Self‐SquaredDragons:TheMandelbrotandJuliaSets

TimothySongCS702Spring2010Advisor:BobMartin

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TABLEOFCONTENTSSection0 Abstract……………………………………………………………………………………………………………….2SectionI Introduction…………………………………………………………………………………………………….…3SectionII BenoîtMandelbrot…………………………………………………………………..…………………………4SectionIII DefinitionofaFractal………………………………………………………………..……………………….6SectionIV ElementaryFractals………………………………………………………………..………………………….8 a)VonKoch………………………………………………………………..…………………………..8 b)CantorDusts………………………………………………………………..……………………..9 1.DisconnectedDusts……………………………………………………………….11 2.Self‐similarityintheDusts……………………………………………………..12 3.InvarianceamongsttheDusts………………………………………………..12SectionV JuliaSets………………………………………………………………..…………………………………………13 a)InvarianceofJuliasets………………………………………………………………..…….17 b)Self‐SimilarityofJuliasets…………………………………………………………………18 c)ConnectednessofJuliaSets……………………………………………………………….19SectionVI TheMandelbrotSet………………………………………………………………..………………………..22 a)ConnectednessoftheMandelbrotSet………………………………………………24 b)Self‐SimilaritywithintheMandelbrotSet………………………………………....25SectionVII Juliavs.Mandelbrot………………………………………………………………..……………………….28 a)AsymptoticSelf‐similaritybetweenthesets……………………………………..30 b)Peitgen’sObservation……………………………………………………………..…….…33SectionVIII Conclusion………………………………………………………………..………………………………………34SectionIX Acknowledgements……………………………………………………..……………………………………35SectionX Bibliography………………………………………………………………..……………………………………36

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SectionI:Abstract

Fractalsaresocomplexthattheybelongintheirownindependentcategory

ofmathematics;fractalgeometry.Fromthetimetheword“fractal”wascreatedby

BenoîtMandelbrot,thissubjecthasgrownfromobscurityandunappreciatedto

partsofoureverydaylives.Twoofthemostfamousfractalsarecreatedfromthe

sameequation.BoththeMandelbrotandJuliasetsareformationsspawnedfrom

thecomplexquadraticmapping

€

z1→ z0 + c .Althoughtheybothusethesame

formula,thesetsareinfinitelydifferent,butstillremaincloselyrelated.

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SectionII:Introduction

Beforethediscussionaboutfractalsbegins,itismostnecessarytorevealthe

manbehindthemadness.Hisworkistheinspirationforthispaper,notonly

becausehisworkcreatedthebasisforstudy,butbecausehisphilosophyonthese

beautifulbeastsmotivatedmetodivedeeperintothisfield,whichapplicationsnow

spanacrossphysics,topology,medicine,anddigitalphotographytonameafew.

Nextfractalswillbedefinedcarefullytocreateunderstandingtoanyreader.

Therearenumerousaspectstowhatdefinesafractal,butwhatisimportanttonote

isthateveryfractaldoesnothavetofiteveryaspect,butsimplyithastomatcha

fewofthem.Thisideabecomesclearerinthereviewofseveralbasicfractalsthat

exemplifytheseaspects.

Afterthebackgroundonfractalshasbeencovered,themainfocusofthe

papercanbegininitsexplorationofthefamousJuliasets.Thissurveyis

mathematicalforthesakeofaccuracy,butalsopictoralforthesakeofintuition.The

picturesarenecessarytoolstounderstandingthedepthofknowledgethatresides

withinthesesets.

InadiscussionoftheJuliasetsitisimpossibletoavoiddiscussingthe

Mandelbrotset,namedafterthegodfatheroffractals.Juliasetsareveryintricately

relatedtotheMandelbrotset,andbyinvestigatingbothofthem,thesesimilarities

shouldbecomeapparent.Througharigorouscomparison,hopefullytheknowledge

ofbothsetswillcomenaturally.

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SectionII:Mandelbrot

BenoîtB.Mandelbrotcoinedtheterm“fractals”ashewasstudyingthetype

ofgeometrythatisn’tperfectcirclesandsquares,rightanglesandforty‐fivedegree

angles,butrather,ageometrythat“describesmanyoftheirregularandfragmented

patternsaroundus.”1Thismakessense,becausehederived‘fractal’fromtheLatin

adjectivefractus,meaning“tobreak”.Whatbeganasajourneyintoanunknown,

disregardedfieldofmathematicsquicklyturnedintothedevelopmentofatotally

newandexcitingfieldofmathematics:

Fractal is aword invented byMandelbrot to bring together under one heading alargeclassofobjectsthathave[played]…anhistoricalrole…inthedevelopmentofpuremathematics. Agreatrevolutionof ideasseparatestheclassicalmathematicsofthe19thcenturyfromthemodernmathematicsofthe20th…Thesenewstructureswere regarded… as ‘pathological’…as a ‘gallery of monsters,’ kin to the cubistpaintingandatonalmusicthatwereupsettingestablishedstandardsoftasteintheartsataboutthesametime.2

Oneissuethatarosewhiledeterminingtheexactfocusofthisessaywasthefactthat

IfearedthatfractalsandtheirrelationtotherealworldandsolidEuclidean

Geometricobjectswouldbetooinexactforthepurposesofmythesis.However,a

quotetakenfromMandelbrot’sbookexplainswhythisisnotanissue:

allpulchritudeisrelative…Weoughtnot…tobelievethatthebanksoftheoceanarereallydeformed,becausetheyhavenottheformofaregularbulwark;northatthemountainsareoutofshape,becausetheyarenotexactpyramidsorcones;northatthe stars are unskillfully placed, because they are not all situated at uniformdistance.3

Thisattitudeactuallyexcitedmeandencouragedmemoreininvestigatingthisfield.

