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Iterated Function SystemsThe fractals are constructed using a fixed geometric replacement
rule: Cantor set, Sierpinski carpet or gasket, Peano curve, Koch snowflake, Menger sponge.snowflake, Menger sponge.
Karl Weierstrass (1872): Nondifferentiable function
Georg Cantor (1883): Cantor set
Giuseppe Peano (1890), David Hilbert (1891):
Space filling curves
Helge Von Koch (1904): Koch snowflake
W l Si i ki (1915) Si i ki t i l d tWaclaw Sierpinski (1915): Sierpinski triangle and carpet
Random FractalsRandom Fractals
Random fractals can be generated by stochastic rather than g ydeterministic processes, for example, trajectories of the Brownian motion, fractal landscapes and random trees.
Fractals as Attractors of Nonlinear Dynamical SSystems
Fractals can be generated as strange attractors of Nonlinear g gDynamical Systems, for example, attractor of trajectories of the Lorenz dynamical system, Rossler attractor, attractor of Ueda systemUeda system.
Rossler attractorLorenz attractor
Escape‐time fractalsEscape time fractals
Escape‐time fractals — These are based on sensitive d d f h hdependence of the trajectories on the starting point or on initial conditions.
Examples of this type are the Julia and Mandelbrot sets (GastonExamples of this type are the Julia and Mandelbrot sets (Gaston Julia, Pierre Fatou, Benoit Mandelbrot), and Newton fractal.
Julia set Newton fractalJulia set Newton fractal
Forthcoming Book: Benoit Mandelbrot A Life in ManyForthcoming Book: Benoit Mandelbrot, A Life in Many
Dimensions
• Contents:• Introduction — Benoit Mandelbrot: Nor Does Lightning Travel in a Straight Line (M Frame)• Fractals in Mathematics — Chapters by Michael Barnsley, Julien Barral, Kenneth Falconer,
Hillel Furstenberg, Stephane Jaffard, Michael Lapidus, Jacques Peyriere & Murad Taqqu• Fractals in Physics — Chapters by Amon Aharony, Bernard Sapoval, Michael Shlesinger,
Katepalli Sreenivasan & Bruce West• Fractals in Computer Science — Chapters by Henry Kaufman & Ken Musgrave• Fractals in Engineering — Chapters by Nathan Cohen & Marc‐Olivier Coppens
F t l i Fi Ch t b M ti Sh bik & N i T l b• Fractals in Finance — Chapters by Martin Shubik & Nassim Taleb• Fractals in Art — Chapters by Javier Barrallo, Ron Eglash & Rhonda Roland Shearer• Fractals in History — Chapter by John Gaddis• Fractals in Architecture — Chapter by Emer O'Daly
F t l i Ph i l Ch t b E ld W ib l• Fractals in Physiology — Chapter by Ewald Weibel• Fractals in Education — Chapters by Harlan Brothers & Nial Neger• Fractals in Music — Chapter by Charles Wuorinen• Fractals in Film — Chapter by Nigel Lesmoir‐Gordon
F t l i C d Ch t b D t i M ti• Fractals in Comedy — Chapter by Demetri Martin
Cantor Set
O h it ti t d l t iddl thi d f h i t lOn each iteration step, delete middle third of each interval.
Properties: C has structure at arbitrary small scales;C has structure at arbitrary small scales;C is self‐similar;The dimension of C is not integer;C hC has measure zero; C consists of uncountably many points.
Cantor Set: Measure zeroCantor Set: Measure zero
Sum up lengths of 221114121 2
⎟⎟⎞
⎜⎜⎛
++++++the deleted sets:
13/21
131
...33
13
...3
43
23 3232
=−
=
⎟⎟⎠
⎜⎜⎝
+++=+++
Measure (length) of the deleted set = 1 Measure of C is zero
3/213
Measure of C is zero.
Cantor Set: Continuum of points
[ ] }2,1,0{,...333
,1,0 33
221 ∈+++=∈ kaaaaxxExpand x in base‐3:
Points in the Cantor set do not have 1 in the base‐3 representation
One‐to‐one correspondence with base‐2 representation of the points in the unit interval
→ Cardinality of Cantor set is continuum !→ Cardinality of Cantor set is continuum !
