Fractals - the ultimate art of mathematics
Adam Kozak
Outline
What is fractal?
Self-similarity dimension
Fractal types
Iteration Function Systems (IFS)
L-systems
Introduction to complex numbers
Mandelbrot sets
Julia and Fatou sets
Mandelbulbs
2
What is fractal? Why should I payattention to it?
Geometric object with property of self-similarity in any scalefactor – in exact manner, approximate or stochastic
Similarity dimension may be not equal to topologic dimension(non-integer value)
Relatively simple recursive definitions
3
Applications:
Fractal compression
Fractal art
Ideas in engineering, electronics, chemistry, medicine, urban planningwhich have self-similarity patterns
Fractal antenna in mobile phones capable ofcapturing much wider scope of frequencies inmuch smaller areas than classic antenna
Fractals in nature
4
Sou
rce:
Wik
iped
ia
Romanesco broccoli
High voltage breakdown within a 4″ block of acrylic
Fern
Coast with rivers
Kolmogorov complexity
Everyting what can be described, can be described as a string of characters overany alphabet of size > 1.
E.g. infinite string Ala ma kota, Ala ma kota, Ala ma kota,…
To encode such a string literally we would need infinite memory, however… we know that we can recreate its any finite substring simply using a computerTHIS STRING IS COMPUTABLE
Kolmogorov complexity of a finite string is a length of the shortest computerprogram which recreates the string (this is an uncomputable function – there isno algorithm to evaluate it!)
It is also called informational complexity
5
Kolmogorov(„Ala ma kota, Ala ma kota, Ala ma kota, Ala ma kota, Ala ma kota, …”)
=
Length(„while (true) print(’Ala ma kota, ’);”)
Hausdorff similarity dimension
Similarity dimension may be not equal to topologic dimension(non-integer value)
For „normal” geometric object if we scale it by factor (0<<1), we need copies of this object to fill the area of originalobject where d is dimension
6
1,58496...2log
3log
2log
3loglim
3,21
1log
loglim1
0
n
n
n
nn
d
d
N
NdN
d1
Fractal types
Fractals may be obtained from different concepts:
Atractors of Iterated Function Systems (IFS)
Julia & Fatou sets
Mandelbrot sets
L-system (Lindenmayer system)
7
Contracting mapping
Let (X, d) be a metric space, then f: X X is a contractingmapping if:
Banach fixed point theorem: There exists exactly one pointpX such, that f(p)=p (fixed point of contracting mapping)
8
212121 ,)(),(:,:1,0 aadafafdXaa
Recursive execution of
contracting mapping:
f(x,y)=(x/3,y/3)
1010101 coscoscoscoscoscoslim
xfxxxxfx n
nnn
Iterated Function Systems (IFS)
9
Recursive transformations of geometric object which sum productof a set of n affine contracting mappings (compositions of rotation, reflection, translation and contracting scaling): {Fi : X X } (1i n)
S is any non empty set of points in a given space X
S is a fractal – an attractor of IFS, it’s independent of initial S (S
is a fixed point of set of contracting mappings {Fi} in metric space(H, h) where H is set of all compact subsets of X and h is Hausdorff distance)
kk
n
i
kik SSSFSSS
lim1
10
Iterated Function Systems (IFS)
10
Any affine contracting mapping Fi in space 2 has the followingformula:
1111
cossin
sincos
'
',
'
'
yx
y
x
yx
yx
y
x
it
t
y
x
y
xyxF
feydxy
cbyaxx
2
0
30
4
1
2
1
y
x
yx
y
x
t
t
x
y
An example of IFS – Sierpiński triangle
IFS: {Fi: 2 2} (i=1..3):
Sierpiński triangle is a fixed point (attractor) of IteratedFunction System {F1, F2 , F3}
11
43
0
10
01
21
21,
0
41
10
01
21
21,
0
41
10
01
21
21,
0
3
2
1
y
xyxF
y
xyxF
y
xyxF
yx
yxF ,1 yxF ,2
yxF ,3
IFS – workshop
12
Barnsley fern
with some clues ;)
[src: Wikipedia]
Sierpiński carpet
[src: Wikipedia]
Sierpiński triangle in 3D space (pyramid)
[src: Wikipedia]
Task: locate, count and define contractig mappings
L-system (Lindenmayer system)
L-systems are based on recursive grammar with definedvariables, constants, rules, axiom and generating parameters; we can assign some operations to each symbol eg.:
variables : X F
constants : + − [ ]
axiom: X
rules : (X → F-[[X]+X]+F[+FX]-X), (F → FF)
parameter - angle: 25°
13
Assigned meaning of symbols for above L-system:
( F ) draw forward
( - ) turn left 25°
( + ) turn right 25°
( X ) does nothing, just controls evolution of the curve
( [ ) saves coordinates and angle on stack (push)
( ] ) recovers coordinates and angle from stack (pop)
Exemplary generator:
http://www.kevs3d.co.uk/dev/lsystems/#
Quick introdution to complexnumbers
There is no a real number x such, that 𝒙2 = −𝟏
Ok, so let’s create a number which is two-dimensional, and put such a number on imaginary axis, let’s call it 𝐢
14
Real numbers
Imaginary numbersComplex plane
i
1
-i
-1
1+i
Let’s preserve addition and multiplication like for real numbers keeping in
mind, that 𝒊2 = −𝟏:𝒂 + 𝒃𝒊 + 𝒄 + 𝒅𝒊 = 𝒂 + 𝒄 + 𝒃 + 𝒅 𝒊
𝒂 + 𝒃𝒊 𝒄 + 𝒅𝒊 = 𝒂𝒄 + 𝒂𝒅 + 𝒃𝒄 𝒊 + 𝒃𝒅𝒊𝟐 = 𝒂𝒄 − 𝒃𝒅 + 𝒂𝒅 + 𝒃𝒄 𝒊
Quick introdution to complexnumbers
But there is another representation!
