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DemonstratingLorenzCurveDistributionandIncreasingGiniCoefficientwiththeIterating(KochSnowflake)FractalAttractor.

BlairD.Macdonald

[email protected]

Abstract

Globalincomehasincreasedexponentiallyoverthelasttwohundredyears;while,and

atthesametimerespectiveGinicoefficientshavealsoincreased:thisinvestigation

testedwhetherthispatternisapropertyofthemathematicalgeometrytermedafractal

attractor.TheKochSnowflakefractalwasselectedandinvertedtobestmodel

economicproductionandgrowth:alltriangleareasizesinthefractalgrewwith

iteration-timefromanarbitrarysize–growingthetotalset.Areaoftrianglethe‘bits’

representedwealth.Kinematicanalysis–velocityandacceleration–wasundertaken,

anditwasnotedgrowingtrianglespropagateinasinusoidalspiral.UsingLorenzcurve

andGinimethods,bitsizedistribution–foreachiteration-time–wasgraphed.The

curvesproducedmatchedtheregularLorenzcurveshapeandexpandedouttotheright

withfractalgrowth–increasingthecorrespondingGinicoefficients:contradicting

Kuznetscycles.The‘gap’betweeniterationtrianglesizes(wealth)wasfoundto

accelerateapart,justasitisconjecturedtodosoinreality.Itwasconcludedthewealth

(andincome)Lorenzdistribution–alongwithaccelerationproperties–isanaspectof

thefractal.FormandchangeoftheLorenzcurveareinextricablylinkedtothegrowth

anddevelopmentofafractalattractor;andfromthis–givenrealeconomicdata–itcan

bededucedaneconomy–whetherculturalornot–behavesasafractalandcanbe

explainedasafractal.Questionsofthediscreteandwavepropertiesandthe

acceleratedexpansion–similartothatoftreesandtheconjecturedgrowthofuniverse

atlarge–ofthefractalgrowth,werediscussed.

Keywords:

Fractalgeometry,Lorenzcurve,GiniCoefficient,Wealthdistribution


BlairD.Macdonald

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1 INTRODUCTION

Theincreasingspreadofincomeandwealthdistribution–anditspossibleacceleration

(Piketty2016)–haslongbeenatthecentreofcontroversyfrombothsidesofthe

economic–andpolitical–spectrum.Thedisparityisalmostalwayspresentedasa

negativeconsequenceofthe‘free’marketandofeconomicgrowthandviewedas‘a

wrong’:somethingthatshouldbecontrolledorstemmed.TheOECDevenpurportitto

bethecauseof‘fallsineconomicsgrowth’(OECD2014).Whatifcouldbeshowntobe

anaspectofagreatermathematicalgeometry:revealingittobeauniversal,natural

phenomenon–apropertyofnature,andindeed‘themarket’itself?Wouldthisrelieve

usofourconcernsandmanyconjectures,muchlikeKepler’sellipticalgeometrydidin

the17thCentaury–completingtheproblemofplanetaryorbits?

Thereisageometrythatmaystandasacandidate,offeringsimilarpotentialtothe

ellipse,itis‘thefractal’andfractalgeometry.Developedduringthelastdecadesofthe

19thCentenary:fractalsgenerallyknowntobeoneofthebestwaystodescribenature.

Thefatheroffractalgeometry,BenoitMandelbrot(1924-1910)–interestingly–saidof

himselftobe‘awouldbeKepler–ofcomplexity’(Wolfram2012).Inhisearlywork

beforefractalsMandelbrotwroteonpowerslawsandshowedtheincomeandwealth

disparitytofollowatypical(‘natural’)Paretopowerlawpattern(B.Mandelbrot1960,

TheGuardian2011),andinhisoriginalworkonfractals,heusedthisgeometryto

describetheshapeandbehaviourofmarketpricesthroughtime(B.B.Mandelbrot

2008).InthispaperIwouldliketocontinuehiswork,andshowthatnotonlyareprices

fractal,butindeedincomedistributionistoo,andbothareaspectsofa(greater)fractal

–oneweterm‘amarket’.Ibelievethereisafundamentalconnectionbetweenpower

laws,fractalsandincomeandwealthdistribution.

Toinvestigatetheabove,anemergentfractalattractorwastestedtowhetherit

producesaLorenzcurveandrespectiveGinicoefficient–andifso,dotheychangewith

growthandwithtime.Isinequalityinextricablylinkedtogrowth?Thehypothesisof

thisstudywasyestheyarelinked:inequalityisanaspectorpropertyofthefractal,and


BlairD.Macdonald

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aneconomy–whetherculturalornot–isafractalphenomenon,bestdescribedby

fractalgeometry.

1.1 TheLorenzCurve

IncomeandwealthdistributionwasfirstrepresentedgraphicallybyM.O.Lorenz’s

1905andthecurveheproduced–theLorenzcurve–namedafterhim(Lorenz1905).

Showninfigure1below:thepercentageofhouseholdsisonthex-axis,andthe

percentageofincomeorwealthonthey-axis.TheLorenzcurveshowsthedistribution

ofindividuals’income(orwealth)foragivenpopulation.TheLorenzcurvealwaysfalls

belowtheline‘perfectequality’(alineofequaldistributionthroughoutthepopulation)

andtheGinicoefficient(termed‘Ginis’forshort)tellstheratiobetweentheLorenz

curveandthelineofequality(areaA),andthetotal(triangle)areaunderthelineof

equality(areasAandB).

Figure 1. Lorenz Diagram. The graph shows that the Gini coefficient is equal to the area marked A divided by

the sum of the areas marked A and B. that is, Gini = A / (A + B). It is also equal to 2*A due to the fact

that A + B = 0.5 (since the axes scale from 0 to 1)(“Gini Coefficient” 2015).


BlairD.Macdonald

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Lorenzdistributionhasalreadybeenshowntobeuniversalorfractal:itisa‘natural’

scaleinvariantpropertyobservedinanywealthorincomedistributionofany

population–whetherculturalornot(DamgaardandWeiner2000),andisasrelevant

tothenaturalsciencesasitistotheclassicaleconomics.EcologistGeeratJ.Vermeij

remarks:“Oneofthemostpervasiveandfar-reachingrealitiesineconomicsystemsis

inequality,thetendencyforonepartyinvolvedinaninteractionoverresourcestogain

more,orless,thanitsrival.”(Vermeij2009)

1.2 HighWealthGiniandExponentialIncomeGrowth

Wealthisastockconceptandincomeaflow,andbothmaybearguedtobecausalto

eachother:incomecreateswealth;wealthcreatesincome.Parkinexplainsthe

differencebetweenthewealthLorenzandtheincomeLorenzcurvesisthewealth

excludeshumancapital:‘wealthandincomearejustdifferentwaysoflookingatthe

samething…becausethenationalsurveyofwealthexcludeshumancapital,income

distributionisamoreaccuratemeasureofincomeinequality’(Parkin,Powell,and

Matthews2008).Notwithstandingthis,bothwealthandincomeproducecomparable

Lorenzcurves–evenifthewealthGinicoefficientsarehigherthanincome.Inthis

studythefocuswasonwealthdistribution–duedirectlytothe‘stock’natureofthe

fractal:forthisreason,therelationshipbetweenwealthandincomeisassumed

inextricableandequivalent.