AsastudentofComputerSciencethatwasnevertoofondofthemathematicsside,

thisrealizationcameasawelcomerelief.Italsoofferedafreshperspective

comparedtotheonethatI’veusuallyexperiencedinmyshortmathematical

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experiences.However,eventhoughthisstudydoesnotinvolve100%theoremsand

formulas,therearemanydefinitionsthatmustbeclarifiedbeforethestudybegins.

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SectionIII:DefinitionofaFractal

Mandelbrotoutlinesthreeimportantconceptsinhisbook,beforehedivulges

intothedeeperintricaciesoffractals.First,thedefinitionofafractal,fromtheman

whocoinedtheterm:“isbydefinitionasetforwhichtheHausdorff‐Besicovitch

dimensionstrictlyexceedsthetopologicaldimension.”4Topologicaldimension,DT

(akaLebesgueDimension)istheminimumnumberofintegersittakestodescribea

shape.So,forapoint,thetopologicaldimensionwouldbe0,foraline:1,foraplane:

2,foracube:3(height,length,width).Hausdorffdimension(D),alsodefined

informally,forself‐similar(meaningeachinstanceoftheobjecthasidenticalcopies

ofitselfatdifferentscales)objectswouldbe:D=log(numberofpiecesyouspliteach

iteration)/log(eachlevelofmagnificationperiteration).5Lauweriergivesamore

precisedefinition:

Weselectanarbitrarilysmallmeasurementunita,theyardstick.Nextwemeasurethelengthofthemeanderinglinebyapproximatingitascloselyaspossiblewithabentlinemadeupofequalline‐segmentsoflengtha.Ifwesupposetheyardstickisused N times, so that the total length measured is N a, then according toMandelbrot’sdefinitionthe“fractaldimension”isgivenby6

€

D = lima→∞

logN

log 1a

Hausdorffdimensionhasmanyothernames,includingfractaldimensionand

similaritydimension.Putinamathematicalinequality:D>DT.

AlthoughMandelbrotoriginallygavetheterm“fractal”averyspecific

definition,throughtheyearstheexactdefinitionoffractalhasbecomealittlemore

lax.Manyassociatethetermfractalwithself‐similarity.Anobjectisself‐similarifit

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looksapproximatelythesameonsmallscalesasitdoesinlargescales.However

self‐similarityisnotlimitedtoastrictdefinitionofsimplylookingthesameon

differentscales.Self‐similarityalsocanfallacrosstranslation,meaningthatyou

don’tseeexactcopies,butsmaller,rotated,copiesaswell.Morediscussionabout

self‐similaritywillfollowinrelevantsections.

Fractalsarealsoknowntohavesimple,recursivedefinitions.Thisaddsto

theircomplexitybecausesuchsimpledefinitionsendupcreatingveryintricate

structures.Theyalsohavethedistinguishingfeatureofbeingindescribablethrough

simpleEuclideanGeometry(e.g.squares,circles,etc.)Sometimesthesestructures

canbeinfinitelycomplex,suchastheMandelbrotset,discussedinSectionVI.This

infinitecomplexitycanborderonchaos.

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SectionIVa:VonKochSnowflakes

ThevonKochsnowflakeisacommonelementaryfractalthatisusedto

familiarizepeoplewiththeworldoffractals.Itssimpleiterativerulesmakeiteasy

toconceptualizeanditspropertiesareverydistinctiveoffractals.Ifyoulookat

Figure1below:

FIGURE17

youwillseethevonKochreplacementrulein4iterations,startingonastraightline.

Eachiterationtakesthemiddlethirdofastraight‐linesegment,andthenaddstwo

morepiecesofequallengthwhichareplacedatanangle,creatingtwosidesofa

triangle.Everytimeyoufollowthisprocess,thelengthofyourlineisincreasedby

4/3.So,ifeachiterationincreasesthelinebythisamount,infinitelymanyiterations

willcreatealineofinfinitelength,whichMandelbrotnamesavonKocharc.This

shouldalsobenotedthatthislineisalsoboundedinafinitespace.

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Accordingtoourdefinitionofdimension,DT=log(4)/log(3)≈1.26,andD=

1,meaningDT<D.So,bydefinition,von‐Kocharcsarefractals.“Itiscontinuous,

andithasnodefinitetangentanywhere–likethegraphofacontinuousfunction

withoutaderivative.”8ThisfeaturemakesthevonKocharcaverydistastefulcurve

toconventionalmathematicians,whoarerecordedas“turningawayinfearand

horrorfromthislamentableplagueoffunctionswithnoderivatives.”9

IfthevonKochreplacementruleisiterateduponatriangle,yourresults

becomethewell‐knownvonKochsnowflake.JustlikethevonKocharc,whenthe

iterationisruninfinitelymanytimes,theborderlengthbecomesinfinite,whichfor

thissnowflakealsomeansthattheareabecomesthecirclethatyoudrawaround

thefirsttriangle.Basically,itapproachesthatbound.

FIGURE210

OneoftheearlycriticismsofthestudyofFractalGeometrythatMandelbrotstarted

wasthatfractalswere“prettypictures”andnothingmore.However,takingthevon‐

Kochfractalasanexample,fractalshavebecomesurprisinglyusefulinavarietyof

practicalpurposes.

