Observed Galaxy Distribution Transition with Increasing Redshift a Property of the Fractal

ObservedGalaxyDistributionTransitionwithIncreasingRedshifta

PropertyoftheFractal

BlairD.Macdonald

FirstPublished:November20th2014.

Updated:June13th2016

Abstract

Istheuniverseafractal?Thisisoneofthegreat–thoughnotoftentalkedabout–

questionsincosmology.Inmyearlierpublication–whereIinvertedthe(Koch

snowflake)fractal–Ishowedthefractaldemonstrated:Hubble’sLaw,accelerating

expansion,allfroma‘singularity’bigbanglikebeginning.Surveysoftheuniverse–the

mostrecentandlargest,the2012WiggleZDarkEnergySurvey–show,galaxy

distributiononsmallscalestobefractal,whileonlarge-scales,homogeneityholds.

Thereappearstobenewanomalytoexplain:thereisagalaxydistributiontransition

fromroughtosmoothoncosmicscales.Frommy‘fractspansion’modelIderiveda

Fractal-Hubblediagram;onthisdiagrammeasurementpointsalongthecurveare

clusteredneartheorigin.Thisclusteringwasnotaddressedindiscussionsorpartof

theconclusionofmyearlierexperiment.Canthisclusteringofthesepointstheon

fractalHubblediagramaccountfortheobservedgalaxydistributiontransition–from

roughtosmooth?Couldthistransitionbeanotherpropertyoffractals,andtherefore

couldtheuniverse–itself–befractal?Itwasfound,yesthey–thepoints–do.

Clusteringofmeasurementpoints(andofgalaxies)isasaresultofobservationposition

inthefractal.Onsmallscales–relativetolargescales–thecosmicsurveysarewhat

onewouldexpecttoseeifonewereviewingfromwithinaniterating–growing–

fractal.Iftrees–naturalfractalsthathavealsobeenfoundtogrowatacceleratingrates

–areusedtodemonstratethisfractal:thelarge-scalesmoothnessmaybeakintoa

tree’strunk;andtherough(fractal)onsmall-scales,toitsbranches.Thisdiscovery

unifiestheanomaliesassociatedwiththestandardcosmologicalmodel.Togetherthey

are–throughthemechanicsofthefractal–inextricablylinked.

Keywords:FractalCosmology,WiggleZ,Hubble’sLaw,GeneralRelativity,GalaxyDistribution,CosmologicalPrinciple

ObservedGalaxyDistributionTransitionwithIncreasingRedshiftaPropertyoftheFractal

BlairD.Macdonald November2014

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1 INTRODUCTION

Theuniversehasbeenarguedtobe,andobservedtobe,fractalonsmall-scales,and

homogenousoflarge-scales.InthispublicationIwouldliketoprovideanexplanation

tothisobservedgalaxydistributiontransition,andshow–basedonmyearlierfindings

–theobservedhomogeneityatlargescalesisapropertyofthefractal.Myexplanation

willunifythegalaxydistributiontransitionswithotherobservedanomaliesofthe

ΛCDMmodel.

Inmyrecentpublication,‘FractalGeometryaPossibleExplanationtotheAccelerating

ExpansionoftheUniverseandOtherStandardΛCDMModelAnomalies’[1]–Irana

simpleexperimentonafundamental(KochSnowflake)fractaltomodelitsexpansion–

asitwouldbeviewedfromwithinit,andfromafixedposition.Ishowedtheiterating

fractalexplainsanddemonstratesmanyoftheobservationalanomaliesassociatedwith

thestandardΛCDMmodelofcosmology–asingularitybeginning,increasing

recessionalvelocity,andacceleratingexpansionoftheobserveduniversewithrespect

todistance.Iconcludedtheuniverseisafractal.Frommy‘fractspansion’(fractal-

expansion)modelIdeveloped–whatItermed–aFractal-Hubblediagram(figure1

below),andconcludedtheclusteringofmeasurementpoints(crosses)neartheorigin

ofthediagram–followedbytheincreasingwideningofthem–demonstratedwhy

universeishomogeneousonlargescales.Thewideningofthespacingbetweenthe

measurementpoints(withdistance)wasconsistentwiththecosmologicalprinciple.