Realdatashowslocal–bycountry–wealthGinisincreaseonlytobecurtailedbya

possible‘Kuznetseffect’–whereGinisriseandthenarefallwitheconomicgrowthand

developmentdueintervention–orasMilanovicpositsit,technology,opennessand

politics(TOP)(Milanovic2016b).ForAmericaalone,Ginishaveincreased(Fortune

2015),andthewealthproportionof‘the1%’alsoincreasedinAmericafrom1980to

2000(Domhof2016)–aperiodofmarketliberalisation.Thisincreaseisalsoevidentin

Chinafromtheperiod1967to2007–seeappendixfigure4below.Mostother

developednationshavealsohadtheirGiniincreaseinrecentyears(“Distributionof

Wealth”2015)asshownbelowinappendixfigure5.

IfMilanovicisright,andtheupanddownGinis–socalledKznetwaves–persist,what

thenistheircause?ThisistoFerreira‘thegreatepistemologicalchallengeof


BlairD.Macdonald

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inequalityanalysis’(Ferreira2016).Atleastforthe‘ups’,canthefractaloffera

solution?

Onlongtermincomegrowth:dataforthelasttwoCenturiesshowglobalnational

incomeincreasedexponentially,whilecorrespondingincomeGinisoverthesame

periodalsoshowincrease(thoughatadecreasingrate)–figure2below(Milanovic

2009).Assumingincomeandwealtharethesame,doesthisexponentialgrowthof

incomeandconcurrentincreasingGinicoefficientsconcurwiththefractal

development?

Figure 2 Global Income and Gini Coefficients for last 2 Centuries. Over the last two centuries global income

growth (green) per capita has been exponential and respective Gini coefficient (red) increasing (at a decreasing

rate). (Milanovic 2009)

1.3 TheClassicalFractal

Fractals–alsodescribedasL-systems–areemergentobjectsthatdevelopand(or)

growwiththeiterationofasimplerule.Theypossessself-similarityatallscalesand

canbeobservedasbeingregularbutirregular(samebutdifferent)objects.Theyare

classicallydemonstratedbytheoriginalMandelbrotSet(B.B.Mandelbrot1980),and–

inoneoftheirmostsimplestforms–theKochSnowflake(Figure1below,AandB

respectively).Familiarexamplesoftheminrealityareclouds,waves,coastlinesand

trees.Allfractalshaveadefined‘fractaldimension’.Stewartinhisbookonchaos(and

fractals)‘Goddoesnotplaydice’–said:‘..coastlinesandKochSnowflakesareequally

0

1000

2000

3000

4000

5000

6000

7000

8000

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

1820 1850 1870 1913 1929 1950 1960 1980 2002

Income$

GiniCoe

fficien

t

Year

WorldGinicoefficients

Globalaverageincomepercapita($PPP)


BlairD.Macdonald

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rough’(Stewart1997).Indeed,theveryclosefractaldimensionvaluesofboththeKoch

Snowflakeandthecoastlinesofislands(GreatBritain)1.26,andbetween1.15and1.2

respectively–standsastestamentoftheirpowertobestmodelnatureandreality.

Figure1Bbelowshowstheclassicalviewoffractalgrowth(ordevelopment)–

achievedbytheiterationofasimplerule,byaddingmore(triangle)bits–of

diminishingsize–totheprevioustriangle,originatingfromaninitial(iteration0)bit–

alsoknownasthe‘axiom’inL-systemtheory.Thesnowflakeshapeisformedatand

around5or6iterations;fromthispointon–totheobserver–itnolongerchanges

shape.Thisgrowthto‘equilibrium’observationisnotlosttothiseconomist,andwillbe

thesubjectofanotherfutureinvestigation;butitisthecomparisonordistributionof

thesizeoftriangles–fromthesmalltotheoriginallarge–thatisthefocusofthisstudy.

Figure 3. (Classical) Fractals. (A) boundary of the Mandelbrot set; (B) The Koch Snowflake fractal from

iteration-time (t) 0 to 3. Reference: (A) (Prokofiev 2007); (B) (“Koch Snowflake” 2014).

Fractalsarebytheirnaturecomplexobjects–butthispropertyofcomplexityposesa

problem,anditwasforthisreasontheKochSnowflakefractalattractorwaschosenfor

analysis.TheKochSnowflakehasregular-regularityratherthanregular-irregularity(or

complexity).

TestingwasdoneusingstandardLorenzcurveandGinicoefficientmethods.The

distributionoftriangleareasweregraphedandGinicoefficientscalculatedforeach

iterationtimeasthefractalgrew(ordeveloped).


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Areaofthetrianglebitswasassumedtostandforanindividual’swealth.

1.4 Production,andtheInvertedFractal

To‘produce’aLorenzcurvewithafractalattractor,wefirstneedtodeterminewhatit

meantbyproductionandthusgrowthofthefractal.Whenweattempttodothis,we

findthereisaparadox–therearetwoviewsthatconflictwitheachother:onewhere

theoriginaltrianglebitsizeremainsconstant,andthenewbitsdiminishinsizeasthe

fractaliterates–thisistermed(A)‘aconsumptionperspective’;and(B)whereitisthe

newtrianglebitssizeremainconstant,andallearliertrianglebitsexpandandgrowas

thefractaliterates–termed‘aproductionperspective’.Aistheclassicalview–as

showninfigure3B(andAinfigure4below)andBthe‘inverted’,showninfigure4B.

Bothoftheseviews,AandB,arerelativeviewsofthesameprocess:botharetrue,but

onlyonereallydescribestheproductionandthegrowthfromproduction,andforthis

studyitisassumedtobetheinvertedviewB.

Figure 4. Expansion of the inverted Koch Snowflake fractal (fractspansion). The schematics above

demonstrate fractal development by (A) the classical Snowflake perspective, where the standard sized thatched

(iteration ‘0’) is the focus, and the following triangles diminish in size from colour red iteration 0 to colour purple

iteration 3; and (B) the inverted, fractspanding perspective where the new (thatched) triangle is the focus and

held at standard size while the original red iteration 0 triangle expands in area – as the fractal iterates.

