penta flakes and fractals for high school students [20th century geometry]

Pentaflakes,DurerTilingandFractals

[forhighschoolstudents]

NKSrinivasanPhD

Introduction

Muchofthematerialinthisarticleis20th

centuryand21stcentury'modern'geometry.

Thefocusison'FractalGeometry'

,introducedthrough'Penta-flakes'.

Wemakelittlediversionsintorelated

topicssuchasFibonaccisequence,golden

ratio,recursiveanditerativeprocessesand

dynamicalsystemsandotherrelatedmath

stuff.Thereisabriefintroductionto

Chaostheoryandits'fractal'

manifestation,leadingto'Mandelbrotset'.

Ihaveattemptedtowritethisarticlewith

highschoolmathstudentsinmind.Read

on!

Penta-flakes

Youarefamiliarwithsnowflakeswithsix

branchesorarms,inaregularhexagonal

pattern--thatis,with6-foldrotational

symmetry.Youcanconstructa'Penta-flake"

with5-foldrotationalsymmetry,starting

withapentagon.

Pentagon

Letusrecallabitaboutpentagonsyouhave

studiedinmiddleschoolgeometry.

Aregularpentagonhasfivesides,all

equal,withfive-foldrotationalsymmetry,

whileasquarehasfourequalsides.The

internalangleofasquareis90degrees,

whiletheinternalangleofapentagonis

108degrees...keepthisinmind.

Theexternalangle,formedbyextendingany

sideofapolygonis,ofcourse,72

degrees,thatis180-108.

Theangle72degreesisfascinating;several

propertiesariseformthis.Youwillcome

across'sin36'andalso'sin54'degrees.

Notethatsin54=0.809

Therefore2sin54=1.618

Doesthisnumberringabell?

Goldenratio

Well,thisnumberisthewell-known

Fibonaccinumber,arisingfromtheFibonacci

sequence:0,1,1,2,3,5,8,13,21,34,55,

89-----

Weformthissequencebyaddingtheprevious

twonumbers,startingwith0and1.

Ifyoutaketheratioofanytwonumbers,

callitF';F'willtendtoFibonacci

number:

F'=8/5=1.6

F'=21/13=1.615

F'=55/34=1.6176

F'=89/55=1.6181818

ThisratiowilltendtowardstheGolden

Ratio=1.61818

Goldenratioisdenotedbyphi:

phi=φ=(1+√5)/2

[Thisisalsooneofthesolutionsofthe

quadraticequation:x2=1+x

orx2-x-1=0]

YoucanrelateapentagontoFibonacci

numbereasily.

Inapentagon,thediagonalisphixside.

[Youcanprovethisforyourselfbydrawing

thediagonalofapentagonandits

perpendicularbisector.Theresultingright

triangleswillhave54degreeangles.Then

thediagonalis2sin54=1.618.]

Theaestheticappealofpentagonisdueto

theinclusionofgoldenratioinitsshape.

AlbrechtDurerTiling

Nowwejumptothistopic.AlbrechtDurer

(1471-1528),thefamousGermanpainterand

whowaswell-knownforhiswood-cutdrawings

,wroteabookcalled"thePainter'sManual'.

Inthatbook,heshowedatilingwithfive

foldorpentagonalsymmetry.Thisbookwas

publishedin1525.'Durertiling',formed

withpentagonsandrhombuses,isthepattern

ofPenta-flakes.

Penta-flakesorDurertilingisalsoa

fractal."Fractalgeometry"wasdiscovered

byBenoitMandelbrot,in1975.

ThusDurerwasthepioneerofFractal

geometry,eventhoughitwasnotknownas

such..Youmayfindanintroductiontothis

geometry,throughKochcurves,inyour

geometrytextbook.

Fractalimagesarealsofoundincertain

tilesandwallsofsomecathedralsand

mosques,includingtheCathedralat

Anagni,Italy.Theywereconstructedinthe

medievalperiod,say10thto13thcentury.

Apentaflakefigurebytheauthor

Whatisafractal?

Afractalisageometricfigurewith

self-similarityproperty.

Thismeansthateverysmallpartofthe

figurewillcontainareplicaorcopyof

thelargerarea..thisislayman'sdefinition

ofafractal.

