# Fractals

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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

## 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
- Computer graphic techniques
- Documentation: Program is as good as it's documentation !

## Mathematics[edit]

"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

- Numbers
- Function
- Numerical methods
- Newton method
- Durand-Kerner method

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]

- Definitions
- Iterations : forward and backward ( inverse ) and critical orbit
- Periodic points or cycle

###### Algorithms[edit]

Algorithms or representation finctions^{[1]} ( for space transformations see here)

- Escape time for drawing: LSM LCM
- Inverse iteration method ( IIM) for drawing:
- Julia set = IIM/J

- zeros of Qn or parabolic checkerboard ( chessboard)
- atom domains
- True shape
- Discrete Langrangian Descriptors
- curves
- DEM = Distance Estimation Method
- Maping component to the unit disk ( Riemann map ):
- Multiplier map and internal ray
- on the parameter plane
- on the dynamic plane

- Boettcher map, complex potential and external ray
- on the parameter plane
- on the dynamic plane

- Multiplier map and internal ray
- histogram colorings
- 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
- Triangle Inequality Average Coloring = TIA and curvature average algorithm ( CAA)
- Stripe Average Coloring = SAC
- Discrete Velocity of non-attracting Basins and Petals by Chris King
- Average distance

- orbit trap
- Julia morphing - to sculpt shapes of Mandelbrot set parts ( zoom ) and Show Inflection
- 2D to 3D : bump maping
- heightmap
- slope
- Embossing and Lighting
- lighting

- wake - combinatorial algorithms
- Perturbation method
- constructing map with desired properities

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

**Julia set**- connected
- Hyperbolic Julia sets
- Parabolic Julia set
- Elliptic Julia set: Siegel disc - a linearizable irrationaly indifferent fixed point
- Cremer Julia sets -a non-linearizable irrationaly indifferent fixed point

- disconnected

- connected
- 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
- exterior of the Mandelbrot set
- Boundary of whole set and it's components
- interior of hyperbolic components

- Mandelbrot set and speed improvements

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

## software[edit]

- AlmondBread
- fractint
- Spider by Yuval Fisher
- Fragmentarium - GLSL
- Kalles Fraktaler
- Mandelbulber ( m3p file holds only the parameters, while .m3i holds also the raw image )
- Mandel - software for real and complex dynamics by Wolf Jung
- Mandel Machine
- gnofract
- Programs by Claude Heiland-Allen
- mandelbrot-perturbator
- mightymandel - GLSL
- gmandel - A Mandelbrot Set explorer implemented in Haskell using GTK/OpenGL/libqd, git repo
- mandelbrot-book program
- emndl - exponential strip visualisation of the Mandelbrot set

- Libraries by Claude Heiland-Allen
- kf-extras programs for manipulating output from Kalles Fraktaler 2 and blog
- 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-book and mandelbrot-book-images
- mandelbrot-text - parsing and pretty printing related to the Mandelbrot set
- ruff = relatively useful fractal functions ( in Haskell)
- emndl - exponential strip visualisation of the Mandelbrot set, git repo and fractalforums article

- 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.
- Programs by Curtis McMullen
- programs by Gert Buschmann
- Fractalzoomer - Java progam by Chris Kalonakis ( with src code)
- Programs by Dmitry Khmelev
- DsTool is a computer program for the interactive investigation of dynamical system
- matcont - is a Matlab software project for the numerical continuation and bifurcation study of continuous and discrete parameterized dynamical systems
- Linas' Art Gallery
- 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!!!
- Dr. Don Spickler - Fractal Generator
- wolfram language guide: Iterated Maps And Fractals
- James Gleick's CHAOS: The Software, version by Rudy Rucker