SectionIVb:CANTORDUSTS

WhileVon‐Kochsnowflakesaresurprisinginitsinfinitelengthandvisual

appealmakesitinteresting,thereareotherfractalsthatdisappearwithfurther

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iterations.TheCantorset,orCantorDusts,isanexampleofsuchaset.Georg

Cantorhadtheideaforaclosedsetthatinitiatesfrom[0,1].“Foreachnumberin

thisinterval,thereisacorrespondingpointintheCantorSet…Thus,thecardinality

oftheCantorSetmustbeatleastaslargeasthecardinalityoftheinterval.”11

Thesetisclosedbecauseitincludestheendpoints.Tocreatetheset,simply

dividethesetintothirds,andremovethemiddlethird.Atthefirststepofthis

iteration,youwillremovetheopenset]1/3,2/3[,openbracketssignifyingthe

exclusionoftheendpoints.Ifthisiterationiscontinuedinfinitetimes,theresult,

accordingtoMandelbrot,istheCantorfractaldusts.Figure3belowillustratesthe

first3iterations.

FIGURE312

TheCantorsetisaverygoodexampleofafractalbecauseit’sHausdorff

dimensionisD=log(2)/log(3)≈0.63,whileit’stopologicaldimensionDTwouldbe

0.SinceDT<D,andthesetisafractal.Itisalsoself‐similar(eachiterationisa

thirdthelengthandtranslated)whichmakesitagoodexampleofafractal.Cantor

dustsalsohaveanotherinterestingfeaturethatmakethemworthyofnote.Cantor

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sets,wheniteratedtowardsinfinity,becometotallydisconnected.AsseeninFigure

3,thelinethatstartsascompletelyconnectedquicklybecomesdisconnected.

SectionIVb1:DisconnectedDusts

Intopology,asetiscalledconnectedifitcannotbedecomposedintotwo

disjoint,non‐emptysubsets(whicharebothopenandclosedinthetopologyofthe

set).13Anothertypeofconnectednessiscalledpathwiseconnected.Thismeansthat

acontinuouspaththatisentirelywithinthesetcanconnectanytwopointsofa

set.14Ontheotherhand,theinverseofaconnectedsetisatotallydisconnectedset,

whichmeansthatasetisconnectedcomponents(i.e.maximalconnectedsubsets)

aresinglepoints.

WhenobservingthehigheriterationsoftheCantorSet,nicknamedThe

CantorDusts,itisclearthatthelinesegmentsbecomesmallerandsmaller.What

cannotbeseenbythenakedeyeisthatthelinesegmentseventuallybecomepoints.

Atthislevelofiteration,theCantorDustsbecometotallydisconnected.Theproof

followsfromNelson:

Proof: Fixany

€

ε > 0 andpoint

€

p∈ C .Let

€

n ∈ N besufficiently largesuchthat

€

13n

< ε .Then,pisguaranteedtobeinoneoftheintervals(Inforsome

€

n ∈ N )that

makeupC,eachoflength

€

13n .TheendpointsoftheCantorsetinthisintervalare

infinite innumber, andall contained in theopen interval (

€

p −ε, p + ε ). So, p is aclusterpointof

€

C,Mε (p) containinganinfinitenumberofpoints.Andsinceweareconsideringany

€

p∈ C ,Cisperfect.Furthermore,thisintervalInisclosedinRandso in theCantorsetCaswell.Since

€

Inc = C \ In consistsofacountablenumberof

closedintervals, it is itselfclosed.WecanthenrepresentCasthedisjointunionoftwo clopen sets, (

€

C∩ In ) and (

€

C∩ Inc), the result being that the Cantor set C is

totallydisconnected.15

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ThispropertyisimportanttoourstudyoftheJuliasets,whichwillbecontinuedin

SectionV.

SectionIVb2:Self‐SimilarityintheDusts

Theideaofself‐similarityisanintuitivelysimpleonetograspwhenyouview

theCantorDusts.TakealookatFigure3.Ifyouviewthesetfromthefirstline

segmentanddown,youcanseethepatternclearlyforming:eachnextiterationhas

it’smiddlethirdremovedtocreatethefollowingiteration.Toshowhowthesetis

self‐similar,simplyremoveallthepiecestotherightfromthepictureandyouwill

haveasmaller,similar,setofCantorDusts.Ifyouelongateeverypieceofthe

remainingsideby3,youwillhaveyouroriginalCantorDusts.Thisexampleshows

usquiteeasilythatthesetisself‐similarthroughalineartransformation,whichis

notthecaseforallfractals.

SectionIVb3:InvarianceamongsttheDusts

Thischaracteristicofself‐similarityalsohelpsvisualizeanother

characteristicoftheCantorDusts,whichisinvariance.Asetwillbeinvariantifit

doesnotchangeundercertaintransformations.Asasimpleexample,weclaimthat

theCantorDustsareinvariant.IfyoutakeanypointwithintheCantorsetand

multiplyitby3,youwillfindanotherpointthatlieswithintheCantorset.This

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property,aswellasthepreviouslydiscussedpropertiesoftheCantorset,iscrucial

tounderstandingtheJuliasets,whichsharealotoftheseproperties.

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SectionV:JuliaSets

GastonJuliawasamathematicianwhostudiedmathematicsduringthelate

19thandearly20thcentury.Atthehumbleageof25,hismostfamouspaper,

"Mémoiresurl'itérationdesfonctionsrationnelles"waspublishedinthefamous

Frenchmathematicsjournal,JournaldeMathématiquesPuresetAppliquées.This

paperdescribedamethodofiterationofarationalfunction.Althoughhisideas

werepublishedin1918andwerehighlypraisedatthetime,itwasn’tuntilBenoit

MandelbrotdiscoveredthepaperandbeganusingJulia’sfindingsinhisownstudies

thatJulia’sworkbecamewellknown.