Themorethefractaliterates,thesmootheritbecomesatlargescales.WhatIneglected

toanalysisordiscussinthediscussionswas:whytherewasclusteringofmeasurement

pointsclosetotheorigininthefirstplace?



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Figure 1. Measurement point clustering on the Fractal-Hubble diagram. Measurement points (crosses) are

clustered near the origin, and spread out as distance between triangles geometric centres increases.cm=

arbitrary length unit. cm = centimetre.

Independentofmywork,thequestionofwhetherornottheuniverseisafractalhas

indeedbeendebated–andstudiedoncosmicscales[2][3],[4]–thoughthisdebateis

notoftenmentionedinmainstreamorevenpopularscience.

Theproponentsoffractalcosmology(LucianoPietronero,FrancescoSylosLabini,and

others)havebeenarguingtheobservedhierarchiesofclusteringandsuperclustering–

basedonsurveyssimilartothe20032dfRedshiftSurveymap(figure2below)–are

directevidenceofauniversethatisfractal[5],[6],[7],[8],[9],[10],[11].Thecurrent

consensus,however,is–afterevendeepercosmicsurveys–theuniverseonlarge

scalessmoothensouttobecomehomogenous–andisoverallnot(a)

fractal[12],[13],[14].

y=0,6667xR²=1

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20 000

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0 10 000 20 000 30 000 40 000 50 000 60 000

velocitycm

-1i-

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distancecm-1



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Figure 2. 2dF Galaxy Redshift Survey [15]

Itwasthe2012WiggleZDarkEnergySurvey[16]–thelargestsurveytodate–that

settledthefractalquestion,andalsocaughtmyattention.Theyconcluded–with

increased,butsimilarconfidenceaspreviousteams(namely,the2004SloanDigitalSky

Survey[17],[18])–theuniverseshowsevidenceoffractalgalaxydistribution–with

clusteringandsuper-clustering–onlyonsmallscales(lessthan70to100Megaparsecs

away)–beyondthisdistance,thepatterngoesthroughtransitiontohomogenous

galaxydistribution.

OnviewingtheWiggleZDarkEnergySurveyresults–particularlyitsfigure13(figure3

below)ofchanginggalaxydistribution(fromfractalatsmallcosmicscalestosmoothat

largecosmicscales)IquestionedwhethertheinvertedFractal-Hubblemeasurement

pointclusteringneartheoriginmayofferexplanationandinsighttothedecreasing

‘ScaledN(r)’transitions(andincreasingsmoothness)withdistance.



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Figure 3. WiggleZ Dark Energy Survey figure 13, page 16. Revealing changing galaxy distributions from

small-scale to large-scale.

InthispublicationIshallinvestigatewhethertheresultsfromtheWiggleZDarkEnergy

Survey(anditscontemporises)matchaviewofwhatonewouldexpecttoseeifthey

viewed,notatafractal–asisimpliedintheirstudies–butfromwithina(growing)

fractal.TotestthisIshallreturntothefractspansionmodelandanalysetheoccurrence

ofthesaidclusteringofmeasurementpoints.Fortheclusteringinthefractspansion

modeltohaveanysignificancetocosmology,itwouldhavetodemonstratehowthe

distributionoftriangles(intheinvertedKochsnowflake)changesoverdistancefrom

theobserver(section4.1and4.2).

2 METHODS

Toanalysisandexplainthepointclusteringonthefractal-Hubblediagram,relevant

tablesinfractspansionspreadsheetmodel[19]wereanalysed–particularlythetable

fromwherethefractal-Hubblediagramwasderived.Calculationsweremadeand

diagramscreated–thesemayalsobeviewedinthespreadsheetmodel:

1. Thequantityoftrianglesizespertotaldistanceincrementonthefractal-Hubble

diagramwascalculatedby:countingthequantityoftrianglesizes(indistance



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columnintable2)anddividingthisbythedistanceincrementsmeasuredinthe

sample.SeeTable2aofspreadsheetmodel.

2. Thequantityoftrianglesateachincrementwascalculatedbytotallingthe

quantityoftriangles(fromtable4)foreachrespectiveiteration-distance.

