevelopedasolidfoundationinthegeometryandmathematicsofthegoldenratio,aswellasitsappearanceinmonumentalworksofartandarchitectureduringmorethantwothousandyearsofcivilization.Assuch,you’rewellonyourwaytoearningyourveryunofficial—butveryvaluable—Doctoratein“Phi”losophy(it’sworthitsweightingoldenratios!).Thefinallegofourjourneyinvolvesthefascinatingstudyofgoldenformsinnatureandtheuniversebeyond.

V

GOLDENLIFE

“Alllifeisbiology.Allbiologyisphysiology.Allphysiologyischemistry.Allchemistryisphysics.Allphysicsismath.”1

—Dr.StephenMarquardt

I n1854,GermanpsychologistAdolfZiesing(1810–1876)publishedNeueLehrevondenProportionendesmenschlichenKörpers(NewDoctrineof

theProportionsoftheHumanBody),inwhichheexpressedhisbeliefthat

thegoldenratioappearedinitsfullestrealizationinthehumanform.

Furthermore,hearguedthatitamountedtoauniversallawthat

representedthe“ideal”inallstructuresandformsoflifeandmatter,2

echoingPlato’sancient“TheoryofForms.”AccordingtoZiesing,the

goldenratiowasanexpressionofbeautyandcompletenessinbothnature

andart,andhisideainspiredthelikesofLeCorbusierandotherswho

wentontocreateparadigm-shiftingdesignsanddiscoveriesaboutour

worldanditsvariedinhabitants.Timeandagain,phihasemergedinthese

investigations,thoughnotallfindingsareasstraightforwardastheymay

seem,orascompleteintheirexplanationoftheevidenceasZeisingand

otherspostulated.

Natureisfulloflogarithmicspirals,butfindingactualgoldenspiralsinnatureisrare.

PHIANDPHYLLOTAXIS

EventhemostardentofphiskepticswillagreethatthegoldenandFibonaccispiralscanbefoundinarangeofplants,pinecones,pineapples,sunflowerseedpods,andmanyothers.Italsoappearsinthepositionofpetalsaroundthecenterofaflowerandinthepositionofleavesandstemsaroundabranch.

ThisspiralpatternwasnoticedasearlyasthefirstcenturyA.D.bytheRomannaturalphilosopherPlinytheElder,butthefirstseriousstudyoftherelationshipbetweenplantspiralsandFibonaccinumberswasmadebytheSwissbotanistandnaturalistCharlesBonnet.In1754,Bonnetrecordedhisobservationofthespiralrotationsofleavesandstems,suchasthearrangementofscalesfoundonpinecones,inhisbookRecherchessurl’usagedesfeuillesdanslesplantes(ResearchontheUseofLeavesinPlants).Bonnetalsocoinedthetermphyllotaxis,fromtheGreekwordsphyllonforleafandtaxisforarrangement,todescribeit.3

SwissnaturalistCharlesBonnetisshowninthisengravingbyJamesCaldwallfromthe1802editionofEnglishphysicianRobertJohnThornton’sANewIllustrationoftheSexualSystemofCarolusvonLinnaeus.

PhyllotaxisisshowninthearrangementofalmondblossomsaroundthestalkinthisbotanicalillustrationfromBirdsandNature(1900).

AsimpleillustrationofthisprincipleofplantspiralsbasedontwosuccessiveFibonaccinumbersappearsinthepinecone.Intheillustrationbelow,eightcounterclockwisespiralsandthirteenclockwisespiralsareclearlydiscernable.

AvisualrepresentationofVogel’sformulaofthepolarcoordinatesofsunflowerfloretsforn=1ton=500.

Thesameprincipleappliestothesunflower—orratherthetinyfive-petaledfloretsatitscenter.Herewefindthattheirarrangementconsistsoffifty-fiveclockwisespiralsandthirty-fourcounterclockwisespirals.Bothfifty-fiveandthirty-fourareFibonaccinumbers—asisthenumberofpetals(five)perfloret!

In1979,GermanmathematicianHelmutVogeldevisedanequationtorepresentthisFibonaccispiralpatternofflorets,whereθisthepolarangleandnistheindexnumberofthefloretinquestion:

Θ=n×137.5º

Fifty-fiveclockwisespiralsandthirty-fourcounterclockwisespiralscanbedifferentiatedintheaboveimageofasunflowerhead.Noticehowthefivepetalsofeachfloretarevisibleattheseedpod’sperimeter.

Inthismodel,137.5ºistheangleofrotation,alsoknownasthegoldenangle.Why137.5?Asitturnsout,whenyoudividethedegreesofacircle(360)bythegoldenratio(1.618),thevalueyouobtainforthisarcis222.5º.Thatmakesthesmallersegmentofthecircle137.5º.

Thegoldenanglecanbeobservedinthearrangementofpetalsaroundaflowerbudaswell.Leavesandstemsalsoarrangethemselvesatthisangle,mostlikelyasameansofoptimizingtheamountoflighttheycanreceiveandasawaytoenablegrowthinthemostefficientmeanspossible.

Thegoldenangle.

ThegoldenangleisreflectedinthearrangementoftheleavesoftheEcheveriasucculentplant(top),aswellasthearrangementofpetalsonthelotusflower(bottom).

THEBEAUTYOFFIVE

Asweobservedinchapters1and2,fiveisaveryspecialnumberinthegeometryandcalculationofthegoldenratio.Notonlyistherearelationshipbetweenphiandthefive-sidedpentagonandpentagram,butthisnumberisalsoconvenientlythefifthdigitofFibonacci’ssequence!LongafterthePythagoreansadoptedthepentagramasasymboloftheirschoolandPlatodiscoveredhisfivePlatonicsolids,LeonardodaVincistudiedfive-petaledviolets,notingtheirunderlyingpentagonalstructure.Indeed,manyofthemostcommonandmostbeautifulplantsandflowers,includingthoseoftherosefamily,exhibitthisperfectgoldensymmetry.

FivecounterclockwisespiralsareclearlydifferentiatedintheAfricanspiralaloeplant.

Asketchofthefive-sidedpentagonappearsatthetopleftcornerofLeonardodaVinci’sc.1490studyofthefive-petaledviolet.

Plumeria

Sacreddatura

Bluepassionflower

Periwinkle

Morningglory.Allhaveeitherfivepetalsor,inthecaseofthepassionflower,fivestamens.

Thisfive-foldsymmetrycanalsobeobservedinthestructureoffruit,includingtheapple,papaya,andaptly-namedstarfruit.It’salsofoundintheedibleseedpodsoftheokraandcacaoplants,amongotherculinaryplants.

Five-nessisfoundintheanimalkingdom,too.Themostobviousexampleisthestarfishanditscousins,thebrittlestarandtheseaurchin.

Cross-sectionsoftheapple.

papaya.

starfruitrevealstar-shapedcores.

Thepentagonalshapeoftheokraseedpodisveryobviousinthisphotograph.

ThisbotanicalillustrationoftheTheobromacocoaplantrevealsthefive-foldarrangementofcocoabeansinsideit

ThesedetailedillustrationsfromGermannaturalistErnstHaeckel’sKunstformenderNatur(ArtFormsinNature)showsthefive-foldsymmetryofthetropicalgiantbasketstarfish(bottom)andvariousspeciesofseaurchin(top).

FRACTALS

Thegoldenratioalsoplaysanimportantpartinthegeometryoffractals,andfractalsplayanimportantpartinthegeometryofnature.Afractalisaninfinitelyself-similargeometricfigureorcurve,eachpartofwhichhasthe

samestructureandpropertiesasthewhole.Fractalsarecreatedbyrepeatingasimpleprocessagainandagain,withascalingfactorappliedtoeachiteration,asisdonewiththenested

samestructureandpropertiesasthewhole.Fractalsarecreatedbyrepeatingasimpleprocessagainandagain,withascalingfactorappliedtoeachiteration,asisdonewiththenestedgoldenrectanglesthatformthebasisofthegoldenspiral.AnotherabstractexampleistheluteofPythagoras,whichiscreatedfromasequenceofpentagramsthatincreaseinsizebyafactorofphi.Someofthemostwell-knownfractalsincludetheKochsnowflakeandtheSierpinski

triangle,whichhavescalingfactorsof4and2,respectively.TheFibonacciwordandgoldendragonfractals,ontheotherhand,expressphiintheirscaling.Recently,AmericanmathematicianEdmundHarrissmadeheadlineswhenhedevelopedtheHarrissSpiral,afractalbasedonthegoldenspiral.4Oneinterestingphenomenonthatoccursinspace-fillingfractalswithascalingfactorequalto

theinversegoldenratio(1/Φorф)isthatthepatternfillstheavailablespacewithoutoverlap,leavingnogaps.Forscalingfactorslessthanф,theresultingpatternappearssparsewithmuchopenspace.Incontrast,atscalingfactorsgreaterthanф,thepatternappearsovergrownwithlittleopenspace.Thefractalswediscussherearetheoretical(i.e.,notfoundinthematerialworld).However,

growthpatternsinnatureoftenapproximatethestructureofthesefractalsintheirself-similarity.AprimeexampleisRomanescobroccoli,althoughfractalgrowthpatternsarealsoobservedinthevascularsystemofplants.

TheluteofPythagorasisrepresentedinthiscolorfulquiltpattern.

Thegoldendragonfractal.

Aboveareexamplesoffractaltreeswithscalingfactorsof0.5,0.618(1/ф),and0.7.Notehowthetreewiththegoldenratioscalingfactoristheonlyonetogrowsuchthatallthesectionstouch,withnoemptyspaceandnooverlap.

THEMARVELOUSSPIRAL

FrenchmathematicianandphilosopherRenéDescartes(1596–1650)wasthefirsttodescribewhatisnowcalledthelogarithmicspiral.However,itwasSwissmathematicianJacobBernoulli(1654–1705)whobecamesufficientlyentrancedbyitsuniquemathematicalpropertiestorefertoitasthespiramirabilis,Latinfor“marvelousspiral.”Asthisspiralincreasesinsize,itsshaperemainsthesamebecauseitexpandsataconstantrateinageometricprogression.Alsoknownasanequiangularorexponentialspiral,thesebeautifulspiralsarefoundthroughoutnature,bothinlivingcreaturesandinhurricanes,galaxies,andothernaturalphenomena.

AnArchimedeanspiralwithaconstantdistancebetweenturningsappears(erroneously)onthebottomofJacobBernoulli’sepitaphintheBaselMinster.TheSwissmathematicianhadintendedtodepictanetchingofhisspiramirabilis,inwhichthedistancebetweenturningsincreasesataconstantrate,instead.

Logarithmicspiralscanbeusedtodescribeacontinuouslyrisingtone(above)orthegrowthpatternofflowers(below).

Thebeautyandcommonappearanceoflogarithmicspiralsis,unfortunately,asourceofmuchconfusion.Manypeopleincorrectlyassumethatalllogarithmicspiralsaregoldenspiralsexpandingcontinuouslybyafactorof1.618.Infact,thegoldenspiralisanunusualexampleofalogarithmicspiral—muchlikeanapplebeingaspecialmemberofthefruitfamily,orapentagonbeingaspecialmemberofthepolygonfamily.Alltruegoldenspiralsarelogarithmicspirals,butnotalllogarithmicspiralsaregoldenspirals,justasallapplesarefruits,butnotallfruitsareapples.

Thenautilusshellgetspulledintothismeleeofconfusionbecauseithasoneofthemostbeautiful,graceful,andrecognizablespiralsinnature.Asaresult,

ofthemostbeautiful,graceful,andrecognizablespiralsinnature.Asaresult,boththenautilusspiralandthegoldenspiralcreatedfromsuccessivegoldenrectangleshavebecometheposterchildrenforthegoldenratio.Inreality,theproportionsofthenautilusspiralaredistinctfromthoseofthegoldenspiral,asshownintheimagebelow.

Thecommonpracticeoflabelingthesetwoverydifferentlogarithmicspiralsasgoldenspiralshasledtoasurprisingamountofsuspicionandireamongmanyscientists,mathematicians,andothersabouttheprevalenceofthegoldenratioinnatureandthearts.ArticlesbyphiskepticsproliferateontheInternet,claimingthatthenautilusconnectionandjustabouteverythingelseyoumayhaveheardaboutthegoldenratio,isjustamyththatwon’tgoaway.Evenprofessionalmathematicianshavejoinedthefray.Accordingtoonemathematician,thenautilusshellspiral’srateofgrowthisactuallycloserto4/3.Anotherrecognizedscientist,knownforhisbrilliantsculpturesofthree-dimensionalgeometricmodels,useda3Dprintertocreateaseashellbasedontheclassicgoldenspiral,proclaimingittobetheonlytruegoldennautilusintheworldandlamentingthatthepoornautilusisalwaysbeingabusedbythegoldenratio“cultleaders.”Thesemenwerecertainlynotwrongaboutthehumblenautilus,butthere’saplottwist.

IneverhadanyaspirationstobeacultleaderwhenIcreatedGoldenNumber.net,andasIlearnedoftheseobjectionsIdecideditwastimetoinvestigateformyself.ItookmytrustygoldenmeangaugetothenautilusshellI’dhadonmybookshelfforyearsandfoundthatitsspiralsalignedreasonablycloselytothegauge,astheyalwayshad.ThenIrealizedthatthereismorethanonewaytocreateaspiralbasedonthegoldenratio.

Intheclassicgoldenspiral,thewidthofeachsectionexpandsby1.618witheveryquarter(90-degree)turn,anditsproportionsbearlittleresemblancetothoseofthenautilusspiral.However,anotherspiralexiststhatisjustasgolden.Thisspiralexpandsbyafactorof1.618withevery180-degreerotation.Notehowitexpandsmuchmoregradually.Clearly,agoldenspiralbasedona180-degreerotationismuchmoresimilartothenautilusspiralthanagoldenspiralbasedona90-degreerotation.

Thespiralontheleftincreasestoawidthof1atpointA.Aone-halfrotation(180-degree)topointBexpandsthewidthofthespiralto1.618orΦ.Anotherone-halfturntopointCincreasesthewidthofthespiralfromthecenterpointto2.618—Φ2.Theredlinesshowtheexpansionofthespiralthroughanotherfullrotation.ThisexpandsthewidthfromBtotheedgeofthespiralbyΦ2again,fromΦtoΦ3!Andsothepatternofexpansionbythegoldenratiocontinues.

ThealignmentofthenautilusspiralinmyofficewiththegoldenmeangaugewasfairlyclosewhenIextendeditfromtheoutsideedgetothecenterofthe

http://GoldenNumber.net

spiral(shownbelow),butIfoundacloseralignmentwhenIextendedthegaugefromtheoutsideedgetotheedgeofthespiralontheoppositeside,asillustratedbelow.

Then,measuringtheexpansionofmynautilusshellatevery30-degreerotation,Ifoundanexpansionratethatrangedfrom1.545to1.627,withanaverageof1.587,avarianceof1.9%fromthegoldenratio.Imeasuredothernautilusshellsandfoundvaluesthatwereslightlylargerthanthegoldenratioaswell.