Juliasetsareaspecialtypeoffractalthatissointricateandcomplexlooking

thatMandelbrotoriginallydeemedthem“self‐squareddragons”.Aself‐squared

dragonisanothernameforquadraticfractals.PerhapsMandelbrotusedthisname

topokefunatthecriticsthatwerecallinghisfractaldiscoveries“monsters”,butthe

nameisactuallyquitesuitableonceyoubegintoseecertaincomputer‐generated

variations.

FIGURE416

Thesesetsarecalledself‐squaredbecausetheiterativeruleusedtocreatethemis

simplyaquadraticmapping:

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zn+1zn2+cFIGURE5

Wherezandcarecomplexnumbers.Thissimplemapping,iterativelyrepeated

resultsinasequenceofcomplexnumbers:

z0z2+c(z2+c)2+c((z2+c)2+c)2+c…

Juliastudiedmanydifferenttypesofrationalpolynomialexpressions,butthis

surveyfocusesonthemostfamousmapping;theoneshownabovewhichclosely

relatestooneofthemostfamousfractalsinhistory,theMandelbrotSet,discussed

indetailinSectionVI.Allabovevariablesarecomplexnumbersintheforma+bi,

whereaandbarerealnumbers.Thecomplexnumbercisaconstantthatisthekey

tothebeautyandcomplexityoftheJuliaset.Itdoesnotchangeduringiteration,but

eachseparateJuliasethasaseparatecvalue.

AlthoughcholdsthekeytotheJuliasetsamazingvisualstructure,the

startingvalueofzisthevariablethatdefineswhetheragivenpointwillbeapartof

theJuliaset.Asforthequestion,howdothevaluesofcandzaffecttheset,Ifindit

mostusefultousepicturestohelpaidintuition.

FIGURE3a17 FIGURE3b

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FIGURE3c FIGURE3d

FIGURE3e

AlltheseJuliasetsarecreatedusinguniquecvalues.Theconstantsusedto

generateeachfigurearelocatedinthebottomfield.Animportantaspecttothese

separateJuliasetsisthatthereexistinfinitelymanyversionsoftheJuliaset.

Becausecisacomplexnumber,andthereareinfinitelymanycomplexnumbers,

thereareinfinitelymanyJuliasets.

Tocreatethevisual“dragons”suchasinFigure4seenabove,theJuliaSet

mustbeiterateduponacomplexplane.Duringtheinfiniteiterationsofthe

quadraticmappingseeninFigure5,thechosenconstantofcwillcausetheJuliaset

todisplayoneoftwotypesofbehavior.Heinz‐Ottoclassifiesthisdichotomyofthe

JuliaSet:

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1. Eitherthesequencebecomesunbounded:theelementsofthesequenceleave

anycirclearoundtheorigin

2. Orthesequenceremainsbounded:thereisacirclearoundtheoriginwhichis

neverleftbythesequence18

Anyvaluethatfallswithinthefirstcategoryisclassifiedastheescapeset(E).These

valuestrendtowardinfinity.Thesecondcategoryisclassifiedastheprisonerset

(P),orthesetofvaluesthatdonottendtowardinfinity,andstaywithinalimits.

Neithersetcanbeempty.TheescapesetEcanbepreciselydefined:Ec={z0:|zn|

∞ asn∞}whiletheprisonersetPcanbepreciselydefinedasPc={z0|z0∉Ec}.

TheJuliasetisdefinedastheboundarybetweentheprisonersetandtheescapeset.

PerhapsitwouldbeeasiertoconceptualizetheJuliasetthroughavisual

representation.GastonJulia’sideaswerefaraheadofhistime,andhedidn’tlive

longenoughtoseecomputersandcomputergraphicsadvancetothepointwhere

theycoulditerateuponthesesets.Today,thereareendlessapplicationsandweb‐

appletsthatdrawthesetforyouandletyouzoomandexplorethesetyourself.

Clearly,humansdoingcalculationsbyhandwouldneverbeabletocalculateinfinite

iterationsonallpossiblepoints.Evenacomputercannotcalculateinfinitenumbers

andinfinitepoints.So,programscreatingimagesoftheJuliasetaresettocompute

acertainnumberofiterationsbeforedeclaringapointaspartoftheescapesetor

theprisonerset.Oncethesetisdeclared,thenthepointisdesignatedacolor.The

mostelementaryschemewouldbeblackandwhite;blackbeingpartoftheprisoner

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setandwhitebeingpartoftheescapeset.Youcanseethecleardistinctionbetween

thesetsinFigure6below.

FIGURE619 Whenlookinguponthese“dragons”isitclearthattheyareinfact,fractals?If

youlookcloselyatFigure6,itisclearthattherearemanysmallercopiesofthe

twistingaspectofthisJuliaset,aswellassmallercopiesofthewholeinthespiked

cornersoftheJuliaset

SectionVa:InvarianceofJuliaSets

DeterminingtheinvarianceoftheJuliasetsisamultiple‐stepprocess.First,

wemustdiscusstheJuliaset’siteration.TheJuliasetmappingiscreatedfromthe

simpleiterationonz0z12+c,meaningeachcomplexnumberziscalculatedasthe

squareofapreviouscomplexnumber,plusacomplexconstantc.So,wecouldre‐

writethisformulaasournewcomplexnumberw=z2+c.Wecanisolatethe

variablestoonesideandget:z2–w+c=0.Becauseweknowthatzisacomplex

number,wecanisolateittoonesideandthentakethecomplexsquareroottoget:

