# Fractals

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
```

## Programming

 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
3. Documentation: Program is as good as it's documentation !

## Mathematics

 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
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
1. zunzun : curve fitting
2. Symbolic methods
4. Group theory
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

• logistic map
• tent map

### Iterations of complex numbers :2D

#### Rational maps

##### Polynomials
###### Algorithms

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) and decomposition of target set, like binary decomposition ( BD)
2. Inverse iteration method ( IIM) for drawing:
1. Julia set = IIM/J
3. zeros of Qn or parabolic checkerboard ( chessboard)
4. atom domains
5. True shape
6. Discrete Langrangian Descriptors
7. curves
8. DEM = Distance Estimation Method
1. DEM/M- for Mandelbrot set
2. DEM/J for Julia set
9. 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
2. on the dynamic plane
10. histogram colorings
11. 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
12. orbit trap
13. 2D to 3D : bump maping
1. heightmap
2. slope
3. Embossing and Lighting
4. lighting
14. wake - combinatorial algorithms
1. tuning
15. Zoom
1. on the parameter plane
###### Dynamical plane Julia and Fatou set
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
###### Parameter plane and Mandelbrot set
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
3. Transformations of parameter plane
4. Sequences and orders on the parameter plane
5. Parts of parameter plane
6. speed improvements

## 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
11. Strange attractore : pyviz: gallery-attractors

## software

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
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
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
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