Fractals

                            “What I cannot create, I do not understand.” — Richard P. Feynman

                          "Just keep in mind that what is obvious for you won't be necessarily obvious for the reader."   :— (cKleinhuis )  

Here you can find algorithms and examples of source code for drawing fractals and some techiques related with it like :

• making images
• numerical and symbolic computations

Multiplatform, open source and free tools are suggested.

Make a good description of programs/algorithms :

• both formal ( strict definition ) and informal description
• equations
• images ( if it is possible put comment into image: EXIF, ... )
• pseudocode
• code in various programming languages.

The program is as good as it's documentation.

Try to separate 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, or prcedures which can be used in other programs.

Introduction

• Introduction
• Introductory Examples

Programming

                     "separate the calculation phase from the colouring phase" Claude Heiland-Allen

• Formula parser
• Computer graphic techniques
  • color
  • Dimension
    • 2D
      • graphic files
      • plane
        • Plane transformations
          • conformal map
      • optimisation
      • 2D algorithms
    • 3D
    • 4D

Mathematics

       “It can be argued that the mathematics behind these images is even prettier than the pictures themselves.” Robert L. Devaney

       "many questions concerning (discrete) dynamical systems are of a number theoretic or combinatorial nature." Christian Krattenthaler

• Numbers
  • sequences
  • Period
• computations
  • Numerical methods
    • Finding roots of equation
      • Newton method
    • Finding function from sequence , curve fitting, model fitting
  • Symbolic methods
    • Kneading sequences
• Group theory
  • Binary adding group
  • Basilica group
  • Kleinian group
• Geometry
  • Hyperbolic geometry
• Vector field
  • polynomial vector field in one complex variable
  • From discrete dynamical systems to continuous dynamical systems
• dynamical system
  • discrete map
  • difference equation
  • differential equation

Fractals made by the iterations

Iterations of complex numbers :2D

Rational maps

Polynomials

Chebyshev polynomials

Complex quadratic polynomials

• Theory
  • Definitions
  • Iterations : forward and backward ( inverse )
    • Fractional iterations
• Algorithms
  • Escape time
  • zeros of Qn or parabolic checkerboard ( chessboard)
  • DEM/M
  • Discrete Velocity of non-attracting Basins and Petals by Chris King
  • orbit trap
  • wake - combinatorial algorithms
    • tuning
      • principle Misiurewicz points for the wake k/r of main cardioid
      • subwake, tuning and internal address
      • roots, islands and Douady tuning
    • Julia morphing - to sculpt shapes of Mandelbrot set parts ( zoom ) and Show Inflection
  • atom domains
• Dynamical plane Julia and Fatou set
  • Julia set
    • Cremer Julia sets
  • Fatou set
    • Basin of attraction of superattracting fixed point (infinity) : exterior of all Julia sets and interior of some Julia sets
      • Escape time
      • Boettcher coordinate
      • Orbit portraits and lamination of dynamical plane
    • Interior of Julia sets:
      • Basin of attraction of attracting periodic/fixed point - Koenigs coordinate
      • Local dynamics near indifferent fixed point/cycle
        • Local dynamics near rationally indifferent fixed point/cycle ( parabolic ). Leau-Fatou flower theorem
          • 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
          • Repelling and attracting directions
          • Rays landing on the parabolic fixed point
          • parabolic checkerboard
        • 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
  • Transformations of parameter plane
  • Parts of parameter plane
    • exterior of the Mandelbrot set
    • Mandelbrot set
      • Boundary
        • root points
        • Misiurewicz points
      • interior of hyperbolic components
        • centers of hyperbolic components = nuclesu of Mu-atoms

Other fractals

• Real-world fractals
• Lyapunov fractal
• L-Systems
• Midpoint displacement algorithm
• Diamond-square algorithm
• a limit set of a Kleinian group
  • Apollonian fractals
• Fractal mountains
• Iterated function systems, Nonlinear IFS
• Flame fractals

Fractal programs

• fractint
• Spider by Yuval Fisher
• Fragmentarium - GLSL
• Kalles Fraktaler
• Mandel - software for real and complex dynamics by Wolf Jung
• Mandel Machine
• gnofract
• Programs by Claude Heiland-Allen
  • mightymandel - GLSL
  • book program
  • mandelbrot-perturbator : http://code.mathr.co.uk/mandelbrot-perturbator/
• Libraries by Claude Heiland-Allen
  • kf-extras - programs for manipulating output from Kalles Fraktaler 2^[1][2]
  • mandelbrot-symbolics - symbolic algorithms related to the Mandelbrot set
  • mandelbrot-numerics - numerical algorithms related to the Mandelbrot set
  • mandelbrot-graphics - CPU-based visualisation of the Mandelbrot set
  • mandelbrot-text - parsing and pretty printing related to the Mandelbrot set
  • ruff = relatively useful fractal functions ( in Haskell)
• Xaos
• Shadertoy - GLSL
• Dynamics - program by Helena E. Nusse and James Yorke
• The Computer Language Benchmarks Game : mandelbrot

Links to other fractal learning resources on the web

1. ↑ kf-extras - git repo
2. ↑ fractalforums : kf-extras-for-manipulating-output-from-kalles-fraktaler-2