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

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
   "whereas mathemathical idea is a timless thing, few things are more ephemeral then computer hardware and software" Tristan Needham in Visual Complex Analysis


  1.   Introduction
  2.   Introductory Examples



To do:
Update completion indicators

"Just keep in mind that what is obvious for you won't be necessarily obvious for the reader."   :— (cKleinhuis )
  1.   Formula parser
  2.   Computer graphic techniques
    1.   color
    2.   image noise
    3. Image editing - Post-processing
    4.   Dimension
      1.   2D
        1.   graphic, data and parameter files
        2.   plane
          1.   grid, mesh, ruler, ...
          2.   Plane transformations
            1.   examples of conformal maps applied to pictures
              1.   Moebius transformation
        3.   optimisation
        4.   2D algorithms
      2.   3D
      3.   4D
  3. Documentation: Program is as good as it's documentation !



To do:
Update completion indicators

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

      "We choose to do mathematics, not because it is easy, but because it is hard." user "Haskell Curry "
       You don't need to be a mathematician to appreciate the impressive beauty of fractals, and many times intuition and curiosity are two of the more important ingredients that drive mathematical discovery. Víctor José García Garrido
  1.   Numbers
    1. by base
      1. binary number
      2. decimal
    2.   sequences
    3.   Period
    4.   Continued fraction
  2.   Function, map, iterated function
    1. Derivative
  3.   computations
    1.   Numerical methods
      1.   Finding roots of equation
        1.   Newton method
        2. Durand-Kerner method
      2.   Finding function from sequence , curve fitting, model fitting
    2.   Symbolic methods
      1.   Kneading sequences
  4.   Group theory
    1.   Binary adding group
    2.   Basilica group
    3.   Kleinian group
  5.   Geometry
    1.   Hyperbolic geometry
  6.   Vector field
    1.   Polynomial vector field in one complex variable
    2.   From discrete dynamical systems to continuous dynamical systems
  7.   dynamical system
    1.   discrete map
    2.   difference equation
    3.   differential equation

Fractals made by the iterationsEdit

  1. How to analyze map ?
  2. How to construct map with desired properities ?

Iterations of real numbers : 1DEdit

  • logistic map
  • real quadratic map
  • tent map

Iterations of complex numbers :2DEdit

Rational mapsEdit

  1. Analysis
  2. Herman rings
Chebyshev polynomialsEdit
Complex quadratic polynomialsEdit
  1.  Definitions
  2. Iterations : forward and backward ( inverse ) and critical orbit
    1. Fractional iterations
  3. Periodic points or cycle
    1. Period

Algorithms, methods of drawing/computing or representation finctions[1] ( for space transformations see here)

  1. escape and attracting time for (level sets method (LSM), level curves method (LCM)
    1. the Julia sets
      1. Target sets and bailout tests
        1. Decomposition of the target set: Binary Decomposition Method ( BDM) / zeros of Qn or parabolic checkerboard ( chessboard)
        2. Esher like tilings
        3. orbit trap
    2. the Mandelbrot set
  2. Inverse iteration method ( IIM) for drawing:
    1. Julia set = IIM/J
  3. atom domains
  4. True shape
  5. Discrete Langrangian Descriptors
  6. curves
    1. boundary trace
    2. equipotential curve
  7. DEM = Distance Estimation Method
    1. DEM/M- for Mandelbrot set
    2. DEM/J for Julia set
  8. Maping component to the unit disk ( Riemann map ):
    1. Multiplier map and internal ray
      1. on the parameter plane
      2. on the dynamic plane
    2. Boettcher map, complex potential and external ray
      1. on the parameter plane
        1. parameter ray
        2. complex potential , external angle
      2. on the dynamic plane
  9. histogram colorings
  10. Average Colorings "are a family of coloring functions that use the decimal part of the smooth iteration count to interpolate between average sums." Jussi Harkonen
    1. Triangle Inequality Average Coloring = TIA and curvature average algorithm ( CAA)
    2. Stripe Average Coloring = SAC
    3. Discrete Velocity of non-attracting Basins and Petals by Chris King
    4. Average distance
  11. 2D to 3D : bump maping
    1. heightmap
    2. slope
    3. Embossing and Lighting
    4. lighting
  12. Parameter plane: combinatorial algorithms
    1. wake
    2. 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
      4. Period doubling cascade and the Myrberg-Feigenbaum point in the 1/2 family. Escape route 1/2
  13. Zoom
    1. on the parameter plane
      1. Perturbation method
      2. Julia morphing - to sculpt shapes of Mandelbrot set parts ( zoom ) and Show Inflection
 Dynamical plane Julia and Fatou setEdit
  1. Julia set
    1. connected
      1. Hyperbolic Julia sets
      2. Parabolic Julia set
      3. Elliptic Julia set: Siegel disc - a linearizable irrationaly indifferent fixed point
      4. Cremer Julia sets -a non-linearizable irrationaly indifferent fixed point
    2. disconnected
  2. Fatou set
    1. exterior of all Julia sets = basin of attraction of superattracting fixed point (infinity)
      1. Escape time
      2.  Boettcher coordinate
      3.  Orbit portraits and lamination of dynamical plane
    2. Interior of Julia sets:
      1.   Basin of attraction of superattracting periodic/fixed point - Boettchers coordinate , c is a center of period n component of Mandelbrot set
        1. Circle Julia set ( c = 0 is a center of period 1 component)
        2. Basilica Julia set ( c = -1 is a center of period 2 component)
      2.   Basin of attraction of attracting periodic/fixed point - Koenigs coordinate
      3.   Local dynamics near indifferent fixed point/cycle
        1.   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.   Local dynamics near irrationally indifferent fixed point/cycle ( elliptic ) - Siegel disc
 Parameter plane and Mandelbrot setEdit
  1. Topological model of Mandelbrot set : Lavaurs algorithm and lamination of parameter plane
  2. structure of Mandelbrot set
    1. real slice and ordering of hyperbolic componnets
      1. periodic part: 1/2 family : period doubling cascade and the Myrberg-Feigenbaum point. Escape route 1/2
      2. chaotic part
  3. Transformations of parameter plane
  4. Sequences and orders on the parameter plane
  5. Parts of parameter plane
    1. exterior of the Mandelbrot set
      1. External (Parameter) Rays of:
        1. the wake ( root point)
        2. the principle Misiurewicz points for the wake k/r of main cardioid
        3. subwake (root points, tuning and internal address)
        4. islands ( root point, Douady tuning)
        5. branch tips of the shrub ( Misiurewicz points)
    2. Boundary of whole set and it's components
      1. parabolic points: root points and cusps
      2. Misiurewicz points
        1. Devaney algorithm for principle Misiurewicz point
    3. interior of hyperbolic components
      1. centers of hyperbolic components = nuclesu of Mu-atoms
      2. Internal rays
  6. speed improvements