Cantor set
Cantor set can be generated iteratively using two transformations:
21)(1)( ff
:intervals nested closed of sequence aConstruct 33
)(,3
)( 21
IIII
xxfxxf
⊃⊃⊃⊃⊃
+==
[ ])()(
1,0.......
010002011
0
210
IIIfIfII
IIII n
∪=∪==
⊃⊃⊃⊃⊃
)()()()(
011001010000
010020100112112
IIIIIIfIIfIfIfI
∪∪∪=∪∪∪=∪=
,)(:in ations transformAffine
...1
bxaxfR
+=shift or on translatiis coeff., scaling is ba
Cantor Set: Continuum of points
C t t i i l t t th t fCantor set is equivalent to the set of all possible sequences of 0 and 1
Affine transformations in 2D
:formthehasation transformaffine 2D
+=⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
2
1
2
1)( txAfe
xx
dcba
xx
wxw⎠⎝⎠⎝⎠⎝⎠⎝ 22
:aswritten becan Matrix A
f
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
2211
2211
cossinsincosθθθθ
rrrr
dcba
A
Examples: Scaling, shift, rotation, reflection.
Affine transformation consists of a linear transformation A
⎠⎝⎠⎝
Affine transformation consists of a linear transformation Afollowed by a translation t.
Affine transformations in 2DHow to find w ?
Use: w(Red_triangle) = Blue_triangle
3,2,1),(
⎞⎛⎞⎛⎞⎛⎞⎛
==
bkawb kk
3,2,1, =⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛k
bb
fe
aa
dcba
ky
kx
ky
kx
Solve for a, b, c, d, e, f.
Metric Space
A metric space (X, d) is a space X together with a real‐valued function d: X x X ‐> R which measures the distancefunction d: X x X > R which measures the distance between pairs of pts x and y X.
A metric space X is complete if every Cauchy sequence has a limit in X.
Forward iterates of f are transformationsForward iterates of f are transformations
)(by defined :
0
→
fXXf n
o
o
210))(()()(),()(
,)(
)1(
1
0
=
=
+ fffffxfxf
xxf
nnn ooo
o
o
...,2,1,0)),(()()()1( ===+ nxffxffxf nnn ooo o
Contraction Mapping
→ dXXXfconstantaisthereifmappingncontractioais
),( space metric aon :mation A transfor
≤<≤fs
yxdsyfxfds .for factor ity contractiv is
),())(),(( such that ,10 constant ais thereifmappingn contractio a is
:Thm Mappingn Contractio The
xfdXf
f ,point fixed oneexactly has Then ).,( space metric complete aon mappingn contractio a be Let
=x
nxfx
f
n
: toconverges ,...}2,1,0:)({ iteratesofsequencethe, any for and o
∞→→ nxxf fn as )}({ o
Contraction Mapping on the Space of Fractals
nonempty ofspaceingcorrespondthebe))(),( let and space, metric a be ),(Let
dhX(HdX
).( metric Hausdorff with of subsetscompact p ypgp))(),(
dhX(
.factor ity contractiv with ),( space metric on the mappingn contractio a be :Let
sdXXXw →
))()(onmappingncontractioais}:)({)(
by defined )()(:,Then
dhX(HBxxwBw
XHXHw∈=
→
.factor ity contractiv with ))(),(on mappingn contractio a is
sdhX(H
Iterated Function System
)(spacemetriccompletea of consists SystemFunction IteratedAn :IFS
dX
}1;{: mappingsn contractio ofset finite aith together w
),( spacemetriccomplete a
NnwXw
dX
n
=}...,1,max{,factor ity contractivwith
}...,1,;{Nnsss
NnwX
n
n
==
)()...()()( 21 BwBwBwBW N∪∪=
satisfiespoint fixed unique Its H. space on the mappingn contractio a is)()...()()( 21 BwBwBwBW N∪∪
.any for as )(lim
),()...()()( 21