Noteverynautilusspiraliscreatedequal,norareanyofthemcreatedwithcompleteperfection.Justaswiththehumanform,nautilusshellshavevariationsandimperfectionsintheirshapesandintheconformityoftheirdimensionstoanideal180-degreegoldenspiral.So,whilemanyinaccurateclaimshavebeenmaderegardingboththeexistenceandnonexistenceofgoldenspiralsinnature,weseethatthenautilusspiraldoesexpandataratequiteclosetophi—itjustdependsonhowyoumeasureit.

Hopefullythisrestoresthehonorandreputationofthenautilus,butwestillmustbecarefultodistinguishbetweengoldenspiralsandthegeneralclassoflogarithmicspiralsthatappearthroughoutnature.Anoccasionalhurricaneorgalaxythatfitspartofagoldenspiraloverlayshouldnotimpelustoconcludethatallhurricanesandgalaxiesarebasedonphi.

Alogarithmicspiralthatincreasesbyafactorof1.618every180-degreerotationalignsmuchmorecloselywiththespiralofanautilusshell.

ANASAsatelliteimageofTyphoonSoncainthePacificOcean,2011.Whileatfirstglancethestormcloudsmayseemtoformagoldenspiral,phi-basedspiralsarerareinnature.

Thespiralsofanestfernfrond.

Ayoungfiddleheadfernfrond.

Aseahorsetail.

Achameleontail.

AChinesespiranthesorchid.

TheWhirlpoolGalaxy.Alloftheseexamplesarenaturallyoccurringlogarithmicspiralswithvariousgrowthfactors.

THEANIMALKINGDOM

UsingPhiMatrix,itisrelativelyeasytofindgoldenratioproportionsinthespiraldimensionsofotherseashells.Itisalsoeasytofindshellswhoseproportionsarenotbasedonthegoldenratio,suchastheoneshownbelow,whichexpandsbyabout1.139witheverycompleterotation.So,althoughweencounterthegoldenratioratherfrequentlywhenexaminingshellspirals,itisdefinitelynotauniversalcharacteristic.

Whilethespiralsofseveralspeciesofseashell,includingthestripedfoxhorseconch(above),expandatarateclosetophi,thisscrewshell(above)hasanexpansionrateofabout1.139.

Thesameistrueforinsects.Thosewithmarkingsorbodyproportionsthatembodythegoldenratioarerelativelycommon,asshownbelow.However,insects—comprisingasmuchas90percentofallmulticellularanimallifeformsonEarth5—comeinsuchanamazingvarietyofbasicshapesandstructuresthatitwouldbeimpossibletoconcludethatthegoldenratioisauniversal,orevendominant,principleoftheirdesign.

Scarabbeetle

Giantsilkmoth

Germanyellowjacket.

Aswemoveupthekingdomsoflife,therearefewerspeciesandmoreconsistentstructuresthatdefineacommonappearance.WithintheFelidae(i.e.,cat)familyintheorderofCarnivora,wefindthegoldenratiointheproportionandpositionoftheeyes,nose,andmouth.Specifically,theinnercornersoftheeyesaligncloselywiththegoldencutofthedistancebetweenthecenterofthenoseandtheoutsideoftheeyes.Furthermore,thetopofacat’snosealignscloselywiththegoldencutofthedistancebetweenacat’spupilsandthemouth.

Boththedomestickitten.

Africanlionhavephi-basedfacialdimensions.

WithintheHominidae(i.e.,greatape,includinghumans)familyinthePrimateorder,weoftenobserveasimilarrelationshipbetweenthepositionsoftheeyes,nose,andmouth.Inparticular,thebottomofthenoseiscloselyalignedwiththegoldencutofthedistancebetweenthepupilsandmouth.Thereisalsoacleargoldenrelationshipbetweenthepositionandproportionoftheeyesinrelationtothewidthoftheface.Notsurprisingly,thesesameproportionsarefoundinhumanfaces.

Goldenproportionsarealsofoundinthefacesofseveralmonkeyspecies,includingthejuvenilemacaque(above)andthechimpanzee(below)

GOLDENHUMANPROPORTIONS

Occam’srazorisaphilosophicalprinciplepopularizedbythefourteenth-centuryfriarWilliamofOckham(c.1285–1347),statingthatamongcompetinghypotheses,theonewiththefewestassumptionsisthemostlikelyexplanation.Morethansevenhundredyearslater,itisstillaguidingprincipleforscientists,andweshouldconsideritwhenexaminingthescientificexplanationsfortheproportionsofthehumanfaceandbody.InLeonardo’sVitruvianMan,wefindevidenceofasystemofhumanproportionsbasedonhalves,thirds,quarters,sixths,sevenths,eighths,andtenths.However,thesesamehumanproportionscanbemoreeasilyexpressedwithaseriesofgoldenratios.Whichsystemmakesmoresense?IfyoucouldaskWilliamofOckham,hemighthavesuggestedthesimpler,parsimoniousgoldenratiotheory.Whenweconsidertheconstantratesoffractalexpansionintheproportionsofotherlivingorganisms,thisexplanationseemsevenmorelikely.

Holdyourhandoutinfrontofyouandlookattheproportionsofyourindexfinger.X-rayimagesshowthateachboneofyourindexfinger,fromitstiptoitsbaseatthewrist,islargerthantheprecedingonebasedontheFibonaccinumbers2,3,5,and8.WealreadyknowthattheratiosofsuccessiveFibonaccinumbersapproachthegoldenratio,soit’snotahugestretchtoconsiderthattheratioofthelengthoftheforearmtothelengthofthehandisapproximately1.618.

WhenarulerisplacednexttoanX-rayoftheindexfinger,therelationshipbetweentheFibonaccisequenceandthelengthsofeachbonebecomesapparent.

THEHUMANFACESo,whataboutthehumanface?Dogoldenratiosexistthere,too?Thebasicstructureofallofourfacesisfundamentallythesame.That’swhatmakesuslookhuman,andnotlikealionorchimpanzee.There’sawidevarietywithinthatbasichumanstructurethough,sohowdowepickafacethatwouldberepresentativeofallhumankind?OneapproachtoansweringtothatquestionwasfoundintheresearchdonebyresearchersLisaDeBruineandBenJonesatFaceResearch.org.5UsingthePsychoMorphsoftwaredevelopedbyDr.BernardTiddeman,theycombinedfull-colorfaceimagesoffiftywhitemenandfiftywhitewomenbetweentheagesofeighteenandthirty-fivetodevelopan“averaged”face.Theresearchersalsousedfourimagesfrommaleandfemaleindividualsofwhite,westAsian,eastAsian,andAfricandescenttocreate“averaged”facesforthoseethnicgroups,withstrikinglysimilarresults.Eventhoughonlysixteenindividualfaceswereused,combiningthesefourethniccompositesintoa“universal”faceyieldedacompositefacenearlyidenticalintheirbasicproportionstotheaveragedmaleandfemalefacesbasedonfiftyindividuals.

http://faceresearch.org

Thesearevisualrepresentationsofthemathematicallyaveragedproportionsoffiftymaleandfiftyfemalefaces,basedon189facialmarkers,providingaverygoodstatisticallyvalidbenchmarkforassessingtheappearanceofthegoldenratioinvariousfeatures.

AfterapplyingasimplePhiMatrixgoldenratiogridtothemaleandfemalecompositefaces,wediscoverthat,aswithotherhominids,theinnercornersofeacheyeareoftenlocatedatthegoldencutofthedistancefromonesideofthefacetotheother,andtheoutsidecornersoftheeyesarelocatedatthegoldencutofthedistancefromtheinnercornerseacheyetothesideoftheface.Measuringtheverticaldistancefromthepupilstothechinrevealsanothercommonlyobservedgoldenproportionatthecenterliplineofthemouth.Examplesofhowthissamebasicgoldenratiostructureapplytodifferentethnicgroupsareshownhereandhere.

Thegoldenratio-basedPhiMatrixgridsillustratekeygoldenratioproportionsthatarecommonlyfoundinhumanfaces.

Examiningthedistancesbetweenvariousfeaturesmorecarefully,wediscoverthatthereareatleastadozenproportionsinthis“averaged”humanfacethatreflectthegoldenratio,includingtheproportionsandpositionsofoureyes,eyebrows,mouth,lips,andnose.Theheight-to-widthproportionoftheheadisagoldenrectangle,asisthefacialfeatureareaboundedbythehairline,chin,andeyebrows.It’sremarkablethedegreetowhichtheaveragehumanfaceembodiesthesame“secretscience”ofharmonizinggoldenratiosthathasbeenappliedintheartsthroughouthistory

Somequestionwhythegoldenratiowouldappearatallinthehumanface.Anequallyappropriatequestionistoaskiswhyitwouldnot.WefindthisratioandtherelatedFibonacciseriesinavarietyoflifeforms.Manywhosaythatthegoldenratiodoesnotappearinthehumanfacesimplyneglecttousethefacialmarkersdefiningtheproportionswhereitcommonlydoesappear.Somemakingthisclaimhavenevereventakenanymeasurementsatall.Mymeasurements,aswellasthosemadebyrecognizedexpertslikeDr.StephenMarquardtandDr.EddyLevin,corroboratenotonlytheappearanceofthegoldenratioinhumanfacialproportions,butalsotheirimpactonourperceptionsofbeautyand

facialproportions,butalsotheirimpactonourperceptionsofbeautyandattractiveness.

Eventhoughthiscompositeoffacesfromfourethnicgroupsisbasedonlyonsixteenindividualfaces,theproportionsofthisfemale’sfacecloselyapproximatethoseofthewhitefemalecompositefaceshownhere.

CollinSpears,anindependentresearcher,usedtheFaceResearch.orgsoftwaretodevelopcompositeimagesofmenofwomenfromoverfortycountries.Theresultswerefascinating.Eventhoughthereareslightdifferencesinfacialshapes,theaveragedfacesallfitthegoldenratiofacialgridpatternquitewell,illustratingthecommonappearanceofthegoldenratioinfacesfromaroundtheworld.

GOLDENDNA?

Ifthegoldenratioseemstoaffecttheproportionsofourbodiesandfaces,whataboutthemostfundamentalbuildingblockofhumanlife,DNA?Theabbreviationstandsfordeoxyribonucleicacid,andthissubmicroscopicdoublehelixcontainsallthenecessaryinstructionsfortheformationanddevelopmentofeverylifeform,includingviruses.JusthowtinyisDNA?Everycellinthehumanbodycontainsninety-twostrandsofDNA

(therearetwenty-threepairsofchromosomesforatotalofforty-six,eachofwhichismadeupoftwoDNAstrands).Accordingtothelatestestimates,humanscontainapproximatelythirtytofortytrillioncells!6Bynecessity,eachofthesecellsistiny,ranginginsizefromafewmicrometers(i.e.,millionthsofameter)toroughly100micrometers,andthewidthoftheDNAstrandscontainedineachcellnucleusisfarsmaller,measuredinnanometers(i.e.,billionthsofameter).Estimatesplacethelengthofasingle360-degreerotationofDNAat3.2nanometers,andthestrand’swidthisestimatedat2.0nanometers.7Thosemeasurementscreatearatioof1.6,whichissurprisinglyclosetothegoldenratio.

http://faceresearch.org

AdigitalillustrationofstrandsofDNAcoiledwithinachromosome.

Thephi-baseddoublehelixstructureofDNAismagnifiedinthisdigitalrendering.

Infact,geneticistshavediscovereddifferentkindsofDNA,butitisB-DNAthatisbelievedtobethemostprevalentinnature.Asithappens,inthisDNAstructure,minorgroovesalternatewithmajorgrooves,andthesealsoappeartohaveaphi-basedrelationship.Furthermore,thedoublehelixofB-DNAhasabouttenbasepairsofDNAper360-degree

rotation.Thiscreatesacross-sectionalconfigurationwithtensides,likethatofadecagon.Canyouspotthepentagonal-likestructuresinthecenterofthatcross-section?Eachdiploidcellofthehumanbody—thatis,mostofourcells,withthenotableexceptionof

ourhaploidreproductivecells—containsatleastsixbillionbasepairsthatprovidetheuniquegeneticprogramforyou,andyoualone!Evenmoreincredibly,itallcoilsintoaspaceofabout6micrometers—1/16thewidthofahumanhair—butifstretched,asingleDNAstrandwouldextendtomorethan6feet(1.8m)long!8

Notethefive-foldsymmetryinthecross-sectionofDNA-B’smolecularstructure,showncenter.

THENATURALBEAUTYOFPHI

Reverenceforthebeautifulhumanformhasinspiredcountlesstalesandworksofartfromancienttimesuntiltoday.Afterherkidnapping,HelenofTroy’sbeautifulfacewassaidtolaunchathousandshipswhentheAcheanssetouttoreclaimherandreturnhertoSparta,sparkingthelegendaryTrojanWar.Beforeandsince,humannotionsofbeautyhavedirectedmankind’shistorywhileinspiringsomeofourgreatestworksofart,literature,andmusic.

THEMARQUARDTBEAUTYMASKStephenR.Marquardt’sfascinationwiththehumanfacewassparkedbyatraumaticchildhoodevent.Whenhewasfouryearsold,heandhisparentswereinanautoaccidentthatbrokeeveryboneinhismother’sface.Fortunately,averyskilledsurgeonperformedaverysuccessfulfacialrestoration,butevenso,herappearancewasnotablyaltered.Theexperiencelefthimwithagnawingdesiretounderstandhowsubtledifferencesaffectthewayweperceiveandrecognizefaces,aswellashowwedecidewhichonesaremostbeautiful.

Dr.Marquardtearnedamedicaldegreewithaspecialtyinoralandmaxillofacialsurgery.Ashesearchedforanswers,hewentontoinvent—or,inhiswords,“discover”—theMarquardtBeautyMask,whichreflectsthegoldenratioinmanyofitsproportions.Thisisbecauseitwascreatedfromasetoften-sideddecagons,which,likethefive-sidedpentagon,hasarelationshipwithphi.Hisfacialimagingresearchisacknowledgedbyprofessionalsworldwideandhasbeenpresentedextensivelyinthepublicmediaindozensofarticlesanddocumentariesonbeauty,includingthe2001BBCdocumentaryTheHumanFace.Hissetofeightmaskscovermaleandfemalefacesinthreedimensionswithfrontalandlateral(side)views,andinsmilingandnonsmilingexpressions.

Dr.Marquardtretiredfromactivesurgicalpracticeafternearlythreedecadestocontinuehisresearchonhumancross-culturalbeauty.Byapplyinghispatentedmasktofacialimagesfromdifferenteras,cultures,andethnicities,hehasrevealedanarchetypalfacialstructurethatdefineshumanbeauty,aninportantprincipleinourunderstandingsofhumanbeauty.Despitechangesinfashionoverthemillennia,basichumanperceptionsofbeautyhaveremainedunchanged.It’shard-wiredintoourDNA,andpartofwhoweare.

AnalyzingfacesrecognizedfortheirbeautyinpastageswithmyPhiMatrixsoftware,Ifoundthatkeyfacialmarkers,includingthepupils,theedgesoftheeyes,nose,lipline,chin,andwidthoftheface,wereallalignedwiththesame

phi-basedgrid.Onthefollowingpages,weseethatthegoldenratioisalsofoundverycommonlyinbeautifulmodelsoftodayacrossallethnicgroups,illustratinginyetanotherwaythatourdeepestperceptionsofbeautyareunchanged,andapplyuniversallytoall.

