Fractals

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TODO

Editor's note
This book is still under development. Please help us

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


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[edit]

  1. 75% developed Introduction
  2. 25% developed Introductory Examples

Programming[edit]

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"Just keep in mind that what is obvious for you won't be necessarily obvious for the reader."   :— (cKleinhuis )
  1. 0% developed Formula parser
  2. 0% developed Computer graphic techniques
    1. 0% developed color
    2. 0% developed Dimension
      1. 0% developed 2D
        1. 0% developed graphic files
        2. 0% developed plane
          1. 0% developed grid, ruler, ...
          2. 0% developed Plane transformations
            1. 0% developed conformal map
        3. 0% developed optimisation
        4. 0% developed 2D algorithms
      2. 0% developed 3D
      3. 0% developed 4D

Mathematics[edit]

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


  1. 0% developed Numbers
    1. 0% developed sequences
    2. 0% developed Period
    3. 0% developed Continued fraction
  2. 0% developed Function
  3. 0% developed computations
    1. 0% developed Numerical methods
      1. 0% developed Finding roots of equation
        1. 0% developed Newton method
      2. 0% developed Finding function from sequence , curve fitting, model fitting
    2. 0% developed Symbolic methods
      1. 0% developed Kneading sequences
  4. 0% developed Group theory
    1. 0% developed Binary adding group
    2. 0% developed Basilica group
    3. 0% developed Kleinian group
  5. 0% developed Geometry
    1. 0% developed Hyperbolic geometry
  6. 0% developed Vector field
    1. 0% developed polynomial vector field in one complex variable
    2. 0% developed From discrete dynamical systems to continuous dynamical systems
  7. 0% developed dynamical system
    1. 0% developed discrete map
    2. 0% developed difference equation
    3. 0% developed differential equation

Fractals made by the iterations[edit]

Iterations of real numbers : 1D[edit]

Iterations of complex numbers :2D[edit]

Rational maps[edit]

Polynomials[edit]
Chebyshev polynomials[edit]
Complex quadratic polynomials[edit]
Theory[edit]
  1. 0% developed  as of  2010.08.01Definitions
  2. Iterations : forward and backward ( inverse )
    1. Fractional iterations
  3. Periodic points
    1. Period
Algorithms[edit]
  1. Escape time
  2. zeros of Qn or parabolic checkerboard ( chessboard)
  3. atom domains
  4. DEM = Distance Estimation Method
    1. DEM/M- for Mandelbrot set
    2. DEM/J for Julia set
  5. Maping component to the unit disk ( Riemann mapping theorem ):
    1. Multiplier map on the parameter plane
  6. Triangle Inequality Average Coloring
  7. Discrete Velocity of non-attracting Basins and Petals by Chris King
  8. orbit trap
  9. wake - combinatorial algorithms
    1. tuning
      1. principle Misiurewicz points for the wake k/r of main cardioid
      2. subwake, tuning and internal address
      3. roots, islands and Douady tuning
    2. Julia morphing - to sculpt shapes of Mandelbrot set parts ( zoom ) and Show Inflection
25% developed  as of  2010.08.01Dynamical plane Julia and Fatou set[edit]
  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. 0% developed  as of  2010.08.01Boettcher coordinate
      3. 0% developed  as of  2010.08.01Orbit portraits and lamination of dynamical plane
    2. Interior of Julia sets:
      1. 0% developed  as of  2010.08.01 Basin of attraction of attracting periodic/fixed point - Koenigs coordinate
      2. 0% developed  as of  2010.08.01 Local dynamics near indifferent fixed point/cycle
        1. 0% developed  as of  2010.08.01 Local dynamics near rationally indifferent fixed point/cycle ( parabolic ). Leau-Fatou flower theorem
          1. Fatou_coordinate
            1. Fatou_coordinate for f(z)=z/(1+z)
            2. Fatou_coordinate for f(z)=z+z^2
            3. Fatou_coordinate for f(z)=z^2 + c
          2. Repelling and attracting directions
          3. Rays landing on the parabolic fixed point
          4. parabolic checkerboard
        2. 0% developed  as of  2010.08.01 Local dynamics near irrationally indifferent fixed point/cycle ( elliptic ) - Siegel disc
25% developed  as of  2010.08.01Parameter plane and Mandelbrot set[edit]
  1. Topological model of Mandelbrot set : Lavaurs algorithm and lamination of parameter plane
  2. Transformations of parameter plane
  3. Parts of parameter plane
    1. exterior of the Mandelbrot set
      1. External Parameter Ray
    2. Mandelbrot set
      1. Boundary of whole set and it's components
        1. root points
        2. Misiurewicz points
          1. Devaney algorithm for principle Misiurewicz point
      2. interior of hyperbolic components
        1. centers of hyperbolic components = nuclesu of Mu-atoms

The Buddhabrot[edit]

exponential families[edit]

trigonometric families[edit]

The Newton-Raphson fractal[edit]

Quaternion Fractals : 3D[edit]

Other fractals[edit]

  1. Real-world fractals
  2. Lyapunov fractal
  3. L-Systems
  4. Midpoint displacement algorithm
  5. Diamond-square algorithm
  6. a limit set of a Kleinian group
    1. 25% developed  as of  2010.08.01 Apollonian fractals
  7. 0% developed  as of  2010.08.01Fractal mountains
  8. Iterated function systems, Nonlinear IFS
  9. Flame fractals
  10. cellular automata

software[edit]

  1. fractint
  2. Spider by Yuval Fisher
  3. Fragmentarium - GLSL
  4. Kalles Fraktaler
  5. Mandel - software for real and complex dynamics by Wolf Jung
  6. Mandel Machine
  7. gnofract
  8. Programs by Claude Heiland-Allen
    1. mightymandel - GLSL
    2. gmandel - A Mandelbrot Set explorer implemented in Haskell using GTK/OpenGL/libqd, git repo
    3. mandelbrot-book program
    4. mandelbrot-perturbator : http://code.mathr.co.uk/mandelbrot-perturbator/
  9. Libraries by Claude Heiland-Allen
    1. kf-extras programs for manipulating output from Kalles Fraktaler 2
    2. mandelbrot-symbolics - symbolic algorithms related to the Mandelbrot set
    3. mandelbrot-numerics - numerical algorithms related to the Mandelbrot set
    4. mandelbrot-graphics - CPU-based visualisation of the Mandelbrot set
    5. mandelbrot-text - parsing and pretty printing related to the Mandelbrot set
    6. ruff = relatively useful fractal functions ( in Haskell)
  10. UltraFractal
  11. Xaos
  12. Shadertoy - GLSL
  13. Dynamics - program by Helena E. Nusse and James Yorke
  14. The Computer Language Benchmarks Game : mandelbrot
  15. lt = a Mac OS X application for researchers in complex dynamical systems.

Links[edit]

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