Fractal GeometryFractal Geometry
Fractals are important becauseFractals are important because they reveal a new area ofthey reveal a new area of
mathematics directly relevant tomathematics directly relevant to the study of nature.the study of nature.
- Ian Stewart- Ian Stewart
an nature be described inan nature be described in
terms of Euclidean Geometry(terms of Euclidean Geometry(
 
Euclidean Geometry)Euclidean Geometry)
• * mountain range using triangles and* mountain range using triangles and pyramids(((pyramids(((
•louds using circles(((louds using circles(((
• +eaves(((+eaves(((
• "oc,s((("oc,s(((
 
+oo, outside o you see+oo, outside o you see
any shapes in Euclideanany shapes in Euclidean
Geometry(Geometry(
 
If so/ they were more thanIf so/ they were more than
li,ely man made. Forli,ely man made. For
e&amplee&ample
 The point is this The point is this
• 'ur world is fashioned with rough'ur world is fashioned with rough
edges and non-uniform shapes.edges and non-uniform shapes.
• Euclidean geometry describes idealEuclidean geometry describes ideal
shapes which rarely occur in nature.shapes which rarely occur in nature.
• 0hy then do we even bother with0hy then do we even bother with
Euclidean geometry(Euclidean geometry(
• *stronomers*stronomers
• Scientists nowScientists now
shapes areshapes are
particles andparticles and
is elliptical notis elliptical not
circular.circular.
relatively easy.relatively easy.
• "elationships between two shapes"elationships between two shapes
For e&ample) S!uare vs. shape ofFor e&ample) S!uare vs. shape of
6ississippi6ississippi
 
7. 6en and women consider a7. 6en and women consider a
house with smooth edges andhouse with smooth edges and
uniform shapes more beautifuluniform shapes more beautiful
than a house with rough edgesthan a house with rough edges
and non-uniform shapes.and non-uniform shapes.• 0hy is that nature is more beautiful with rough0hy is that nature is more beautiful with rough
edges and non-uniform shapes and man madeedges and non-uniform shapes and man made
ob8ects are less beautiful with rough edges andob8ects are less beautiful with rough edges and
non-uniform shapes( *nd vice versa(non-uniform shapes( *nd vice versa(
 
• Geometry of irregular shapes whichGeometry of irregular shapes which
are characteri#ed by in2nite detail/are characteri#ed by in2nite detail/
in2nite length/ and the absence ofin2nite length/ and the absence of
smoothness.smoothness.
 
* rectangle is not a fractal.* rectangle is not a fractal.
• 0hen we loo, through a microscope0hen we loo, through a microscope
at the rectangle do we see any newat the rectangle do we see any new
details.details.
 
• The teacher may want to put fractal The teacher may want to put fractal
pictures here.pictures here.
 
The Koch snowflake is a Mathematical curve and one of the earliest fractal curves to have been described. It appeared in a 1904 paper entitled "n a continuous curve without tan!ents constructible from elementar# !eometr# b# the $wedish mathematician %el!e von Koch. The lesser known Koch curve is the same as the snowflake e&cept it starts with a line se!ment instead of an e'uilateral trian!le. ne can ima!ine that it was created b# startin! with a line se!ment then recursivel# alterin! each line se!ment as follows(
•)ivide the line se!ment into three se!ments of e'ual len!th. •)raw an e'uilateral trian!le that has the middle se!ment from step 1 as its base and points outward. •*emove the line se!ment that is the base of the trian!le from step +.
 
,et be the number of sides be the len!th of a sin!le side be the len!th of the perimeter and the snowflakes area after the
th iteration. /urther denote the area of the initial trian!le and the len!th of an initial side 1. Then
The capacit# dimension is then
 
Fractal dimensionFracta mens on • 9fractal dimension9fractal dimension
: *mount of variation in the structure*mount of variation in the structure
: 6easure of roughness or fragmentation of the ob8ect6easure of roughness or fragmentation of the ob8ect • Small d-less 8aggedSmall d-less 8agged
• +arge d-more 8agged+arge d-more 8agged
• Self similar ob8ectsSelf similar ob8ects : nsnsdd91 ;Some boo,s write this as ns91 ;Some boo,s write this as ns-d-d91<91<
• s9scaling factors9scaling factor
• n number of subparts in subdivisionn number of subparts in subdivision
• d9ln;n<=ln;1=s<d9ln;n<=ln;1=s< : >d9ln;n<=ln;s< however s is the number of segments versus>d9ln;n<=ln;s< however s is the number of segments versus
how much the main segment was reducedhow much the main segment was reduced I.e. line divided into 7 segments. Instead of saying the lineI.e. line divided into 7 segments. Instead of saying the line
is 1=7/ say instead there are 7 sements. ?otice that 1=;1=7<is 1=7/ say instead there are 7 sements. ?otice that 1=;1=7< 9 7@9 7@
: If there are diAerent scaling factorsIf there are diAerent scaling factors • • SS,,
dd9191
B91
n
I prefer) nsI prefer) ns-d-d9191
)d9ln;n<=ln;s<)d9ln;n<=ln;s<• imension is a ratioimension is a ratio of the ;new si#e<=;oldof the ;new si#e<=;old si#e<si#e< : ivide line into nivide line into n
identical segmentsidentical segments
• n9sn9s
: ivide lines on s!uareivide lines on s!uare into small s!uares byinto small s!uares by dividing each line intodividing each line into n identical segmentsn identical segments
• n9sn9s44 small s!uaressmall s!uares
: ivide cubeivide cube
• Boch3s snowCa,eBoch3s snowCa,e
segmentssegments
• Fractal imensionFractal imension
: 9lnD=ln7 9 1.449lnD=ln7 9 1.44
: For your reference) oo,For your reference) oo,
methodmethod
• n9Dn9D
• s91=7s91=7
• d9lnD=ln;1=;1=7<<d9lnD=ln;1=;1=7<<
 