JuliaTitiFlavia(64–91CE)wasthedaughterandonlychildtoRomanEmperorTitus.ThismarblestatuereflectsclassicbeautyintheeraoftheRomanempire.

Nefertiti,anEgyptianqueenrenownedforherbeauty,ruledwithherhusband,PharaohAkhenaten,around1350BCE.Hernameliterallymeans“abeautifulwomanhascome,”andherbeautifullyproportionedfeaturesstillintriguepeopletoday.

Ethnicdifferencesexistintheaveragedimensionsandproportionsoffinerfacialfeaturessuchastheeyes,eyebrows,lips,andnose,butthefundamentalfacialstructurebasedonthegoldenratiosdefinesanarchetypeforbeautyacrossthesemoresubtledifferences.

Thisattractivewoman’sfacehasastrikingnumberofphi-basedproportions.

Throughouthistory,caricaturistshaveplayedwithfacialproportionsinordertocomicallyorgrotesquelyexaggeratethepeculiaritiesordefectsofaperson’sface.Insomecasestheydepictedindividualstheydespisedasexceedinglyunattractive.Inessence,theyoftentranslatedaperson’sperceivednegativeinnerqualitiestotheirouterappearancebyshrinkingthespacebetweentheeyesandthenose,forexample,orlengtheningthespacebetweenthenoseandthemouth,asshowninFlemishartistQuentinMatsys’ssatiricalpaintingTheUglyDuchess.Caricaturesillustratehowsensitivewearetowhatweperceiveasnormsinfacialproportions,andhowunnaturalafacecanlookwhenthose

proportionsarechangedevenslightly,whilestillleavingthesubjectinstantlyrecognizable.

TheUglyDuchess(1513)byFlemishpainterQuentinMatsis.

THEGOLDENMEANDENTALGAUGE

WhendentalaestheticspioneerDr.EddyLevinwasstartingoutinhisownpractice,hebecameintriguedwiththequestionofwhy,afterallhishardworktomakecrookedordamagedteethlooknatural,theteethstilloftenlookedfalse.Then,inaninstant,hehadanepiphany:thegoldenratiocouldhelphimmaketheappearanceofaperson’steethmorenaturalandbeautiful!Heputthiseurekamomentintopractice,firsttestinghisnewideaonayounggirlinahospitalwherehewasteaching.Herfrontteethwereinaterriblestateandneededcrowning,anddespitetheskepticismoftheotherstaffmembersandtechnicians,hecrownedofallherfrontteethusingtheprinciplesofthegoldenratio.Everybodyagreedthatitwasamagnificentsuccess.

frontteethusingtheprinciplesofthegoldenratio.Everybodyagreedthatitwasamagnificentsuccess.ThetechnicianonDr.Levin’steamwentontogivelecturesontheapplicationofthegolden

ratiotodentistry,andDr.Levinwentontoinventthegoldenmeandentalgaugeandgridsystem.Basedonaseriesofgoldenratiosthatshowthepreferredproportionsoftheteethwhenviewedfromthefront,hisdiagnosticgridsallowedotherdentiststoevaluatetheirpatients’teethandtoadjustthemaccordingly.Forexample,theratioofthewidthoftheuppercentralincisorstothewidthoftheupperlateralincisorsshouldequalphi,1.618.Dr.Levin’ssystemidentifiedseveralothergoldenfacialrelationships,includingtheratioof

thedistancebetweenthenoseandthebottomofthechintothedistancebetweentheteethandthebottomofthechin.9HissystemiscompulsorystudyinmanyUSuniversities,andhisresearchandpracticerevealjusthowusefulthegoldenratiocanbeincosmeticdentistry.

Anattractivesetofteethreflectsthegoldenratio.

Thesimilaritybetweenaveragedfacialproportionsandthoseofindividualsthatsocietyrecognizesasextraordinarilybeautifulleadstoanotherinsightaboutattractiveness:Afaceofaveragedproportionsissurprisinglyattractive,evenbeautiful!Thoseperceivedashavingextraordinarybeautygenerallyalsohaveexceptionalcharacteristicsbeyondthebasicproportionsinthefinerfeaturesoftheirfaces,suchasintheeyes,lips,eyebrows,nose,andmore.Thisiswhytheuseofmakeuptoenhancecertainfacialfeaturescanmakeaverynoticeabledifferenceinhowattractiveoneisperceivedtobe.Enhancementsaside,it’sremarkabletounderstandhowcompletelyourfacesembodyaninterrelatedsetofgoldenratios—thesameproportionsusedtogenerateexceptionallybeautifulworksofartandarchitecture.

So,thenexttimeyoulookatyourselfinthemirror,takeanextramomenttosmileandexamineallyourgoldenproportions.Andthenthinkforamoment

smileandexamineallyourgoldenproportions.Andthenthinkforamomentabouthowitconnectsyoutoeveryotherhumanontheplanetandtothebeautyoflifeintheplantsandanimalsthataboundinnatureallaroundyou.

T

VI

AGOLDENUNIVERSE?

“Wherethereismatter,thereisgeometry.”1

—JohannesKepler

T hefrequentappearancesofthegoldenratioinlivingorganismsis

intriguing,buttherearestillotherinstancesthatarefarmoreunexpected,

evenastounding.Asdiscussedinchapter1,theeminentmathematician

JohannesKeplerrepresentedthecosmosasaseriesofnestedPlatonic

solids,withthephi-baseddodecahedronandicosahedronoccupyingthe

spacesbetweenEarth’sorbitandthatofVenusandMars.Althoughan

elegantattemptatcapturingthe“harmonyofthespheres,”hismodeldid

notalignwithobservedplanetarymotions.However,intheend,hedid

successfullydiscoveranddescribethemotionoftheplanetsaroundthe

Sun,completelytransformingourunderstandingofthecosmosinthe

process.Healsoheldfasttohisreverenceforthegoldenratio.Couldthis

genius,whosparkedtheScientificRevolution,haverevealedmoresecrets

abouttheuniversehadhelivedtoaripeoldage?

Thisillustrationofthe(fictitious)relationshipbetweentheorbitsofthefirstsixplanetsofoursolarsystemandthefivePlatonicsolidsappearsinJohannesKepler’s1619bookHarmonicesMundi.

THEGOLDENCOSMOS

Nearly2,500yearsago,PlatopostulatedwithinTimeausthatthephysicaluniversewasmadeofearth,water,air,andfire,andthateachoftheseelementscouldbelinkedtoaparticularpolyhedron.Thefifthsolid,thedodecahedron,wasthoughttorepresenttheshapeoftheuniverse.Modernscienceshowsthattheseassociationsarefictitious,butPlato’sextensiveinquiryintothenatureofrealityrevealedotherimportanttruthsandquestionsthatwouldeventuallyleadtonewdiscoveries.Forexample,a2003analysisoftheWMAPcosmicbackgroundradiationdatabyJean-PierreLuminetandhisteamshowedthatthedodecahedronshapecouldexplainsomeoftheobserveddatabetterthanothermodels.2Thejuryisstilloutonthishypothesis,butthereareothercompellingfindingsaboutthestructureofouruniverse.OnethatamazesmethemostinvolvestherelativesizesoftheEarthandMoon.

Asweobservedinchapter4,theKeplertrianglerepresentstheproportionsoftheGreatPyramidwithavarianceoflessthan0.2percent.ThissametriangledefinesaratheramazingrelationshipbetweentheradiioftheEarthandMoon.ConsiderthefollowingmeasurementsprovidedbytheNationalAeronauticalSpaceAgency(NASA)3:

Earthradius(km):6,371.00Moonradius(km):1,737.40

Tovisualizetheirrelativesizes,imaginetheMoonsittingdirectlyontopoftheEarth,withalineconnectingtheEarth’scentertotheMoon’scenter.Now

theEarth,withalineconnectingtheEarth’scentertotheMoon’scenter.NowimaginealineextendedhorizontallytoEarth’seasternmostpointatitsperimeter,andthenconnectthatpointwiththecenterpointoftheMoontoformatriangle.

IfthistrianglereflectedthegoldenratioastheKeplertriangledoes,theheightofthetriangle(thedistancebetweenthecentersoftheEarthandMoon,equaltotheircombinedradius)tolengthofthebase(Earth’sradius)wouldequal√Ф,approximately1.27202.Butdoesit?

There’saneasywaytofindout,simplyaddtheradiioftheEarthandMoon,thendividethisnumberbytheEarth’sradius:

6,371.00+1,737.40=8,108.408,108.40/6,371.00=1.27270

Thevariancebetweenthisnumberand√Фisamere.0538percent.

PLANETARYORBITSEarthhasanotherunusualrelationshipwithVenus,itsnextclosestneighborintheSolarSystem.EarthandVenushaveanorbitalresonancethatbringsthemtothesamepositionsinspacefivetimesduringeightEarthorbitsandthirteenVenusorbits.Fibonaccistrikesagain!Now,imaginealinedrawnfromtheorbitalpositionsofVenustoEarthatregularintervalsoftime.Asshownbelow,theresultingpatternisabeautifulsetofnestedpentagonalflowers.

theresultingpatternisabeautifulsetofnestedpentagonalflowers.

AverysimilarnestedpentagonalpatternemergesfromageocentricviewpointlookingattherelativepositionsofVenusandtheSun.Additionally,theorbitalperiodofVenusis224.7days,about0.6152ofoneEarthyear(365.256days).4,5Thisnumbervariesonly0.5percentfrom1/Φ.

GOLDENSTARSBackattheotherendofthecosmologicalscale,a2015researcharticle6byJohnLindneroftheUniversityofHawaiiandteamreportedthediscoveryofaclass

ofwhite-bluevariablestarsthatpulsateinafractalpatternatfrequenciesclosetothegoldenratio.ThestarsareoftheRRLyraevariableclass,auniquestarclasswhichareatleast10billionyearsoldandwhosebrightnesscanvaryby200percentinaslittleastwelvehours.Onestarwasobservedatthirty-minuteintervalsoverafour-yearperiodwiththeKeplertelescope,andwasfoundtohavecharacteristicfrequenciesina4.05-houranda6.41-hourcycle,whichhavearatioof1.583,within2.2percentofthegoldenratio.Thesestarsarereferredtoas“golden”becausetheratiooftwooftheirfrequencycomponentsisnearthegoldenratio,andtheapparentirrationalqualityoftherelativefrequenciesisacluethatthepulsingisfractalintime.

Toconfirmthis,theLindnerteamperformedafractalanalysisoftheirplotsatdifferentmagnifications.Thiswasdonebyconvertingtheirplotstofrequencyspectra.Theythencountedthenumberofspikesintheconvertedplotswhoseheightssurpassedacertainthreshold,withapowerlawdependenceonthethresholdthatwasasignoffractalbehavior.Thepulsatingfrequenciesconformedtofractalpatterns,andwhentheoscillationswereseparatedintoparts,additionalweakerfrequencieswereidentified.Researchersdescribedtheweakerfrequenciesasfollowingapatternsimilartoshorelinesthatappearjaggedatanydistancefromwhichtheyareviewed.Theauthorsbelievethatthisfractalpulsationmaycarryinformationaboutcharacteristicsofthestar’ssurface,suchaschangesinopacity.

Itisstillnotclearwhetherornotthestar’sfractalpatternbehaviorhappensforareason.Ifitdoes,thenthereareothercluesregardingthephysicsofstarsawaitingdiscovery.

ThisgraphicshowsthelocationoftheRRLyraevariablestarsontheHertzsprung–Russelldiagramthatcomparesthecolorandbrightnessofdifferentclassesofstars.

BLACKHOLESIn1958,AmericanphysicistDavidFinklesteindescribedblackholesasregionsinspacewherethegravitationalpullissostrongthatnothing—notevenlight—canescapeit.Theyarebelievedtooccurwhenmassivestarscollapse,andafterswallowingotherstarsandmergingwithotherblackholes,theybecomesupermassive.Manyphysicistsbelievethesemonstrous,supersizedblackholesexistatthecentersofmostgalaxies,includingourownMilkyWay,andovertheyearstheyhaveattemptedtodescribetheuniqueandpowerfulphysicalpropertiesofblackholes,includingtheirmassandangularmomentum(i.e.,speedofrotation),usingmathematics.

speedofrotation),usingmathematics.Ina1989paperpublishedinClassicalandQuantumGravity,7English

astrophysicistPaulDaviessuggestedthataphi-basedrelationshipexistsatthetransitionpointofaspinningblackholefromonestatetoanother,suchaswhenitchangesfromastateheatingupasitlosesenergytoastateofcoolingdown.Specifically,heclaimedthatthetransitionoccurswhenthesquareofitsmassisequalto1/Фtimesthesquareofitsangularmomentum,althoughotherphysicistshavechallengedhisfinding.

Otherresearchersofblackholeshavecomeupwithnumerousequationsinvolvingphiasaconstant.AmongthemareNormanCruz,MarcoOlivares,andJ.R.VillanuevaoftheUniversityofSantiagoinChile.Intheir2017paper“TheGoldenRatioinSchwarzschild-KottlerBlackHoles,”8theypresentedevidencethatphiappearsinthemovementofparticleswithinablackhole—specificallytheratiobetweenthefarthestdistanceandnearestdistancebetweentwophotonsorbitingatmaximalradialacceleration.

A2011researchpaper9byJ.A.NietoattheAutonomousUniversityofSinaloainMexicorevealedasurprisinglinkbetweenblackholesandthegoldenratiowhenheattemptedtodescribetheirpropertiesinhigherdimensions.Specifically,whendescribingblackholesinfourdimensions,heuncoveredthisformula:

Nietoinstantlyrecognizedthefamousformula,andinadditiontoformallyestablishingaconnectionbetweenthegoldenratioandblackholes,hehelpedtoclarifythecharacteristicsofablackhole’seventhorizon,whichisthepointofnoreturnatwhichthegravitationalpullofamassiveobjectbecomessogreatastomakeescapeimpossible.

Anartistrenderingofasupermassiveblackholeatthecenterofagalaxy.

PHI-BASEDMATTER?

Journeyingfromtheexpansivescaleofouterspacetothemicroscopicworldofmolecularstructures,weencounterquasicrystals,buckyballs,andotherformsofmatterthatappeartoreflectthegoldenratiointheirarrangementofatomsandmolecules.

Thisdigitalillustrationdepictsacollectionoftinygraphene“buckyballs”(seehere).

QUASICRYSTALSIn1982,scientistDanShechtmancapturedanimagewithascanningelectronmicroscopethatseemedtocontradictbasicassumptionsinthefieldofcrystallography,abranchofchemistrythatstudiescrystallinesolids.Tenbrightdotsappearedineachcircle,revealingadiffractionpatternoften-foldsymmetry.Theprevailingwisdomatthetimeheldthatcrystalscouldonlypossesstwo-fold,three-fold,four-fold,andsix-foldrotationalsymmetry,butShechtman’sdiscoverychangedallthat.Infact,itwassounbelievablethathewasaskedtoleavehisresearchgroupinthecourseoftryingtodefendhisfindings.Thebattleragedon,andeventuallyotherscientistswereforcedtore-examinetheirunderstandingofthenatureofmatter.WiththehelpofPenrose’stilingmosaics,thescientificworldgraduallybegantoacceptShechtman’sfindings.