Thereisapracticalreasonforthis:the‘inverted’productiongrowthcanbe‘observed’

inreallifenaturefractals,trees.Withtrees(thusallplants–inprinciple)itisthetrunk

ofthetreethatgrows,andthenewbranchsizethatremainsconstant.Thisisconverse


BlairD.Macdonald

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to‘our’staticobservationoftrees,wherethe(larger)constantsizedtrunkisobserved

withdiminishingsizedbranches‘protruding’–selfsimilar–fromit.

1.5 Acceleration

Withinversion(figure4B)itcanbeshownasthefractalgrowslargerandlargerwith

respecttoiterationtime,butasitdoesthedistancebetweencentre-pointsofthe

originalthatchedtriangleandtheolder(fromiteration0toiteration3)accelerate

apart.Tosupportthisconjecture–andtheinversionview–arecentstudyoftreeshas

showntree’sgrowth–andthusallplants–acceleratewithage(Stephensonetal.2014)

–againmatchingthisinversionperspective.Totestandanalysetheconsequencesof

exponential(incomeorwealth)growthandtheaffectof‘groups’kinematicanalysis

wasconductedonthefractal.Ifthefractalbehavesasaneconomyandproducesa

Lorenzcurve,groupswillaccelerateapart.

Therelationshipbetweenthese‘observation’andinverted‘production’viewsofthe

fractal–withreferencetoeconomictheory–mustbeaddressed,butmuchofthisis

outsidethescopeofthisinvestigation,andsoforthepurposesofthisinvestigationthe

fractalwas‘inverted’andanalysed.Itshouldbemadeclear,inversionornot,itisthe

samegeometryanalysed,justwithdifferentperspectives:onefromthetoplooking

down,theotherfromthebottomlookingup.

2 METHODS

Toanalysetheinvertedfractaltwospreadsheetmodelsweredeveloped,andstoredin

arepository:onetoanalysetheinvertedgrowthbehaviouroftheKochSnowflake

fractal(Macdonald2016);andanothertomodeltheLorenzandGinibehaviour

(Macdonald2015).

Tocreateaquantitativedataseriesforanalysis(usedinbothspreadsheetmodels)the

classicalKochSnowflakeareaequationswereadaptedtoinvertthefractal.Adatatable

wasproducedtocalculatetheareagrowthateachandeveryiterationofasingle

triangle.Areawascalculatedfromthefollowingformula(1)measuredinstandard

(arbitrary)centimetreunits(cm)


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𝐴 =𝑙!√34 (1)

where(A)wastheareaofasingletriangle,andwherelwasthetriangle’sbaselength.

LengthinTable1wassetto15.1967128766173cm-1sotheareaoftheinitial(base)

triangle(i=0)approximatedanarbitraryareaof100cm-2.Toexpandthetrianglewith

iteration(andthusinvertthefractal)thebaselengthwasmultipliedby3(insteadof

dividingitby3asusedintheclassicalsnowflakemethod).Theiteration-time(i)

numberwasplacedinacolumn,followedbythebaselengthoftheequilateraltriangle,

andinthefinalcolumntheformulatocalculatetheareaofthetriangle.Iteration‘0’

referstoapreiterationsize;thefirstposition.Calculationsweremadetothearbitrary

12thiteration,andtheresultsgraphed.

2.1 LorenzCurve

AmethodtoconstructaLorenzcurvebyspeadsheetisdemonstratedby(arnoldhite

2016):thismethodwasfollowed.Atablewascreatedrankingtrianglesbytheirsizeat

eachiterationtime–inascendingorder.Ateachrankedquantitythefollowingwas

calculated:

1. apercentagequantity(Quantity/TotalQuantity)–forthelineofequality;

2. apercentagearea(Area/TotalArea)–fortheLorenzcurve;

3. CumulativepercentageArea;

4. andfinally–forthecalculationoftheGiniCoefficient–theareaunderthe

LorenzCurve.

Toclarify,thefirsttrianglerepresentstheinitialconditionforiteration:itisthefirstindividual.

2.2 GiniCoefficient

AnelementarymethodtocalculatetheGinicoefficientbyspreadsheetisdemonstrated

by(arnoldhite2014)–thismethodwasfollowed.TheGiniCoefficientisacalculatedby

dividingtheareabetweenthelineofequalityandtheLorenzcurve(areaAinfigure1

above)bythetotalareaunderthelineofequality:


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𝑨(𝑨+ 𝑩).

(2)

SummingalltheareasundertheLorenzCurvegivestheareaofB.

GiniCoefficientswerecalculatedforeachiteration-time,andforeachiteration-timea

newtablewascreated.

2.3 AnalysisofAreaExpansionoftheTotalInvertedFractal

Toanalysethevelocityandaccelerationofexpansionofinvertedfractaltoanobserver

withinthefractalanotherspreadsheetmodelwasdeveloped,andstoredina

repository(Macdonald2016).

Withiterationnewtrianglesare(indiscretequantities)introducedintotheset–atan

exponentialrate.Whiletheareasofnewtrianglesremainconstant,theearliertriangles

expand,andbythisthetotalfractalsetexpands.Tocalculatetheareachangeofatotal

invertedfractal(asititerated),theareaofthesingletriangle(ateachiterationtime)

wasmultipliedbyitscorrespondingquantityoftriangles(ateachiterationtime).

Twodatatables(tables3and4inthe‘invertedfractal’spreadsheetfile)were

developed.Table3columnswerefilledwiththecalculatedtriangleareasateachofthe

correspondingiterationtime–beginningwiththebirthofthetriangleandcontinuing

toiterationten.Intable4,triangleareasoftable3weremultipliedbythenumberof

trianglesintheseriescorrespondingwiththeiriterationtime.

Valuescalculatedintable3and4weretotalledandanalysedinanewtable(table5).

Analysedwere:totalareaexpansionperiteration,expansionratio,expansionvelocity,

expansionacceleration,andexpansionaccelerationratio.Calculationsinthecolumns

usedkinematicequationsdevelopedbelow.

2.4 Kinematics

Classicalphysicsequationswereusedtocalculatevelocityandaccelerationof:the

recedingpoints(table2)andtheincreasingarea(table5).

2.4.1 Velocity

Velocity(v)wascalculatedbythefollowingequation


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𝒗 =∆𝒅∆𝒊

(3)

whereclassicaltimewasexchangedforiterationtime(i).Velocityismeasuredin

standardunitsperiterationcm-1i-1forrecedingpointsandcm-2i-1forincreasing

area.

2.4.2 Acceleration

Acceleration(a)wascalculatedbythefollowingequation

Accelerationismeasuredinstandardunitsperiterationcm-1i-2andcm-2i-2.

2.5 Velocity-Distance(‘gap’)Diagram

Totestforincreasingwealthvelocityagainstdistance(thegap)betweengroups–

analogisttocosmology’sHubble’sLaw–aHubble‘like’ascattergraphdiagramtitled

the‘Fractal/Hubble-Wealthdiagram’wasconstructedfromtheresultsoftherecession

velocityanddistancecalculations(intable2ofthe‘invertedfractal’spreadsheetfile).

