# 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.

## Contents

## IntroductionEdit

## ProgrammingEdit

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

- Formula parser
- Computer graphic techniques
- color
- Dimension
- 2D
- graphic files
- plane
- Plane transformations

- optimisation
- 2D algorithms

- 3D
- 4D

- 2D

## MathematicsEdit

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

- Symbolic methods
- Kneading sequences

- Numerical methods
- Group theory
- 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 iterationsEdit

### Iterations of **real numbers : 1D**Edit

### Iterations of complex numbers :2DEdit

#### Rational mapsEdit

##### PolynomialsEdit

###### Chebyshev polynomialsEdit

###### Complex quadratic polynomialsEdit

- Theory
- Algorithms
**Dynamical plane**Julia and Fatou set**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**- 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

- Boundary

#### The BuddhabrotEdit

#### exponential familiesEdit

#### trigonometric familiesEdit

#### The Newton-Raphson fractalEdit

**Quaternion Fractals : 3D**Edit

## Other fractalsEdit

- 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

## Fractal software : programs and librariesEdit

- 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)

- kf-extras - programs for manipulating output from Kalles Fraktaler 2
- 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.