YoucandrawalinesayABof3inchesin

length.Cutandremove1/3ofitslength,

thatisoneinch,fromthecentralpartof

theline.Keeprepeatingtheprocess,by

takingtheremainingtwosmalllinesand

cuttingout(1/3)ofitslengthatthe

center...Youwillgetsmallerandsmaller

linesegments...thiswillleadtoa

fractal.[ThisfollowsCantorset.]

YoumaybefamiliarwithKochcurvesorKoch

snowflakes,asimplefractalconstruction,

givenintextbooks.Letusexplorethis.

KochcurveorKochsnowflakes

HelgevonKoch,[1870-1924],aSwedish

mathematician,developedthisfractalin

1904---startingwithanequilateral

triangle.Cutmiddleonethirdofeachside

ofthetriangle.Constructanisosceles

triangleatthecutportion,withtheside

lengthleftover..keeprepeatingit....you

willgetsmallerandsmallersides,buta

beautifulfigurewillemergecalled"Koch

snow-flake".

[Aswewillseelater,jaggedcoastlineof

MainestateinUSAorthecoast-linesinUK

havebeenmodeledwithfractals.]

Youcanworkouttheperimeterandareaof

thisKochcurve.

Perimeterandareaofafractal

Youcanworkouttheperimeterandareaof

anyshape[triangle,square,pentagon...]as

theshapeiscutintosmallerfiguresand

leadstoafractal.

OneBritishmathematician,LewisFry

Richardson,wasintriguedbythetotal

distanceofthejaggedcoastlineof

Britain.Thesimpleanswerwouldbe:"drive

aroundandtotalthedistance".Hemademany

measurementsandbasedonthese,Mandelbrot

developedafractalimageofthiscoastline

andworkedouttheperimeter.

ConsiderKochcurvefirst.

Forthiscurve,witheachstep,weare

increasingthesidesfromthreetofour.Ifs

isthelengthoftheside,

perimeteraftern:

P(n)=3.s.(4/3)n

wherenisthenumberofstepsor

iterationwehavedoneontheoriginal

triangle.

SupposeIdrawaKochcurvewiths=1inch,

thentheinitialperimeterofthe

equilateraltraingleis:

P(0)=3.1=3in

Inthenextiteration,P(1)=3.1.(1.33)=

3.99in

After5iterations,theperimeterincreases

to:P(5)=3.(1.33)5

=3x4.162=12.485in

After10iterations,wegetP(10)=51.96in.

Youcanalsoderiveanexpressionforthe

areaofaKochcurveorflake:

A=a/5{8-3(4/9)n}

wherenisthenumberofiterations.

Notethattheperimeterincreasesrapidly

witheachiteration.Thereforeifyouwishto

increasethelengthofalinearobject,you

canemployafractalcurve.Forinstance,to

increasethelengthofanantennaina

mobilephone,occupyingasmallarea,some

engineersdevelopedafractaldesign.

Pentaflakes

Penta-flakesarefractalsconstructedoutof

pentagons,withrhombusesbetweenthe

pentagons.Therhombuswillhavethetwo

angles:144degand36degrees.Apicture

ofpenta-flakeisshownhere.Inthis,Iuse

differentcolorsforthepentagonstoadda

colorpatterntotheflakes.

Sierpinskitriangleandcarpet.

WaclawSierpinski[1882-1969]wasapolish

mathematicianworkingonsettheoryand

numbertheory.Hediscoveredasimple

fractalformedfromanequilateraltriangle

asfollows:Jointhemidpointsofthethree

sidestoformanothersmallerequilateral

triangle.Cutoutandremovethisinner

triangle.Nowyouareleftwiththree

smallerequilateraltriangles.Repeatthe

sameprocessineachofthosesmaller

triangles..Keeprepeating;youwillget

smallerandsmallertriangles.

ThisiscalledaSierpinskitriangle.

"SierpinskiCarpet"isformedbytakinga

squareand,thencutandremoveasquare,

halfthesize,fromthemiddleandrepeating

theprocesswiththeremainingsquares.

Youcandothesamethinginthreedimension

withacube,scoopingoutasmallercubefrom

thecenter.

RecursiveProcess

Inthemathematicaljargon,therepeated

processofconstructingafractalwith

smallerandsmallerpiecesisa'recursive

process'anditcanbeprogrammedeasily.

Youstartwitharecursionformula.

Torecallwhatarecursionformulais,I

giveashortnotehere.

