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 pay attention to it?

 Geometric object with property of self-similarity in any scale factor – in exact manner, approximate or stochastic  Similarity dimension may be not equal to topologic dimension (non-integer value)  Relatively simple recursive definitions

Applications:  Fractal compression  Fractal art  Ideas in engineering, electronics, chemistry, medicine, urban planning which have self-similarity patterns

Fractal antenna in mobile phones capable of capturing much wider scope of frequencies in much smaller areas than classic antenna

3 Fractals in nature

Romanesco broccoli Fern

Wikipedia : :

High voltage breakdown within a 4″ block of acrylic

Source Coast with rivers 4 Kolmogorov complexity

 Everyting what can be described, can be described as a string of characters over any 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 computer THIS STRING IS COMPUTABLE

 Kolmogorov complexity of a finite string is a length of the shortest computer program which recreates the string (this is an uncomputable function – there is no algorithm to evaluate it!)

Kolmogorov(„Ala ma kota, Ala ma kota, Ala ma kota, Ala ma kota, Ala ma kota, …”) = Length(„while (true) print(’Ala ma kota, ’);”)

 It is also called informational complexity

5 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  1  d copies of this object to fill the area of original object where d is dimension

d log N  N   1    d  lim  0 log1    1 2n , N   3n  log 3n log 3  d  lim  1,58496... n log 2n log 2

6 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 contracting mapping if:  0,1:a1,a2  X : d f (a1), f (a2 ) da1,a2   Banach fixed point theorem: There exists exactly one point pX such, that f(p)=p (fixed point of contracting mapping)

Recursive execution of contracting mapping: f(x,y)=(x/3,y/3)

   xn1  f xn  lim xn  x x  f x  cos10   coscos 10  coscoscos 10  n

8 Iterated Function Systems (IFS)

 Recursive transformations of geometric object which sum product of a set of n affine contracting mappings (compositions of rotation,

reflection, translation and contracting scaling): {Fi : X  X } (1i n) n

S0  S Sk  Fi Sk 1  S  lim Sk  k i1

 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)

9 Iterated Function Systems (IFS)

2  Any affine contracting mapping Fi in space  has the following formula:

x' ax  by  c x'  x cosx sin  y x tx    Fi x, y            y' dx  ey  f y'  y sin x cos y y t y 

1  x 1  1  y 1

 y

1  x  2  x 1   y 4

 x   y  30

tx  0

t y  2

10 An example of IFS – Sierpiński triangle

2 2  IFS: {Fi:    } (i=1..3): F3x, y

x   y  0 1 21 0x 1 4 F1x, y         1 20 1y 0  1 21 0x 1 4 F2 x, y         1 20 1y 0  1 21 0x 0  F3 x, y         1 20 1y  3 4 F2 x, y F1x, y  Sierpiński triangle is a fixed point (attractor) of Iterated

Function System {F1, F2 , F3}

11 IFS – workshop

Task: locate, count and define contractig mappings

Sierpiński triangle in 3D space (pyramid) Sierpiński carpet [src: Wikipedia] [src: Wikipedia] Barnsley fern with some clues ;) [src: Wikipedia]

12 L-system (Lindenmayer system)

 L-systems are based on recursive grammar with defined variables, 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°

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/# 13 Quick introdution to complex numbers

 There is no a real number x such, that 풙2 = −ퟏ  Ok, so let’s create a number which is two-dimensional, and put such a number on imaginary axis, let’s call it 퐢 Complex plane Imaginary numbers 1+i i

Real numbers -1 1

-i

Let’s preserve addition and multiplication like for real numbers keeping in mind, that 풊2 = −ퟏ: 풂 + 풃풊 + 풄 + 풅풊 = 풂 + 풄 + 풃 + 풅 풊 풂 + 풃풊 풄 + 풅풊 = 풂풄 + 풂풅 + 풃풄 풊 + 풃풅풊ퟐ = 풂풄 − 풃풅 + 풂풅 + 풃풄 풊

14 Quick introdution to complex numbers

 But there is another representation!