Shechtman(farleft)discussestheatomicstructureofthequasicrystalataNationalInstituteofStandardsandTechnology(NIST)meetingin1985.

Mostcrystalsinnature,includingsugar,salt,anddiamonds,areperfectlysymmetricalandperiodic,withstructuresarrangedinthesameorientationthroughouttheentirecrystal.Quasicrystals,however,areasymmetricalandaperiodic.Theirdiscoverypresentsanewstateofmatterthatwascompletelyunexpected,combiningthepropertiesofcrystalsandwithpropertiesofnoncrystallinematter,suchasglass.WhileShechtmanfirstobserved

quasicrystalsinanaluminum-manganesealloy(Al6Mn),hundredsofquasicrystalshavesincebeenobservedinothersubstances,manyofwhicharealuminum-basedalloys.Thefirstnaturallyoccurringquasicrystal,icosahedrite,wasdiscoveredin2009inRussia.10

Athree-dimensionalgoldenrhombusformsthestructuralbasisofsomequasicrystals.

ThisphotographcomparesthesizeofaHo-Mg-Znquasicrystaltothatofapenny.AccordingtotheUSDepartmentofEnergy,thisnewmaterialhashighpotentialforuseasalow-frictioncoatingforautomotivemechanicalparts.

ThePenrosetilingsolutiontofive-foldsymmetryintwodimensionsrequirestwoshapes:thedartandthekite.Inthreedimensions,thiscanbeaccomplishedwithjustoneshape:asix-sided,three-dimensionaldiamondwithgoldenproportions.

Otherquasicrystalstakedifferentforms.Intheimagebelow,aHo-Mg-Znquasicrystalhasformedintotherelatedpentagonaldodecahedron,withtrueregularpentagonsasitsfaces.

Almostthreedecadesaftertheirdiscovery,theNobelPrizeinChemistrywasfinallyawardedtoShechtmaninrecognitionofhisquasicrystaldiscovery.SciencehassinceturnedtothemedievalIslamicAlhambrapalaceinSpainandthefuneraryofshrineDarb-iImaminIran,whichdisplaymagnificentaperiodicphi-basedmosaics.WithSchechtman’sdiscoveryofquasiperiodicity,anentirenewclassofsolidsispossible,andsymmetryinanynumberofdimensionsbecomesattainable!

TheelectrondiffractionpatternoftheHo-Mg-Znquasicrystalrevealsitsfive-foldsymmetry.Noticetheproliferationofpentagrams,pentagons,andotherphi-basedshapesintheoverlayingdiagram.

ThesefivegirihtileshavebeenusedtocreateaperiodicgeometricpatternsinIslamicarchitectureforalmostathousandyears.Notetheinclusionofthefive-sidedpentagonandthegoldenrhombus.

ThisgirihpatternappearsonthewallsoftheTumanAkamausoleumwithintheShah-i-ZindanecropolisinSamarkand,Uzbekistan.

Quasiperiodicity5-foldsymmetry.


BUCKYBALLSAswesawinchapter3,LucaPacioli’sseminalpublicationonthe“divineproportion”featureddaVinci-drawnillustrationsofthree-dimensionalskeletonicsolids,includingthephi-baseddodecahedronandicosahedron.TheseskeletonicsolidsalsoincludedthethirteenArchimedeansolids,oneofwhichresemblesthemodernsoccerball(seehere).Thisthree-dimensionalsolidisformallycalledatruncatedicosahedron,anditconsistsoftwelvepentagonsandtwentyhexagons.

In1985,chemistsRobertCurl,HarryKroto,andRichardSmalleyannouncedtheirdiscoveryofacarbonmolecule(C60)withtheexactstructureofArchimedes’truncatedicosahedron,namingitafterAmericanarchitectandfuturistBuckminsterFuller,whopopularizedthegeodesicdome.Likethedodecahedronandicosahedron,thebuckminsterfullerene(aka“buckyball”)

reflectsthegoldenratioinitsdimensions.Forexample,whenyoumapthemolecule’ssixtypointscenteredattheoriginofathree-dimensionalCartesiancoordinatesystem,allsixtycoordinatesarebasedonmultiplesofΦ,asfollows11:

X(0,±1,±3Φ)Y(±1,±[2+Φ],±2Φ)Z(±2,±[1+2Φ],±Φ)

Thestructureofbuckminsterfullerenecarbonmoleculemirrorsthatofthephi-basedtruncatedicosahedronArchimedeansolid.

QUANTUMPHIInJanuaryof2010,Dr.RaduColdeaofOxfordUniversitypublishedapaperdiscussingtheappearanceofagoldenratiosymmetryinsolidstatematter.12Thepaperexplainedthatparticlesontheatomicscaledonotbehaveasthoseinthemacro-atomicworld,displayingnewpropertiesthatemergeasaresultofHeisenberg’sUncertaintyPrinciple.Byartificiallyintroducingmorequantumuncertaintyintheirexperimentswithcobaltniobate,achainofatomsactinglikeananoscaleguitarstringcreatedaseriesorscaleofresonantnotes,thefirsttwoofwhichhadafrequencyrelationshipof1.618.Coldeaisconvincedthatthis

wasnocoincidence,andthatitreflectedabeautifulpropertyofhiddensymmetryofthisquantumsystem,knownasE8.TheE8,anexceptionalsimpleLiegroup,hasabeautifulrelationshiptothegoldenratio,asillustratedbythegoldenratioconcentrichalfcirclesoverlayingtheupperhalfofthestructureinblue,red,gold,andwhite,revealingapatternmuchlikethebeautifulrosewindowofNotre-DameCathedral.

AnillustrationoftheE8Coxeterplaneprojectionofthe421semiregularpolytope,discoveredin1900byEnglishmathematicianThoroldGosset,whichdisplaysthirty-foldsymmetryandgoldenproportions.

TheCoxeterplaneprojectionofthe421polytoperemindsoneofthephi-baseddimensionsofthegorgeousnorthrosewindowofNotre-DameCathedral.

BETTINGONPHI

SomepeoplehopethatFibonaccinumbersprovideanedgeinpickinglotterynumbersorbetsingambling.ThetruthisthattheoutcomesofgamesofchancearedeterminedbyrandomoutcomesandhavenospecialconnectiontotheFibonaccisequence.Thereare,however,bettingsystemsusedtomanagethewaybetsareplaced,andthe

Fibonaccisystem,basedontheFibonaccisequence,isavariationontheMartingaleprogression,abettingstrategyoftenusedforgameswherethepossibilityofoneoutcomeoranother,asinthetossofacoin,approaches50percent.Theplayerdoublestheirbetaftereachturnuntileventuallytheywinbackalltheirlosses.IntheFibonaccisystem,oftenusedforcasinoandonlineroulette,thepatternofbetsplacedfollowstheFibonaccisequenceinthateachbetplacedisthesumoftheprevioustwobetsuntilawinismade,atwhichpointthebetgoesbacktwonumbersinthesequencebecausetheirsumwasequaltothepreviousbet.UsingtheFibonaccisystem,thebetsstaylowerthanthoseinaMartingaleprogression,butitdoesnotcoverallofthelossesinabadstreak.Animportantcautionisthatbettingsystemsdonotalterthefundamentaloddsofagame,

whicharealwaysinfavorofthecasinoorthelottery.Theymayjustbeusefulinmakingtheplayingofbetsmoremethodical,asillustratedintheexampleabove.

ROUND

Bet1

Bet2

Bet3

Bet4

Bet5

Bet5

NETRESULT

Scenario1

Bet1andlose

Bet1andlose

Bet2andwin

-

-

Evenat0

Scenario2

Bet1andlose

Bet1andlose

Bet2andlose

Bet3andwin

-

Downby1

Scenario3

Bet1andwin

Bet1andwin

Bet1andlose

Bet1andlose

Bet2andwin

Aheadby2

Intheworldofcomputerscience,theFibonaccisearchtechniqueisusefulforsearchinga

sortedlistofentriesinanarraytofindaparticularone.AFibonacciheapisadatastructureforpriorityqueueoperationsthatensurehigh-priorityelementsareservedbeforelow-priorityelements,andithasbetterperformancethanmanyotherpriorityqueuedatastructures,helpingtoimprovecomputerprogramruntimeperformanceandsolvecomplexroutingproblemsforcommunicationsnetworks.There’sanotherpopulationthatusesthegoldenratioandFibonaccisequenceforavery

differentpurposethananyI’vediscussedsofar.TheyapplythesamemathematicalrelationshipsfoundinthespiralsofplantsandworksofRenaissancearttotheiranalysisof

relationshipsfoundinthespiralsofplantsandworksofRenaissancearttotheiranalysisofstockmarkets,foreigncurrencyexchanges,andotherfinancialinstruments.Financialmarketshavepatternsofeconomiccyclesthatoccuronalargescale,overaperiodofyears.PatternsthatalignwithgoldenratiosandFibonaccinumberssometimesappearonthisscale,attimesmirroringthepatternsofindividualstocksorcurrenciestradingwithasingleday.Inthatlight,themovementsonadailyorweeklybasismaybeseenasafractalexpressionofthesamemovementsoverlargerperiodsoftime.Sometechnicaltradersbelievethesewavepatternsdefinethetimingofhighsandlows,aswellaspriceresistancepoints.

Phi-basedpatternsinthetimingofhigh-lowcyclesareshowninthischartIrecreatedoftheDowJonesIndustrialAveragedailyclosesfor2004.13

BelowIrecreatedachartshowingtheDowJonesIndustrialAveragedailyclosesforallof2008.14Theredrectangleoutlinesthehighestandlowestpricepointsoftheyear,andthetwogoldencutsdefinethepriceresistancelines.Asshown,thefallfromApriltomid-Julystoppedrightattheuppergoldenratiopriceresistancepointandthenbouncedback.OnceitbrokethroughbothresistancelinesinSeptember,itroseagainonlytopeakexactlyatthelowergoldenratiopriceresistancelinebeforefallingagain.Ofcourse,suchpatternsaremucheasiertodetectretrospectively,butanalystsfrequentlyuseindicatorslikethesewhentryingtoidentifyfuturetrends.Animportantcaution:Justasthegoldenratioaloneisnota“silverbullet”forsuccessinthe

arts,itisalsoonlyonetoolofmanyintheanalysisoffinancialmarkets.Prudentinvestorsuseavarietyoftoolsandtechniquestooptimizereturnsandmanagerisks.Bycombiningtheimprovedknowledgeofthelikelymarketinflectionpointsinpricewithotheranalytics,manytradersbelievetheycanimprovetheirsuccessrateforprofitabletrades,andthusimprovetheiroverallfinancialperformance.ResearchdonebymathematicalpsychologistVladimirA.Lefebvresuggeststhatpatternswe

seeinthefinancialmarketsmaybemorethanafluke.His1992bookAPsychologicalTheoryofBipolarityandReflexivity15presentedfindingsthathumansexhibitpositiveandnegativeevaluationsoftheopinionstheyholdinaratiothatapproachesФ—62percentpositiveand38percentnegative.Changesinstockpriceslargelyreflecthumanopinions,valuations,andexpectations,sothiscouldexplaintheconnection.

THEGOLDENQUESTION

Aswelookbackatmankind’smanydiscoveriesthroughthemillennia,itbecomesclearthatweliveinauniversegovernedbymathematicallaws,goldenornot.WhetherinKepler’slawsofplanetaryorbits,Einstein’stheoryofrelativity,orthemathematicsoftheopticsinyoureyethatallowyoutoreadthispage,everythingweexperienceinthephysicaluniversecanbemeasuredanddescribedbymathematics.

Asforthegoldenratio,we’veseenhowithascapturedtheimaginationofcountlessmathematicians,artists,designers,polymaths,biologists,chemists,andeveneconomistswithitssingularbeauty.Itisreflectedinsomeofthegreatestworksofartandarchitectureevercreatedinmankind’shistory.Noteverythingisbasedonthegoldenratio,butthenumberofplacesinwhichitseemstoappearistrulyamazing,andwearesuretouncoveritmoreandmoreastechnologyadvancesandourknowledgeofthephysicaluniverseexpands.

Ifyouexplorethistopicinmoredepth,you’llfindsomepeoplewhowilltellyouthatthegoldenratioisauniversalconstantthatdefineseverything.You’llfindotherssayingtheeventheevidencethatI’vepresentedinthisbookdoesnotexistatall.Thisisyourgoldenopportunitytocarefullyconsiderwhatyou’veseenandlearned,cometoyourownthoughtfulconclusions,andthenpondertheimplications.

Onequestionyoumightaskiswhythecontroversyexists.HowandwhycouldthissinglenumberfoundinasimplegeometricconstructioninthewritingsofanancientGreekmathematiciancausesuchwidespread,passionatediscussionanddisagreement?Theanswermaybefoundinthesimplefactthatinitsownuniqueway,phitouchesuponsomeofthemostfundamentalquestionsofphilosophyandthemeaningoflife.Whenwediscovercommonthreadsinthemathematicaldesignofthingsinourworld,especiallywhereitseemsunexpectedorunexplained,itcanbegthequestionofwhethertherecouldbesomethingmorethanchanceatwork—agranderplanofdesignwithsomeguidingpurpose,orevenadesigner.Othersmayseektoexplainthesesameobservationsascoincidencesarisingfromnaturalprocessesinadaptionsandoptimizations.Everyonehasanunderlyingbeliefsystemthatinfluencestheirinterpretationofeverythingtheyseeandhear,nomatterhowmuchevidenceispresentedtothecontrary.Thesefundamentalquestionsofwherewecamefrom,whywearehere,andwherewearegoing,aremysteriesthatweallmustponderwithanopenmindandanopenheart.

Thereisanotherimportantaspecttothegoldenratio,however,thatbringsamuchmoreuniversallycommonresponse,andthatistotouchuponourperceptionsofbeauty.Forsome,thatbeautyiscenteredonitsuniquepropertiesinmathematicsandgeometry,oritsabilitytocreateaperfectlyformedfractalpattern.Forsome,itisperceived,whetherconsciouslyornot,inthebeautyofnatureandinthehumanfaceandform.Forothers,intentionallyornot,itisexpressedintheircreativeworksofartanddesign.

Atwhateverlevelthisbeautyisperceived,amoreimportantquestionneedstobeasked:Howandwhydoweperceivebeautyatall?Whydowehaveaninnateabilitytoseebeauty,andwhydowealsohaveaneedtoexpressit?Fromanevolutionaryperspective,onecouldarguethatbeautyisanindicatorofhealth,andthatbeingdrawntothingsthatarehealthyresultsinbetterdecisionsforsurvival,whetheritbewhichfruittoeatorwhichmatetoselectforpropagationofthespecies.That’slogicalenough,butwhatevolutionaryadvantageistheretotheappreciatingbeautyinasunset,astarrynight,aninspiringworkofart,orasongthattouchessomethingdeepinsideyou?Ithinkifwe’rehonest,mostpeoplewillrecognizethatthereisanotheraspecttothehumanexperiencethatgoesbeyondthefactsfoundsolelyinscientific,naturalisticexplanationsofourphysicalexistence.Forme,andformanyothersthroughouthistory,thegoldenratiohasbeenalightinthedarknessthatdrawsustoadifferentperspectiveandadeeperunderstandingofallthatwefindaroundus—andwithinus.

