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
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
2 Compute xt = x t + 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
2 Compute xt = xt + 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