FractalGeometry.docx

FractalGeometry.docx

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FractalGeometry

Fractals:

UsefulBeauty

（GeneralIntroductiontoFractalGeometry）

ReturntoindexBBM

"Cloudsarenotspheres,mountainsarenotcones,coastlinesarenotcircles,andbarkisnotsmooth,nordoeslightningtravelinastraightline."

BenoitMandelbrot

EdytaPatrzalek,StanAckermansInstitute,

IPO,CentreforUser-SystemInteraction,EindhovenUniversityofTechnology

Abstract

Fractalsisanewbranchofmathematicsandart.Perhapsthisisthereasonwhymostpeoplerecognizefractalsonlyasprettypicturesusefulasbackgroundsonthecomputerscreenororiginalpostcardpatterns.Butwhataretheyreally?

MostphysicalsystemsofnatureandmanyhumanartifactsarenotregulargeometricshapesofthestandardgeometryderivedfromEuclid.Fractalgeometryoffersalmostunlimitedwaysofdescribing,measuringandpredictingthesenaturalphenomena.Butisitpossibletodefinethewholeworldusingmathematicalequations?

Thisarticledescribeshowthefourmostfamousfractalswerecreatedandexplainsthemostimportantfractalproperties,whichmakefractalsusefulfordifferentdomainofscience.

Introduction

Manypeoplearefascinatedbythebeautifulimagestermedfractals.Extendingbeyondthetypicalperceptionofmathematicsasabodyofcomplicated,boringformulas,fractalgeometrymixesartwithmathematicstodemonstratethatequationsaremorethanjustacollectionofnumbers.Whatmakesfractalsevenmoreinterestingisthattheyarethebestexistingmathematicaldescriptions

ofmanynaturalforms,suchascoastlines,mountainsorpartsoflivingorganisms.

Althoughfractalgeometryiscloselyconnectedwithcomputertechniques,somepeoplehadworkedonfractalslongbeforetheinventionofcomputers.ThosepeoplewereBritishcartographers,whoencounteredtheprobleminmeasuringthelengthofBritaincoast.Thecoastlinemeasuredonalargescalemapwasapproximatelyhalfthelengthofcoastlinemeasuredonadetailedmap.Theclosertheylooked,themoredetailedandlongerthecoastlinebecame.Theydidnotrealizethattheyhaddiscoveredoneofthemainpropertiesoffractals.

Fractals’properties

Twoofthemostimportantpropertiesoffractalsareself-similarityandnon-integerdimension.

Whatdoesself-similaritymean?

Ifyoulookcarefullyatafernleaf,youwillnoticethateverylittleleaf-partofthebiggerone-hasthesameshapeasthewholefernleaf.Youcansaythatthefernleafisself-similar.Thesameiswithfractals:

youcanmagnifythemmanytimesandaftereverystepyouwillseethesameshape,whichischaracteristicofthatparticularfractal.

Thenon-integerdimensionismoredifficulttoexplain.Classicalgeometrydealswithobjectsofintegerdimensions:

zerodimensionalpoints,onedimensionallinesandcurves,twodimensionalplanefiguressuchassquaresandcircles,andthreedimensionalsolidssuchascubesandspheres.However,manynaturalphenomenaarebetterdescribedusingadimensionbetweentwowholenumbers.Sowhileastraightlinehasadimensionofone,afractalcurvewillhaveadimensionbetweenoneandtwo,dependingonhowmuchspaceittakesupasittwistsandcurves.Themoretheflatfractalfillsaplane,thecloseritapproachestwodimensions.Likewise,a"hillyfractalscene"willreachadimensionsomewherebetweentwoandthree.Soafractallandscapemadeupofalargehillcoveredwithtinymoundswouldbeclosetotheseconddimension,whilearoughsurfacecomposedofmanymedium-sizedhillswouldbeclosetothethirddimension.

Therearealotofdifferenttypesoffractals.InthispaperIwillpresenttwoofthemostpopulartypes:

complexnumberfractalsandIteratedFunctionSystem（IFS）fractals.

Complexnumberfractals

Beforedescribingthistypeoffractal,Idecidedtoexplainbrieflythetheoryofcomplexnumbers.

Acomplexnumberconsistsofarealnumberaddedtoanimaginarynumber.Itiscommontorefertoacomplexnumberasa"point"onthecomplexplane.Ifthecomplexnumberis

thecoordinatesofthepointarea（horizontal-realaxis）andb（vertical-imaginaryaxis）.

Theunitofimaginarynumbers:

.

TwoleadingresearchersinthefieldofcomplexnumberfractalsareGastonMauriceJuliaandBenoitMandelbrot.

GastonMauriceJuliawasbornattheendof19thcenturyinAlgeria.Hespenthislifestudyingtheiterationofpolynomialsandrationalfunctions.Aroundthe1920s,afterpublishinghispaperontheiterationofarationalfunction,Juliabecamefamous.However,afterhisdeath,hewasforgotten.

Inthe1970s,theworkofGastonMauriceJuliawasrevivedandpopularizedbythePolish-bornBenoitMandelbrot.InspiredbyJulia’swork,andwiththeaidofcomputergraphics,IBMemployeeMandelbrotwasabletoshowthefirstpicturesofthemostbeautifulfractalsknowntoday.