InthisbookI’veonlytouchedonafewoftheplaceswherethegoldenratiocanbefound,andafewwaysinwhichitcanbeapplied.Moreappearancesandapplicationsarebeingdiscoveredallthetime,inasurprisingnumberofplaces.Thebestwaytoknowforyourselfwherethegoldenratioappearsandwhereitisonlyimaginedistoexplorewithanopenmind,learnallyoucan,andcometounderstandingsthataretrulyyourown.

Asyouundertakethisjourneyofdiscovery,considerthelivesandcontributionsofthosewhohavemadethisjourneybeforeyou.Eucliddiscoveredtheprinciplesofgeometrythattaughtandinspiredpeopleforthousandsofyears.LeonardodaVinciandotherartistsoftheRenaissancecreatedaunionofmathematicsandartthatstillinspiresustoday.JohannesKeplerdiscoveredfundamentaltruthsaboutthesolarsystemthathadeludedothersforgenerationsbeforehim.LeCorbusierusedtheinherentharmonyofthegoldenratiotodesigntheUNSecretariatbuildingthatservesasthehomeoftheleadingorganizationtoconfrontcommonglobalchallengesandbringharmonytotheworld’snations.DanShechtmandiscoveredanewstateofmatterthatwaspreviouslythoughtimpossible.Thegoldenratiocontinuestofindapplicationsineverythingfrom

impossible.Thegoldenratiocontinuestofindapplicationsineverythingfromlogodesigntoquantummechanics.

ChartresCathedralilluminatedatnight.

Acloseupofthesunflower’sfive-petaledrosettesshowsjusthowubiquitousthenumber5isinnaturallifeforms.

LucaPacioli’sappellationofthegoldenratioasthe“divineproportion”is,indeed,fitting:Itisseenandexperiencedbymanyasadoortoadeeperunderstandingofbeautyandmeaninginlife,unveilingahiddenharmonyorconnectednessinsomuchofwhatwesee.That’sanincredibleroleforasingle

numbertoplay,butthenagain,thisonenumberhasplayedanincredibleroleinhumanhistory,andperhapsinthefoundationsoflifeitself.

Thespiralinthisfractalillustrationgrowshorizontallybyafactorofphi.

APPENDICES

“Foreveryonewhoasksreceives;theonewhoseeksfinds;andtotheonewhoknocks,thedoorwillbeopened.”1

—Matthew7:7(NewInternationalVersion)

APPENDIXA

FURTHERDISCUSSION

Aswehaveseen,thegoldenratioisatopicthatspansthousandsofyearsandimpactsaverydiverserangeofdisciplines.Forthisreason,itisdifficulttofullygraspevenasmallpercentageofallthatcanbeknownaboutit,which,inturn,leadstomuchmisinformationandmisunderstanding.Ihavestudiedthegoldenratiofortwentyyears,andthroughouttheprocessofwritingthisbookIhavelearnedmorethanIwouldhaveexpected.

Thegoldenratiohasanunusualandunexpectedabilitytocreatecontroversy.Theincrediblebreadthofinformationaboutitmeansthatmostpeoplehavelimitedinformationwithwhichtoformtheiropinionsandconclusions.However,thecontroversialaspectaroundthegoldenratioisrelatedtoitsabilitytotouchonthequestionofwhethertheevidenceofdesignweseeinourworldarebestexplainedbyefficienciesandoptimizationinnaturalprocesses,orbyagreaterplanofdesign,oraDesigner.Thisisaverypersonalandimportantquestionforallofus,andourpersonalbeliefsystemsgreatlyinfluencethewaywefilterandinterpretgoldenratioevidence,thusleadingveryeducatedandintelligentpeopletocometoverydifferentconclusions.Inthisbook,Ihavetriedtomaintainabalancebetweentheseoften-polarizingextremes,presentingthesimplegeometryandmathematicalfactsandevidencesinboththeartsandnaturethatIfeltwasaccurateandmeaningfultoabetterunderstanding.

Thebestwecandotoobtainthemostaccurate,truthful,andmeaningfulanswersistostudythegoldenratiomoredeeplyandtocometoourownconclusionsratherthanblindlyacceptingthosefromotherswithaparticularviewpoint,includingmine.Somewillsaythatthegoldenratioappears—when,infact,itdoesnot—andjumptotheconclusionthatitrepresentscompleteprooffortheexistenceofGod.Conversely,otherswillsaythatthegoldenratiodoesnotappear—when,infact,itreallydoes—anddenyallevidenceofitsexistence.Iwouldliketoaddressinthisfirstappendixsomeoftheobjectionalargumentsthatarefrequentlyofferedbythosewhobelievethatappearancesofthegoldenratioinnatureandtheartsaresimplyfallacy,ormyth,sothatreaderscanbetterassesstheargumentsforthemselves.

“Youwerejustlookingforpatternsandfoundthem.”Somesources,mostnotablymathematicians,arguethegoldenratiodoesnotexistoutsideofmathematicsandgeometry,except,perhaps,inthespiralsandphyllotaxisofplantsbasedonFibonaccinumbers.Theysaythatifwethinkweseeagoldenratio,wearesimplyexperiencingthehumanneedtofindmeaningwithinpatternsaroundus.Thescientificnameforthisisapophenia,whichperMerriamWebster’sdictionaryisdefinedas“thetendencytoperceiveaconnectionormeaningfulpatternbetweenunrelatedorrandomthings(suchasobjectsorideas).”Althoughourtendencytofindpatternsisareasonableconcern,theflipsideoftheproverbialcoinisthatthereisriskinignoringpatternsandmeaningswheretheydo,infact,exist.Humansarepattern-seekingbeings;itishowwedoeverythingfromlearningtospeaktoapplyingthescientificmethodtodiscoverthenatureoftheuniverse.Thequestionisnotwhetherweseek—ordonotseek—andfindpatterns,whichofcoursewedo,but,instead,whetherwehavereasonablemethodsandstandardstoevaluatethepatternswefind.Withthis,wesimplyneedtofindbalancebetweenblindlyignoringthepatternsandtheirsignificanceswheretheydoexistandover-zealouslyfindingpatternsandsignificanceswheretheydonot.

“Nothingcanbethegoldenratiobecauseit’sirrational.”Somearguethatitisimpossibletoapplythegoldenratiobecauseit’sanirrationalnumberthathasaninfinitenumberofdecimalplaces.Onerecognizedskepticargues“it’simpossibleforanythingintherealworldtofallintothegoldenratio.”Itturnsoutthatthisargumentitselfisequallyirrational,oratleastexcessivelytheoreticalandpedantic.Whilenothingcan“fallinto”thegoldenratio,it’squiteeasytodrawalineinwhichthegoldenratiowill“fallinto.”Anyonecanthuseasilyapplythegoldenratioindesign.Itthenbecomesamatterofhowmuchprecisiononeneeds,whichthendetermineshowthickthedefininglinecanbe.Thisargumentcompletelymissesanotherveryimportantpoint:Nodimensionthatwemeasure,orconstruct,canbeanexactrepresentationofanynumber,whetherthatnumberisrationalorirrational;itisthenatureoftheuniverseinwhichwelive.Youcouldtrytodrawa1-inchcircle,buteventhough1isaninteger,thecirclewillneverbeexactly1.00000000000000000000inchesindiameter.Theapplicationoftheconceptofnumbersiswhathasmeaningintherealworld,andaccuracytomorethanfourorfivedecimalplacesisrarelyrequiredforanything.Withintheconceptitself,anynumbercanbeappliedwithenoughaccuracyforallpracticalpurposes.

“Youcannotdetermineifthegoldenratiowasappliedafterthefact.”Thisargumentisoftenusedasanattempttodismisseventhemostlegitimateinvestigationsoftheappearanceandapplicationsofthegoldenratio.Incaseswhereaconclusionisbasedonjustasingle,closeapproximationofthegoldenratio,itmaybeavalidpoint.However,incaseswhereweseenumerousinstancesofthegoldenratiowithahighdegreeofaccuracy,itquicklylosesitsvalidity.Ifwefindanoccasionalgoldenratioinahumanface,itisnotmuchofabasisforaconclusion.Ifwefindadozenormorespecificgoldenratiosthatcommonlyappearinhundredsofattractivefaces,wearethenlikelytohavefoundsomethingofsignificance.Determiningtruththroughanalyticalinvestigationafterthefactisthenatureofscienceitself,inmanycases,andiscertainlythebasisformanyverdictsrenderedincourtsoflaw.Asdiscussedinmy“rulesofengagement”forgoldenratioanalysisinthearts,thevalidityofconclusionscanbemaintainedbyfocusingonthefollowing:

•Relevance:Usingfeaturesthatanyreasonablepersonwouldseeasthemostobviousorimportantplacesinwhichdesignandcompositiondecisionswouldhavebeenmade

•Commonality/Repeatability:Usingfeaturesthatappearinanumberofinstancestodemonstrateknowledgeandintent

•Accuracy:Acceptingonlythosemeasurementsthatarewithin+/-1%oftheexactgoldenratio,andwithimagesusingthehighestresolutionavailable

•Simplicity:Basingmeasurementsonthesimplestpossiblemeasures—thosethatmostlikelywouldhavebeenappliedbytheartistordesigner

“Itcouldhavebeenanyoneofaninfinitenumberofothernumbers.”Someskepticsbelievethatobservedappearancesaren’tnecessarilythegoldenratiobecauseitcouldhavebeenanyoneofaninfinitenumberofothernumbersclosetothegoldenratio.Thisapproachturnsourinvestigationintolookingforaneedleinaninfinitelylargehaystack.Theoddsoffindinganythingthatfitsthegoldenratioexactly,withitsinfinitenumberofdecimalplaces,becomeinfinitelysmall.Thisisnotthecaseintherealworld,whereweroutinelyusephysicalmeasurementsthataremeaningful,discernible,andfinite.Therearephysicalandengineeringlimitationsonourabilitytobuildsomethingwithmuchmorethanfourorfivedecimalplacesofaccuracy,andthereisusuallynoneedforanymoreprecision.IfwemeasuretheGreatPyramidandfindthegoldenratiowithinmereinchesofits481.4-foot(147m)height,thisshouldbecloseenoughtocometoareasonableconclusionthatitmightwellhavebeenafactor

initsdesign.Itonlytakesfoursignificantdecimalplacestodothat,notaninfinitenumber.

Thereareonlythirty-threenumberswithfoursignificantdecimalplacesthathaveavariancefromphioflessthan1percent,not“infinitelymanynumbers.”Theserangeinincrementsof0.001from1.602to1.634,sothosearereallytheonlyrelevantratiostouse.Additionally,theremaybeinfinitelymanynumbersnearphi,buttherearepreciousfewsimple,integerratiosandgeometricconstructionswhichcloselyapproximatethegoldenratio.

Ifwetakeallpossibleratiosoftheintegernumbersfrom1to50,thereare1,275thataregreaterthan,orequal,to1.Onlytenoftheseresultinauniqueratiowithavariancefromphioflessthan1percent.Onthenextpage,Fibonaccisequencenumbersareinbold,andbecomethemostaccurateofanyoftheseratiosveryearlyintheseries:

Ratio Decimal VariancefromФ

13/8 1.625 0.43%

21/13 1.615 -0.16%

29/18 1.611 -0.43%

31/19 1.632 0.84%

34/21 1.619 0.06%

37/23 1.609 -0.58%

44/27 1.630 0.72%

45/28 1.607 -0.67%

47/29 1.621 0.16%

49/30 1.633 0.95%

Ifwetakeallthepossiblerighttrianglesbasedonintegernumbersof1to50foranytwoofthethreesides,thereare2,550uniquecombinations.Onlyfiveoftheresultingtriangleshaveavarianceoflessthan1percentfromphi:

SideA(1) SideB(√Ф) HypotenuseC(Ф) Variance

8.660 11 14 -0.09%

11 14 17.804 +.02%

26 33 42.012 -.08%

28.983 37 47 +.22%

37 47 59.816 -.05%

IftheancientEgyptiansdidinfactusethesekedmethodwithagradientof5.5/7(equalto11/14)todeterminetheproportionsoftheGreatPyramid,thismeanstheysomehowselectedtheonesetofintegerswhoseratioshastheveryleastvariancefromaprecisegoldenratio.Thedifferenceisonly0.02percent.Whywouldtheyselectaratiothatissouniqueinmathematicsandgeometryandsoprevalentinnatureandbeauty?

Thegoldenratioisthusinfinitelymoreprobabletoappearthan“infinitelymanyothernumbers.”Therefore,inreality,the“infinitelymanynumbers,”orratios,thatartistsandarchitectscanchoosefromincreatingsomethingthatlookslikephi,butisnotphi,isverysmall.Takeanothergoodlookatthelistsaboveandseeifanyofthenumbersinvolvedappearaslikelycandidates—numbersthathavesomespecialsignificanceontheirownthatwouldhavemadethemworthierofselectionthanphiitself.

Anotherproblemwiththe“infinitelymanynumbers”premiseisthatphisimplyisnotjust“anothernumber”amonganinfinitesetofothers,orevenamongthetwentysetsofnumbersabovethatareclosetoit.Tothecontrary,phiisoneofthemostuniquenumbersingeometry,mathematics,life,andnature,withpropertiesthatnoothernumbercontains.Itspropertiescreateefficiencyindesign,aswellasvisualharmonyandbeautylikenoothernumber.It’sbeenrecognizedbymankindsincethetimeoftheancientsforitsrelationshiptonatureanditsvalueinaesthetics.So,whenconsideredinthisway,whenweseesomethingthatislessthan1percentawayfromphi,thelikelihoodthatanartistorarchitectusedphiinsteadoftheseclosesubstitutesisreallyquitehigh.Iftheychoseanothernumber,wewouldhavetoaskwhatmadethemselectaproportionsoveryclosetophiratherthananentirelydifferentproportionaltogether(e.g.,1.414asthesquarerootof2,1.5,1.732asthesquarerootof3,etc.).

SincewedonothaveasignedaffidavitfromtheancientEgyptiansandGreeks,LeonardodaVinci,GeorgesSeurat,MotherNature,orGod,Himself,toverifythatphiwasusedinanyoftheircreations,wemustmakethemostreasonableassumptionspossiblegiventheevidenceavailable.Thephysicaluniverseisbasedonmathematics.Phiappearsextensivelythroughoutmathematicsandgeometry.Whatbasisinreasonistheretothensuspectthataninfinitenumberof

otherverycloseratioswithnoparticularsignificanceinmathematicsandgeometrywouldexpressthemselvesinthesimple,fundamentalpatternswithintheuniverse?