3. AnamendedFractal-Hubblediagram–combining(recessional)velocitywiththe

quantityoftrianglesateverydistance–wascreated.Seetable7ofspreadsheet

model.

3 RESULTS

Theresultsareasfollows:

1. 8ofthe10measurementpointsarelocatedinsidethefirst(1.20E+4cm-1)

incrementdistance.Theremaining2measurementpointsareoutsidethisrange.

2. Figure3belowshowsthequantityoftrianglesbydistance–betweengeometric

centresfromtheobserver.Thequantityoftrianglesdecreasedexponentially

from7.86E+05,atiteration-distance0,toaquantityof1atdistance51800cm-1

(iteration-10).

Figure 3. Quantity of triangles at each distance (point) from the observer on the inverted Koch Snowflake fractal. As the distance between triangle geometric centres increases (exponentially) with iteration, and so

increasing the distance from the observer, the quantity of triangles per iteration decreases exponentially to a

quantity of one – at time 0. cm = centimetre.

y=3E+06e-1,373xR²=0,99973

0

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200 000

300 000

400 000

500 000

600 000

700 000

800 000

900 000

0 2 7 23 70 212

639

1920

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51800

quan

tityoftriang

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distancecm-1



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4 DISCUSSIONS

4.1 ClusteringofMeasurementPoints

Atthepointofobservation(theoriginontheFractal-Hubblediagram)thereisa

quantityof786,432triangles,allofwhicharethesamesizeastheobserverstriangle

viewingposition.Theclusteringofthemeasurementpointsneartheoriginofthe

diagramisduetothelocationtheobserverwithinthe(inverted)fractal,andthe

relativesizeofthesetrianglesneartheobserver.Theobserverwill,inprinciple,be

surroundedbythesesizedtriangles.Theobserverwillnotseeallthesetriangles,how

manytheywillseeisbeyondofthescopeofthisinvestigation,butitwillbemany.As

weviewfurtherout,thequantityoftrianglesdecrease–whiletheareaofthe

respectivetrianglesincrease.Thispropertyofclusteringneartheoriginisscale

invariant:nomatterthedistance,thispatternofclusteringneartheoriginwillremain.

4.2 ClusteringandtheFractal-HubbleLaw

Figure4(below)combines–ontheoriginal(Kochsnowflake)fractal-Hubblediagram–

thequantityoftrianglesateachdistancepointwiththe(recessional)velocityatthe

samedistancepoint.Thediagramrevealstherelationshipbetweentheclusteringof

measurementpointsclosetothe(lowrecessionalvelocity)origin,andthesmooth

distribution(highrecessionalvelocity)atlargedistances.



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Figure 4. The inverted (Koch snowflake) fractal, Fractal-Hubble Diagram combined with the quantity of triangles at each distance from the observer . As distance increases with respect to iteration-time: the

recession velocity of distance between geometric points increases; while the quantity of triangles at each

distance decreases. cm = centimetre.

4.3 Treemetaphor

Iftheobservationfromdeepwithinasnowflakefractalissubstitutedwithobservation

fromhighwithinacommonbranchingtree(alsoanaturalfractal):theclusteringof

pointsontheFractal-Hubblediagramwouldequallycorrespondtotheclusteringof

self-similar(sized)branches–inthetree–surroundingtheobserver.Iftheobserver

weretolookdown,inwardsfromtheouterbranches–towardsthetrunkofthetree–

thebranch(nodes)quantitywoulddecrease,thevolumeofthesinglebrancheswould

increase,andthebranch‘clustering’wouldsmoothout.

4.4 TheUniverse

Thedistributionofmeasurement-pointclusteringalongtheFractal-Hubblediagram

matchesthetransitionsfromroughtosmoothasrevealedin(recent)galaxysurveys.