Complex plane Imaginary numbers 1 + 푖 = 푟 푐표푠 + 푖푠푖푛 = 2 푐표푠45 + 푖푠푖푛45 i 2 2 = 2 + 푖 r 2 2  Real numbers -1 1

-i

Now applying the rules for trygonometric functions we see that multiplication is actually related to rotation on a plane! Complex plane is a field.

풂 + 풃풊 풄 + 풅풊 = 푟1 푐표푠1 + 푖푠푖푛1 푟2 푐표푠2 + 푖푠푖푛2 = 푟1푟2 cos(1 + 2) + 푖푠푖푛(1 + 2)

15 Riemann sphere

 Let’s map whole complex plane onto a spehere, where infility corresponds to a noth pole

16 Mandelbrot sets

1. Mandelbrot sets are defined for rational functions over closed set of complex numbers C  C {z*} (z* corresponds to infinity) cC  c  a bi where i2  1 2. Rational function W :C  C is a division of two polynomials:

k k1 w(z) ak z  ak1z  a1z  a0 W (z)   m m1 l(z) bm z  bm1z  b1z  b0

3. Let Wc denote a rational function dependent on parameter cC n n1 4. Let Wc (z) Wc Wc (z)

5. Mandelbrot set M(Wc) of a rational function Wc is a set of such n points cC that Wc (0) is not convergent to z*:

n M Wc  {cC : lim Wc (0)  z*} n

17 Mandelbrot sets

n M Wc  {cC : lim Wc (0)  z*} n  This may be satisfied in two ways:

. Recursion is convergent to some point c0 n lim Wc (0)  c0 where c0 C n c c Orbit of point c 0 . Recursion finally falls into a cycle (number of stable cycles is related to degree of W)

Orbit of point c

18 Mandelbrot set - example

 The first and best known Mandelbrot set was defined for

polynomial function n 2 Wc (z)  z  c  Thus we need to check for each point in cC if sequence c, c2+c, (c2+c)2+c2+c, … goes to infinity or not

 Workshop: Check, if point c=0+i belongs to Mandelbrot set for this function 1 2 Wc (0)  0  i  i  2 2 Wc (0)  i  i  i 1  3 2 Wc (0)  i 1  i  1 2i 1 i  i  4 2 Wc (0)   i  i  1 i  i 1

Orbit of point c point of Orbit ..... 

19 Mandelbrot set journey

http://www.youtube.com/watch?v=9G6uO7ZHtK8

20 Does Mandelbrot set exist? Take a look  „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); } }

21 Newton method for finding function root

https://commons.wikimedia.org/wiki/File:NewtonIteration_Ani.gif

f (xn ) xn1  xn  f '(xn )

22 Julia and Fatou sets

Are based on the same rational functions as Mandelbrot sets and are strictly related to them (Julia set is connected for parameters 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 is

attracted by infinity point (z*). 02k i 3 3 3 n f (z)  z 1  z 1  zk0,1,2  1e

z0 1, z1  1 i 3 2, z2   1 i 3 2, f (z) z3 1 W (z)  z   z   2z  z 2  3 f '(z) 3z 2 W(z)  2z  z2  3

Here is Julia/Fatou set for function W(z) obtained from Newton’s method for function f(z) = z3-1. Thus attracting points for Wn(z) correspond to roots of f(z). http://www.youtube.com/watch?v=nczm0jdyWps

Green color is attracting basin of z0, red of z1, and blue of z2.

23 Mandelbulbs – Mandelbrot sets in 3D

 Defined by Daniel White and Paul Nylander using double rotation transformation for spherical coordinates, since there is no 3D equivalence to 2D complex numbers having all properties of field

http://www.youtube.com/watch?v=rEhWtQfx5nw

24 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 Virtual Laboratory Series, Springer 1996.  B. Mandelbrot. The fractal geometry of nature. W.H. Freemen and Co. New York, 1982.  http://www.skytopia.com/project/fractal/man delbulb.html  http://bugman123.com/Hypercomplex/

Background source: http://www.skytopia.com/project/fractal/mandelbulb.html 25