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7/27/2019 Fractals -1- Dr Christoph Traxler
1/22
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
7/27/2019 Fractals -1- Dr Christoph Traxler
2/22
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
Euclidean vs. Fractal Geometry
Euclidean Geometry Fractal Geometry
similar
scaled by 3
Christoph Traxler 4
7/27/2019 Fractals -1- Dr Christoph Traxler
3/22
1. Introduction
1.3
Objects defined by recursiveObjects defined by analytical
Euclidean vs. Fractal Geometry
Euclidean Geometry Fractal Geometry
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
7/27/2019 Fractals -1- Dr Christoph Traxler
4/22
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;
7/27/2019 Fractals -1- Dr Christoph Traxler
5/22
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
7/27/2019 Fractals -1- Dr Christoph Traxler
6/22
1. Introduction
1.6
The Cantor Set
Escapee vs.prisoner
1
x1
x2
Christoph Traxler 11
x0x0
x1 x2
The Cantor Set
Properties:
Self similar
Cant be described analytically
Cross section of Saturn rings is similar tothe Cantor Set
Christoph Traxler 12
errors can be described by Cantor Set(Mandelbrot 62)
7/27/2019 Fractals -1- Dr Christoph Traxler
7/22
1. Introduction
1.7
X4
Helge von Koch, 1904
The Koch Curve
X2
X3
nn
Christoph Traxler 13
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
Christoph Traxler 14
7/27/2019 Fractals -1- Dr Christoph Traxler
8/22
1. Introduction
1.8
Circumference consists of 3 Koch Curves
length(Koch Island) =
The Koch Island
T/3 T/9 T/27
Christoph Traxler 15
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
Christoph Traxler 16
22
1 35
2
5
9
12
3aaAA
7/27/2019 Fractals -1- Dr Christoph Traxler
9/22
1. Introduction
1.9
The Koch Island
23
5
2lim aA
Christoph Traxler 17
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
Christoph Traxler 18
100 km 50 km
7/27/2019 Fractals -1- Dr Christoph Traxler
10/22
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
Christoph Traxler 19
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
Christoph Traxler 20
scale 1/9
3
7/27/2019 Fractals -1- Dr Christoph Traxler
11/22
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
Christoph Traxler 21
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
Christoph Traxler 22
4 1/3
1/9
4k 1/3k
16
7/27/2019 Fractals -1- Dr Christoph Traxler
12/22
1. Introduction
1.12
Self Similarity of Fractals
Scaling factors are characteristic for thedecomposition of fractals into self similar parts
n number of self similar pieces
s
nD
sn
D 1log
log1
Christoph Traxler 23
s scaling factor
D self similarity dimension D = 1 + d
Self Similarity of Fractals
Line: log3/log3 = 1
Square: log9/log3 = 2
Cube: log27/log3 = 3
Cantor Set:
log2k /log3k = log2/log3 0.6309
Christoph Traxler 24
Koch Curve:
log4k /log3k = log4/log3 1.2619
7/27/2019 Fractals -1- Dr Christoph Traxler
13/22
7/27/2019 Fractals -1- Dr Christoph Traxler
14/22
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
Christoph Traxler 27
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
Christoph Traxler 28
Dimension: log3 /log2 = log3/log2 1.5849
The limit object consists of branching points
7/27/2019 Fractals -1- Dr Christoph Traxler
15/22
1. Introduction
1.15
The Sierpinski Gasket
Corner pointAllows the branching orders:
Touching points 2 (corner of initial triangle)
4 (touching point)
3 (any other point)
Christoph Traxler 29
The Sierpinski Carpet
Initiator X0 Generator X1 X2 X3
Dimension: log8k /log3k = log8/log3 1.892
Christoph Traxler 30
Univeral: it contains a topological versionof any 1-dimensional object
7/27/2019 Fractals -1- Dr Christoph Traxler
16/22
1. Introduction
1.16
Contains branching points with any order
Order 4 branchin
The Sierpinski Carpet
structure
Cantor set
Christoph Traxler 31
Square
Line
Fractals in 3D
Sierpinski Tetrahedron,D = log4/log2 = 2
Menger Sponge, D = log20/log3 2.726
Christoph Traxler 32
7/27/2019 Fractals -1- Dr Christoph Traxler
17/22
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
Christoph Traxler 33
Fractals in 3D
Folding the Sierpinski tetrahedron
Christoph Traxler 34
7/27/2019 Fractals -1- Dr Christoph Traxler
18/22
1. Introduction
1.18
Fractals in 3D
Christoph Traxler 35
The Sierpinski Gasket
Haptic fractals creation without computers
Christoph Traxler 36
7/27/2019 Fractals -1- Dr Christoph Traxler
19/22
7/27/2019 Fractals -1- Dr Christoph Traxler
20/22
1. Introduction
1.20
Wada Basin Fractals
Complex inter-reflection pattern of mirrorspheres shows fractal properties
Christoph Traxler 39
Wada Basin Fractals
Can be simulated by ray tracing
Christoph Traxler 40
7/27/2019 Fractals -1- Dr Christoph Traxler
21/22
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
Christoph Traxler 41
Application of Fractal Geometry
Computer Graphics, - 3D modeling of naturalphenomena, textures, animation
Electronics and signal processing
Material engineeringFlow simulation
-
Christoph Traxler 42
,theory), measuring dimensions, selforganisation patterns
7/27/2019 Fractals -1- Dr Christoph Traxler
22/22
1. Introduction
1 22
Application of Fractal Geometry
Fractal antenna for a cellular phone
More efficient, - less space
Christoph Traxler 43
Application of Fractal Geometry
Indigenous people in several African regionsarrange their villages in fractal patterns
Ron Eglashs African fractals web site:
csdt.rpi.edu/african/afractal/afractal.htm
Christoph Traxler 44