# Fractals

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“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 ( in other words : "separate the calculation phase from the colouring phase" Claude Heiland-Allen) 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

## Introduction[edit]

## Programming[edit]

- Formula parser
- Computer graphic techniques
- color
- Dimension

## Mathematics[edit]

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

- Numbers
- computations
- Numerical methods
- Symbolic methods
- Kneading sequences

- Group theory
- Geometry

## 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
- Algorithms
- Escape time
- zeros of Qn or parabolic checkerboard ( chessboard)
- DEM/M
- wake - combinatorial algorithms
- tuning
- Julia morphing - to sculpt shapes of Mandelbrot set parts ( zoom )

- atom domains

**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

#### 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

## Fractal programs[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
- Xaos
- Shadertoy - GLSL