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