Fractals
Elena Cherkaev
elena@math.utah.edu
Iterated Function Systems
The 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 Sierpinski
Waclaw
Si i ki (1915):
(1915) Sierpinski
Si i ki triangle
t i l andd carpett
Random Fractals
Random Fractals
Random fractals can be generated by stochastic rather than g
y
deterministic processes, for example, trajectories of the Brownian motion, fractal landscapes and random trees. Fractals as Attractors of Nonlinear Dynamical Systems
Fractals can be generated as strange attractors of Nonlinear g
g
Dynamical Systems, for example, attractor of trajectories of the Lorenz dynamical system, Rossler attractor, attractor of Ueda system
Ueda system. Lorenz attractor
Rossler attractor
Escape‐time
Escape
time fractals
fractals
Escape‐time fractals — These are based on sensitive d
dependence of the trajectories on the starting point or on d
f h
h
initial conditions. Examples of this type are the Julia and Mandelbrot sets (Gaston
Examples of this type are the Julia and Mandelbrot sets (Gaston Julia, Pierre Fatou, Benoit Mandelbrot), and Newton fractal. Julia set
Julia set
Newton fractal
Newton fractal
Forthcoming Book: Benoit Mandelbrot, A Life in Many Forthcoming
Book: Benoit Mandelbrot A Life in Many
Dimensions
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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
Fractals in Finance —
Ch t b M ti Sh bik & Nassim
Chapters by Martin Shubik
& N i Taleb
T l b
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
Fractals in Physiology —
Ch t b E ld Weibel
Chapter by Ewald
W ib l
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
Fractals in Comedy —
d
Ch t b D
Chapter by Demetri
t i Martin
M ti
Cantor Set
O
On each iteration step, delete middle third of each interval. h it ti
t d l t
iddl thi d f
hi t
l
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;
Ch
C has measure zero; C consists of uncountably many points.
Cantor Set: Measure zero
Cantor Set: Measure zero
Sum up lengths of the deleted sets: ⎞
1
1
1
1⎛
2 22
⎜1 + + + ... ⎟⎟
+ 2 2 + 4 3 + ... =
3
3
3
3 ⎜⎝ 32 33
⎠
1
1
=
=1
3 1− 2 / 3
Measure (length) of the deleted set = 1 Measure of C is zero
Measure of C is zero. Cantor Set: Continuum of points
Expand x in base‐3: x ∈ [0, 1],
x=
a1 a2 a3
+ 2 + 3 + ... , ak ∈ {0, 1, 2}
3 3 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: 1
1
2
f1 ( x) = x , f 2 ( x) = x +
3
3
3
Construct a sequence of closed nested intervals :
I 0 ⊃ I1 ⊃ I 2 ⊃ .... ⊃ I n ⊃ ...
I 0 = [0, 1]
I1 = f1 ( I 0 ) ∪ f 2 ( I 0 ) = I 00 ∪ I 01
I 2 = f1 ( I1 ) ∪ f 2 ( I1 ) = f1 ( I 00 ∪ I 01 ) ∪ f 2 ( I 00 ∪ I 01 )
= I 000 ∪ I 010 ∪ I 001 ∪ I 011
...
Affine transformations in R1 :
f ( x) = a x + b,
a is scaling coeff., b is translation or shift
Cantor Set: Continuum of points
Cantor set is equivalent to the set of C
t
ti
i l t t th
t f
all possible sequences of 0 and 1
Affine transformations in 2D
2D affine transformation has the form :
⎛ x1 ⎞ ⎛ a b ⎞ ⎛ x1 ⎞ ⎛ e ⎞
⎟⎟ ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ = A x + t
w( x) = w ⎜⎜ ⎟⎟ = ⎜⎜
⎝ x2 ⎠ ⎝ c d ⎠ ⎝ x2 ⎠ ⎝ f ⎠
Matrix A can be written as :
⎛ a b ⎞ ⎛ r1 cos θ1
⎟⎟ = ⎜⎜
A = ⎜⎜
⎝ c d ⎠ ⎝ r1 sin θ1
− r2 sin θ 2 ⎞
⎟⎟
r2 cos θ 2 ⎠
Examples: Scaling, shift, rotation, reflection.