Onthex-axiswasthetotaldistance(notdisplacementfromobserver)oftrianglecentre

pointsateachiterationtimefromi0;andonthey-axistheexpansionvelocityateach

iterationtime.Abestfittinglinearregressionlinewascalculatedandanequation(5)

wasderived

𝒗 = 𝑲𝟎𝑫 (5)

whereK0the(present)Hubble(like)wealthconstant(thegradient),andDthe

distance.

2.6 Acceleration-DistanceDiagram

UsingthesamemethodsasusedtodeveloptheFractal/Hubble-Wealthdiagram(as

describedabovein2.5)anaccelerationvs.distancediagramwascreatedandan

expansionconstantderived.

𝒂 =∆𝒗∆𝒊

(4)


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2.7 SpiralPropagationthoughTime

Thepropagationoftrianglesinthe(inverted)KochSnowflakefractal,isnotlinearbut

intheformofalogarithmicspiral–asshowninFigure2B(above),andinAppendix

Figure1.Themethodgiventhusfarassumes,andcalculatesthelinearcircumferenceof

thisspiralandnotthetruedisplacement(theradius).Thismethodwasjustifiedby

arguingtherequiredradius(ordisplacement)ofthelogarithmicspiralcalculationwas

toocomplextocalculate,(andbeyondthescopeofthisinvestigation),andthat

expansioninferencesfrominvertedfractalcouldbemadefromthelinear

circumferencealone.Thisbeensaid,aspiralmodelwascreatedindependently,and

radiimeasuredtotestwhetherspiralresultswereconsistentwiththelinearresultsin

theinvestigation.MeasurementsweremadeusingTI–Nspire™geometricsoftware

(seeAppendixIFigure1).Displacements,andthederivedHubblediagramfromthis

radiusmodelwereexpectedtoshowsignificantlylowervaluesthantheabove

(calculated)circumferencenon-vectormethod,butnonethelesssharethesame

(exponential)behaviour.AppendixFigure1showsinthedistancebetweencentre

points,andinbluethedisplacement.

SeeAppendixFigures1,and2,andTable1forresults.

3 RESULTS

Figure5belowisacompositediagramofLorenzcurvesfromiteration-time1(theline

ofperfectequality)toiterationtime5.Allcurvesarederivedfromthespreadsheet

model.

3.1 LorenzCurves

Fromfigure5below:asthefractaliterated,thederivedLorenzcurvesexpendedoutor

shiftedtotherightofthe‘lineofequality’.Atiteration-time1–thelineofequality–

distributionishomogenous;onetrianglewithonearea.Withiteration-timeandgrowth

theseconditerationtime(i=2)Lorenzshowed25%oftheareaisfoundin75%ofthe

triangles;andconverselytheremaining75%oftheareaisfoundintheremaining25%

ofthetriangles.Atiteration-time5(i=5)thisdistributionhasexpandedandshowsonly

asmallpercentageofthetrianglesaccountforsome99%ofthearea.


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Figure 5. Fractal (Koch Snowflake) Lorenz Curves Expansion. Lorenz Curves are produced for the Koch

Snowflake fractal from iteration (i) 2 to iteration 5. The Lorenz curve expands out to the right with growth and

time as a result of increasing spread and quantity of triangles sizes.

3.2 GiniCoefficient

Ginicoefficientsateachiterationareshownbelowinfigure6.TheGinicoefficientincreasedatadecreasingrate–approximatingavalueof1–withiteration-time.

Figure 6. Koch Snowflake Gini Coefficient by Iteration-time (i). The Gini coefficient is a measure of the area

between the line of equality and the Lorenz curve in relation to an area of perfect equality. As the (Koch

Snowflake) fractal grows – and/or develops – with iteration time, the Gini Coefficient increases.

0%

10%

20%

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60%

70%

80%

90%

100%

0%

6%

12%

18%

23%

29%

35%

41%

47%

53%

59%

64%

70%

76%

82%

88%

94%

100%

Cum.%

AreaofTria

ngles

Cum.%QuanDtyofTriangles

LineofEquality

i=2

i=3

i=4

i=5

0

0.2

0.4

0.6

0.8

1

2 3 4 5

GiniCoe

fficien

t

i


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3.3 ExpansionofInitialTriangle

TheareaoftheinitialtriangleoftheinvertedKochSnowflakefractalincreasedexponentially–shownhereinFigure7.

Figure 7. Area Expansion of a single triangle. iteration time (i). cm = centimetres.

Thisexpansionwithrespecttoiterationtimeiswrittenas

𝑨 = 𝟏𝒆𝟐.𝟏𝟗𝟕𝟐𝒊 (6)

3.4 TotalFractalExpansion

Theareaofthetotalfractal(Figure8A)andthedistancebetweenpoints(Figure8B)of

theinvertedfractalalsoexpandedexponentially.

A

B

y = 1e2.1972x R² = 1

0

200

400

600

800

0 1 2 3 4

Are

a cm

-2

i

y=1.1081e2.3032xR²=0.99947

0

4,000

8,000

12,000

0 1 2 3 4

areacm-2

i

y=0.5549e1.2245xR²=0.99755

0

20

40

60

80

0 1 2 3 4

distan

cec

m -1

i


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Figure 8. Inverted Koch Snowflake fractal expansion per iteration time (i). (A) total area expansion and (B)

distance between points. cm = centimetres.

Theexpansionofthetotalarea (𝑨 𝑻)isdescribedas

𝑨𝑻 = 𝟏.𝟏𝟎𝟖𝟏𝒆𝟐.𝟑𝟎𝟑𝟐𝒊 (7)

Theexpansionofdistancebetweenpoints(D)isdescribedbytheequation

𝑫 = 𝟎.𝟓𝟓𝟒𝟗𝒆𝟏.𝟐𝟐𝟒𝟓𝒊 (8)

3.5 Velocity

The(recession)velocitiesforbothtotalareaanddistancebetweenpoints(Figures9A

and9Brespectively)increasedexponentiallyperiterationtime.

A

B

Figure 9. Inverted Koch Snowflake fractal (expansion) velocity. Expansion velocity of the inverted fractal at

each corresponding iteration time (i): (A) expansion of total area, and (B) distance between points. cm =

centimetres.

