Fractal
Geometry
"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
Fractal tree branch made by algorithmic drawing
Traditional
Japanese
woodblocks (left)
and their modern
fractal
counterparts
(right).
Hokusai , The Great Wave Off Kanagawa
Fractal generated landscape by Vistapro
Star Trek II: The Wrath of Khan, Genesis Effect
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.”
spiral phyllotaxis
spiral arrangement leaves, florets, or seeds.
modeling lilac inflorescence
Generation 1
Generation 2
Generation 4
Generation 6
Generation 12
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
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
Conic Sections
Ptolemaic View of
the Universe
Dante’s Divine Comedy
Clockwork Universe
a0
r=
1 + e cosq
a0
a=
1 - e2
e > 1 hyperbola
e = 1 parabola
e < 1 ellipse
e = 0 circle
Spherical Triangle
Escher’s Circle Limit IV
The First Monster Curve
Giuseppe Peano, 1890
Koch Snowflake
Helge von Koch
1870 - 1924
The Ultimate Fractal
Benoit Mandelbrot
Mandelbrot Set
zn+1 = zn2 + c
EUCLIDEAN
FRACTAL
- traditional ( > 2000 - modern monsters
yr old )
( 100 years old )
- deals with lines,
spheres, cylinders
and cones
- these forms can be
scaled
- no specific size or
scale
- self similarity is
apparent
with magnification
- shapes described
by formula
- shapes defined by a
recursive algorithm
- shapes evolve
- appropriate for
nature
- suits man-made
objects
Julia Set
GASTON MAURICE JULIA
1893-1978
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