TakethecaseofFibonaccisequencegiven

earlier;therecursionformulaisbasedon

thefactthatFibonumbernisthesumof

previoustwonumbersinthesequence

startingfrom0and1.ThenthFibonumber

F(n)=F(n-1)+F(n-2).

Thisisarecursionformula.

Againconsiderthecalculationoffactorials

recursively:

n!=(n)n-1!

Wecanusethisformulatocalculate

factorialofanynumber.Forinstance:

5!=5.4!

4!=4.3!

3!=3.2!

2!=2.1!=2---terminatehere!

Therefore5!=5.4.3.2.1=120

Suchrecursionprocessesareeasytoprogram

foracomputer.

Inthesameway,webuildfractalimagesby

recursivelygoingintosmallerandsmaller

sizeunits.

Therepeatedmethodoriteration,usinga

computersoftwarecalled"IterativeFunction

Systems'orIFScanalsobedeveloped.

ComputergraphicsusetheseIFSmethodsor

algorithms.BenoitMandelbrotemployedsuch

IFSmethodscreatinghisfractalimages

around1975whileworkingforIBM.

Fractaldimensions

Considerasquare.Ifyoucutupthesquare

withhalfitsside,thenyouget4squares.

Thereforewewrite:4=2nwheren=2.

Thereforethedimensionofasquare,'n'is

2.

Consideracube;supposeyoucutitupinto

smallercubes,eachhalfitsside,thenhow

manycubesyouwillget?Theansweris8.

Now8=2nwheren=3.Thedimensionofa

cubeis3.

Here2iscalledthescalingfactor;since

wedividethesidebyhalf,thescalefactor

isthereciprocalof(1/2).

WhatisthedimensionofaSierpinski

triangle?

Foreachiterationweget3similar

triangles,aftertakingthemidpointofthe

sides.Thescalefactoris2again.

Therefore3=2n

Takinglogarithmonbothsides,

log3=n.log2

orn=log(3)/log(2)≈1.585(nearly)

so,thisfractalhasadimensionwhichisa

"fraction".

[Thisdefinitionof'similaritydimension'

isduetoFelixHausdorffandiscalled

"Hausdorffdimension".Thereareother

definitionsormeasuresofdimensionwhich

wewillnotdiscusshere!]

ConsiderthePentaflake:eachpentagonis

surroundedby6pentagons;thescalefactor

is1/(1+phi)=1/2.618=0.382

sothedimensionisobtainedfromthe

formula:

6=0.382n

orthedimensionn=log(6)/log(0.382)

n=0.7782/0.4179=1.862

[Note:YoucanconstructaPentaflake

withoutthecentralpentagontoo.]

WhatisthedimensionofaKochcurve?

Itis:

n=log(4)/log(3)=0.602/0.477=1.2619.

Whataboutn-flakes?

Yes,wecanconstruct

hexa-flakes,octa-flakesandsoon.

Applicationsoffractals

Fractalsarenotjustmathematical

curiosities,butarehelpfulin

understandingmanyphenomena,bothnatural

andman-made.Fractalgeometryisusedto

studythecoastlines,earthquakes,stock

marketfluctuations,heartfibrilations

andweather.

Amajorapplicationhasbeenin'image

compression'incomputergraphics.Somenice

computerimageshavebeendevelopedusing

fractalsforsomemoviestoo.

Mandelbrotcalleditsimply:"theoryof

roughness".[Abriefbiographicalsketchof

Mandelbrot,thecreatorof"Fractal

geometry"isgivenattheendofthis

article.]

ScalingLaws

Scalingoffiguresorthreedimensional

objectsisafascinatingsubjectandis

centraltofractalgeometry.

Wesawthatwhenweconstructafractalby

cuttingintosmallerandsmallerobjects,

theperimeterkeepsincreasing.Atwhatrate

thisincreaseoccurs?

Tounderstandscalinglaws,Iillustrate

withasimpleexample.

Supposeyoumakeboxesofdifferentsizes;

youareconcernedwithtwofactors:costand

volumeofthebox.Costoftheboxwouldbe

proportionaltothesurfaceareaofmaterial

requiredtoconstructthebox.Assumingyou

aremakingboxesintheformofcubes,the

surfaceareais6s2wheresisthesideof

thebox.Thevolumeoftheboxwouldbes3

.Animportantcriteriafortheselectionof

aboxisthecostperunitvolume.