Velocityisdescribedbythefollowingequationsrespectively

𝒗 = 𝟏.𝟏𝟗𝟎𝟖𝒆𝟐.𝟑𝟎𝟑𝟐𝒊 (9)

𝒗𝑻 = 𝟎.𝟓𝟓𝟒𝟗𝒆𝟏.𝟎𝟗𝟖𝟔𝒊 (10)

y=1.1908e2.2426xR²=0.99995

0

2,000

4,000

6,000

8,000

10,000

0 1 2 3 4 5

velocitycm

-2 i

-1

i

y=0.5849e1.0986xR²=1

0

10

20

30

40

50

0 1 2 3 4 5

velocitycm

-1 i-

1

i


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where𝒗𝑻isthe(recession)velocityofthetotalarea;and𝒗the(recession)velocityof

distancebetweenpoints.

3.6 TheFractal/Hubble-WealthDiagram

Asthedistancebetweencentrepointsincreases(ateachcorrespondingiterationtime),

sotoodoestherecessionvelocityofthepoints–asshowninFigure10below.

Figure 10. The Fractal/ Hubble-Wealth diagram. As distance between triangle geometric centres increases

with iteration time (i), the recession velocity of the points increases. cm = centimetres.

Recessionvelocityvs.distanceofthefractalisdescribedbytheequation

𝒗 = 𝟎.𝟔𝟔𝟕𝟗𝑫 (11)

wheretheconstantfactorismeasuredinunitsofcm-1i -1 cm-1.

3.7 AccelerationofAreaandDistanceBetweenPoints

Theaccelerationsforbothtotalareaand(recession)distancebetweenpoints(Figure

11Aand11Brespectively)increasedexponentiallyperiterationtime.

y=0.6667xR²=1

0

10,000

20,000

30,000

40,000

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

velocitycm

-1i

-1

distancecm-1


BlairD.Macdonald

17

A

B

Figure 11. Inverted Koch Snowflake fractal (expansion) acceleration. Acceleration of the inverted fractal at

each corresponding iteration time (i): (A) expansion of total area, and (B) distance between points. cm =

centimetres.

Accelerationisdescribedbythefollowingequationsrespectively

𝒂𝑻 = 𝟏.𝟏𝟗𝟓𝟖𝒆𝟐.𝟐𝟎𝟕𝟑𝒊 (12)

𝒂 = 𝟎.𝟓𝟖𝟒𝟗𝒆𝟎.𝟗𝟕𝟕𝒊 (13)

whereaTisthe(recession)accelerationofthetotalarea;athe(recession)acceleration

ofdistancebetweenpoints.

Asthedistanceofcentrepointsincreases(ateachcorrespondingiterationtime)from

anobserver,sodoestherecessionaccelerationofthepoints(expandingaway)–as

showninFigure12below.

y=1.1958e2.2073xR²=0.99998

0

2

4

6

8

10

0 1 2 3 4

acceleraDo

nc

m -2

i -2

i

y=0.5849e0.977xR²=0.98977

0

10

20

30

40

0 1 2 3 4 5

acceleraDo

ncm

-1i

-2

i


BlairD.Macdonald

18

Figure 12. Recessional acceleration vs. distance on the inverted Koch Snowflake fractal. As distance

between triangle geometric centres increases with iteration time (i), the recession acceleration of the points

increases. cm = centimetres.

Therecessionaccelerationofpointsateachiterationtimeatdifferingdistancesonthe

invertedfractalisdescribedbytheequation

𝒂 = 𝟎.𝟒𝟒𝟒𝟕𝑫 (14)

wheretheconstantfactorismeasuredinunitsofcm-1i -2cm-1.a =acceleration;D =distance.

4 DISCUSSIONS

Inthisstudy,theKochSnowflakefractalattractorwasmodelledtodescribeanaspect

ofaneconomy–specificallygrowthandwealthdistribution.Therearehowever,many

otheraspectsandinsightstobetakenfromthesamefractal–withrespecttothe

marketandknowledge–andallwithbroadimplications.Someoftheseinsightshave

beentoucheduponhere,thoughinasuperficialmanner,becausetheyareimportantto

thetopic–othersnot.Thefollowingdiscussionsarewhatareconsideredmostrelevant

tothetopicofdistribution;buttogether,intime–withtheotherpotentialinsightson

thefractalexposed–thefractalmaysoundthewhatweterm‘themarket’tobeafractal

y=0.4445xR²=1

0

5,000

10,000

15,000

20,000

25,000

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

acceleraDo

ncm

-1 i-

2

distancecm-1


BlairD.Macdonald

19

object,onlyseparatedinourrealitybyitsintangibilityandabstractness,butnonethe

less,real.

4.1 AFractalLorenzCurve

Thestudydemonstratedthedistributionofbitareasizesonafractalisnotequal,and

becomesmoreunequalasbitsareproducedwithiteration-time:thisappearstomatch

reallifeLorenzincomedistributioncurvesandhowtheyarederived.

TheiteratinginvertedKochSnowflakefractalattractorproducedaclearLorenzcurve

(figure5):onesimilartoanyproducedfromrealeconomicwealthandincomedata.As

thispatterncanalsobeproducedfromthestaticclassicalfractal,thestudyshowedthe

Lorenzdistributionanalysis–theLorenzcurve–isanaspectorpropertyofafractal

attractorandcanbederivedfromallthingsfractalinnature.Fromthisitwasdeduced

thecultural/humanmarketeconomybehavesasafractal.Inequalityofincomeor

wealthdistributionisoftenseenasanundesirabletrade-offofthe‘freemarket’;

however,asrevealedinthisfractalexperiment,itmaybemoretruetosaythis

inequalityisanaturalphenomenon,apropertyofthe‘market’–notunusualorspecial,

buttypical.

Insightsfromthismodelwillbepresentinallstructuresthatiterate(withtime)and

showfractal/hierarchicalstructure–fromclouds,totrees,toinformation,andevento

theabundanceoftheelementsuniverse.Asanasidetofurthersupportthemodel,

Lorenzcurveswerecreatedfromthebranchweightdistributionofatypicaltree

(Macdonald2016a)andfortheabundanceofelementsinthenear13.8billionyearold

universe(seeappendixfigure3)(Macdonald2016b).Bothoftheseexperiments

producedcurvesmatchingthefractalLorenzcurveandtheeconomicsLorenzcurves.

Treesareclearfractals;andtheuniverseisconjecturedtobefractal.Fractalcosmology

isarecognisedbodyofsciencewithit’sownuniquetheoryonthestructureofthe

universe(Pietronero1987).

4.1.1 Regular Regularity Assumption

Theunrealistic‘regular-regular’assumption(asdescribedintheintroduction)

demonstratedbytheKochSnowflake–asopposedtoamorerealisticrough‘regular-

irregular’fractalliketheMandelbrotset–revealsonlyonestructuralaspectofthe


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fractalattractor:itdoesnotshowtheirregularchaoticordiverseaspectsasrevealedin

individualcountriesLorenzcurves,andrespectiveGinicoefficients–bothstaticand

temporal.Itshowsanaspectorpropertyofallfractals,partoftheirstructure.Realityis

bynomeansregular;forinstance,wealthisnotaddedinequalbitsasassumedby

equaltriangles,andwealthdoesnotalwaysgrowastrianglesgrow,itcandecrease.