Affine transformation consists of a linear transformation A
Affine transformation consists of a linear transformation A
followed by a translation t. Affine transformations in 2D
How to find w ? Use: w(Red_triangle) = Blue_triangle
bk = w(ak ), k = 1, 2, 3
⎛ a b ⎞ ⎛ akx ⎞ ⎛ e ⎞ ⎛ bkx ⎞
⎜⎜
⎟⎟ ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟, k = 1, 2, 3
⎝ c d ⎠ ⎝ aky ⎠ ⎝ f ⎠ ⎝ bky ⎠
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 distance function d: X x
R which measures the distance
between pairs of pts x and y ‫ א‬X. A metric space X is complete if every Cauchy sequence has a limit in X.
Forward iterates of f are transformations Forward iterates of f
are transformations
f o n : X → X defined by
f o 0 ( x ) = x,
f o1 ( x) = f ( x),
f o( n +1) ( x) = f o f o n ( x) = f ( f o n ( x)),
)) n = 0, 1, 2, ...
Contraction Mapping
A transformation f : X → X on a metric space ( X , d )
is a contraction mapping if there is a constant
0 ≤ s < 1, such that d ( f ( x), f ( y )) ≤ s d ( x, y )
s is contractivity factor for f .
The Contraction Mapping Thm :
Let f be a contraction mapping on a complete
metric space ( X , d ). Then f has exactly one fixed point x f ,
and for any x , the sequence of iterates { f on ( x) : n = 0, 1, 2,...}
converges to x f :
{ f on ( x)} → x f as n → ∞
Contraction Mapping on the Space of Fractals
Let ( X , d ) be a metric space, and let
((H ( X ), h(d )) be the correspond
p ingg space
p of nonempty
py
compact subsets of X with Hausdorff metric h(d ).
Let w : X → X be a contraction mapping on the metric
space ( X , d ) with contractivity factor s.
Then, w : H ( X ) → H ( X ) defined by
w( B) = {w( x) : x ∈ B}
is a contraction mapping on (H ( X ),
) h(d ))
with contractivity factor s.
Iterated Function System
IFS : An Iterated Function System consists of
a complete metric space ( X , d )
together with a finite set of contraction mappings wn :
{ X ; wn , n = 1, ...N }
with contractivity factor s, s = max{sn , n = 1, ...N }
W ( B) = w1 ( B) ∪ w2 ( B)... ∪ wN ( B)
is a contraction mapping on the space H.
Its unique fixed point satisfies
A = W ( A) = w1 ( A) ∪ w2 ( A)... ∪ wN ( A),
A = lim W on ( B) as n → ∞ for any B ∈ H .
A is attractor of IFS.
Example: Sierpinski Triangle
W ( B) = w1 ( B) ∪ w2 ( B) ∪ w3 ( B)
Calculate iterations of W :
An = W on ( A0 ) , n = 1,2,...
⎛ 0.5 0 ⎞ ⎛ x1 ⎞
⎟⎟ ⎜⎜ ⎟⎟
w1 ( x) = ⎜⎜
⎝ 0 0.5 ⎠ ⎝ x2 ⎠
⎛ 0.5 0 ⎞ ⎛ x1 ⎞ ⎛ 0 ⎞
⎟⎟ ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟
w2 ( x) = ⎜⎜
⎝ 0 0.5 ⎠ ⎝ x2 ⎠ ⎝ 0.5 ⎠
⎛ 0.5 0 ⎞ ⎛ x1 ⎞ ⎛ 0.5 ⎞
⎟⎟ ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟
w3 ( x) = ⎜⎜
⎝ 0 0.5 ⎠ ⎝ x2 ⎠ ⎝ 0 ⎠
Deterministic and Random Algorithms
Deterministic and Random Algorithms
Deterministic fractal :
IFS : { X ; w1 , w2 ,...wN }
W ( B) = w1 ( B) ∪ w2 ( B) ∪ w3 ( B)
Choose a compact set A0 . Compute iteratively
An = W on ( A0 ) , n = 1, 2,...