Thereforecostofthebox/volume=C/v

ratioisrelatedasfollows:

R=c/v=6/s

OrR=ks-1

Thisequationcanalsobewrittenas

follows:

R=k(1/s)1

Thisisascalinglawwhichmeansthatas

sizeincreases,thecostperunitvolumeof

aboxdecreases--aninverserelation.

Thisresultiswellknowntoallofusand

thereforewebuyitemsinlargercontainers.

Herewecangeneralizesuchlawsas'power

laws':

y=kxn

wheretheexponentnorpoweristhescaling

power,kaconstantandxisthescale

factor.

Forinstance,wecanask:"Howthe

perimeterPincreasesaswereducethesize

inaKochcurve?"

Notethatnisthesameasthesimilarity

dimensionorHausdorffdimensionmentioned

earlier.ncanbeafractionoran

integer.Therearesomefractalswithinteger

dimensionstoo.

[Forinstance'space-fillingcurves'like

SierpinskicurveorHilbertcurvehas

dimensionof2only.'Kolams"orline

drawingsbuiltarounddotsinSouthIndian

homesisafractalwithintegerdimensions.]

BenoitMandelbrot,aPolishmathematician

,whosettledinUSAandworkedinIBMfor

nearlythreedecades,discoveredthe

"FractalGeometry"around1975,andalso

'Mandelbrotset'.Hecoinedtheterm

"fractal'fromtheLatinword'fractus'

meaning'broken'or'fractured'.Hisclassic

book"TheFractalGeometryofNature"was

publishedin1982.

Heusedextensivelycomputergraphicsto

createthefractalimages.Withouttheuse

ofthecomputers,fractalgeometrywouldnot

havebeendiscoveredatall---thenumber

crunchinginvolvedistoomuchformanual

computing.

Notethattherewereearlierworkssuchas

JuliasetsorfractalafterGastonJulia

(1918)whowasamathprofessorinParis.

3-Dimensionalfractals

Youcanbuildthreedimensionalobjectswith

fractalgeometry.Considerbuildingboxes

withinboxes,halftheoriginalsize.

Mandelbrotusedtoexplainthefractal

geometryinCauliflowers!Fernleaves

exhibitfractalimages.

Fractalsandchaostheory

Does'chaos'relatetothegeometryof

fractals?

Letusstartwithasimpleiterative

formula:

xn+1=rxn(1-xn)--------(1)

whererisapositiveconstant.

[Thisequationiscalled"Logistic

map"equation.]

Youmaybefamiliarwiththisformula,widely

usedinmodelingpopulationgrowth,limited

byresources[likefishpopulationinalake

orbacteriainaPetridishwhenthe

populationreachesamaximum--asteady

level.]Theresultingcurveiscalled

'logistic'curveorgrowthcurveor

sigmoidalcurve,whenxreaches1astimet

tendstoinfinity.Theequationisas

follows:

N(t)=N0[1/1-exp(-ct)]wherecisa

constantandtistimeandN0isthefinal

(limited)population:

Letx(t)=N/N0

Notethatxtendsto1asttendsto

infinity.

[Thisgrowthcurveisexplainedintext

bookson'precalculus'.]

[Thelogisticfunctionissimplydefinedas

follows:f(x)=1/(1+e-x)]

Thisiterativeformula,equation1,isvery

sensitivetothevalueofr.

Ifr=2forinstance,theiterationsoon

reachesastablevalue,lessthan1.

Whenrisbetween3and[1+√6]or

(3.44949),thevalueofxwilloscillate

betweensomevaluesandisagainstable.If

risincreasedandgetscloseto3.56995,

thexvaluesvary'chaotically'witheach

iteration.

[Toillustrate,letusconsiderafew

cases:

Case1:r=2,x0=0.2

Thenx1=2(0.2)(0.8)=0.32

x2=2(0.32)(0.68)=0.435

x3=2(0.435)(0.565)=0.491

----------

x5=0.4999

Notethattheiteratesstayat0.5,whichis

calleda'fixedpoint'.

Case2:x0=0.2butr=3

Theiteratesareasfolllows:

x1=0.48

x2=0.749

x3=0.564

x4=0.738

x5=0.580

x6=0.731

Theiteratesoscillatebetween0.58and

0.73.Thevalueofxvsrbranchesoutto

twovalues.

Case3:letusfollowtheiteratesforx0=

0.2,butr=4

x1=0.64

x2=0.922

x3=0.2877

x4=0.822

x5=0.585

x6=0.971

Theiteratesfluctuatebetween0.97and

0.28.Chaoticbehaviorhasstartedatthis

valueofr.

