Fractals

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

Introduction

1. Introduction
2. Introductory Examples

Programming

"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. Dimension
      1. 2D
         1. graphic files
         2. plane
            1. grid, ruler, ...
            2. Plane transformations
               1. conformal map
         3. optimisation
         4. 2D algorithms
      2. 3D
      3. 4D
3. Documentation: Program is as good as it's documentation !

Mathematics

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

—Robert L. Devaney

1. Numbers
   1. sequences
   2. Period
   3. Continued fraction
2. Function
3. computations
   1. Numerical methods
      1. Finding roots of equation
         1. Newton method
      2. Finding function from sequence , curve fitting, model fitting
         1. zunzun : curve 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 iterations

Iterations of complex numbers :2D

Rational maps

Polynomials

Theory

1. Definitions
2. Iterations : forward and backward ( inverse )
   1. Fractional iterations
3. Periodic points
   1. Period

Algorithms

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 map ):
   1. Multiplier map
      1. on the parameter plane on the parameter plane
      2. on the dynamic plane
   2. Boettcher map and complex potential
      1. on the parameter plane
      2. on the dynamic plane
6. histogram colorings
7. 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
   2. Stripe Average Coloring = SAC
   3. Discrete Velocity of non-attracting Basins and Petals by Chris King
   4. Average distance
8. orbit trap
9. Julia morphing - to sculpt shapes of Mandelbrot set parts ( zoom ) and Show Inflection
10. 2D to 3D : bump maping
    1. heightmap
    2. slope
    3. Embossing and Lighting
    4. lighting
11. 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

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
         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 set

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 and speed improvements
      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

Other fractals

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

software

1. fractint
2. Spider by Yuval Fisher
3. Fragmentarium - GLSL
4. Kalles Fraktaler
5. mandelbulber
6. Mandel - software for real and complex dynamics by Wolf Jung
7. Mandel Machine
8. gnofract
9. 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/
10. 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)
11. UltraFractal
12. Xaos
13. Shadertoy - GLSL
14. Dynamics - program by Helena E. Nusse and James Yorke
15. The Computer Language Benchmarks Game : mandelbrot
16. lt = a Mac OS X application for researchers in complex dynamical systems.
17. Programs by Curtis McMullen
18. programs by Gert Buschmann
    1. RatioField
    2. Ratio
19. Fractalzoomer - Java progam by Chris Kalonakis ( with src code)
20. Programs by Dmitry Khmelev
21. DsTool is a computer program for the interactive investigation of dynamical system
    1. pyDsToo/
22. matcont - is a Matlab software project for the numerical continuation and bifurcation study of continuous and discrete parameterized dynamical systems