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€

z1,2 = ± w − c .Becausethissolutioncouldbepositiveornegative,thatmeansthere

aretwosolutions,orpreimages(z1,z2)foreachw,withtheexceptionofw=c.This

givesustwotransformations:

€

w→ + w − c

€

w→− w − c FIGURE7

ThesetransformationsarederivedfromouroriginalJuliasetmapping,

€

z→ z2 + c ,

whichsignifiesthatanypointintheJuliasetwundergoingthesetransformations

willcreateanotherpointontheJuliaset.Inotherwords,theJuliasetisaninvariant

setwithrespecttotheinversetransformationsinFigure7.Thisalsomeansthatthe

Juliasetisaninvariantsetwithrespectto

€

z→ z2 + c .20Ouroriginalmappingis

classifiedasforwarditerationbecauseyoustartatthebeginningandkeepiterating

untilyourreachtheend(orinthiscase,amaximumnumberofiterations).The

mappingswecreatedinFigure7aremethodofbackwardsiteration,andbecause

theJuliasetisinvariantunderboththeseconditions,theJuliasetsaresaidtohave

completeinvariance.

SectionVb:Self‐SimilarityofJuliaSets

BecauseJuliasetshavecompleteinvarianceimpliesthatthe“globalstructure

oftheJuliasetsmustalsoappearintheimagesandpreimagesoftheJuliaset,which

explainstheapparentself‐similarity.”21UnliketheCantorDustssimilarity

describedinSectionIV‐b2,thisself‐similarityoccursthroughanon­linear

transformation.Insimplerterms,self‐similarcopiesofthesetoccurindistorted,

twistedversionsofthewhole.TheBestvisualrepresentationofthisideaisDouady

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andHubbard’srabbit,seeninFigure8below.Eachboxedsectionisareplicaofthe

others,andthelineinthemiddleseparatetwoequalhalvesthataretranslatedand

flipped.

FIGURE822

Thisimagehasbeenkeptlargesoyoucanseethesimilaritiesinitsstructure.

SectionVc:ConnectednessofJuliaSets

Sofar,wehavediscussedsomeinterestingaspectsoftheJuliasets.Allthese

pointsculminateintothemostimportantcharacteristicoftheJuliasets:their

connectivity.AdefinitionofconnectivityisofferedinSection

ThefoundersoftheoriginaltheoriesthatleadtothediscoveryoftheJulia

set,FatouandJulia,cameupwiththestructuraldichotomyoftheJuliasets.This

statedthat:

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1. TheJuliasetisaCantorsetifandonlyif,theiterationofthecriticalpoint0

leadstoinfinity(inabsolutevalue)

2. TheJuliasetisonepiece(i.e.connected)ifandonlyiftheiterationofthe

criticalpoint0isbounded.23

Thecriticalpointisthex‐coordinateonyourgraph,andthey‐coordinateisthe

criticalvalue.Thefateofthecriticalpoint,orthecriticalorbitdeterminesifyour

Juliasetbelongstoeitheroneofthesetwocategories.WhetheraJuliasetbecomes

atotallydisconnectedCantorsetoraconnectedsetisimportantinitsrelationtothe

Mandelbrotset.,whichwillbediscussedinSectionVII.

FIGURE10a FIGURE10b

AsyoucanseeinFigures10aand10b,Juliasetscandisplayverydifferent

typesofbehavior.In10a,wehaveaJuliasetthatistotallydisconnected.Atfirst

glance,thestructuremightseemtohavesomeislandsscatteredabout,butifyou

zoomintothedrawingcloseenough,andwithenoughcomputationalaccuracy,you

willfindthatthesetismadeupofdistinct,disconnectedpointsandnothingelse.It

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carriesthesamequalitiesoftheCantordustsdiscussedintheearliersection.On

theotherhand,Figure10bdisplaystheoppositetypeofbehaviorinthatitremains

connected.Simplyput,theentiresetisonepiece,andallpointswithinthesetcan

reacheachotherwhiletravellingthroughtheset.

ThefollowingtheoremfromJuliaandFatou:“Let

€

Ωp denotethesetofcritical

pointsforapolynomialP.Then:

•

€

Ωp ⊂ Kp ⇔ Jp isconnected•

€

Ωp ∩Kp =∅⇒ Jp isaCantorset24

ThistheoremstatesthateveryJuliasetmustbeeitherconnected,ortotally

disconnected(i.e.aCantorset).Theproofofthetheoreminvolvestheclosedcurves

surroundingfilled‐in,connectedJuliasets.Forthisproof,notcontainedhereforits

mathematicalcomplexity,lookitupinDevaney1989(Reference24).Thisproofis

importantnotonlytotheconnectednessofthefilled‐inJuliasets,butthesame

strategyisusedtoprovetheconnectednessoftheMandelbrotset.

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SectionVI:TheMandelbrotSet

WhendiscussingtheJuliaset,isitinevitabletogravitatetowardsdiscussion

oftheMandelbrotSet.IftheJuliasetwastheparent,theMandelbrotsetismost

definitelythechild.NowthatwehaveextensivelyexploredtheJuliasets,and

coveredmanydifferentinterestingfeaturesofsaidset,whatisthenextstep?From

thelastsection,wehaveresolvedthatthereareinfinitelymanyJuliasets,andthat

allJuliasetsfallintotwocategories:

1. Connected

2. Totallydisconnected(i.e.adust)

ThisiscalledthestructuraldichotomyoftheJuliasets,aslabeledbyitsdiscoveries,

FatouandJulia.Howeverinfinitelyinterestingthisfactmaybe,itsusefulnessand

practicalitywasinquestionuntilBenoitMandelbrotdecidedtoexplorethis

dichotomyis1979.Inbothoftheserelatedsets,thesameiterationisused:

€

z→ z2 + c .RecallthatfortheJuliasets,wekeepcfixedanditerateuponthe

complexztodeterminewhetherthecriticalpointtrendstowardsinfinity.Inthe

caseoftheMandelbrotset,wekeeptheinitialz0fixed,andeachcriticalpointis

changedaccordingtoc.MandelbrotusedhisunderstandingoftheJuliasetsand

computergraphicstogenerateapictoralrepresentationofthedichotomyinthe

complexplane.Throughhisexperiment,theMandelbrotsetMwasborn.

