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 x_t = x² + c

t

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 x_t = x_t² + c If |x_t| > 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