4.1.2 Growth and Recession

Whiletheiteratingfractaldemonstratesgrowth,recedingorrecessionwouldimplythe

processisreversedandallbitsizesdecreaseinsizewithiteration-time.

4.2 GiniCoefficients

ThecalculatedareaGinicoefficientsfromthefractalmodelalsomatchrealwealthGini

values–noticeablygreaterthanrealincomeGinicoefficientsandthusmoreskewedto

theright.

4.3 ShiftingLorenzCurveandIncreasingGinicoefficientwithGrowth

Areagrowthofthefractalandinequalitybetweenareasizesareinextricabilitylinked

asafunctionoftime:thisappearstobealawofthefractal.Ifaneconomyisassumedto

behaveasafractalattractor,wecanexpectitsrespectiveLorenzcurvewillshiftoutto

therightandthedistributionofwealth,andorincomewillbecome–inextricableto

growth–moreunequalwithtime.

TheoutwardshiftingoftheLorenzcurvesandthusincreasingfractalGinicoefficients

(figure6)isduetotheexponentialgrowthofthefractal(figure8A).Thismaydirectly

explaintheincreaseinglobalincomeGinicoefficientswithtimealongwiththe

exponential(income)growth–observedoverthelasttwoCenturies(asintroducedin

figure2).

4.4 Gini–DevelopmentContradictions

Asaresultofgrowth,thefractalproducedincreasingfractal-Ginicoefficientsat–

approachingavalueof1(figure6)–withrespecttoiteration-time.Fromthis,usingthis

fractalmodelitcanbeinferredthedegreetowhicharealeconomyisdevelopedbyits

Ginicoefficientalone–givenanfractalbecomesmore‘developed’andunequalasit

grows–withtime.Thispropertysuggestsalowiteration-time(young)economy


BlairD.Macdonald

21

pertainstolessinequality,andconversely,aneconomywithlargeriteration-timea

highinequality.ThismayhoweverbecontradictedbyrealincomeGinidata(thoughit

ismoresupportedforwealthdistributionalone).Manylesseconomicallydeveloped

economics(LEDCs),havehigherincomeGinicoefficientsthanmoreeconomically

developedeconomies(MEDCs).LEDGs;however,maybeatypical,andmaynothave

achievedthese‘high’Ginivaluesbygrowing(oremerging)inafractal‘economic’

mannerwithrespecttoiteration-time,butinsteadhadtheirGinicoefficients

anomalouslydistorted–tothehigh–onlyasaresultofhighdegreesofcorruption,or

lowdegreesoftechnology,opennessaspositedbyMilanovic.

4.5 KuznetsCurveCycles

Continuingdirectlyfromtheabove,thefractal-Lorenzcurvemodel,anditsincreasing

Giniwithtime,directlycontradictstheincomeKuznetscurve–andKuznetscycles–

wheretheKuznetscurveshows(income)Ginisfirstrisewitheconomicgrowthand

thenfall.Thisresultmaysuggest(againasmentionedabove)the–decreasingGini–

KuznetscurveandKuznetscyclesbehaviourisatypical–aculturalphenomenon,

broughtaboutbyintervention.Indeedthis–intervention–ishowtheKuznetscurveis

generallyreasoned:‘re-distributionisachievedthroughprogressivetaxesandwelfare

payments,andjobcreation’(economicsonline2016).Thissustainedincreaseofthe

fractalisindisagreementwithMilanovicandothers.Milanovic’sputsthedecrease–

anddipinthemiddleclass(appendixfigure6)–downtotechnologyopennessand

politics:however,politicsmaywellbethesolereasonastechnologyandopennesseach

ontheirownshouldleadtogrowthandthuswealthdiversion,notthereverse.Itmaybe

concludedthedownwardsideoftheKuznetscyclesaresolelyaproductofintervention

andtheupwardsideaproductof‘freer’liberalisedmarketsassociatedwithgrowth.

4.5.1 Matching Real Ginis

RealdatashowingincreasingwealthandincomeGinicoefficientspointsto–or

matches–freermarketsandthismatcheshowthefractalGinigrows.Ginishavegrown

astheworld’sincomeeconomyhasgrownoverthelasttwoCenturies(figure2);and,

inperiodsof‘freermarkets’orduringtherecenttimeofglobalisation(appendixfigure

5),Ginishavealso–onthewhole–increased,thisisespeciallyevidentinChina

(appendixfigure4).


BlairD.Macdonald

22

4.6 GrowthandWealthfromanInvertedPerspective

UnliketheclassicalKochSnowflakefractalformation(asdescribedintheintroduction),

theinvertedSnowflakeisnotastaticmodel;itisdynamicanditdemonstratesgrowth.

Thefollowingarethekeyinsightsfromthemodelasaconsequenceofthisgrowth.

4.6.1 Bit Size – Aggregate – Growth

Themodeldemonstratedthesimultaneousandinextricablegrowthofallbitsizes

relativetoeachotherwithrespecttotime:nomattertheir(arbitrary)startingsizeto

growthewhole(aggregate)geometricsetsize,alltheparts(alltriangles)grow.In

principle,anysinglebitsizewillintimegrowtobeequaltoitspredecessors:abranch

onatree–arealfractal–willintimegrowtobethesizeofthetrunkofthesametree.

Ofcoursewhenthe‘single’branchachievesthissize,thetrunk,andthepredecessors

willhaveagaingrown.

4.6.2 Relative Position and Distribution

TheinvertedKochSnowflakeisinprincipleinfiniteinsize–andthusintime.Fromthis

itcanbeinferred:whereverone’sobservationpointiswithinthefractalset–thisisto

say,ifatrianglebitischosenatrandomasaobservingpositionintheset,andan

observerobserves,fromthisbit,andlooksforwardandback–therewillalwaysbe

small‘bits’ahead,andalwayslarge‘bits’behind.Withintheinfiniteset,theobserver

willneverbethelargest.Thisinsighthasrelevancetoone’spositionwithinan

economicsystem:whereveroneisonthe‘wealth/incomeladder’,therewillalwaysbe

someone‘wealthier’above,andsomeone‘poorer’below–unlessyouarefirstandthe

largest.Ofcourseatthescaleandinthecontextofall‘complex’lifeonEarth,wealthis

finite,buttherangeofsizesofwealthareextremelydiverse–fromarguablywe

humans,throughtomicrobes,maybemolecules,andeven–asdiscussedabove–the

elements(andbeyond).Whatevertherange,ifsomethingisfractalbynature(and

natureappearstobefractalwhereverwelook),thedistributionofsizeswillfita

Lorenzdistributionstructure.

