Fractals

This book is still under development. Please help us 25% This wikibook is about : how to make fractals (:-)) It covers only topics which are important for that (:-)) "What I cannot create, I do not understand." Richard P. Feynman Introduction 75% /Introduction/ 25% /Introductory Examples/ Mathematics for computer graphic: numbers, sequences, functions, numerical methods, fields, ... Programming computer graphic: files, plane, curves, ... plane transformations Fractal software Fractal links Fractals made by the iterations Theory 0% 2010.08.01Definitions Iterations : forward and backward ( inverse ) and critical orbit Fractional iterations critical orbit Periodic points or cycle periodic points of complex quadratic map Period How to analyze map ? How to read location from the image? How to construct map with desired properities ? Algorithms ( graphical (coloring, transformations), numerical, symbolic, other) Iterations of real numbers : 1D (angle) doubling map logistic map real quadratic map tent map Iterations of complex numbers :2D complex-analytic formulas (like Mandelbrot set and Julia set) non-complex-analytic formulas (like Mandelbar and Burning Ship) Rational maps Analysis Herman rings Polynomials Chebyshev polynomials Complex quadratic polynomials Dynamic plane: Julia and Fatou sets coloring the dynamic plane and the Julia and the Fatou sets Julia set with an non-empty interior ( connected ) Hyperbolic Julia sets attracting : filled Julia set have attracting cycle ( c is inside hyperbolic component ) superattracting : filled Julia set have superattracting cycle( c is in the center of hyperbolic component ). Examples : Airplane Julia set, Douady's Rabbit, Basillica. Parabolic Julia set Elliptic Julia set: Siegel disc - a linearizable irrationaly indifferent fixed point with empty interior disconnected ( c is outside of Mandelbrot set ) connected ( c is inside Mandelbrot set ) Cremer Julia sets -a non-linearizable irrationaly indifferent fixed point dendrits or Dendrite Julia sets ( Julia set is connected and locally connected ). Examples : Misiurewicz Julia sets (c is a Misiurewicz point ) Feigenbaum Julia sets ( c is Generalized Feigenbaum point: the limit of the period-q cascade of bifurcations and landing points of parameter ray or rays with irrational angles ) others which have no description Fatou set exterior of all Julia sets = basin of attraction of superattracting fixed point (infinity) Escape time 0% 2010.08.01Boettcher coordinate 0% 2010.08.01Orbit portraits and lamination of dynamical plane Dynamic external rays Interior of Julia sets: 0% 2010.08.01 Basin of attraction of superattracting periodic/fixed point - Boettchers coordinate , c is a center of period n component of Mandelbrot set Circle Julia set ( c = 0 is a center of period 1 component) Basilica Julia set ( c = -1 is a center of period 2 component) 0% 2010.08.01 Basin of attraction of attracting periodic/fixed point - Koenigs coordinate 0% 2010.08.01 Local dynamics near indifferent fixed point/cycle 0% 2010.08.01 Local dynamics near rationally indifferent fixed point/cycle ( parabolic ). Leau-Fatou flower theorem petal of the Leau-Fatou flower Repelling and attracting directions Rays landing on the parabolic fixed point parabolic checkerboard parabolic perturbation Fatou_coordinate Fatou_coordinate for f(z)=z/(1+z) Fatou_coordinate for f(z)=z+z^2 Fatou_coordinate for f(z)=z^2 + c 0% 2010.08.01 Local dynamics near irrationally indifferent fixed point/cycle ( elliptic ) - Siegel disc Parameter plane and Mandelbrot set Topological model of Mandelbrot set : Lavaurs algorithm and lamination of parameter plane structure of Mandelbrot set and ordering of hyperbolic components family: real slice of Mandelbrot set. periodic part: period doubling cascade. Escape route 1/2 the Myrberg-Feigenbaum point of family chaotic part main antenna is a shrub of family Transformations of parameter plane Sequences and orders on the parameter plane Parts of parameter plane exterior of the Mandelbrot set: escape time, Level Set Method ( LSM/M), Binary Decomposition Method (BDM/M) External (Parameter) Rays of: the wake ( root point) the principle Misiurewicz points for the wake k/r of main cardioid subwake (root points, tuning and internal address) branch tips of the shrub ( Misiurewicz points) islands ( root point, Douady tuning) Interior and the boundary : components Number of the Mandelbrot set's components Boundary of whole set and it's components parabolic points: root points and cusps unroll a closed curve and then stretch out into an infinite strip Misiurewicz points Devaney algorithm for principle Misiurewicz point interior of hyperbolic components centers of hyperbolic components = nuclesu of Mu-atoms Internal rays Islands the biggest island of the wake distortion of mini Mandelbrot sets islands ( root point, Douady tuning) Points ( parameter of the iterated function) speed improvements coloring algorithm The Buddhabrot exponential families trigonometric families /The Newton-Raphson fractal/ /Hopalong/ Quaternion Fractals : 3D Other fractals Mathematical analysis University level mathematics books F