Sequence of iterates converges to the attractor
of IFS - - deterministic fractal.
Random Iteration Algorithm : " Apply wi with probability pi "
Start with x0 ∈ X ;
Choose recursively xn +1 ∈ {w1 ( xn ), w2 ( xn ),...wN ( xn )}
with
ith probabilit
b bility pi .
3D IFSs and 3D Fern
3D IFSs and 3D Fern
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Instead of a 2x2 real matrix A and a column vector Instead
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 Fern
3D Fern
The IFS for the 3D Fern
0
0 ⎤ ⎡0 ⎤
⎡0
w1 ( x) = ⎢⎢0 0.18 0⎥⎥ x + ⎢⎢0⎥⎥,
⎢⎣0
0
0⎥⎦ ⎢⎣0⎥⎦
0
0 ⎤ ⎡0⎤
⎡0.85
0.85 0.1 ⎥⎥ x + ⎢⎢1.6⎥⎥,
w2 ( x) = ⎢⎢ 0
⎢⎣ 0
− 0.1 0.85⎥⎦ ⎢⎣ 0 ⎥⎦
⎡0.2 0.2 0 ⎤ ⎡ 0 ⎤
w3 ( x) = ⎢⎢0.2 0.2 0 ⎥⎥ x + ⎢⎢0.8⎥⎥,
⎢⎣ 0
0 0.3⎥⎦ ⎢⎣ 0 ⎥⎦
⎡− 0.2 0.2 0 ⎤ ⎡ 0 ⎤
w4 ( x) = ⎢⎢ 0.2 0.2 0 ⎥⎥ x + ⎢⎢0.8⎥⎥
⎢⎣ 0
0 0.3⎥⎦ ⎢⎣ 0 ⎥⎦
Fractal Dimension
Fractal Dimension
Box dimension: log( N )
D = lim
ε →0
log 1
( ε)
ε
3D IFS fractals: Menger sponge
0 ⎤ ⎡2 / 3⎤
⎡1 / 3 0
w1 ( x) = ⎢⎢ 0 1 / 3 0 ⎥⎥ x + ⎢⎢ 0 ⎥⎥,
⎢⎣ 0
0 1 / 3⎥⎦ ⎢⎣ 0 ⎥⎦
log(
g(20)
D=
≅ 2.73
log(3)
0 ⎤ ⎡ 0 ⎤
⎡1 / 3 0
w2 ( x) = ⎢⎢ 0 1 / 3 0 ⎥⎥ x + ⎢⎢ 0 ⎥⎥,...
⎢⎣ 0
0 1 / 3⎥⎦ ⎢⎣2 / 3⎥⎦
Sierpinski pyramid
log(5)
D=
≅ 2.32
log(2 )
Self similar fractals
Self 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 Parameters
Continuous 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 parameters.
t
Animation : Dancing fern Simulations by Eric Heisler
Deterministic and random trees
Tree Fractals: Transformations
Tree Fractals: Transformations
r ⎡r cos θ
w1 (v ) = ⎢
⎣ r sin θ
(x1,y1)
r ⎡ r cos θ
w2 (v ) = ⎢
⎣− r sin θ
− r sin
i θ ⎤ r ⎡ x1 ⎤
v+⎢ ⎥
⎥
r cos θ ⎦ ⎣ y1 ⎦
r sin θ ⎤ r ⎡ x1 ⎤
v+⎢ ⎥
⎥
r cos θ ⎦ ⎣ y1 ⎦
Each iteration takes a line segment and creates two branches. Each
iteration takes a line segment and creates two branches
Transformations w rotate trunk by θ or ‐ θ , y ,, and move to new position p
shrink by r
Tree Fractals: IFS with condensation Let C be the trunk of the tree : w0 ( B) = C , B ∈ H
W = w0 ∪ w1 ∪ w2
A0 = C , C is a condensation set.