€

M = {c ∈ C | Jc : is_connected}

€

M = {c ∈ C | c→ c 2 + c→ ...remains_bounded}25

24

Mandelbrotgraphedhisideaonthecomplexplanefrom‐2to2,because

when|c|>2,thecriticalpointescapestoinfinity.Inotherwords,thecomplex

iterationzwillbeattractedtoinfinity,andbepartoftheescapesetE.Thisproof

canbefoundin[Peitgen794].Thisalsomeansthatitfallsunderthesecondofthe

twocategorieslistedabove,andtheresultingJuliasetwillbeaCantordust,atotally

disconnectedset.TheJuliasetcontainsaboundedcriticalpointfortheinterval[‐2,

0.25],meaningeverycomplexnumberwithinthisrangecreatesaconnectedJulia

set.TheseconnectedJuliasetsarewhatmakeuptheMandelbrotset.

FIGURE1126

ToreallyappreciatethecomplexityoftheMandelbrotset,youhavetoseeit.

Creatingavisualrepresentationofthesetentailscountingthenumberofiterations

thatittakesforthemappingtoescapetoinfinityforeachpixelonthescreen.By

calculatingtheboundforeachpixel,youarecheckingtoseeifforthatcvalue,the

quadraticiterationescapestoinfinityornot.Insteadofcalculatinginfinite

iterations(whichwouldbequiteimpossible,evenonacomputer),programmers

usuallyusedelineatesomesortofnumberofmaximumiterationswherethepoint

willbeassumedtotrendtowardsinfinityandassignitacolor.InthecaseinFigure

25

10,thosenumbersthatfollowthistrend(i.e.areintheescapesetE)arecolored

darkblueandthosethatarebounded,(i.e.intheprisonersetP)arecoloredblack,

TheadditionalcolorsareincludedtomakethesmallerintricaciesoftheMandelbrot

setmorevisibleandaestheticallypleasing.

Atthemostbasiclevel,theMandelbrotsetiseverythingthatfallsintheblack

areaoffigure11.

SectionVIa:ConnectednessoftheMandelbrotSet

TheintricatepicturesoftheMandelbrotSetareasbeautifulastheyare

complicated.Itisthisinfinitecomplexitythatmakesthesetsointriguing.One

importantaspectofMisthatitisconnected.Originally,afterMandelbrotprinted

outthefirstimageoftheMandelbrotset,heinitiallythoughtthatMwas

disconnected,becauseoutsideofthemain“bulbs”,thereseemedtobesome

scattereddusts.Theerrorwasnotthemathematician,butthemachine.Printers

justcouldnotprintatresolutionshighenoughfortheconnectednesstobevisible.

ThisisanunderstandablemistakeonceyouseewhatMandelbrotsawsomany

yearsago:

FIGURE12

26

However,itisinfactsimplyconnected,astothedefinitionofferedinSection

IV‐b2.DouadyandHubbardprovedthislongago,intheirfamouspaper“Iteration

despolynomesquadratiques”or“TheIterationofQuadraticPolynomials”.The

proofstatesthat“theencirclementoftheMandelbrotsetalwaysgeneratesdomains

whichareboundedbycircle‐likecurves.Iftheencirclementisproperly

manufactureditcanbeshownthattheboundingcurvesareinfactequipotentialsof

theMandelbrotset.”27Moreprecisely:“Let

€

VM (r) denotetheclosed,simply

connecteddomainboundedby

€

ΓM (r) .Then:

€

M = VMr>1 (r) .Hence,Misconnected.”28

SectionVIIb:Self‐SimilaritywithintheMandelbrotset

ThisaspectoftheMandelbrotsetistrulyinterestingwhenyouobservethe

smallerislandsthataresurroundingtheMandelbrotset.Thenextsetofpictures

wasobtainedusingtheMacintoshPowerFractalprogram.

FIGURE13a

27

FIGURE13b

FIGURE13c

FIGURE13d

28

Figure13aisclearlythefullMandelbrotset.Therestofthepicturesaretakenfrom

differentspacessurroundingthemain“bulb”.DouadyandHubbardhaveatheorem

thatthereareinfinitelymanycopiesoftheMandelbrotsetwithintheMandelbrot

set,hereweseeafewofthem.TheimportantthingtonoteisthattheMandelbrot

setisquasi‐selfsimilar:itisdoesnotfitasimpledefinitionofself‐similarity.Rather,

theMandelbrotsetisdescribedasbeingasymptoticallyself‐similaratMisiurewicz

points.ThesepointshaveaspecialrelationtotheJuliasetaswell,andwillbe

discussedinSectionVII‐b.

29

SectionVII:Juliavs.Mandelbrot

NowthatboththeJuliasetsandtheMandelbrotsethavebeendefinedand

analyzedtothepointoffamiliarity,letusexaminethedifferences,similarities,and

relationshipsbetweentwoofthemostfamousfractalsinexistence.Thefirst

observation,thathasalreadybeenmentioned,isthatthetwosetsbothusethesame

iteration( ).Howthen,arethestructuresofthesetssodifferent?

