Fractals -1- Dr Christoph Traxler

7/27/2019 Fractals -1- Dr Christoph Traxler

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1. Introduction

1.1

Mathematical Monsters

Guiseppe PeanoCantor Set, 1870

Christoph Traxler 1

Georg Cantor

Peano Curve, 1890

Mathematical Monsters

Koch Curve, 1904

Waclaw Sierpinski

Helge von Koch

Christoph Traxler 2

Sierpinski Triangle, 1916


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1. Introduction

1.2

> 2000 years old Around 30 years old

Euclidean Geometry Fractal Geometry

Euclidean vs. Fractal Geometry

Applicable for

artificial objects

Applicable for natural

objects

Christoph Traxler 3

Invariant under scaling,selfShapes change with scaling



similar

scaled by 3

Christoph Traxler 4


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1. Introduction

1.3

Objects defined by recursiveObjects defined by analytical



algorithms

Localy rough, not differentiable

Elements: iteration of functions

equations

Localy smooth, differentiable

Elements: vertices, edges,

surfaces

Christoph Traxler 5

Fractals

Classes of Fractals

not linear

Julia Sets

linear

IFS

-

Fractal Brownian

Motion

Christoph Traxler 6

Strange attractors

Bifurcation diagrams

Diffusion Limited

Aggregation

L-Systems


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1. Introduction

1.4

The Cantor Set

Georg Cantor (1845-1918), founder of settheory

0 1 X0 = [0,1]

X1 = [0,1/3] [2/3,1]X2 = [0,1/9] [2/9,1/3]

[2/3,7/9] [8/9,1]...

Christoph Traxler 7

length(C) = length(limXn) = lim(2/3)n = 0

C ... limit objectC limn

Xn

Initiator

Construction of the Cantor Set

enera or

X[0] = initiator;

for(n=1;n=;n++)X[n] = substitute each span

Christoph Traxler 8

o X n- w t generator;


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1. Introduction

1.5

The Cantor Set

Arithmetic description of the Cantor Set withtriadic numbers {0,1,2}

,contain the digit 1 in their triadic expansion

C = {x = 0.x1x2x3...xn | xi #1, i = 1,2,...} uncountable set of points ( Cantor Dust )

Christoph Traxler 9

0.0 0.1 0.2

0.00 0.01 0.02 0.20 0.21 0.22

The Cantor Set

The Cantor Set as fix point of afeedback system:

0Two categories of points: prisoner

& escapeePrisoner set is the Cantor Set

Christoph Traxler 10

Xn Xn+1

5.0

5.0

33

3

xif

xif

x

xX


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1. Introduction

1.6

The Cantor Set

Escapee vs.prisoner

1

x1

x2


x0x0

x1 x2

The Cantor Set

Properties:

Self similar

Cant be described analytically

Cross section of Saturn rings is similar tothe Cantor Set


errors can be described by Cantor Set(Mandelbrot 62)


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1. Introduction

1.7

X4

Helge von Koch, 1904

The Koch Curve

X2

X3

nn


X0

X1

Initiator

Generator

The Koch Curve

Self similar

Length(Xn) = (4/3)n

Each part of K has

infinite lengthContinuous but notdifferentiable

scaled by 3



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1. Introduction

1.8

Circumference consists of 3 Koch Curves

length(Koch Island) =

The Koch Island

T/3 T/9 T/27


T T 3T/3 T 3T/3 12T/9

T 3T/3 12T/9 48T/27

The Koch Island

T/3 T/9 T/27

21

21 431343 aAaAAk

k


22

1 35

2

5

9

12

3aaAA


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1. Introduction

1.9

The Koch Island

23

5

2lim aA


Infinite

circumference

but finite area

compass setting length

500 km 2600 km

Measuring the Coast of Britain

100 km

54 km

17 km

3800 km

5770 km

8640 km


100 km 50 km


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1. Introduction

1.10

Log/Log Diagrams

Description of the relation between compasssetting and measured length

log(u)4.0

3.8

3.6

bs

du log1

loglog

b


d = 0.36log(1/s)

3.4

-2.7 -2.3 -1.9 -1.5 -1.1

log(b)

ower aw: dsu

scale 1

log3(u)

2

Measuring the Koch Curve

scale 1/3

log3(1/s)

4

1

2691.04

lo d


scale 1/9

3


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1. Introduction

1.11

3 1/3

number ofpieces

reductionfactor

Self Similarity of Line, Square, Cube

1/6

9=32

36=62

42 1/42

1764=422

6

1/3

1/6

1/42


27=33

216=63

74088=423

1/3

1/61/42

Scaling factors are not arbitrary

Self Similarity of Fractals

num er opieces

re uc onfactor

2 1/31/9

2k 1/3k4


4 1/3

1/9

4k 1/3k

16


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1. Introduction

1.12


Scaling factors are characteristic for thedecomposition of fractals into self similar parts

n number of self similar pieces

s

nD

sn

D 1log

log1


s scaling factor

D self similarity dimension D = 1 + d


Line: log3/log3 = 1

Square: log9/log3 = 2

Cube: log27/log3 = 3

Cantor Set:

log2k /log3k = log2/log3 0.6309


Koch Curve:

log4k /log3k = log4/log3 1.2619


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1. Introduction

1.14

The Peano Curve

Parametrization of the square

Each point of the square can be adressed by

0.2 0.28

0.82

Contradiction to classic notion of dimension


0.0. .

0.4

0.5

0.6

0.7

0.8.

0.77

The Sierpinski Gasket

Waclaw Sierpinski, 1916

Initiator X0 Generator X1 X2 X3


Dimension: log3 /log2 = log3/log2 1.5849

The limit object consists of branching points


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1. Introduction

1.15


Corner pointAllows the branching orders:

Touching points 2 (corner of initial triangle)

4 (touching point)

3 (any other point)


The Sierpinski Carpet

Initiator X0 Generator X1 X2 X3

Dimension: log8k /log3k = log8/log3 1.892


Univeral: it contains a topological versionof any 1-dimensional object


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1. Introduction

1.16

Contains branching points with any order

Order 4 branchin

The Sierpinski Carpet

structure

Cantor set


Square

Line

Fractals in 3D

Sierpinski Tetrahedron,D = log4/log2 = 2

Menger Sponge, D = log20/log3 2.726



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1. Introduction

1.17

Fractals in 3D

The Sierpinski tetrahedron can be seen asnetwork of branching points with spatial

Since its dimension is 2 it should be possibleto fold it into the plane

In the plane it becomes a space filling networkof branching points


Fractals in 3D

Folding the Sierpinski tetrahedron



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1. Introduction

1.18

Fractals in 3D



Haptic fractals creation without computers



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1. Introduction

1.20

Wada Basin Fractals

Complex inter-reflection pattern of mirrorspheres shows fractal properties


Wada Basin Fractals

Can be simulated by ray tracing



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1. Introduction

1.21

Wada Basin Fractals

Web sites:

www.miqel.com/fractals_math_patterns/visual- - - - .

local.wasp.uwa.edu.au/~pbourke/fractals/wada/index.html

www.youtube.com/watch?v=C1VZkP2dNXM


Application of Fractal Geometry

Computer Graphics, - 3D modeling of naturalphenomena, textures, animation

Electronics and signal processing

Material engineeringFlow simulation

-


,theory), measuring dimensions, selforganisation patterns


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1. Introduction

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Fractal antenna for a cellular phone

More efficient, - less space



Indigenous people in several African regionsarrange their villages in fractal patterns

Ron Eglashs African fractals web site:

csdt.rpi.edu/african/afractal/afractal.htm


Fractals -1- Dr Christoph Traxler

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