Canwedetermineifartistshaveappliedthisratiointheircreations?Weonlyneedtolookattheevidenceandapplyreason.Aswithanygooddetectivenovel,weneedtoaskiftherewasmeans,motive,andopportunity.Themeanscanbeassimpleasamarkerandapieceofstring.Themotivecanbesimplytoappreciateandre-createthebeautyandharmonyofallweseeintheworldaroundus,consciouslyornot.Theopportunitiestocreatewithphiarethen“infinitelymany.”Ifphiisnotthelikelyor“preferred”number,andifsomeonesaysthatanobservationisnotbasedonthegoldenratio,thenthemostscientificapproachwouldrequirethatoneproposeabetterorcompetingtheoryratherthansimplyplayingtheskepticanddismissingthebestavailableexplanation.

ConsidertheSourceWhatevertheargumentsorrationalepresented,intheendweshouldalsoconsiderthesource.Whatistheperson’smotivations?Whatarethepersonalviewpointsorideologiesthatheorshewantstopromote?Probefurthertoaskiftheirviewsarebasedinverifiableevidence,orjustareflectionoftheirownbeliefsaboutlife.Ask,too,whatqualificationstheyhavetospeakonthetopic,andinwhichspecificareas.Thegoldenratio,aswehaveseen,isaverybroadanddeeptopic,andittakesquiteabitofin-depthstudytofullyappreciateitsappearancesandapplicationsinanyonearea.Tolearnaboutthegoldenratioinmathematics,seekoutamathematicianwhocandescribeitspropertieswithequationsandproofs.Tolearnaboutthegoldenratiointhearts,seekoutanartist,architect,designer,orphotographerwhousesitforcompositiondecisionstocreatevisualharmonieswithintheirworkstoenhancedesignaesthetics.Tolearnaboutthegoldenratioinbeauty,seekoutaprofessionalincosmeticmedicalapplicationswhousesittosuccessfullyenhancethebeautyofhisorherpatients.Itisgoodtokeepinmindtoothatmostmathematiciansarenotexpertsinthedesignarts,mostartistsanddesignersarenotexpertsincosmeticsurgery,andmostplasticsurgeonsarenotexpertsinadvancedmathematics.

It’seasytobeaskeptic,toholdstrongopinions,tocriticizeideaswithoutdoingtheresearchandanalysisrequiredtounveilthehiddentruthsthatremaintobefound,andtopresentnewideasandinsightsofone’sown.Thescientificmethodhasgivenusatoolforincredibleadvancement,andsomebelievethatsciencewillprovidealltheanswers.Scienceisawonderfultoolanddiscipline.Scientists,however,arepeoplewiththeirownshortcomingslikeanyoneelse.

Scientists,however,arepeoplewiththeirownshortcomingslikeanyoneelse.It’scommoneveninscienceforthosewithnewideastobeharshlycriticizedforthinkingoutsidethecurrentlyacceptedparadigmordogma.It’seasytolookbackonehundred,fivehundred,ortwothousandyearsandjudgethescientificknowledgeofthosetimesasprimitivebytoday’sstandards.It’snotsoeasytocometogripswiththeverylikelypossibilitythatmuchofwhatweholdastruetodaymaybeviewedjustasprimitivebythosewhowillliveonehundred,fivehundred,ortwothousandyearsfromnow.Weshouldbeopentonewideasandnewwaysofthinking,whichwillleadusintonewadvancesinknowledgeratherthanserveasaroadblocktoourselvesandothers.

ConclusionIfyouchoosetostudythisveryfascinatingtopicfurther,youwillbeintheverygoodcompanyofsomeofthegreatestmindsinhistory,andyouwillfindthatitconnectsyouwithmoreareasofinterestandabroaderrangeofpeopleandideasthanimaginable.Thisprovidesanincredibleopportunityforeducationalenrichmentandforpersonalgrowth.Itwillalsoexposeyoutomanyconflictingopinions,whichareheldwithgreatpassionbymanypeople.Keepanopenmindandanopenheart,analyzethemeritsandshortcomingsofwhateverevidenceandviewpointsarepresented,andenjoythejourney.

APPENDIXB

GOLDENCONSTRUCTIONS

InthetimeofEuclid,geometricconstructionswererestrictedtotheuseofacompassandstraightedgeonly.These“pure”constructionsarewhatappearedintheancientGreekmathematician’sfamousbookTheElements,whichlaidthecornerstoneformathematicseducationforthenexttwomillennia.Asitturnsout,therearealotofwaystoderivethegoldenratiousingonlythesimpletoolsmentionedabove.Herearetwoofthemostcommonconstructionsthatcanbemadewitharulerandcompass:

1.DrawalineAB.2.DrawaperpendicularverticallineBCthatishalftheheightofAB.3.DrawalineACtocompletethetriangle.4.DrawanarcatpointCfrompointBtopointDonthehypotenuse.5.DrawanarcatpointAfrompointDtopointSonlineAB.

Inthisconstruction,AB/AS=Ф

Here’sanothergoldenratioconstructionthatcaneasilybedonewithacompassandruler:

1.DrawalineAS.2.DrawaverticallineSCthatisequalinlengthtoAS.3.DividelineASatitsmidpointM.4.DrawanarcatmidpointMfrompointCtopointBontheextensionoflineAS.

Inthisconstruction,AB/AS=Ф.

AGeometricExpressionof(1+√5)/2Shownhere,wesawthatthemathematicalexpressionofthegoldenratiois(1+√5)/2.GeometerScottBeachdevelopedawaytorepresentthisexpressiongeometrically:

Toconstructthis,followthesesteps:1.Asyoudidinthefirstconstructionofthisappendix,createrighttriangleABCwiththelengthofsideABequalto1andthelengthofsideACequalto2.(ThePythagoreanTheoremcanbeusedtodeterminethatthelengthofsideBCis√5.)

2.ExtendsideBCby1unitoflengthtoestablishpointD.3.BisectlinesegmentCDtoestablishpointE.

Intheresultinggraphicrepresentationof(1+√5)/2,thesegmentCDrepresents1+√5,whichmeansthatthelengthofsegmentCEisequalto(1+√5)/2,orФ.Furthermore,DB/BE=Ф.

CircleConstructionsAmongmathematicians,there’sabitofacompetitiontoseehowfewlinescanbeusedtocreateagoldenratio,orhowmanygoldensectionscanbecreatedwiththeleastnumberoflines.(Okay,soascompetitionsgo,it’snotexactlytheSuperBowl,butthenontheotherhandnobodywillbeponderingthewinningteamsoftheNFLtwothousandyearsfromnow.)Belowyou’llfindafewingeniousconstructionsinvolvingcircles.

Threeadjacentcircles1.Usingacompass,drawthree1-inchcirclesontopofaline,makingsurethattheytouchoneanotherbutdonotoverlap.ThesegmentextendingbetweenthebottomofthefirstandthirdcirclesisAC,withalengthof2.

2.Drawalinethatconnectsthetopofthefirstcircletothebottomofthethirdcircle,formingsegmentBC.

3.CompletethetrianglebyconnectingpointBtopointA.4.AddpointDwhereBCintersectswiththeleftsideofthesecondcircle.5.AddpointEwhereBCintersectswiththerightsideofthesecondcircle.

Inthisconstruction,bothDE/BDandDE/EC=Ф.

Threeconcentriccircles1.Drawthreeconcentriccircleswhoseradiusesareinaratioof1:2:4.2.DrawlineAGtangenttothetopoftheinnercirclethatconnectsthemiddlecircletotheoutercircle.

OverlappingcirclesIn2002,AustrianartistandcomposerKurtHofstetterpublishedthisconstruction,involvingonlyfouroverlappingcirclesandaline,inForumGeometricorum:

1.Usingacompass,drawtwooverlappingcirclessuchthatthecenterofeachcircle(pointsCandD)definestheradiusoftheother.

2.CentertwolargercirclesonpointsCandD,withradiithataretwicethatofthesmallcircles.

3.DrawalineAGfromtheleft-mostintersectionpointsofthetwosmallercircles(A)totheleft-mostintersectionpointofthetwolargecircles(G),asshown.

BelowisanotherconstructionbyHofstetter:

1.DrawacirclearoundcenterpointA.

2.DrawapointBontheright-mostedgeofthiscircle.FromcenterpointB,drawacirclethatpassesthroughpointA.

3.Ontheleft-mostpointofthefirstcircle,drawapointC.4.FrompointC,drawanarcthatpassesthroughpointB.6.Wherethearcintersectsthetophalfofthesecondcircle,drawapointE.Extendalinefromthispointtothebottomintersectionpointofthetwocircles(D)toformlineED.

7.Finally,drawlineABbetweenpointsAandB,andthendrawapointGattheintersectionofABandED.

Inthisconstruction,AB/AG=Ф.

ORTHOGONS

Theaboveconstructionofagoldensectionisthemostcommonlyknownoftwelveorthogons(aka“dynamicrectangles”)whichinessencearerectangulargeometricstructuresthatareconstructedfromasquareusingonlyacompassandstraightedge.Amongorthogons,theonethatyieldsagoldenrectangle(ratio:1/2+√5/2)isknownastheauron,comingfromtheLatinrootaur,meaning“gold.”

Orthogonsprovideasystemofdesignthat,forcenturies,hasallowedartistsandartisanstocreateconsistent,harmoniousfigureswithouttheneedforcomplicatedcalculationsormeasuringdevices.Examplesoforthogons,withtheirheighttowidthratios,includethediagon(√2),quadriagon(1/2+√2/2),andhemidiagon(√5/2).Informationontheapplicationoforthogonstotheprinciplesofdesignisavailableatwww.timelessbydesign.org,awebsiterunbyprofessionalartistValrieJensen.

http://www.timelessbydesign.org

Theauron.

Thediagonisshowntop,center,inthisscanofthis1575editionofthefirstGermantranslationofVitruvius’swork,VitruviusTeutsch.

NOTES&FURTHERREADING

Theinformationinthisbookcomesfromamixtureoforiginalresearch,contributionsbyvisitorstomywebsitesatwww.goldennumber.netandwww.phimatrix.com,originalinterviews,onlinesources,andbooks.Wikipedia.comisagreatstartingpointforfurtherresearchonavarietyofmathematics-relatedsubjects,buttherearealsoanumberofgoodsourcesfocusedmoresolidlyonmathematicsanditshistory,includingtheMacTutorHistoryofMathematicsarchivefromtheUniversityofSt.Andrews,Scotland(http://www-groups.dcs.st-and.ac.uk/~history/index.html),andWolframMathWorld(http://mathworld.wolfram.com/).

GENERALREADINGHerz-Fischler,Roger.AMathematicalHistoryoftheGoldenNumber.NewYork:DoverPublications,1998.Huntley,H.E.,TheDivineProportion:AStudyinMathematicalBeauty.NewYork:DoverPublications,1970.Lawlor,Robert.SacredGeometry:PhilosophyandPractice.London:ThamesandHudson,1982.Livio,Mario.TheGoldenRatio:TheStoryofPhi.TheWorld’sMostAstonishingNumber.NewYork:BroadwayBooks,2002.Olsen,ScottA.TheGoldenSection:Nature’sGreatestSecret.Glastonbury:WoodenBooks,2009.Skinner,Stephen.SacredGeometry:DecipheringtheCode.NewYork:Sterling,2006.

INTRODUCTION1.“Internetusersper100inhabitants1997to2007,”ICTIndicatorsDatabase,InternationalTelecommunicationUnion(ITU),http://www.itu.int/ITU-D/ict/statistics/ict/.

2.“ICTFactsandFigures2017,”TelecommunicationDevelopmentBureau,InternationalTelecommunicationUnion(ITU),https://www.itu.int/en/ITU-D/Statistics/Pages/facts/default.aspx.

3.“HistoryofWikipedia,”Wikipedia,https://en.wikipedia.org/wiki/History_of_Wikipedia.

http://www.goldennumber.net

http://www.phimatrix.com

http://Wikipedia.com

http://www-groups.dcs.st-and.ac.uk/~history/index.html

http://mathworld.wolfram.com/

http://www.itu.int/ITU-D/ict/statistics/ict/

https://www.itu.int/en/ITU-D/Statistics/Pages/facts/default.aspx

https://en.wikipedia.org/wiki/History_of_Wikipedia

4.RogerNerz-Fischler,AMathematicalHistoryoftheGoldenNumber(NewYork:Dover,1987),167.

5.MarioLivio,TheGoldenRatio:TheStoryofPhi.TheWorld’sMostAstonishingNumber(NewYork:BroadwayBooks,2002),7.

6.DavidE.Joyce,“Euclid’sElements:BookVI:Definition3,”DepartmentofMathematicsandComputerScience,ClarkUniversity,https://mathcs.clarku.edu/~djoyce/elements/bookVI/defVI3.html.

CHAPTERI1.AsquotedbyKarlFink,GeschichtederElementar-Mathematik(1890),translatedas“ABriefHistoryofMathematics”(Chicago:OpenCourtPublishingCompany,1900)byWoosterWoodruffBemanandDavidEugeneSmith.AlsoseeCarlBenjaminBoyer,AHistoryofMathematics(NewYork:Wiley,1968).

2.“TimaeusbyPlato,”translatedbyBenjaminJowett,TheInternetClassicsArchive,http://classics.mit.edu/Plato/timaeus.html.

3.ThesepassagesandillustrationswererecreatedandeditedbasedonthetranslationsandcontentatDavidE.Joyce,“Euclid’sElements,”DepartmentofMathematicsandComputerScience,ClarkUniversity,https://mathcs.clarku.edu/~djoyce/elements/elements.html.

4.RogerNerz-Fischler,AMathematicalHistoryoftheGoldenNumber(NewYork:Dover,1987),159.

5.EricW.Weisstein,“IcosahedralGroup,”MathWorld—AWolframWebResource,http://mathworld.wolfram.com/IcosahedralGroup.html.

6.Ibid.

CHAPTERII1.Asquotedat“Quotations:Galilei,Galileo(1564-1642),”Convergence,MathematicalAssociationofAmerica,https://www.maa.org/press/periodicals/convergence/quotations/galilei-galileo-1564-1642-1.

2.JacquesSesiano,“Islamicmathematics,”inSelin,Helaine;D’Ambrosio,Ubiratan,eds.,MathematicsAcrossCultures:TheHistoryofNon-WesternMathematics(Dordrecht:SpringerNetherlands,2001),148.

3.J.J.O’ConnorandE.F.Robertson,“TheGoldenRatio,”SchoolofMathematicsandStatistics,UniversityofStAndrews,Scotland,

https://mathcs.clarku.edu/~djoyce/elements/bookVI/defVI3.html

http://classics.mit.edu/Plato/timaeus.html

https://mathcs.clarku.edu/~djoyce/elements/elements.html

http://mathworld.wolfram.com/IcosahedralGroup.html

https://www.maa.org/press/periodicals/convergence/quotations/galilei-galileo-1564-1642-1

http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Golden_ratio.html.4.French-bornmathematicianAlbertGirard(1595-1632)wasthefirsttoformulatethealgebraicexpressionthatdescribestheFibonaccisequence(fn+2=fn+1+fn)andlinkittothegoldenratio,accordingtoScottishmathematicianRobertSimson,“AnExplicationofanObscurePassageinAlbertGirard’sCommentaryuponSimonStevin’sWorks(VideLesOeuvresMathem.deSimonStevin,aLeyde,1634,p.169,170),”PhilosophicalTransactionsoftheRoyalSocietyofLondon48(1753-1754),368-377.