set
A1 = W ( A0 ) = C ∪ w1 (C ) ∪ w2 (C )
A2 = W ( A1 ) = C ∪ w1 (C ) ∪ w2 (C ) ∪ ( w1 ∪ w2 )( w1 (C ) ∪ w2 (C ))
A3 = W ( A2 ) = C ∪ w1 (C ) ∪ w2 (C ) ∪ ...
Random Trees
Random Trees
Examples of random trees calculated with different Examples
of random trees calculated with different
parameters of the contraction (different angles)
Baker’s Map
Baker’ss map: Attractor
Baker
map: Attractor
Stretching and folding are two main mechanisms of forming an attractor Baker’ss map: Attractor
Baker
map: Attractor
At the crossection: topological Cantor set!
topological Cantor set! (all possible sequences of 0 and 1)
Baker’s map: Fractal Dimension
Let a < 1/ 2. The attractor A is approximated by B on ( S ), which consists of 2 n
strips of height a n and unit length. One can cover A with 2 n a − n squares
of side ε = a n
Then, N = (a / 2) − n , and box dimension can be calculated as limit
when
h ε goes to
t zero :
ln( N )
ln(1 / 2)
d = lim
= 1+
<2
ln((1 / ε )
ln((a )
Ueda Attractor
Ueda Attractor
Start with a patch of initial conditions which experiences stretching and folding
which
experiences stretching and folding
Animation of forming an attractor
Simulations by Quishi Wang
The Escape Time Algorithm: Julia Set
The Escape Time Algorithm: Julia Set Suppose, f: C → C is a polynomial.
Start with
ith z0 = z ,
z1 = f ( z0 ) = f o1 ( z ),
z 2 = f ( z1 ) = f o 2 ( z ),
...
z n = f ( z n −1 ) = f ( f (...( z )...)) = f on ( z ),
...
Let F f be the set of points in C whose orbits do not converge to ∞
Ff
= {z ∈ C : {| f
on
( z ) |}
∞
n =0
is bounded
}
Then F f is a Julia set, its boundary J f is Julia set of f .
The Escape Time Algorithm: The
Escape Time Algorithm:
Numerical Approach
W Discretize computational domain W .
For each z = z p ,q , iterate f ( z ) :
z0 , z1 , z 2 ,...
V
What is the number of iterations needed to escape from V ?
Color points in V according to the number of iterations needed
==> Julia Set
Julia Set
Color points in W according to the number of iterations needed for an orbit starting at point z to
an orbit starting at point z, to escape. Here, f(z) = λλ cos(z), λ
f(z) cos(z), λ = 0.75
0.75+i*0.85
i 0.85
Simulations by Brandon Olson Newton fractal
f ( x) = 0
Newton’s method of solution fast (quadraticaly) converges when starting point is close to the solution.
f ( xn )
xn +1 = xn −
= g ( xn )
f ' ( xn )
For solution of the equation: z 4 − 1 = 0
the Newton’ss method gives:
the Newton
method gives:
z 4 −1
g ( z) = z −
4z3
The roots are 1, ‐1,
The roots are 1, 1, i, and
i, and –i,i, there are 4 attracting points.
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
Sensitive dependence on initial conditions d
i iti l
diti
Outside the region of quadratic convergence the Newton’s method can be very sensitive to the choice of starting point
can be very sensitive to the choice of starting point.
z 4 −1 = 0
z5 −1 = 0
Simulations of Aryn Roth
Generalized Newton fractal
Generalized Newton fractal
z n +1 = g ( z n )
z 4 −1
g ( z) = z − a
4z3
a = (1 + i )
Participants
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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
Q
Quishi Wang
g