HBnBWA
AwAwAwAWAn
N
∈∞→=
∪∪==o
IFS.ofattractor is A
Example: Sierpinski Triangle
:ofiterationsCalculate)()()()( 321 ∪∪=
WBwBwBwBW
,...2,1 , )( :ofiterations Calculate
0 == nAWAW
nn
o
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
5.0005.0
)(2
11 x
xxw
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎠⎝⎠⎝
5.00
5.0005.0
)(2
12
2
xx
xw
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎠⎝⎠⎝⎠⎝
05.0
5.0005.0
)(2
13
2
xx
xw⎠⎝⎠⎝⎠⎝ 2
Deterministic and Random AlgorithmsDeterministic and Random Algorithms
)()()()(},...,;{: :fractal ticDeterminis 21 N
BBBBWwwwXIFS
21)(
yiterativel Compute . set compact a Choose)()()()(
0
321
n nAWA
ABwBwBwBW
==
∪∪=
o
fractal.ticdeterminis -- IFS ofattractor the toconverges iterates of Sequence
,...2,1 ,)( 0n nAWA ==
"y probabilit with Apply " :AlgorithmIteration Random ii pw
b bilitith )}(),...(),({y recursivel Choose
; Start with
211
0
nNnnn xwxwxwx Xx
∈∈
+
.y probabilit with ip
3D IFSs and 3D Fern3D IFSs and 3D Fern
• Instead of a 2x2 real matrix A and a column vectorInstead of a 2x2 real matrix A and a column vector
t (*,*), we have a 3x3 real matrix A and a column vector t (*,*,*) for a 3D IFS. Again, it can be expressed as w(x)= Ax+t.
• As an example of 3D, we introduce a 3D Fern, which is the attractor of an IFS of affine maps in 3D.
3D Fern3D Fern
The IFS for the 3D Fern
0000 ⎤⎡⎤⎡,
000
000018.00000
)(1
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡= xxw
,06.1
0
85.01.001.085.00
0085.0)(2
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−= xxw
,08.0
0
3.00002.02.002.02.0
)(3
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡= xxw
03.000 ⎥⎦⎢⎣⎥⎦⎢⎣
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
08.0
0
300002.02.002.02.0
)(4 xxw⎥⎦⎢⎣⎥⎦⎢⎣ 03.000
Fractal DimensionFractal Dimension
Box dimension:
( )ε 1log)log(lim
0
ND→
= ( )εlog
ε
3D IFS fractals: Menger sponge
,...3/2
00
3/10003/10003/1
)(2
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡= xxw,
00
3/2
3/10003/10003/1
)(1
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡= xxw
3/23/100 ⎥⎦⎢⎣⎥⎦⎢⎣03/100 ⎥⎦⎢⎣⎥⎦⎢⎣
732)20log(D ( ) 73.23log
)g(≅=D
Sierpinski pyramid
( ) 32.22log)5log(≅=D ( )2log
Self similar fractalsSelf similar fractals
Looks the same no matter how close you get.
A smaller copy of the fern
A copy within the copy
Continuous Dependence on ParametersContinuous Dependence on Parameters
If the contraction w continuously depends on a parameter p, y p p p,then the fixed point depends continuously on p.
The attractor changes continuously as you change the tparameters.
Animation : Dancing fern
Simulations by Eric Heisler
Deterministic and random trees
Tree Fractals: TransformationsTree Fractals: Transformations
⎤⎡⎤⎡ i θθ⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡ −=
1
11 cossin
sincos)(
yx
vrrrr
vw rr
θθθθ
(x1,y1)
⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡−
=1
12 cossin
sincos)(
yx
vrrrr
vw rr
θθθθ
Each iteration takes a line segment and creates two branches
⎦⎣⎦⎣ 1cossin yrr θθ
Each iteration takes a line segment and creates two branches.
