# Fractals

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]

## Programming[edit]

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

- Formula parser
- color
- 2D
- graphic files
- plane
- grid, ruler, ...
- Plane transformations

- optimisation
- 2D algorithms

- 3D
- 4D

Dimension
- 2D

Computer graphic techniques

## Mathematics[edit]

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

- Numbers
- Function
- Numerical methods
- Finding roots of equation
- Finding function from sequence , curve fitting, model fitting

- Kneading sequences

Symbolic methods

computations
- Numerical methods
- Group theory
- Geometry
- polynomial vector field in one complex variable
- From discrete dynamical systems to continuous dynamical systems

Vector field
- discrete map
- difference equation
- differential equation

dynamical system

## 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]

###### Algorithms[edit]

- Escape time
- zeros of Qn or parabolic checkerboard ( chessboard)
- atom domains
- DEM = Distance Estimation Method
- Maping component to the unit disk ( Riemann mapping theorem ):
- Multiplier map on the parameter plane

- Triangle Inequality Average Coloring
- Discrete Velocity of non-attracting Basins and Petals by Chris King
- orbit trap
- wake - combinatorial algorithms

###### Dynamical plane Julia and Fatou set[edit]

**Julia set**- Fatou set
- Basin of attraction of superattracting fixed point (infinity) :
**exterior of all Julia sets**and interior of some Julia sets **Interior of Julia sets**:- Basin of attraction of
**attracting**periodic/fixed point - Koenigs coordinate - Local dynamics near indifferent fixed point/cycle

- Basin of attraction of

- Basin of attraction of superattracting fixed point (infinity) :

###### Parameter plane and Mandelbrot set[edit]

- Topological model of Mandelbrot set : Lavaurs algorithm and lamination of parameter plane
- Transformations of parameter plane
- Parts of parameter plane

#### The Buddhabrot[edit]

#### exponential families[edit]

#### trigonometric families[edit]

#### The Newton-Raphson fractal[edit]

### Quaternion Fractals : 3D[edit]

## Other fractals[edit]

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

## software[edit]

- 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
- gmandel - A Mandelbrot Set explorer implemented in Haskell using GTK/OpenGL/libqd, git repo
- mandelbrot-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
- 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)

- UltraFractal
- Xaos
- Shadertoy - GLSL
- Dynamics - program by Helena E. Nusse and James Yorke
- The Computer Language Benchmarks Game : mandelbrot
- lt = a Mac OS X application for researchers in complex dynamical systems.