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Fractals
Self Similarity and Fractal Geometry
presented by Pauline Jepp601.73
Biological Computing
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Overview
History
Initial Monsters
Details
Fractals in Nature
Brownian Motion
L-systems
Fractals defined by linear algebra operators
Non-linear fractals
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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 sameside less than two right angles, if
produced indefinitely, meet on that side on which arethe angles
less than the two right angles.
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History
Euclid ~ "formless" patterns
Mandlebrot's Fractals
"Pathological" "gallery of monsters"
In 1875:
Continuous non-differentiablefunctions, ie no tangent La Femme
Au Miroir 1920
Leger, Fernand
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Initial Monsters
1878 Cantor's set
1890 Peano'sspace filling curves
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Initial Monsters
1906 Koch curve
1960 Sierpinski'striangle
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Details
Fractals :
are self similar
fractal dimension
A square may be broken into N^2 self-similar pieces,each with
magnification factor N
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Details
Effective dimensionMandlebrot: " ... a notion that should not be
defined precisely. It is an intuitiveand potent throwback to the
Pythagoreans' archaic Greek geometry"
How long is the coast of Britain?Steinhaus 1954, Richardson
1961
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Brownian Motion
Robert Brown 1827
Jean Perrin 1906
Diffusion-limited aggregation
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L-Systems and Fractal Growth
Packing efficiency
Axiom &production rulesAxiom: BRules: B->F[-B]+B
F->FFB
F[-B]+B
FF[-F[-B]+B]+F[-B]+B
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L-Systems and Fractal Growth
Turtle graphicsSeymour Papert
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L-Systems and Fractal Growth
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L-Systems and Fractal Growth
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L-Systems and Fractal Growth
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Affine Transformation Fractals
"It has a miniature version of itself embedded inside it, but
the smaller versionis slightly rotated."
Transofrmations:
translation,
scale,
reflection
rotation
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Affine Transformation Fractals
Michael Barnsley
Multiple Reduction Copy Machine Algorithm(MRCM)
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Affine Transformation Fractals
Multiple Reduction Copy Machine Algorithm(MRCM)
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Affine Transformation Fractals
Iterated Functional Systems (IFS)
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Affine Transformation Fractals
Iterated Functional Systems (IFS)
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Affine Transformation Fractals
Iterated Functional Systems (IFS)
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Affine Transformation Fractals
Iterated Functional Systems (IFS)
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The Mandelbrot & Julia sets
Iterative Dynamical Systems
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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
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The Mandelbrot & Julia sets
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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
)
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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
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The Mandelbrot & Julia sets
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The Mandelbrot & Julia sets