Soifoneplotsxversusr,onesees

branchingandthevaluesofxmayoscillate

betweentwoorfouroreightvaluesandso

on.Thisdiagram,calleda'bifurcation

diagram';agraphofxversusr,whichhas

self-similarityandis,indeed,afractal.

ThiswasshownbyFeigenbaumandothers,

around1978.Itwasidentifiedasalink

betweenChaostheoryandfractalsby

Mandelbrot.

[Youmayseenicesimulationdiagramsin

somewebsites.]

Dynamicalsystems

Hereisanotherdiversion,tolinkupwith

logisticmap.

Considerthesimplepopulationgrowthmodel

withthelinear(firstorder)differential

equation:

dy/dt=ky----------(2)

Heretisthetimeanddy/dtisthe

populationgrowthrate.

Thesolutiontotheequationisthe

exponentialfunctionforthegrowthofthe

population:

y(t)=y(0)ekt

wherekispositive.

Inreality,populationsdonotgrow

exponentiallyforeverandsoonthegrowth

ratedecreases,oftenduetolimited

resourcessuchasfoodoroxygendissolved

inalakeforfishpopulation.

Toaccountforthiswemayaddanegative

termtoequation(2),asafunctionofy2.

Thusdy/dt=ky-ly2------(3)

Thisequationisa'non-linear'differential

equationandcanbesolvedeasily.

Intermsofdifferenceequations,fora

fixedtimeinterval,sayoneyear,wecan

write:

yn+1-yn=kyn-lyn2

Thisequationismodifiedintothelogistic

equation:

Ifthemaximumpopulationthatcanbe

sustained,calledcarryingcapacity,isN,

dividingbyNthroughout,wecanwrite:

xn+1=k'xn-l'xn2

=k'xn(1-[l'/k']xn)

wherex=y/Nandk'=k+1

Thisequationisthesameas'logisticmap'

equation:

xi+1=rxi(1-xi)

Thereforethelogisiticequation,

bifurcationdiagramandMandelbrotsetare

relatedandformpartofdynamicalsystems.

Thisisthefoundationof"iteratedfunction

systems."[IFS}

MandelbrotSet

BenoitMandelbrotusedasimilariterative

formulaforthesetnamedafterhim:

zi+1=z2i+c

wherecisaconstantandzisacomplex

number.Itisfascinatinghowasimple

formulalikethisonecouldleadtoan

understandingofchaosandfractals.This,

indeed,isthediscoverythatensureda

placeforBenoitMandelbrotinthehallof

fameamongmathematicians.

----------------------------------------

BenoitMandelbrot[1924-2010]wasbornin

Warsaw,PolandinaLithuanianJewish

family.Hehadgreatdifficultyingetting

schooleducationwhenPolandwasruledby

Russia.ThefamilymovedtoPariswherehis

unclewasamathprofessor.Mandelbrot

studiedatEcolePolytechnique(1945-47)

Paris.Whenjewishpersecutionstarted,he

cametoUSAandstudiedformaster'sdegree

inAeronauticsatCaltech,Pasadena.Hedid

postdoctoralfellowshipsinmanyplaces

includingMITandPrinceton.AtPrinceton,

heworkedwithJohnvonNeumann.

HejoinedIBMinitspuremathdivision

[ThomasWatsonLab,YorktownHeights,NY,]

andservedtherefrom1958to1993.Hewas

lateraSterlingprofessoratYale

University.

Hecreatedthefractalgeometry,'the

theoryofroughness',andiswell-knownfor

'Mandelbrotset'.

Interestingly,hediscoveredthefractal

geometrysometimein1979,attheageof55,

aremarkableachievementsincemost

mathematiciansreachtheirpeakcreativity

beforetheageof30----(thoughtherehad

beenexceptionslikeCarlFrederichGauss

andJohnvonNeumann.)Hesucceeded,he

writes,dueto"extraordinarilygood

fortuneandachinglycomplicated

professionallife".Helecturedextensively

onfractalpatternsin

art,architecture,music,poetryand

literature.

His'memoirs'waspublishedin2012,after

hisdeathandistitled"TheFractalist-

Memoirofascientificmaverick".

-----------------------------------------

References:Webpagesof"wolfram"mathand

booksonFractals.