5.JamesJosephTattersall,ElementaryNumberTheoryinNineChapters(2nded.),(Cambridge:CambridgeUniversityPress,2005),28.

6.MarioLivio,TheGoldenRatio:TheStoryofPhi.TheWorld’sMostAstonishingNumber(NewYork:BroadwayBooks,2002),7.

7.ManyinterestingpatternsassociatedwiththeFibonaccisequencecanbefoundatDr.RonKnott,“TheMathematicalMagicoftheFibonacciNumbers,”DepartmentofMathematics,UniversityofSurrey,http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html#section13.1.

8.Jain108,“DivinePhiProportion,”Jain108Mathemagics,https://jain108.com/2017/06/25/divine-phi-proportion/.

9.ThispatternwasfirstdescribedandillustratedbyLucienKhan,andthegraphicbelowwasrecreatedbasedonhisoriginaldesign.

10.J.J.O’ConnorandE.F.Robertson,“TheGoldenRatio.”

CHAPTERIII1.Thisispossiblyaparaphraseofhisphilosophicalreflectionsontheprimeimportanceofmathematics.

2.AsquotedinMarioLivio,TheGoldenRatio:TheStoryofPhi.TheWorld’sMostAstonishingNumber(NewYork:BroadwayBooks,2002),131.

3.RichardOwen,“PierodellaFrancescamasterpiece‘holdsclueto15th-centurymurder’,”TheTimes,January23,2008.

4.“TheTenBooksonArchitecture,3.1,”translatedbyJosephGwilt,Lexundria,https://lexundria.com/vitr/3.1/gw.

http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Golden_ratio.html

http://www.maths.surrey.ac.uk/hosted-sites/r.knott/fibonacci/fibmaths.html#section13.1

https://jain108.com/2017/06/25/divine-phi-proportion/

https://lexundria.com/vitr/3.1/gw

5.JackieNortham,“MysterySolved:SaudiPrinceisBuyerof$450MDaVinciPainting,”TheTwo-Way,December7,2017,https://www.npr.org/sections/thetwo-way/2017/12/07/569142929/mystery-solved-saudi-prince-is-buyer-of-450m-davinci-painting.

6.J.J.O’ConnorandE.F.Robertson,“QuotationsbyLeonardodaVinci,”SchoolofMathematicsandStatistics,UniversityofStAndrews,Scotland,http://www-history.mcs.st-andrews.ac.uk/Quotations/Leonardo.html.QuotedinDesMacHale,Wisdom(London:Prion,2002).

7.“NascitadiVenere,”LeGalleriedegliUffizi,https://www.uffizi.it/opere/nascita-di-venere.

CHAPTERIV1.“Georges-PierreSeurat:Grandcamp,Evening,”MoMA.org,https://www.moma.org/collection/works/79409.

2.deIde,“allRGB,”https://allrgb.com/3.MarkLehner,TheCompletePyramids(London:Thames&Hudson,2001),108.

4.H.C.Agnew,ALetterfromAlexandriaontheEvidenceofthePracticalApplicationoftheQuadratureoftheCircleintheConfigurationoftheGreatPyramidsofGizeh(London:R.andJ.E.Taylor,1838).

5.JohnTaylor,TheGreatPyramid:WhyWasItBuilt?AndWhoBuiltIt?(Cambridge:CambridgeUniversityPress,1859).

6.ThePalermoStone,whichisdatedtotheFifthDynastyofEgypt(c.2392–2283BCE),containsthefirstknownuseoftheEgyptianroyalcubittodescribeNilefloodlevelsduringtheFirstDynastyofEgypt(c.3150–c.2890BCE).

7.D.I.Lightbody,“BiographyofaGreatPyramidCasingStone,”JournalofAncientEgyptianArchitecture1,2016,39–56.

8.GlenR.Dash,“Location,Location,Location:Where,Precisely,aretheThreePyramidsofGiza?”DashFoundationBlog,February13,2014,http://glendash.com/blog/2014/02/13/location-location-location-where-precisely-are-the-three-pyramids-of-giza/.

9.LelandM.Roth,UnderstandingArchitecture:ItsElements,History,andMeaning(3rded.)(NewYork:Routledge,2018).

https://www.npr.org/sections/thetwo-way/2017/12/07/569142929/mystery-solved-saudi-prince-is-buyer-of-450m-davinci-painting

http://www-history.mcs.st-andrews.ac.uk/Quotations/Leonardo.html

https://www.uffizi.it/opere/nascita-di-venere

http://MoMA.org

https://www.moma.org/collection/works/79409

https://allrgb.com/

http://glendash.com/blog/2014/02/13/location-location-location-where-precisely-are-the-three-pyramids-of-giza/

10.ChrisTedder,“GizaSiteLayout,”lastmodified2002,https://web.archive.org/web/20090120115741/http:/www.kolumbus.fi/lea.tedder/OKAD/Gizaplan.htm

11.Henutsenwasdescribedasa“king’sdaughter”bytheInventorySteladiscoveredin1858,butmostEgyptologistsconsideritafake.

12.TheodoreAndreaCook,TheCurvesofLife(NewYork:DoverPublications,1979).

13.“StatueofZeusatOlympia,Greece,”7Wonders,http://www.7wonders.org/europe/greece/olympia/zeus-at-olympia/

14.GuidoZucconi,Florence:AnArchitecturalGuide(SanGiovanniLupatoto,Italy:ArsenaleEditrice,2001).

15.PBS,“BirthofaDynasty,”TheMedici:GodfathersoftheRenaissance,March30,2009,https://www.youtube.com/watch?v=9FFDJK8jmms.

16.MatilaGhyka,TheGeometryofArtandLife(2nded.)(NewYork:DoverPublications,1977),156.

17.MichaelJ.Ostwald,“ReviewofModulorandModulor2byLeCorbusier(CharlesEdouardJeanneret),”NexusNetworkJournal,vol.3,no.1(Winter2001),http://www.nexusjournal.com/reviews_v3n1-Ostwald.html.

18.“UnitedNationsSecretariatBuilding,”Emporis,https://www.emporis.com/buildings/114294/united-nations-secretariat-building-new-york-city-ny-usa.

19.RichardPadovan,Proportion:Science,Philosophy,Architecture(NewYork:Routledge,1999).

20.“FactSheet:HistoryoftheUnitedNationsHeadquarters,”PublicInquiries,UNVisitorsCentre,February20,2013,https://visit.un.org/sites/visit.un.org/files/FS_UN_Headquarters_History_English_Feb_2013.pdf

21.“DB9,”AstonMartin.Lastmodified2014.https://web.archive.org/web/20140817055237/http:/www.astonmartin.com/en/cars/the-new-db9/db9-design.

22.“StarTrek:DesigningtheEnterprise,”Walter“Matt”Jeffries,http://www.mattjefferies.com/start.html.

23.DarrinCrescenzi,“WhytheGoldenRatioMatters,”Medium,April21,2015,https://medium.com/@quick_brown_fox/why-the-golden-ratio-matters-583f6737c10c.

24.Ibid.

CHAPTERV

https://web.archive.org/web/20090120115741/http:/www.kolumbus.fi/lea.tedder/OKAD/Gizaplan.htm

http://www.7wonders.org/europe/greece/olympia/zeus-at-olympia/

https://www.youtube.com/watch?v=9FFDJK8jmms

http://www.nexusjournal.com/reviews_v3n1-Ostwald.html

https://www.emporis.com/buildings/114294/united-nations-secretariat-building-new-york-city-ny-usa

https://visit.un.org/sites/visit.un.org/files/FS_UN_Headquarters_History_English_Feb_2013.pdf

https://web.archive.org/web/20140817055237/http:/www.astonmartin.com/en/cars/the-new-db9/db9-design

http://www.mattjefferies.com/start.html

https://medium.com/@quick_brown_fox/why-the-golden-ratio-matters-583f6737c10c

CHAPTERV1.StephenMarquardt,LecturetotheAmericanAcademyofCosmeticDentistry,April29,2004

2.RichardPadovan,Proportion:Science,Philosophy,Architecture(NewYork:Routledge,1999).

3.ScottOlsen,TheGoldenSection:Nature’sGreatestSecret(Glastonbury:WoodenBooks,2009).

4.AlexBellos,“Thegoldenratiohasspawnedabeautifulnewcurve:theHarrissspiral,”TheGuardian,January13,2015,https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/jan/13/golden-ratio-beautiful-new-curve-harriss-spiral.

5.“Insects,Spiders,Centipedes,Millipedes,”NationalParkService,lastupdatedOctober17,2017,https://www.nps.gov/ever/learn/nature/insects.htm.

6.EvaBianconi,AllisonPiovesan,FedericaFacchin,AlinaBeraudi,etal,“Anestimationofthenumberofcellsinthehumanbody,”AnnalsofHumanBiology40,no.6(2013):463-471,https://www.tandfonline.com/doi/full/10.3109/03014460.2013.807878.

7.RichardR.Sinden,DNAStructureandFunction(SanDiego:AcademicPress,1994),398.

8.“Chromatin,”modENCODEProject,lastupdated2018,http://modencode.sciencemag.org/chromatin/introduction.

9.EdwinI.Levin,“Theupdatedapplicationofthegoldenproportiontodentalaesthetics,”AestheticDentistryToday5,no.3(May2011).

CHAPTERVI1.AriSihvola,“Ubimateria,ibigeometria,”HelsinkiUniversityofTechnology,ElectromagneticsLaboratoryReportSeries,No.339,September2000,https://users.aalto.fi/~asihvola/umig.pdf.

2.J.P.Luminet,“Dodecahedralspacetopologyasanexplanationforweakwide-angletemperaturecorrelationsinthecosmicmicrowavebackground,”Nature425(October9,2003)593-595.

3.Dr.DavidR.Williams,“MoonFactSheet,”NASA,lastupdatedJuly3,2017,https://nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html.

4.Dr.DavidR.Williams,“VenusFactSheet,”NASA,lastupdatedDecember23,2016,

https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/jan/13/golden-ratio-beautiful-new-curve-harriss-spiral

https://www.nps.gov/ever/learn/nature/insects.htm

https://www.tandfonline.com/doi/full/10.3109/03014460.2013.807878

http://modencode.sciencemag.org/chromatin/introduction

https://users.aalto.fi/~asihvola/umig.pdf

https://nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html

https://nssdc.gsfc.nasa.gov/planetary/factsheet/venusfact.html.5.Mercury,theinnermostplanet,hasanorbitalperiodof87.97days,about.2408ofoneEarthyear.Thisnumbervariesonly2.0%from1/Ф3.Saturn,theoutermostvisibleplanet,hasanorbitalperiodof10759.22days,whichis29.4567timesoneEarthyear.Thisnumbervariesonly1.5%fromФ7.Theseare,perhaps,justcoincidences,butwhilewe’reatithere’sonemore:TaketheratioofthemeandistancefromthesunofeachplanetfromMercurytoPluto(yes,weknow)totheonebeforeit.StartwithMercuryas1andthrowinCerestorepresenttheasteroidbelt.Theaverageoftheserelativedistancesis1.6196,avarianceoflessthan0.1%fromФ.

6.JohnF.Lindner,“StrangeNonchaoticStars,”PhysicalReviewLetters114,no.5(February6,2015).

7.P.C.W.Davies,“ThermodynamicphasetransitionsofKerr-NewmanblackholesindeSitterspace,”ClassicalandQuantumGravity6,no.12(1989):1909-1914.DOI:10.1088/0264-9381/6/12/018.

8.N.Cruz,M.Olivares,&J.R.Villanueva,EuropeanPhysicalJournalC,no77(2017):123.https://doi.org/10.1140/epjc/s10052-017-4670-7

9.J.A.Nieto,“Alinkbetweenblackholesandthegoldenratio”(2011),https://arxiv.org/abs/1106.1600v1.

10.L.Bindi,J.M.Eiler,Y.Guanetal.,“Evidencefortheextraterrestrialoriginofanaturalquasicrystal,”ProceedingsoftheNationalAcademyofSciences109,no.5(January1,2012):1396-1401,https://doi.org/10.1073/pnas.1111115109.

11.EricW.Weisstein,“IcosahedralGroup,”MathWorld—AWolframWebResource,http://mathworld.wolfram.com/IcosahedralGroup.html.

12.R.Coldea,D.A.Tennant,E.M.Wheeleretal.,“QuantumcriticalityinanIsingchain:experimentalevidenceforemergentE8symmetry,”Science327(2010):177-180.

13.See“2004DowJonesIndustrialAverageHistoricalPrices/Charts”athttp://futures.tradingcharts.com/historical/DJ/2004/0/continuous.html.

14.See“2008DowJonesIndustrialAverageHistoricalPrices/Charts”athttp://futures.tradingcharts.com/historical/DJ/2008/0/continuous.html.

15.VladimirALefebvre,APsychologicalTheoryofBipolarityandReflexivity(Lewiston,NY:EdwinMellenPress,1992).

APPENDIXA

https://nssdc.gsfc.nasa.gov/planetary/factsheet/venusfact.html

https://doi.org/10.1140/epjc/s10052-017-4670-7

https://arxiv.org/abs/1106.1600v1

https://doi.org/10.1073/pnas.1111115109

http://mathworld.wolfram.com/IcosahedralGroup.html

http://futures.tradingcharts.com/historical/DJ/2004/0/continuous.html

http://futures.tradingcharts.com/historical/DJ/2008/0/continuous.html

1.“Apophenia,”Merriam-WebsterOnline,https://www.merriam-webster.com/dictionary/apophenia.

https://www.merriam-webster.com/dictionary/apophenia

ACKNOWLEDGMENTS

IfirstwroteafewpagesonthegoldenratioinconjunctionwithanotherwebsiteIdevelopedin1997,initiallyasawaytolearnsomethingaboutpublishingontheInternet.Inevercouldhaveimaginedhowitwouldtakeonalifeofitsown,asIbegantobecontactedbypeoplearoundtheglobeandinallwalksoflifewhosharedacommoninterestinthistopic.In2001,Iacquiredaseparatedomain,goldennumber.net,andcontinuedtobeamazed.Thesitetoppedthesearchenginerankingsandreceivedmorethanamillionvisitsperyear,asvisitorsaskedquestionsandcontributedideastocreateanonlinecommunityofinformationexchangeonthistopic.Manynewfriendshipswereformedfromthis,andmyfamilysawmoreofmytimeandinterestfocusingonthispursuit.ItwouldbeimpossibleformetonameandthankallwhohavecontributedtowhatIhavelearnedaboutthisfascinatingtopic,andthesmallpartofallthereistoknowaboutitthathasbeenincludedinthisbook.SomearenamedontheContributorspageofmysite,butIwouldliketorecognizethosewhosesupporthasmeantthemostinmakingthisbookareality:

KathyMeisner,mywife,partner,andbestfriend,whoseloveandsupportwereessentialandsoappreciatedduringthesurprisingnumberofhoursthatwerededicatedtothisbook.Kathy,anexcellentwriterandpublishedauthor,wasmycontinualsoundingboardforallaspectsofthisproject,andprovidedinvaluablecounsel,ideas,andguidancetome.

JulieMeisnerandKatieLeggett,mydaughters,fortheirloveandappreciationofafatherwhowasfrequentlyengagingwithkindredspiritsontheInternet,andforgivingtheirencouragementandoccasionalposesforphotostobeanalyzedforgoldenratios.

