Fractals

Introduction

Here you can find algorithms and examples of source code for drawing fractals.

Multiplatform, open source and free tools are suggested.

Try to disjoin computing parameters from creating images ( it can slow the program but makes it easier to understand the algorithm ).

If it is possible make one-file programs.

Contents

1. Introduction
   1. Introductory Examples
   2. Computer graphic techniques
      1. Computer graphic techniques - 2D
         1. graphic files
         2. color
         3. optimisation
      2. Computer graphic techniques - 3D
      3. 4D
   3. Mathematics
      1. Numerical analysis of dynamical systems : numerical methods
         1. Newton method
      2. Period
      3. Group theory
         1. Binary adding group
         2. Basilica group
2. Iterations of real numbers : 1D
3. Iterations of complex numbers :2D
   1. Rational maps
      1. Polynomials
         1. Iterations of Chebyshev polynomials
         2. Complex quadratic polynomials
            1. Theory
               1. Definitions
               2. Iterations : forward and backward ( inverse )
            2. Dynamical plane : Julia and Fatou set
               1. Julia set
                  1. Cremer Julia sets
               2. Fatou set
                  1. Basin of attraction of superattracting fixed point (infinity) : exterior of all Julia sets and interior of some Julia sets
                     1. Escape time
                     2. Boettcher coordinate
                     3. Orbit portraits and lamination of dynamical plane
                  2. Interior of Julia sets:
                     1. Basin of attraction of attracting periodic/fixed point - Koenigs coordinate
                     2. Local dynamics near indifferent fixed point/cycle
                     3. Local dynamics near rationally indifferent fixed point/cycle ( parabolic ). Leau-Fatou flower theorem
                        1. Fatou_coordinate
                     4. Local dynamics near irrationally indifferent fixed point/cycle ( elliptic ) - Siegel disc
            3. Parameter plane and Mandelbrot set
               1. Topological model of Mandelbrot set : Lavaurs algorithm and lamination of parameter plane
   2. The Buddhabrot.
   3. exponential families
   4. trigonometric families
   5. The Newton-Raphson fractal
4. Quaternion Fractals : 3D
5. Iterated function systems
   1. Nonlinear IFS
   2. Flame fractals
6. Misc.
   1. Real-world fractals
   2. Lyapunov fractal
   3. L-Systems
   4. Midpoint displacement algorithm
   5. Diamond-square algorithm
   6. Apollonian fractals
   7. Fractal mountains
7. Links to other fractal learning resources on the web