Theanswerisassimpleastheiterationitself.RecallthatJuliasetsare

measurediterationsoftheaboveequationforconstantcvalues.Foreveryuniquec

value,therewillbeauniqueJuliaset.Thisiteration,withaconstantc,ismappedon

thecomplexplane.Yourcomplexplaneobviouslyholdscomplexvalues,andfor

eachofthesevaluesisadifferentstartingz0value.Youiterateuponthedifferentz

valueswiththesamec.Soyouwillgetiterationsthatlooklikethis:

€

z→ z2 + c→ (z2 + c)2 + c...

TheMandelbrotsettakestheoppositeapproach.Theiterationusesa

standardstartingz0=0valueforeveryiteration.Itwilliterate,instead,ona

variationofc.So,onthecomplexplane,againimagineallofthecomplexnumbers

thatfallonit.Eachofthesepointswillbeusedasaseparatecvalueforthe

iteration,andeachpointwilleitheriteratetowardsinfinityorremainbounded.

Sincewestartat0,iterationswilllooklikethis:

€

0→ 02 + c→ (02 + c)2 + c...

Thissimplevariationcausesanaestheticallypleasingresult.TheMandelbrot

setisquitebeautiful,anditsbeautyisaccentedbyitsintricacies.Firstly,theentire

Mandelbrotsetis“aroadmapforJuliaSets.”29Definedcasually,butaccurately,

!

z" z2

+ c

30

everypointontheMandelbrotsetmappingcorrespondstoit’sownuniqueJuliaset.

Intuitively,everyMandelbrotpointiscreatedusingadifferentcvalue.IfeveryJulia

setiscreatedusingadifferentcvalue,thentheintuitionshouldfollowthat

MandelbrotpointscreateJuliasets.Toaidintuition,thefollowingfigureshowsthe

correspondencebetweenthesets.

FIGURE1430

EachJuliasetseensurroundingtheMandelbrotsetcorrespondstothecpoint

indicated.Ifthereexistsmoreinterestinthisrelation,therearemany,manyweb

appletsthatallowuserstoclickonaMandelbrotsetandgeneratethe

correspondingJuliaset.

TheseJuliasetsalsofollowatrendwithregardstotheMandelbrotset.For

everyJuliaSetcreatedwithacvaluethatfallswithintheMandelbrotset,youhavea

connectedJuliaset.IfthecvalueusedtocreateaJuliasetfallsoutsidethe

boundariesoftheMandelbrotset,youthenhaveatotallydisconnectedJuliaset.

ObserveFigure14.Ontheleft,youcanseeacpointthatischosenoutsideofthe

31

Mandelbrot’sbarriers.Thissetisclearlyadust,withitsbarelyvisibleportionsand

verygrayappearance.

SectionVIIa:AsymptoticSelf‐SimilaritybetweentheSets

Therelationbetweenthetwotypesofstructuresdoesnotstopattheir

iteration.Asshownearlier,theMandelbrotsethasaninfinitelevelofdetail,only

limitedbythecalculationsacomputercancompleteinatimelymannertodisplay

imagesofgreatdetail.Thispropertyisshownwhenexaminingthesmaller

Mandelbrotsetsthatliewithinthedetailsofthefullset.Attheminutelevel,

magnificationlevels300,000,000foldfromtheoriginal,wecanfindpointsthatare

indistinguishablefrommagnificationsofrelatedJuliasets.Againtheeasiestwayto

understandtheseconceptsisthroughpictures:

FIGURE15.1‐a FIGURE15.2‐a

32

FIGURE15.1‐b FIGURE15.2‐b

FIGURE15.1‐c FIGURE15.2‐c

FIGURE15.1‐d FIGURE15.2‐d

Clearlythefirsttwofigures(15.1‐aand15.2‐a)aresimilar.Theybothshowthe

samespiralpatternandhavethesamesmallerbudssurroundingthem.The

sequenceintheleftcolumnisfromtheseahorsevalleyoftheMandelbrotset.The

33

right‐columnsequenceistheJuliasetcreatedfromthecvalue(‐0.7746899+

0.1242248i)atthecenterofFigure15.1‐a.Eachtimeyoumovedownanimage,the

framehaszoomedoutslightly.AllfiguresweredrawnwithMacOSXsoftware

FractalDomains2.0.9(©DennisC.DeMars).

Thissimilarityisnotself‐similaritythatexistsinmoresimplefractals

structuresliketheCantorDusts.Theirsimilaritiesarisethroughmorecomplex

mathematicsthansimplerotationandmagnification.Forthisspecifictypeof

similarity,weassignthetermasymptoticallyself­similar.“WecallIasymptotically

self‐similaratthepointz0ifthereare:

• Acomplexscalingfactorp,calledmultiplier,with|p|>1

• Asmallradiusr>0

• AndalimitobjectL(asubsetofthecomplexplane)whichisself‐similarat

theorigin

Suchthattherelation

€

limn→∞

Dr(0)∩ pn (I − z0) = L∩Dr(0)holds”31,wherepisa

complexnumbercalledthescalingfactor,andDr(z)isadiskcenteredatz.Themore

extensiveproofscanbefoundinTanLei’spaperonMisiurewiczpoints32.