RobertMeisnerandKathleenMeisner,myparents,fortheloveandsupport,andtheverylifetheygaveme.

Dr.StephenR.Marquardt,thegloballyrecognizedexpertonfacialbeauty,forhisinvaluablecontributionsintheunderstandingofhumanattractivenessthroughhisdevelopmentofthegoldenratio-basedMarquardtBeautyMask,andforhisfriendship,intellectualcamaraderie,inspiration,counsel,andsupport.

Dr.EddyLevin,recognizedforhiscontributionsonthegoldenratioincosmeticdentistry,forhisfriendship,insights,andsupport.

MelanieMadden,myeditor,forherexcellenteditorialexpertiseandguidance,andevenmoreforherintellectandintellectualcuriositythatledtobringingnewcontenttothebookandchallengingmetoinvestigateareasthatIhadnotyetexploredtomakethepresentationmoreaccurateandcomplete.I

http://goldennumber.net

hadnotyetexploredtomakethepresentationmoreaccurateandcomplete.IlearnedmoreonthistopicinthelastyearofwritingthisbookthanIeverwouldhaveexpectedattheoutset.

QuartoPublishingGroupandRacePointfortheirinterestandconfidenceinmeastheirchosenauthorfortheirbookonthistopic,andfortheteamtheydedicatedtoprovidingtheuniquecreativevisionforaveryartisticandprofessionalpresentationofthisthegoldenratio.

RafaelAraujo,forhisbeautifulillustrationsonthecoverandchapteropeningsofthebook.

God,foropeningmyheart,mind,andeyestoseethebeautyandwonderthatisallaroundusandwithinus.

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INDEX

Фsymbol,103

AbuKamilShujaibnAslam,39TheAdorationoftheMagi(Botticelli),77Africanspiralaloeplant,149Agnew,H.C.,96Alhambrapalace,193Alma-Tadema,Lawrence,106AltarpieceofSaintMark(Botticelli),80animalscats,164chameleons,161conchshells,162five-foldsymmetry,152primates,165screwshell,162seahorses,161seaurchins,152,153starfish,152,153

Annunciation(LeonardodaVinci),66Annunciation(SandroBotticelli),77,78,79,80apples,152Archimedeanspiral,156Archimedeantruncatedicosahedron,61Archimedes,195ArtFormsinNature(ErnstHaeckel),153AstonMartinautomobiles,138AthenaParthenos(Phidias),104

TheBaptismofChrist(PierodellaFrancesca),62,63Barr,Mark,103BaselMinster,156BathersinAsnières(GeorgesSeurat),120Bernoulli,Jacob,156BirdsandNature,145TheBirthofVenus(SandroBotticelli),75,77blackholes,188,189bluepassionflower,151Bonnet,Charles,145BookofAlgebra(AbuKamilShujaibnAslam),39Botticelli,Sandro,75,77BPlogo,137Brady,Oliver,13

BridgeofCourbevoie(GeorgesSeurat),124Bronnikov,Fyodor,16Brunelleschi,Filippo,115buckyballs,190,195

Caldwall,James,145Cambio,Arnolfodi,115caricatures,177Cartesianspace,32,33CathedralofSaintMaryoftheFlower,115,116cats,164CattedralediSantaMariadelFiore,115,116TheCestelloAnnunciation(Botticelli),78chameleons,161TheChannelofGravelines,PetitFortPhilippe(GeorgesSeurat),124ChartresCathedral,111,112,113,114,203ChristasSavioroftheWorld(LeonardodaVinci),72Chrystal,George,9Citéradieuse(RadiantCity),127Clark,Carmel,13Clark,Kenneth,62Codea,Radu,196Coltellini,Michele,74TheCompendiousBookonCalculationbyCompletionandBalancing(MuhammadibnMusaal-

Khwarizmi),38conchshells,162Cook,TheodoreAndrea,103Corbusierhaus,127cosmosblackholes,188,189Earth,183,185Moon,183planetarymotion,182,185stars,186,187Sun,185Venus,185

Coxeterplaneprojections,196,197TheCreationofAdam(Michelangelo),84,85TheCreationofEve(Michelangelo),86Crescenzi,Darrin,139,141Cruz,Norman,189Curl,Robert,195TheCurvesofLife(SirTheodoreAndreaCook),103

Dambrun,Jean,16Darb-iImamshrine,193

Dash,Glen,99Davies,Paul,189TheDaVinciCode(DanBrown),66daVinci,Leonardo,12,59,60,61,66,68,69,74,149,150DeArchitectura(Vitruvius),108DeBruine,Lisa,167DeDivinaProportione(LucaPacioli),59,61,62,66,74Dell,Susan,140DeProspectivaPingendi(PierodellaFrancesca),62Descartes,René,68,156discovery,16Disneylogo,137DNA,170dodecahedron,17,25,31,32,33,58,61,182,183,195DowJonesIndustrialAverage,200TheDrunkennessofNoah(Michelangelo),87Dürer,Albrecht,58

E8Coxeterplaneprojections,196,197Earth,183,185Echeveriasucculentplants,148Einstein,Albert,22Elements(Euclid)Arabictranslationof,20BookII,Proposition5,8BookIII,Propositions8,19BookVI,Proposition11,22BookVI,Proposition30,22BookXIII,Proposition1,23BookXIII,Proposition2,23BookXIII,Proposition3,23BookXIII,Proposition4,24BookXIII,Proposition5,24BookXIII,Proposition6,24BookXIII,Proposition8,24,26BookXIII,Proposition9,25BookXIII,Proposition17,25circulationof,58descriptionofФ,10

EncyclopediaBritannica,9EpitomeAstronomiaeCopernicanae(JohannesKepler),30ethnicity,175Euclid,10,19,24“EuclidAloneHasLookedonBeautyBare”(EdnaSt.VincentMillay),19extremeandmeanratio,8,17,22,24

faces,164,165,167,173,179TheFallandExpulsionfromtheGardenofEden(Michelangelo),85“FamousMen”series(JustusofGhent),19TheFashionCode,140fashiondesign,140Fibonacciheap,199Fibonacci,Leonardo,39,42,103Fibonaccinumbers,146,166,169Fibonaccisearchtechnique,199Fibonaccisequence,43,47,48,199Fibonaccispiral.Seegoldenspiral.Fibonaccisystem,199Fibonaccitriangle,48financialmarkets,199fingers,166Finklestein,David,189five-petaledviolets,149,150TheFlagellationofChrist(PierodellaFrancesca),62,65forearm,166fractalpatterns,186,199,205fractals,154Francesca,Pierodella,62,74fruit,152Fuller,Buckminster,195

galaxies,161GalileoGalilei,37gambling,199gaugecalipers,57Germanyellowjackets,163Ghyika,Matila,120giantsilkmoths,163Gizasite,93goldenangle,147,148goldengnomon,29goldenratiocalculating,51constructing,26discovery,16

goldenrectangle,31,99,100,107,130,131,132goldenrhombus,192,193goldenspiral,13,46,47,68,146,154,157,160goldentriangle,29,34Googlelogo,136Gosset,Thorold,196GreatPyramidofGiza,93,99,184

GreatSphinx,102Greekletters,9Gutenberg,Johannes,19

Haeckel,Ernst,153hand,166HarmonicesMundi(JohannesKepler),30,182Herodotus,96Hertzsprung–Russelldiagram,187HeydayofGreece(AugustAhlborn),106Hollar,Wenceslaus,74Ho-Mg-Znquasicrystal,192,193HouseofWisdom(Baghdad),38TheHumanFacedocumentary,173humanbodycaricatures,177DNA,170ethnicity,175face,167,173,179fashion,140forearm,166hand,166head,168indexfinger,166teeth,178

icosahedrite,192icosahedron,31,182,195indexfinger,166insects,163IntroductiontoAlgebra(GeorgeChrystal),9IslamicGoldenAge,39,40

Jahan,Shah,118Jefferies,Matt,138,139JohntheBaptist(LeonardodaVinci),12Jones,Ben,167JuliaTitiFlavia,174JustusofGhent,19

Kepler,Johannes,15,17,27,30,53,95,181,182Keplertriangle,27,95,184KhafrePyramid,99KhufuPyramid,93,99,184al-Khwarizmi,MuhammadibnMusa,38,39Kochsnowflake,154Kroto,Harry,195

KunstformenderNatur(ErnstHaeckel),153al-Khwarizmi,38

Lagrange,JosephLouis,50Lahori,UstadAhmad,118TheLastSupper(LeonardodaVinci),69LeCorbusier,126,134Lefebvre,VladimirA.,200Levin,Eddy,169,178Levy,Ruth,140Levy,Sara,140LiberAbaci(LeonardoFibonacci),39,42,45TheLighthouseatHonfleur(GeorgesSeurat),121LimburgCathedral,114Lindner,John,186logarithmicspiral,156lotusflowers,148Lucas,Edouard,45Luminet,Jean-Pierre,183luteofPythagoras,154

MadonnadellaMisericordia(PierodellaFrancesca),64Maestlin,Michael,51Mahal,Mumtaz,118MarquardtBeautyMask,173Marquardt,StephenR.,143,169,173Martingaleprogression,199Mästlin,Michael,28Matsys,Quentin,177MenkaurePyramid,99Michelangelo(DanieledaVolterra),84Millet,Aimé,104Modestini,DianneDwyer,72Modulardesignsystem,126,127,128,132molecularstructuresbuckyballs,190,195quasicrystals,191MonaLisa(LeonardodaVinci),68Moon,183morningglory,151moths,163MysteriumCosmographicum(Kepler),17,30,31

NationalGeographiclogo,137nautilusshell,157TheNavvies(GeorgesSeurat),122Nefertiti,174

nestfern,161NeueLehrevondenProportionendesmenschlichenKörpers(AdolphZiesing),144ANewIllustrationoftheSexualSystemofCarolusvonLinnaeus(RobertJohnThornton),145NewDoctrineoftheProportionsoftheHumanBody(AdolphZiesing),144Niemeyer,Oscar,127,131Nieto,J.A.,189Nissanlogo,137Notre-DameCathedral,109,110,111,196,197numericreduction,49

Occam’srazor,166Ohm,Martin,9okra,152Olivares,Marco,189OnPerspectiveforPainting(PierodellaFrancesca),62orchids,161origamitriangle,29Ostwald,MichaelJ.,126Oxyrhynchuspapyri,8

Pacioli,Luca,55,58,59,74,195,205Padovan,Richard,130Paganucci,Giovanni,43papaya,152Parthenon,103,104,105,106,107Pascal,Blaise,44Pascal’striangle,44PeasantwithaHoe(GeorgesSeurat),122Penrose,SirRoger,34Penrosetiles,29,34,192Pentagonconstruction,26,193pentagram,17,29,154people.Seehumanbody.periwinkle,151“ThePhiCollection”(SusanDell),140Phidias,103,104,106PhiMatrixsoftware,11,57,62,82,162,168,174phyllotaxis,145pinecones,144,146planetarymotion,182,185plantsAfricanspiralaloeplant,149apples,152bluepassionflower,151Echeveriasucculent,148five-foldsymmetry,152

five-petaledviolet,149,150fractals,154fruit,152goldenangle,147lotusflower,148morningglory,151nautilusshell,157nestfern,161okra,152orchids,161papaya,152periwinkle,151pinecones,146Plumeria,151Romanescobroccoli,154sacreddatura,151starfruit,152sunflowers,146,147,204Theobromacocoaplant,152

Plato,17,183Platonicsolids,17,31,182PlinytheElder,145Plumeriaflower,151polyhedrons,61PolyptychoftheMisericordia(PierodellaFrancesca),62,65primates,165TheProcessionoftheYoungestKing(BenozzoGozzoli),76Proportion:Science,Philosophy,Architecture(RichardPadovan),130APsychologicalTheoryofBipolarityandReflexivity(VladimirA.Lefebvre),200PsychoMorphsoftware,167ThePureElementaryMathematics(MartinOhm),9PyramidsofQueens,101Pythagoras,17,27,29,30Pythagorean3–4–5triangle,28

quasicrystals,191quasi-periodicity,194

RadiantCity,127Raphael,82Recherchessurl’usagedesfeuillesdanslesplantes(CharlesBonnet),145DieReineElementar-Mathematik(MartinOhm),9RemarksonRuralScenery(JohnThomasSmith),134ResearchontheUseofLeavesinPlants(CharlesBonnet),145RhindMathematicalPapyrus,97Roddenberry,Gene,138

Romanescobroccoli,154RRLyrae-classstars,186,187ruleofthirds,134

sacreddaturaplant,151SacredGoldenRatio(OliverBradyandCarmelClark),13SainteMariedelaTouretteconvent,127SalvatorMundi(LeonardodaVinci),72,73,74al-Saud,BaderbinAbdullahbinMohammedbinFarhan,72scarabbeetles,163TheSchoolofAthens(Raphael),82screwshell,162seahorses,161seaurchins,152,153Secretariatbuilding,127TheSeineatLaGrandeJatte(GeorgesSeurat),125sekedmeasurements,97Self-portrait(Raphael),82Semper,Godfried,108TheSeparationoftheEarthfromtheWaters(Michelangelo),86Seurat,Georges,91,120,134Shah-i-Zindanecropolis,193Shechtman,Dan,191,192,193ShortBookon(the)FiveRegularSolids(PierodellaFrancesca),62Sierpinskitriangle,154Simon,Robert,72Simson,Robert,45SistineChapel,84,85,87,88,89skeletonicsolids,60,195Smalley,Richard,195Smith,JohnThomas,134Solomon’sTemple(Jerusalem),61Sphinx,102Squareconstruction,26starfish,152,153starfruit,152StarTrektelevisionshow,138,139Stiftskirche(“CollegiateChurch”),114stockmarkets,199St.Peter’sBasilica,56,89Stradivariusviolins,136St.VincentMillay,Edna,19Sulley,James,9Sully,Mauricede,111SummaryofArithmetic(LucaPacioli),59Sun,185

ASundayAfternoonontheIslandofLaGrandeJatte(GeorgesSeurat),120,121sunflowers,146,147,204

TajMahal,118Talenti,Francesco,115Tedder,Chris,99teeth,178TempleofZeus(Olympia),103Theobromacocoaplant,152Thornton,RobertJohn,145ThreeLinesconstruction,26Tiddeman,Bernard,167Timaeus(Plato),17,183Toyota,137Triangleconstruction,26truncatedicosahedron,195TumanAkamausoleum,193al-Tusi,Nasiral-Din,20TyphoonSonca,160

TheUglyDuchess(QuentinMatsys),177Unitéd’HabitationofBerlin.SeeCorbusierhaus.UNSecretariatbuilding,128,130

Venus,185Vernet,Horace,56Villanueva,J.R.,189Vinci,Leonardoda,134VitruvianMan(LeonardodaVinci),69,72,166Vitruvius,108Vogel,Helmut,146Volterra,Danieleda,84volutes,47

WhirlpoolGalaxy,161WilliamofOckham,166WMAPradiationdata,183WomanwithUmbrella(GeorgesSeurat),123

yellowjackets,163

Ziesing,Adolf,144

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