Thisprincipleiswelldescribed–indirectly–byProfessorHansRoslingofGapminder

inthisBBClectureonpopulation.At(time35:42)heshowstherelativeperspectivesof


BlairD.Macdonald

23

lookinguptothewealthy,anddowntothepoor:theverypoorestthinkofthenext

groupjustasthemiddleclassdothemostrichest(BBC2016).

Thisrelativepositionwillbereturnedtowhenaccelerationisdiscussed–below.

4.6.3 Refuting the ‘Zero Sum Game’ and Other Myths

Fromtheabove(4.6.1),thefractalmodelrefutesconjectureslike‘therichgetricher

andthepoorgetpoorer’,or‘thezerosumgame’ofwealthgrowth.Forthefractalto

grow,allbitsizesgrow,andthisonlyhappenswhennewbitsareadded–ornew

branchesareformed.

Ifbitsizeswereofequalsize–andthusequallydistributed–growthwouldbelimited

tooneortwoiteration-times,andarguablythefractalsetwouldnotgrowatall.Further

tothis,abranchonatreecannotholdbranchesofasimilarsize.Nowhereisthis

observedinnature,ifitwere,itwouldnotbefractal.Forthesamereasons,aneconomy

cannotbeequallydistributed.

4.6.4 Symbiosis

A(fractal-structured)treedemonstratesanactofinternalsymbiosis:theunequal

branchsizestructureofbranchesallowsforgreaterlightcapture–fortheproductive

leaves.Ifthewoodystemofatreesupportsandservicestheleavestogainmore

productivelight,thewealthstructureinaneconomymayserveasimilarrole

supportingproductiveenterprises:withoutthisitwillnot‘holdup’.Fromthisitis

deducedthefractal‘market’structureisthemostefficientmeanstogrow.Ifthetrunk

ofthetreeweretoredistributeitsareaorvolumetotheouter(smaller)branchesin‘an

actoffairness’,thefractalgeometry(tree)wouldcollapse.Thesamewouldresultifthe

outerbranchesweretocannibalise(ortax)thetrunk.Itmaybeinteresting–giventhe

fractalhasnoboundsforexamplesin‘nature’–toinvestigatethedegreetowhichother

biologicalsystemsredistributewealthorincomewithintheirownsystembymeansof

taxorregulation–examplesarenotobvious,andthereisnomentionofthisin

Vermeij’s:Nature,AnEconomicHistory.Inhisbookhegoestogreatlengthdescribing

theuniversalityoftradeandexchange–sharedinalleconomies.

4.6.5 Age Order


BlairD.Macdonald

24

ThefractalderivedLorenzcurverevealsanageorderstructureofthewealth

distributionwithgrowth:theoldesthavethelargestproportion.Thisagreeswithreal

treedevelopmentandtheageoftheelementsintheuniverseontheperiodictable:the

trunkistheoldest‘bit’;andheliumandhydrogenwerethefirstelementsformed

straightafterthebigbang.Subatomicparticlescamebeforeheliumandhydrogen;and

thelarger–andfewer–elementslikegold,cameafter.Whetherthisageorderis

prevalentintheeconomymaybeuncleartoseeatfirst,buttheruleshouldholdon

examination.

4.7 VelocityandAccelerationofWealth

Theinvertedfractalmodeldemonstratedexponentialvelocityofareagrowth(figure9

above);thisconcurswiththelong-termglobalincomegrowth.

Wealthorincomeacceleration(shownfigure11above);however,atleastinthefieldof

economics,isnotsocleartoseebyrealdata–thoughitisoftencasedasaconjectureor

asthreatfromthemoreleftsidedcommentators.Fromthisstudy,thisconjecturecan

berelegatedtoaprediction.Wecanpredictwealthand/orincomewillacceleratewith

time,aswillthe‘gap’betweenpopulationgroups(moreonthisbelow).

Appendixfigure6(Milanovic2016a)showstheglobalpercentagegrowthateach

percentileoftheglobalpopulationbetween1988and2008.Apartfromthe(USA)

middleclass(whichdemandsanexplanationastowhynot),allpercentilesaregrowing

bypositivepercentageandthusare–bycompoundinggrowth–acceleratingnotonly

together,butalsoacceleratingapartfromoneanotherinnominalincometerms.

Iflong-termgrowthaccelerationweretrue–withtime,thispatternwouldnotbe

uniquetoaneconomy:theplants,andpossiblyeventheuniverseasawholehavebeen

foundtoaccelerateinsimilarmanner.Trees,97%ofthem–takenfromastudyofsome

600,000,onsixcontinents,and100yearofdatacollection–havealsobeenshownto

acceleratewithgrowth(Stephensonetal.2014).Treesaretheobviousfractal

structure;theyhaveafractaldimensionandareregularlyusedasfractalexamples.I

havebeenableclaim,usingtheinvertedfractalmodel,plants–intheirtreeform–also

complytofractalgeometry,andwillacceleratewithgrowth.Withtheuniverse,current

observationshaverevealedittootobenotonly(Hubble)expanding,butalso–doingso


BlairD.Macdonald

25

–atanacceleratingrate.Bothoftheseobservations–expandingandaccelerating–

havenoobviousexplanation.Theuniverseissaidtobe:‘beenpulledapart’bya

mysterious‘darkenergy’.InmyoriginalstudyontheinvertedfractalIappliedthis

modelsametothisproblemclaimingtheuniverse’sacceleratingexpansion(the‘dark

energy’)andassociatedproblems,maybebestdescribedby(inverted)fractalgeometry

(Macdonald2014).Ifthesebiologicalandcosmologicalaccelerationobservationsare

thesameasthiswealthandincomeacceleration,theymayallbeunifiedandexplained

byfractalgeometry.

4.8 FractalHubble-WealthDiagram

InthecosmologyinvestigationusingthesameexpandinginvertedfractalIwasableto

produceaHubblediagram(showingincreasingrecessionvelocityofgalaxieswith

distance).AsthemodelderivingthisfractalLorenzcurveisthesameastheinverted

fractalmodel,Ibelieve,giventhematchingprinciplesofthemodel,thisrecessional

velocityandaccelerationbetweenpointscanequallyapplytoaneconomy.TheFractal

Hubble-Wealthdiagram(figure10above)predictsasthevelocityofwealthcreation

increaseswith‘distance’(orthegap)betweentheobserverorindividualandanother

grouporindividual.Whenvelocity(v)isplottedagainstdistanceofpoints(D)(figure

10,andAppendixfigure2)theinvertedfractaldemonstratesHubble’sLawdescribed

bytheequation

𝒗 = 𝑭𝒗𝑫 (15)

where(Fv)istheslopeofthelineofbestfit–thefractal(Hubble-Wealth)recession

velocityconstant.