Thedistributionofgalaxiesintheuniverse(shownaboveinfigure1)isofanature

expectedifonewasviewingfromwithinafractaluniverse:lookingbackthroughthe

y=0.6667xR²=1

0

5000

10000

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universe–intermsofdistanceandtime–toits–nowexpanded–origin.Thesmooth

outerreachesoftheuniverse–outneartheCMBoriginoftheuniverse–isthe‘trunk’

oftheuniverse,andtheroughfractalclustersare‘thebranches’oftheuniverse.Using

atreeasametaphoroftheuniverseisnottosaytheuniverseisafractaltreestructure:

itistosay,justasatreeisafractalstructure,theuniverseisafractalstructure.It

shouldbenotedtreesgrowthhaverecentlybeenfoundtoalsoincreaseataccelerating

rates[20],[21].

4.5 RaisedQuestions

Therearemanyquestionsandissuesarisingfromthisfinding–allofwhich,atthis

point,arebeyondthescopeofthisinvestigation,butnotbeyondthescopeofreason.

4.5.1 Super Clusters

Proponentsoffractalcosmologyareexpectingtoseeevenlargergalacticclusters

furtheroutintothelarge-scalehomogeneousregion.Thefractspansionmodelwould

concurwiththis,onlythatthedistance(inprinciple)tothenextcluster(nextlarger

branchornode)maybebeyondtheageoftheuniverse–andormaynotexistatall.

4.5.2 Emergent Structure

Afractaluniversewouldimplyanemergentstructure–thewholemadeofmanyparts

–justasthetreeismadeofmanybranches.Itmayforceustoquestiontheinitial

conditionsofthebigbangbeginning.Namely,whetherallmass(intheuniverse)was

togetherinoneplaceandatonetime.Itcouldnowbeargued–fromtheprinciplesof

fractalemergence–theuniversedeveloped/evolvedmassfromthebottomup,withthe

passingoftime.

4.5.3 Growth Explanation

Ifthebranchesofthetreeareakintothesmall-scalegalaxyclustersoftheuniverse,we

mayfinditprofitabletosearchforgrowthexplanationstotheuniverseinthebranches

too.Givenitisatthebrancheswheretreesgrowformandthetrunkandfirstbranches

areonlyinfrastructuretothetotalemergentstructure.Thiswouldsuggestgrowth

beginsatthesmallestofscales:atthesub-atomiclevel,thePlanckscale.



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Thequantumnatureofthefractalhasbeenaddressedinmyoriginalpublication–and

mustbefurtherinvestigated[22].

4.5.4 Dark Matter

Beyondthescopeoftheinvestigationiswhetherthisfractalstructureoffersinsightto

thedarkmatterstructureoftheuniversealso?Giventhecurrentmappedstructureof

thedarkmatter,andthestructureoftheobserved‘fractal’universe,Iwouldsuggest

yesitdoes.

4.5.5 General Relativity

WhatafractaluniversemeansforthefutureofGeneralRelativitytheoryisunclearand

beyondthescopeoftheauthor–thoughitisconceivableitmayhavetobeadaptedto

takeaccountthegeometryofthefractal.Workhasalreadybeguninthisarea:from

notedtheoristLaurentNottale[23],[24]andothers[25].

4.5.6 Fractal Dimension

Recentstudieshaveshownfractaldimensiondecreaseswithincreasedzvalues[26].

Thismayalsocomplementmystudy.

4.6 Conclusion

Themodeloffractspansion(fractal-expansion)demonstratestheuniverse’sgalaxy

distributiontransitionfromrough(fractal)onsmall-scales,tosmooth(homogeneous)

onlarge-scales.Thisdemonstrationcannowbecombinedwiththemodelsoriginal

demonstrations:asinglebeginning;aCMB;Hubble’slaw;andisexpansionatan

acceleratingrate–lambda.TheresultsshowstrongagreementwiththeWMAP+ΛCDM

modelsoftheuniverse.

Thepropertieslistedareallpropertiesofthefractal,andareinextricablylinkedwith

eachother.Fromobservationsoftheuniverse–atallscales–itcanbeconcludedthe

universeis–byitsnature–fractal.Itlookslikeafractal,andactslikeafractal–itisa

fractal.

Acknowledgments



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Iwouldliketothankmywifeandchildrenfortheirpatienceandsupport.This

publicationonlycameaboutafteranemailexchangewithDr.TamaraDavis.

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Observed Galaxy Distribution Transition with Increasing Redshift a Property of the Fractal

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