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 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.