15
Real numbers
Imaginary numbersComplex plane
i
1
-i
-1
1 + 𝑖 = 𝑟 𝑐𝑜𝑠 + 𝑖𝑠𝑖𝑛 = 2 𝑐𝑜𝑠45+ 𝑖𝑠𝑖𝑛45
= 22
2+ 𝑖
2
2
Now applying the rules for trygonometric functions we see that multiplication is actually
related to rotation on a plane! Complex plane is a field.
𝒂 + 𝒃𝒊 𝒄 + 𝒅𝒊 = 𝑟1 𝑐𝑜𝑠1 + 𝑖𝑠𝑖𝑛1 𝑟2 𝑐𝑜𝑠2 + 𝑖𝑠𝑖𝑛2 = 𝑟1𝑟2 cos(1 + 2) + 𝑖𝑠𝑖𝑛(1 + 2)
r
Riemann sphere
Let’s map whole complex plane onto a spehere, whereinfility corresponds to a noth pole
16
Mandelbrot sets
1. Mandelbrot sets are defined for rational functions over closedset of complex numbers (z* corresponds to infinity)
2. Rational function is a division of two polynomials:
3. Let Wc denote a rational function dependent on parameter
4. Let
5. Mandelbrot set M(Wc) of a rational function Wc is a set of suchpoints that is not convergent to z*:
17
01
1
1
01
1
1
)(
)()(
bzbzbzb
azazaza
zl
zwzW
m
m
m
m
k
k
k
k
)()( 1 zWWzW n
cc
n
c
)0(n
cW
*})0(lim:{ zWCcWM n
cn
c
12 iwherebiacCc
*}{zCC
Cc
CCW :
Cc
0cc
Mandelbrot sets
This may be satisfied in two ways:
Recursion is convergent to some point c0
Recursion finally falls into a cycle(number of stable cycles is related to degree of W)
18
*})0(lim:{ zWCcWM n
cn
c
CcwherecW n
cn
00)0(lim
c
Orbit of point c
Orbit of point c
Mandelbrot set - example
The first and best known Mandelbrot set was defined for polynomial function
Thus we need to check for each point in cC ifsequence c, c2+c, (c2+c)2+c2+c, … goes toinfinity or not
Workshop: Check, if point c=0+i belongs toMandelbrot set for this function
19
czzW n
c 2)(
.....
11)0(
1211)0(
1)0(
0)0(
24
23
22
21
iiiiW
iiiiiW
iiiW
iiW
c
c
c
c
Orb
it o
f poin
t c
Mandelbrot set journeyhttp://www.youtube.com/watch?v=9G6uO7ZHtK8
20
Does Mandelbrot set exist? Take a look
21
„visual” complexity
very low Kolmogorov complexity of its image for (int y = 0; y < HEIGHT; y++) {
for (int x = 0; x < WIDTH; x++) {
double zx = zy = 0;
double cX = (x - WIDTH/2) / ZOOM;
double cY = (y - HEIGHT/2) / ZOOM;
for (int it = MAX_ITER; zx * zx + zy * zy < 4 && it > 0; it--)
{
tmp = zx * zx - zy * zy + cX;
zy = 2.0 * zx * zy + cY;
zx = tmp;
}
image[x][y] = color(it);
}
}
Newton method for finding functionroot
22
)('
)(1
n
nnn
xf
xfxx
https://commons.wikimedia.org/wiki/File:NewtonIteration_Ani.gif
Julia and Fatou sets
23
Are based on the same rational functions as Mandelbrot setsand are strictly related to them (Julia set is connected forparameters belonging to Mandelbrot set).
Fatou sets are areas in C which are attracted by some points(here colors red, blue and green) for rational function W(z)
Julia set is a ,,border’’ between Fatou set areas which isattracted by infinity point (z*).
32)( 2 zzzW
323
1
)('
)()(
,231,231,1
111)(
2
2
3
210
20
32,1,0
33
zzz
zz
zf
zfzzW
izizz
ezzzzf n
ki
k
Here is Julia/Fatou set for function W(z) obtained fromNewton’s method for function f(z) = z3-1. Thus attractingpoints for Wn(z) correspond to roots of f(z).
Green color is attracting basin of z0, red of z1, and blue of z2.
http://www.youtube.com/watch?v=nczm0jdyWps
Mandelbulbs – Mandelbrot sets in 3D
Defined by Daniel White and Paul Nylander using double rotation transformation for sphericalcoordinates, since there is no 3D equivalence to 2D complex numbers having all properties of field
24
http://www.youtube.com/watch?v=rEhWtQfx5nw
Thank you for attention
References: T. Martyn. Fraktale i obiektowe algorytmy ich
wizualizacji. Nakom, Poznań, 1996.
J. Kudrewicz. Fraktale i chaos. WNT, Warszawa, 2007.
P. Prusinkiewicz and A. Lindenmayer. The Algorithmic Beauty of Plants. The VirtualLaboratory Series, Springer 1996.
B. Mandelbrot. The fractal geometry of nature. W.H. Freemen and Co. New York, 1982.
http://www.skytopia.com/project/fractal/mandelbulb.html
http://bugman123.com/Hypercomplex/
25Background source: http://www.skytopia.com/project/fractal/mandelbulb.html
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