# 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

## IntroductionEdit

## ProgrammingEdit

"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 !

## MathematicsEdit

"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
- by base
- binary number
- decimal

- sequences
- Period
- Continued fraction

- by base
- Function, map, iterated function
- Numerical methods
- Finding roots of equation
- Newton method
- Durand-Kerner method

- Finding function from sequence , curve fitting, model fitting

- Finding roots of equation
- 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 iterationsEdit

### Iterations of real numbers : 1DEdit

- logistic map
- real quadratic map
- tent map

### Iterations of complex numbers :2DEdit

#### Rational mapsEdit

##### PolynomialsEdit

###### Chebyshev polynomialsEdit

###### Complex quadratic polynomialsEdit

###### TheoryEdit

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

###### AlgorithmsEdit

Algorithms, methods of drawing/computing or representation finctions^{[1]} ( for space transformations see here)

- escape and attracting time for (level sets method (LSM), level curves method (LCM)
- the Julia sets
- Target sets and bailout tests
- Decomposition of the target set: Binary Decomposition Method ( BDM) / zeros of Qn or parabolic checkerboard ( chessboard)
- Esher like tilings
- orbit trap

- Target sets and bailout tests
- the Mandelbrot set

- the Julia sets
- Inverse iteration method ( IIM) for drawing:
- Julia set = IIM/J

- 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

- 2D to 3D : bump maping
- heightmap
- slope
- Embossing and Lighting
- lighting

- Parameter plane: combinatorial algorithms
- Zoom

###### Dynamical plane Julia and Fatou setEdit

**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
**exterior of all Julia sets**= basin of attraction of superattracting fixed point (infinity)**Interior of Julia sets**:- Basin of attraction of
**superattracting**periodic/fixed point - Boettchers coordinate , c is a center of period n component of Mandelbrot set- Circle Julia set ( c = 0 is a center of period 1 component)
- Basilica Julia set ( c = -1 is a center of period 2 component)

- Basin of attraction of
**attracting**periodic/fixed point - Koenigs coordinate - Local dynamics near indifferent fixed point/cycle

- Basin of attraction of

###### Parameter plane and Mandelbrot setEdit

- Topological model of Mandelbrot set : Lavaurs algorithm and lamination of parameter plane
- structure of Mandelbrot set
- real slice and ordering of hyperbolic componnets

- Transformations of parameter plane
- Sequences and orders on the parameter plane
- Parts of parameter plane
- exterior of the Mandelbrot set
- Boundary of whole set and it's components
- parabolic points: root points and cusps
- Misiurewicz points

- interior of hyperbolic components
- centers of hyperbolic components = nuclesu of Mu-atoms
- Internal rays

- speed improvements

#### The BuddhabrotEdit

#### exponential familiesEdit

#### trigonometric familiesEdit

#### The Newton-Raphson fractalEdit

### Quaternion Fractals : 3DEdit

## 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
- cellular automata
- Strange attractors

## softwareEdit

- 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-book and mandelbrot-book-images
- mandelbrot-perturbator
- mightymandel - GLSL
- gmandel - A Mandelbrot Set explorer implemented in Haskell using GTK/OpenGL/libqd, git repo
- emndl - exponential strip visualisation of the Mandelbrot set, git repo and fractalforums article

- 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-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.
- 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
- fractalstream-1.0 and home page
- Polynomial Julia Sets online visualisation with formula parser by Mark McClure
- Fractalshades by Geoffroy Billotey