imensionimension
• ivide each side by 4ivide each side by 4 : 6a,es D triangles6a,es D triangles
: 0e ,eep 70e ,eep 7
:  Therefore n97 Therefore n97 • Get 7 new triangles from 1Get 7 new triangles from 1
old triangleold triangle
: s94 ;4 new segments froms94 ;4 new segments from one old segment<one old segment<
 
: ivide each side by 7ivide each side by 7
: ?ow push out the middle face of each cube?ow push out the middle face of each cube
: ?ow push out the center of the cube?ow push out the center of the cube
• 0hat is the fractal dimension(0hat is the fractal dimension( : 0ell we have 4 cubes/ where we used to have 10ell we have 4 cubes/ where we used to have 1
• n94n94
: 0e have divided each side by 70e have divided each side by 7 • s97s97
: Fractal dimension ln;4<=ln;7< 9 4.J4JFractal dimension ln;4<=ln;7< 9 4.J4J
Image from *ngel boo,
generating imagesgenerating images
• "ules"ules : *9M ***9M **
segments in graphsegments in graph structurestructure
: ranch with brac,etsranch with brac,ets
•
• *>@**>@*>@**>@
• **>*>@**>@@****>*>@**>@**>*>@**>@@****>*>@**>@
@@
*
* *


*
* *
**

* * * *
*

**

 
generating images con3dgenerating images con3d
• 6odify *lphabet K*//6odify *lphabet K*// >/@/;/<L>/@/;/<L
• "ules"ules : *9M ***9M **
: 9M *>@**;<9M *>@**;<
: >@ 9 left branch ;< 9>@ 9 left branch ;< 9 right branchStartingright branchStarting asis9asis9
• Generate wordsGenerate words : "epresents se!uence of"epresents se!uence of
segments in graphsegments in graph structurestructure
: ranch with brac,etsranch with brac,ets
•
• *>@**;<*>@**;<
• **>*>@**;<@****;*>@**;<<**>*>@**;<@****;*>@**;<<
*
* *


*
* *
**

* * * *
*

**

 
no inherent geometryno inherent geometry • Grammar based modelGrammar based model
re!uiresre!uires : GrammarGrammar
: Geometric interpretationGeometric interpretation
• Generating an ob8ect from theGenerating an ob8ect from the word is a separate processword is a separate process : e&amplese&amples
• ranches on the tree drawn atranches on the tree drawn at upward anglesupward angles
• hoose to draw segments of tree ashoose to draw segments of tree as successively smaller lengthssuccessively smaller lengths
:  The more it branches/ the smaller The more it branches/ the smaller the last branch isthe last branch is
*
* *
**

* * * *
*

**

 
Grammar and GeometryGrammar and Geometry
•hange branch si#e according to depthhange branch si#e according to depth
of graphof graph
 
$article Systems$article Systems • System is de2ned by a collection of particles thatSystem is de2ned by a collection of particles that
evolve over timeevolve over time : $articles have Cuid-li,e properties$articles have Cuid-li,e properties
• Flowing/ billowing/ spattering/ e&panding/ imploding/ e&plodingFlowing/ billowing/ spattering/ e&panding/ imploding/ e&ploding
: asic particle can be any shapeasic particle can be any shape • Sphere/ bo&/ ellipsoid/ etcSphere/ bo&/ ellipsoid/ etc
: *pply probabilistic rules to particles*pply probabilistic rules to particles • generate new particlesgenerate new particles
• hange attributes according to agehange attributes according to age : 0hat color is particle when detected(0hat color is particle when detected(
: 0hat shape is particle when detected(0hat shape is particle when detected(
:  Transparancy over time( Transparancy over time(
• $articles die ;disappear from system<$articles die ;disappear from system<
• 6ovement6ovement : eterministic or stochastic laws of motioneterministic or stochastic laws of motion
BinematicallyBinematically  forces such as gravityforces such as gravity
 
