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Fractals
Self Similarity and Fractal Geometry
presented by Pauline Jepp
601.73
Biological Computing
Overview
History
Initial Monsters
Details
Fractals in Nature
Brownian Motion
L-systems
Fractals defined by linear algebra operators
Non-linear fractals
History
Euclid's 5 postulates:
1. To draw a straight line from any point to any other.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any centre and distance.
4. That all right angles are equal to each other.
5. That, if a straight line falling on two straight lines make the interior angles on the same
side less than two right angles, if produced indefinitely, meet on that side on which are
the angles less than the two right angles.
History
Euclid ~ "formless" patterns
Mandlebrot's Fractals
"Pathological" "gallery of monsters"
In 1875:
Continuous non-differentiable
functions, ie no tangent
La Femme Au Miroir 1920
Leger, Fernand
Initial Monsters
1878 Cantor's set
1890 Peano's
space filling curves
Initial Monsters
1906 Koch curve
1960 Sierpinski's
triangle
Details
Fractals :
are self similar
fractal dimension
A square may be broken into N^2 self-similar pieces,
each with magnification factor N
Details
Effective dimension
Mandlebrot: " ... a notion that should not be defined precisely. It is an intuitive
and potent throwback to the Pythagoreans' archaic Greek geometry"
How long is the coast of Britain?
Steinhaus 1954, Richardson 1961
Brownian Motion
Robert Brown 1827
Jean Perrin 1906
Diffusion-limited aggregation
L-Systems and Fractal Growth
Packing efficiency
Axiom &
production rules
Axiom: B
Rules: B->F[-B]+B
F->FF
B
F[-B]+B
FF[-F[-B]+B]+F[-B]+B
L-Systems and Fractal Growth
Turtle graphics
Seymour Papert
L-Systems and Fractal Growth
L-Systems and Fractal Growth
L-Systems and Fractal Growth
Affine Transformation Fractals
"It has a miniature version of itself embedded inside it, but the smaller version
is slightly rotated."
Transofrmations:
translation,
scale,
reflection
rotation
Affine Transformation Fractals
Michael Barnsley
Multiple Reduction Copy Machine Algorithm
(MRCM)
Affine Transformation Fractals
Multiple Reduction Copy Machine Algorithm
(MRCM)
Affine Transformation Fractals
Iterated Functional Systems (IFS)
Affine Transformation Fractals
Iterated Functional Systems (IFS)
Affine Transformation Fractals
Iterated Functional Systems (IFS)
Affine Transformation Fractals
Iterated Functional Systems (IFS)
The Mandelbrot & Julia sets
Iterative Dynamical Systems
The Mandelbrot & Julia sets
For each number, c, in a subset of the complex plane
Set x0 = 0
For t = 1 to tmax
Compute xt = x2t + c
If t< tmax, then colour point c white
If t = tmax, then colour point c black
The Mandelbrot & Julia sets
The Mandelbrot & Julia sets
M-Set and computability
cardoid:
x = 1/4(2 cos t - cos 2t )
y = 1/4(2 sint t - sin 2 t )
The Mandelbrot & Julia sets
The M-Set as the Master Julia set.
Set c to some constant complex value
For each number, x0 in a subset of the complex plane
For t = 1 to tmax
Compute xt = xt2 + c
If |xt| > 2 then break out of loop
If t < tmax then colour point c white
It t = tmax thencolour point c black
The Mandelbrot & Julia sets
The Mandelbrot & Julia sets