Transformations w rotate trunk by θ or ‐ θ ,
shrink by r , and move to new position y , p
Tree Fractals: IFS with condensation
setoncondensatiais
,)( :treetheof trunk thebe Let
210
0
∪∪=∈=
CCAwwwW
HBCBwC
))()()(()()()()()()(set.on condensatiais,
21212112
2101
0
∪∪∪∪∪==∪∪==
=
CwCwwwCwCwCAWACwCwCAWA
CCA
...)()()())()()(()()()(
2123
21212112
∪∪∪== CwCwCAWA
Random TreesRandom Trees
Examples of random trees calculated with differentExamples of random trees calculated with different parameters of the contraction (different angles)
Baker’s Map
Baker’s map: AttractorBaker s map: Attractor
Stretching and folding are two main mechanisms
of forming an attractor
Baker’s map: AttractorBaker s map: Attractor
At the crossection:
topological Cantor set!topological Cantor set!
(all possible sequences of 0 and 1)
Baker’s map: Fractal Dimension
2 of consists which ),(by edapproximat isA attractor The .21Let < SB/a non
side ofsquares2A with cover can Onelength.unit and height of strips
=
−
aaa
n
nnn
ε
thlimit as calculated becan dimension box and ,)2/( Then, = − aN n
2)ln()2/1ln(1
)/1ln()ln(lim
:zero togoes when
<+==a
Ndε
ε
)()(
Ueda AttractorUeda Attractor
Start with a patch of initial conditions
which experiences stretching and foldingwhich experiences stretching and folding
Animation of forming an attractor
Simulations by Quishi Wang
The Escape Time Algorithm: Julia SetThe Escape Time Algorithm: Julia Set
Start ith.polynomialais Suppose, Cf: C →
),()(
),()(
, Start with
212
101
0
zfzfz
zfzfz
zz
o
o
==
==
=
),()...))(...(()(...
),()(
1
12
zfzffzfz
zfzfz
onnn − ===
{ }{ } toconvergenot do orbits whosein points ofset thebe Let
...CFf
∞
∞
{ }{ }. ofset Julia is boundary its set, Julia a is Then
boundedis |)(|: 0
fJF
zfCzF
ff
non
f∞
=∈=
The Escape Time Algorithm:The Escape Time Algorithm: Numerical Approach
W
: )( iterate ,each For .domain nalcomputatio Discretize
,=
zzz
zfzzW
qp
V
,... , , 210 zzz
SetJulia needed iterations ofnumber the toaccording Vin pointsColor
? from escape toneeded iterations ofnumber theisWhat
==>
V
Set Julia ==>
Julia Set
Color points in W according to the number of iterations needed for an orbit starting at point z toan orbit starting at point z, to escape.
Here, f(z) = λ cos(z), λ = 0.75+i*0.85f(z) λ cos(z), λ 0.75 i 0.85Simulations by Brandon Olson
Newton fractal
Newton’s method of solution fast (quadraticaly) converges when starting point is close to the solution.
0)( =xf
)()(')(
1 nn
nnn xg
xfxfxx =−=+
For solution of the equation:
the Newton’s method gives: 4
014 =−zthe Newton s method gives:
The roots are 1, ‐1, i, and –i, there are 4 attracting points.
3
4
41)(
zzzzg −
−=
The roots are 1, 1, i, and i, there are 4 attracting points.
Points of the complex plane are colored by a different color, depending on the root to which the Newton method converges.
Newton fractal: S iti d d i iti l ditiSensitive dependence on initial conditions
Outside the region of quadratic convergence the Newton’s method can be very sensitive to the choice of starting pointcan be very sensitive to the choice of starting point.
014 =−z 015 =−z
Simulations of Aryn Roth
Generalized Newton fractalGeneralized Newton fractal
1)(4z −
)(1 nn zgz =+
)1(4
)( 3
iaz
azzg
+=
−=
Participants
• Brandon Olson
• Roxanne Brinkerhoff
• Bill Clark
• Gregory Danner
• Eric Heisler
• Masaki Iino
J d J dki• Jordan Judkins
• Carl Tams
• Liz Doman• Liz Doman
• Aryn Roth
• Quishi WangQ g