Misiurewiczpointsareverycomplexmathematically,butvisuallyitisquite

simpletoseethecorrelationbetweenthesetsatMisiurewiczpointsifyouobserve

thepictures.Theabovesequence,isinfact,oneofthesepoints.Interestingly

enough,thesepointsexistallovertheboundaryoftheMandelbrotset;theyare

“denseattheboundaryoftheMandelbrotset.Thismeansthatifwetakeanypoint

34

ontheboundaryofMandanarbitrarilysmalldiskaroundthatpoint,thenthere

existsaMisiurewiczpointinthatdisk”33.AsummaryofTanLei’sproofsaysthat

• TheJuliasetJcandtheMandelbrotsetarebothasymptoticallyself‐similarin

thepointz=cusingthesamemultiplierp.

• TheassociatedlimitobjectsLjandLMareessentiallythesame;theydiffer

onlybysomescalingandarotation(LM=λLj,whereλisasuitablecomplex

number)

SectionVIIb:Peitgen’sObservation

Peitgenalsofindsasimilarsimilaritybetweenthesets.Forcertainvaluesof

c, andcertainmagnifications,M and Jcproducevery similar images. Although the

imagesarenotasidenticalasMisiurewiczpoints,thereismostdefinitelysimilarity

betweenthetwosets.ThesefigureswerealsodrawnusingFractalDomains:

FIGURE16a FIGURE16b

35

FIGURE16c FIGURE16d

Thesimilarityisblatant,hereseenbetweentheMandelbrotsetontheleftandJulia

setcreated(16b)usingthec=‐0.745429+0.113008ifromthecenterofFigure16a.

Thearrowsdisplaywhere the zoomedpictureoriginates from, the samepointon

thecomplexplaneinbothfigures,whichisthesamevalueasthecgivenearlier.

36

SectionVIII:Conclusion

Throughthissurvey,wehavediscussedthemanwhoI’vedeemed“The

Godfatheroffractals”,BenoîtMandelbrot.Hisstudieshaveopeneduptheentire

fieldofgeometrytoincludethesenewstructures.Hopefullythispaperwaseffective

inhelpingreadersexperiencetheirbeautymathematicallyandvisually.

FromthemostsimplefractalsliketheCantorDusts,tomoreadvanced

fractalsliketheMandelbrotset,allthesebeautifulfigureshavemostdefinitely

changedmyopinionofmathematics.

ThroughtheexaminationoftheJuliasetsandtheMandelbrotset,wehave

justbeguntoscratchthesurfaceoftheintricaciesoffractals.Uptothispoint,

fractalshavebeenusedincarantennas,wirelesscellphones,aswellasUSMarine

camouflagedesign.Thepracticalityoffractals,liketheirstructure,isinfinite.

37

SectionIX:Acknowledgements

I’dliketothankmyfamilyforsupportingmeineverywaythroughoutmy

life.Ifitwasn’tforthem,Iwouldmostdefinitelynotbeabletogetthisfarinlife,let

alonecollege.Second,I’dliketothankBobMartin,mythesisadvisor,forhaving

patiencewithmeaswestruggledtofindatopic.Eventuallywefoundsomething

thatpeakedourinterests,andIappreciatehispatienceandguidancethroughout

thisprocess.Finally,IwanttothanktheMiddleburyComputerSciencedepartment

forallowingmetostudythewonderfulworldofcomputers.ThanksandIhopeto

keepintouchwithyouall.

38

SectionX:Bibliography

1(p1)Mandelbrot,BenoitB.“TheFractalGeometryofNature”W.H.FreemanandCompany,NewYork19832(p3)Mandelbrot3(p6)Mandelbrot4(p15)Mandelbrot5(p19)McGuire,Michael.“AnEyeforFractals”Addison‐WesleyPublishingCompanyInc.RedWoodCity,CA19916(p33)Lauwerier,Hans“Fractals”PrincetonUniversityPress19917Falconer,K.1990,FractalGeometry:MathematicalFoundationsandApplications(Chichester:JohnWileyandSons).8(p35)Mandelbrot9(p36)Mandelbrot10JefferyJohnson,COMPUTERWEEKLY,March30,1989via:http://www.fortunecity.com/emachines/e11/86/newmath.html11(p75)Peitgen,Heinz‐Otto“ChaosandFractals:NewFrontiersofScience”Springer200412Noel,Griffin“CantorDust”Taken04‐28‐10via:http://spanky.triumf.ca/www/fractal‐info/cantor.htm13(p803)Peitgen14(p803)Peitgen15Nelson,Dylan“TheCantorSet–AbriefIntroduction”UCBerkley16Eequor,Wikipedia,November25,2004via:http://en.wikipedia.org/wiki/Julia_set17JuliaSetAppletvia:http://www.shodor.org/interactivate/activities/JuliaSets/18Peitgen19Ashlock,Daniel.UniversityofGuelph.April2010via:http://eldar.mathstat.uoguelph.ca/dashlock/ftax/Julia.html20(p823)Peitgen21(p824)Peitgen22Winter,Dale,“PatternsinIterationandtheGraphofaFunction”http://www.math.lsa.umich.edu/mmss/coursesONLINE/chaos/chaos2/index.html23(p834)Peitgen24(p80)Devaney,RobertL“ChaosandFractals:TheMathematicsbehindtheComputerGraphics”AmericanMathematicalSociety198925(p843)Peitgen26Beyer,Wolfgang“MandelbrotSet”2005via:http://en.wikipedia.org/wiki/File:Mandel_zoom_00_mandelbrot_set.jpg27(p849)Peitgen28(p91)Devaney198929(p855)Peitgen

39

30Miqel,taken04‐28‐10via: http://www.miqel.com/images_1/fractal_math_patterns/mandelbrot‐set/31(p885)Peitgen32Lei,Tan“SimilaritybetweentheMandelbrotsetandtheJuliasets”Springer‐Verlag199033(p886)Peitgen