Mandelbrotset

TheMandelbrotsetisthesetofpointsonacomplexplain.TobuildtheMandelbrotset,wehavetouseanalgorithmbasedontherecursiveformula:

separatingthepointsofthecomplexplaneintotwocategories:

∙pointsinsidetheMandelbrotset,

∙pointsoutsidetheMandelbrotset.

Theimagebelowshowsaportionofthecomplexplane.ThepointsoftheMandelbrotsethavebeencoloredblack.

ItisalsopossibletoassignacolortothepointsoutsidetheMandelbrotset.TheircolorsdependonhowmanyiterationshavebeenrequiredtodeterminethattheyareoutsidetheMandelbrotset.

HowistheMandelbrotsetcreated?

TocreatetheMandelbrotsetwehavetopickapoint（C）onthecomplexplane.Thecomplexnumbercorrespondingwiththispointhastheform:

Aftercalculatingthevalueofpreviousexpression:

usingzeroasthevalueof

weobtainCastheresult.Thenextstepconsistsofassigningtheresultto

andrepeatingthecalculation:

nowtheresultisthecomplexnumber

.Thenwehavetoassignthevalueto

andrepeattheprocessagainandagain.

Thisprocesscanberepresentedasthe"migration"oftheinitialpointCacrosstheplane.Whathappenstothepointwhenwerepeatedlyiteratethefunction?

Willitremainneartotheoriginorwillitgoawayfromit,increasingitsdistancefromtheoriginwithoutlimit?

Inthefirstcase,wesaythatCbelongstotheMandelbrotset（itisoneoftheblackpointsintheimage）;otherwise,wesaythatitgoestoinfinityandweassignacolortoCdependingonthespeedatwhichthepoint"escapes"fromtheorigin.

Wecantakealookatthealgorithmfromadifferentpointofview.Letusimaginethatallthepointsontheplaneareattractedbyboth:

infinityandtheMandelbrotset.Thatmakesiteasytounderstandwhy:

∙pointsfarfromtheMandelbrotsetrapidlymovetowardsinfinity,

∙pointsclosetotheMandelbrotsetslowlyescapetoinfinity,

∙pointsinsidetheMandelbrotsetneverescapetoinfinity.

Juliasets

JuliasetsarestrictlyconnectedwiththeMandelbrotset.TheiterativefunctionthatisusedtoproducethemisthesameasfortheMandelbrotset.Theonlydifferenceisthewaythisformulaisused.InordertodrawapictureoftheMandelbrotset,weiteratetheformulaforeachpointCofthecomplexplane,alwaysstartingwith

.IfwewanttomakeapictureofaJuliaset,Cmustbeconstantduringthewholegenerationprocess,whilethevalueof

varies.ThevalueofCdeterminestheshapeoftheJuliaset;inotherwords,eachpointofthecomplexplaneisassociatedwithaparticularJuliaset.

HowisaJuliasetcreated?

WehavetopickapointC）onthecomplexplane.ThefollowingalgorithmdetermineswhetherornotapointoncomplexplaneZ）belongstotheJuliasetassociatedwithC,anddeterminesthecolorthatshouldbeassignedtoit.ToseeifZbelongstotheset,wehavetoiteratethefunction

using

.WhathappenstotheinitialpointZwhentheformulaisiterated?

Willitremainneartotheoriginorwillitgoawayfromit,increasingitsdistancefromtheoriginwithoutlimit?

Inthefirstcase,itbelongstotheJuliaset;otherwiseitgoestoinfinityandweassignacolortoZdependingonthespeedthepoint"escapes"fromtheorigin.ToproduceanimageofthewholeJuliasetassociatedwithC,wemustrepeatthisprocessforallthepointsZwhosecoordinatesareincludedinthisrange:

;

ThemostimportantrelationshipbetweenJuliasetsandMandelbrotsetisthatwhiletheMandelbrotsetisconnected（itisasinglepiece）,aJuliasetisconnectedonlyifitisassociatedwithapointinsidetheMandelbrotset.Forexample:

theJuliasetassociatedwith

isconnected;theJuliasetassociatedwith

isnotconnected（seepicturebelow）.

IteratedFunctionSystemFractals

IteratedFunctionSystem（IFS）fractalsarecreatedonthebasisofsimpleplanetransformations:

scaling,dislocationandtheplaneaxesrotation.CreatinganIFSfractalconsistsoffollowingsteps:

1.definingasetofplanetransformations,

2.drawinganinitialpatternontheplane（anypattern）,

3.transformingtheinitialpatternusingthetransformationsdefinedinfirststep,

4.transformingthenewpicture（combinationofinitialandtransformedpatterns）usingthesamesetoftransformations,

5.repeatingthefourthstepasmanytimesaspossible（intheory,thisprocedurecanberepeatedaninfinitenumberoftimes）.

ThemostfamousISFfractalsaretheSierpinskiTriangleandtheKochSnowflake.

SierpinskiTriangle

Thisisthefractalwecangetbytakingthemidpointsofeachsideofanequilateraltriangleandconnectingthem.Theiterationsshouldberepeatedaninfinitenumberoftimes.ThepicturesbelowpresentfourinitialstepsoftheconstructionoftheSierpinskiTriangle:

1）

2）

3）

4）