: Fire/ fog/ smo,e/ 2rewor,s/ trees/ grass/Fire/ fog/ smo,e/ 2rewor,s/ trees/ grass/
waterfall/ water spray.waterfall/ water spray.
• GrassGrass
: 6odel clumps by setting up tra8ectory paths6odel clumps by setting up tra8ectory paths
for particlesfor particles
• 0aterfall0aterfall
: $articles fall from 2&ed elevation$articles fall from 2&ed elevation
•eCected by obstacle as splash to groundeCected by obstacle as splash to ground : Eg. drop/ hit roc,/ 2nish in poolEg. drop/ hit roc,/ 2nish in pool
 
$hysically based modeling$hysically based modeling • ?on-rigid ob8ect?on-rigid ob8ect
: "ope/ cloth/ soft rubber ball/ 8ello"ope/ cloth/ soft rubber ball/ 8ello
• escribe behavior in terms of e&ternal and internal forcesescribe behavior in terms of e&ternal and internal forces : *ppro&imate the ob8ect with networ, of point nodes connected by Ce&ible connection*ppro&imate the ob8ect with networ, of point nodes connected by Ce&ible connection
• E&ample springs with spring constant ,E&ample springs with spring constant ,
: %omogeneous ob8ect%omogeneous ob8ect • *ll ,3s e!ual*ll ,3s e!ual
• %oo,e3s +aw%oo,e3s +aw : FFss9-, &9-, &
• &9displacement/ F&9displacement/ Fss 9 restoring force on spring9 restoring force on spring
• ould also model with putty ;doesn3t spring bac,<ould also model with putty ;doesn3t spring bac,<
,
,
,
,
 
: F96ove forward a unitF96ove forward a unit
: +9Turn left+9Turn left
: F"F"F"F"F"F"
• 0hat if change angle to0hat if change angle to degrees degrees : F9M F+F""F+FF9M F+F""F+F
: asis Fasis F
 
Psing turtlePsing turtle
graphics forgraphics for
treestrees • Pse push and pop for sidePse push and pop for side
branches >@branches >@
• *ngle 94J*ngle 94J
readabilityreadability
• F>"F@F>+F@F >"F>"F@F>+F@F@F>"F@F>+F@F >"F>"F@F>+F@F@
F>"F@F>+F@F >+F>"F@F>+F@F@F>"F@F>+F@F >+F>"F@F>+F@F@
F>"F@F>+F@FF>"F@F>+F@F
 
0hat is a Fractal(0hat is a Fractal(
•  * fractal is a mathematical ob8ect that is both self-similar and chaotic.
•self-similar) *s you magnify/ you see the ob8ect over and over again in its parts.
•chaotic) Fractals are in2nitely comple&.
 
 The most famous of all fractals is the 6andelbrot set.
 
...*nd we can continue to #oom in. *s we magnify the ob8ect/ we see the same thing over and over again.....This is
Self Similarity
 
 These two pictures are interesting because they show the same portion of the mandelbrot set colored diAerently.
• The choice of color scheme really inCuences what we see in the picture.
•Is this mathematics or art(
 
0hy is geometry often described as cold and dry( 'ne reason lies in its inability to describe the shape of a cloud/ a mountain/ a coastline/ or a tree. louds are not spheres/ mountains are not cones/ coastlines are not circles/ and bar, is not smooth/ nor does lightning travel in a straight line...
...?ature e&hibits not simply a higher degree but an altogether diAerent level of comple&ity.  The number of distinct scales of length of patterns is for all purposes in2nite.
 The e&istence of these patterns challenges us to study those forms that Euclid leaves aside as being formless/ to investigate the morphology of the amorphous. 6athematicians have disdained this challenge/ however/ and have increasingly chosen to Cee from nature by devising theories unrelated to anything we can see or feel.
 The Fractal Geometry of ?at
 
See the NfractalO he is holding(
 
• In the following slides we will see someIn the following slides we will see some
landscapes that are progressively morelandscapes that are progressively more
comple&. These landscapes arecomple&. These landscapes are NOT NOT   drawings. They are created entirely by adrawings. They are created entirely by a
computer using Nfractal interpolation.Ocomputer using Nfractal interpolation.O
• This is the procedure that is used by This is the procedure that is used by
special eAects artists to create computerspecial eAects artists to create computer
generated scenes for the big screen.generated scenes for the big screen.
;NIndependence ayO was full of them.<;NIndependence ayO was full of them.<
'f course/ mine are much cruder than'f course/ mine are much cruder than
theirs. They were produced in about antheirs. They were produced in about an
hour on my ;ancient< pentium computer athour on my ;ancient< pentium computer at
home by a program called 5IST*$"'.home by a program called 5IST*$"'.
Fractal +andscapesFractal +andscapes
 
 
 The triangles are 2ner still and the mountains in the d
begin to loo, a bit better.
 
loo, li,e real mountains.
 
 
landscape and the clouds.
 
0e can ma,e some artistic choices of color to give us the same
landscap e in diAerent seasons..
Summ er
Sprin g
*utum n