Fractal

1
Fractal Geometry
2
"Clouds are not spheres, mountains are not cones,
coastlines are not circles, and bark is not
smooth, nor does lightning travel in a straight
line. Benoit Mandelbrot
3
Fractal tree branch made by algorithmic drawing
4
Traditional Japanese woodblocks (left) and their
modern fractal counterparts (right).
5
Hokusai , The Great Wave Off Kanagawa
6
(No Transcript)
7
(No Transcript)
8
(No Transcript)
9
(No Transcript)
10
(No Transcript)
11
Fractal generated landscape by Vistapro
12
Star Trek II The Wrath of Khan, Genesis Effect
13
(No Transcript)
14
(No Transcript)
15
(No Transcript)
16
(No Transcript)
17
(No Transcript)
18
(No Transcript)
19
(No Transcript)
20
(No Transcript)
21
PRZEMYSLAW PRUSINKIEWICZ Professor, Department of
Computer Science University of Calgary I apply
methods of computer science to gain a better
understanding of the emergence of forms and
patterns in nature.
22
spiral phyllotaxis spiral arrangement leaves,
florets, or seeds.
23
modeling lilac inflorescence
24
(No Transcript)
25
Generation 1 Generation 2 Generation
4 Generation 6 Generation 12
26
Iterated Function System
Rule 1 x' 0 y' 0.16y Rule 2 x'
0.85x 0.04y y' 0.04x 0.85y
0.16 Rule 3 x' 0.20x 0.26y y'
0.23x 0.22y 0.16 Rule 4 x' 0.15x
0.28y y' 0.26x 0.24y 0.08
27
(No Transcript)
28
Fractals are mathematical hydras. They replicate
themselves by fragmentation. You look at a
fractal pattern and you see a form then you look
closely at a particular region of the pattern,
and you see the same form all over again, only
much smaller this time. A repetition of the same
structural form, a self-similar solution that
goes on ad infinitum.
Paul W. Carlson, fractal artist
29
(No Transcript)
30
(No Transcript)
31
(No Transcript)
32
(No Transcript)
33
(No Transcript)
34
Conic Sections
35
Ptolemaic View of the Universe
Dantes Divine Comedy
36
Clockwork Universe
37
Spherical Triangle
38
Eschers Circle Limit IV
39
The First Monster Curve
Giuseppe Peano, 1890
40
Koch Snowflake
Helge von Koch 1870 - 1924
41
The Ultimate Fractal
Benoit Mandelbrot
Mandelbrot Set zn1 zn2 c
42
(No Transcript)
43
(No Transcript)
44
Julia Set
GASTON MAURICE JULIA 1893-1978
45
(No Transcript)
46
(No Transcript)
47
Euclidean geometry, in which every line has
exactly one parallel through any
point hyperbolic geometry, in which infinitely
many parallels to a line can go through the same
point