4.8.1 Exponentially Accelerating Gap

Inthesamecosmologicalstudyusingtheinvertedfractal,acomplementarydiagramto

thefractalHubblediagramwasproducedshowingaccelerationbydistance:

accordinglyfigure12showsthe‘gap’acceleratesapart:theaccelerationofwealth

creationincreaseswith‘distance’(orthegap)betweentheobserverandanothergroup

orindividual.Groups–or‘thegap’–willaccelerateapartwithgrowth/time.

4.8.2 Non-Inertial Frames of Reference.


BlairD.Macdonald

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Continuingwiththistopicofaccelerationandfurtherrefutingoftheclaim‘therichget

richer,whilethepoorgetpoor’asdiscussedabove(4.7.1):the‘poorgettingpoorer’

maybeonlyperceivedtobesoandisanillusionoftheacceleration.Observations

withinthefractalsetoreconomyaremadefrom–whatistermedinphysics–a‘non-

inertialoracceleratingframeofreference’.Thismayaccountfortheillusion–froma

fixedpositionintheset–therichappeartobegettingricher(aheadintheset),while

(behind)thepoorappeartobegettingpoorer;butinreality,thepoorarealso

accelerating,onlytheyappear–falsely–toberecedingaway.

4.9 WaveProperties

Anotherpropertyofthefractalgrowth–demonstratedinmethods2.1–isthe

sinusoidalspirallingofthediscretecomponentbits‘though’time–trianglesinthecase

oftheKochSnowflakedemonstratedinfigures4Bandappendixfigure1.This(vector)

spiralling–whichisnotobvioustotheobserverasitisthelinierdisplacementthatis

observed–suggestseconomicgrowthisasaresultofapropagatingtransversewave.

Thefractaleconomymayhaveawavefunctionproperty.Furtheranalysisofthis

propertyisbeyondthescopeofthisinvestigationbutwillbeaddressedina

complementarystudy.Sufficetosay,thisoccurrenceofwavepropagationofdiscrete

‘bits’alongwithconceptsoftimeandgrowthisnotlostontheauthorandmaystandas

acandidateorwindowintoquantummechanicaltheory–whereallthingsare

describedbyawavefunction.Putanotherway,todescribethegrowthbehaviourofthe

fractalmayrequiremathematicsderivedfromquantummechanics,anditmaybethat

the‘market’structure–withitsLorenzwealthandincomedistribution–isastanding

wavephenomenon,andallaconsequenceoraspectofiteration–oftime.

5 CONCLUSIONS

ThisinvestigationfoundtheLorenzcurvedistributionisapropertyofafractaland

fromthisthemarketmaybebestdescribedbyfractalgeometry.Theeconomybehaves

asafractal.Thispropertyisrevealedinallthingsfractalsuchastrees,clouds,possibly

theuniverse,andeconomies–whetherculturalornot.Lorenzdistributionis

inextricablylinkedtogrowthanddevelopmentofthefractal(oreconomy).Asthe


BlairD.Macdonald

27

fractalgrowsanddevelops–withtime–theareaofcomponentbitsizes–itswealth–

distributionincreasesresultingwiththeLorenzcurveshiftingouttotherightandthe

respectiveGinicoefficientincreasing.ExponentialincomegrowthandGinicoefficient

globallyoverthelasttwoCentauriesconformtomodelledfractalgrowth–given

incomeisrelatedtowealthcreation.Thewealthgap–ordistancebetweenarea

sizes/wealthgroups–expandsapartatanacceleratingrate(exponentially)becoming

increasinglymoreunequalwithtime.Thevelocityandaccelerationofareaincrease

increasesasthegapordistanceincreases–similar,ifnotidenticaltothecosmological

Hubble’slawand‘darkenergy’(conjecture).Thiscosmologicalsimilarity,alongwith

thewavelikepropertiesofthegrowingfractal,mayberevealingpropertiesbeyondthe

scopeofeconomicstudies,andtheseareyettobeexplored.Theiteratingfractal,

growingoremerging–bit-by-bit–mayofferinsightsinthequantumaspectsofour

reality.

6 APPENDIX

Figure 1. Displacement measurements from radii on the iterating Koch Snowflake created with TI-Nspire ™ software. Displacement is measured between (discrete) triangle centres and used in the calculation of the


BlairD.Macdonald

28

fractal/Hubble constant. The red line traces the circumference (the distance) of the fractal spiral, and the blue

line the displacement of the fractal spiral from an arbitrary centre of observation. cm = centimetres.

Table 1. Displacement records taken from direct radius measurements and calculations from the iterating Koch

Snowflake fractal spiral (Appendix Figure 1).

i Displacement:

cm

Total Displacement:

cm

Expansion Ratio:

Velocity:

𝒄𝒎 𝒊!𝟏

Acceleration:

𝒄𝒎 𝒊!𝟐

0

-

1 1.68 1.68 - 1.68 1.68

2 4.66 6.34 3.77 4.66 2.98

3 12.16 18.5 2.92 12.16 7.50

4 35.4 53.9 2.91 35.40 23.24

cm = centimetres. i = iteration time.

Figure 2. The Fractal Hubble-Wealth Diagram. Recessional velocity vs. distance from radius measurements

(Appendix Figure 1). From an arbitrary observation point on the inverted (Koch Snowflake) fractal: as the

distance between triangle geometric centres points increases, the recession velocity of the points receding away

increases. cm = centimetres. i = iteration time.

y=0.6581xR²=0.99918

0

5

10

15

20

25

30

35

40

0 10 20 30 40 50 60

velocitycm

i-1

displacementcm


BlairD.Macdonald

29

Figure 3. Lorenz Curve of the (118) Elements from the Periodic Table. Using Lorenz curve method the

distribution of elements curve was produced. The Gini coefficient was calculated: it is equal to 0.9864 (4sf)

(Macdonald 2016b)


BlairD.Macdonald

30

Figure 4. Gini (Kuznets curve) with GDP per capita for China 1967-2007(Milanovic 2012)

Figure 5. Income Gini Change from 1988 to 2008. (Milanovic 2012)


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31

Figure 6. Global growth by percentile 1988 to 2008. (Milanovic 2016a)

Acknowledgments

FirstlyIwouldliketothankmyfamilyfortheirpatienceandsupport.Thankyoutomy

economicsstudent’sandcolleague’sintheInternationalBaccalaureateprogrammesin

StockholmSweden–Åva,Sodertalje,andYoungBusinessCreatives–fortheirhelpand

support.FortheirdirectbeliefandmoralsupportIwouldalsoliketothankMaria

WaernandDr.IngegerdRosborg.MathematiciansRolfOberg,andTosunErtanhelped

andguidedmenoend.


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32

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fractal lorenz 161110 - vixravixra.org/pdf/1505.0125v6.pdf · 19th centenary: fractals generally...

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