The BuddhabrotEdit

exponential familiesEdit

trigonometric familiesEdit

The Newton-Raphson fractalEdit

Quaternion Fractals : 3DEdit

Other fractalsEdit

  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.   Apollonian fractals
  7.  Fractal mountains
  8. Iterated function systems, Nonlinear IFS
  9. Flame fractals
  10. cellular automata
  11. Strange attractors


  1. AlmondBread
  2. fractint
  3. Spider by Yuval Fisher
  4. Fragmentarium - GLSL
  5. Kalles Fraktaler
  6. Mandelbulber ( m3p file holds only the parameters, while .m3i holds also the raw image )
  7. Mandel - software for real and complex dynamics by Wolf Jung
  8. Mandel Machine
  9. gnofract
  10. Programs by Claude Heiland-Allen
    1. mandelbrot-book and mandelbrot-book-images
    2. mandelbrot-perturbator
    3. mightymandel - GLSL
    4. gmandel - A Mandelbrot Set explorer implemented in Haskell using GTK/OpenGL/libqd, git repo
    5. emndl - exponential strip visualisation of the Mandelbrot set, git repo and fractalforums article
  11. Libraries by Claude Heiland-Allen
    1. kf-extras programs for manipulating output from Kalles Fraktaler 2 and blog
    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)
  12. UltraFractal
  13. Xaos
  14. Shadertoy - GLSL
  15. Dynamics - program by Helena E. Nusse and James Yorke
  16. The Computer Language Benchmarks Game : mandelbrot
  17. lt = a Mac OS X application for researchers in complex dynamical systems.
  18. Programs by Curtis McMullen
  19. programs by Gert Buschmann
    1. RatioField
    2. Ratio
  20. Fractalzoomer - Java progam by Chris Kalonakis ( with src code)
  21. Programs by Dmitry Khmelev
  22. DsTool is a computer program for the interactive investigation of dynamical system
    1. pyDsToo/
  23. matcont - is a Matlab software project for the numerical continuation and bifurcation study of continuous and discrete parameterized dynamical systems
  24. Linas' Art Gallery
    1. original pages
    2. - fork with new c code
  25. kandid "s a java-based genetic art program from 2002 that features several kinds of algorithms including an Iterated Function System Affine Transformation; Voronoi Diagram; Cellular Automata and a bunch of other things. By far my favorite is the iIFS Affine Transformation in Grayscale mode. It can operate in color modes but the results are always awful." Tim Hodkinson: Kandid beats Apophysis, Chaotica and JWildfire with millions of colors tied behind its back!!!
  26. Dr. Don Spickler - Fractal Generator
  27. wolfram language guide: Iterated Maps And Fractals
  28. James Gleick's CHAOS: The Software, version by Rudy Rucker
  29. fractalstream-1.0 and home page
  30. Polynomial Julia Sets online visualisation with formula parser by Mark McClure
  31. Fractalshades by Geoffroy Billotey


Wikibook Development Stages
Sparse text   Developing text   Maturing text   Developed text   Comprehensive text  
  1. muency : representation function From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2020