# 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

## Introduction[edit | edit source]

- Introduction
- Introductory Examples
- Mathematics for computer graphic: numbers, sequences, functions, numerical methods, fields, ...
- Programming computer graphic: files, plane, transformations, curves, ...
- Fractal software
- Fractal links

## Fractals made by the iterations[edit | edit source]

**Theory**

- Definitions
- Iterations : forward and backward ( inverse ) and critical orbit
- critical orbit
- Periodic points or cycle
- How to analyze map ?
- How to construct map with desired properities ?

**Algorithms**: Algorithms, methods of drawing/computing or representation finctions ( for space transformations see here)

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

- Target sets, trap 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
- boundary - scanning
- unroll a closed curve and then stretch out into an infinite strip
- equipotential curve
- external ray ( parameter and dynamic) trace
- internal ray
- path: escape route

- 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
- parameter ray = field lines
- complex potential , external angle

- on the dynamic plane

- on the parameter 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
- wake : How to find angles of external rays that land on the root point of the wake ( = angles of the wake)?
- principle Misiurewicz points of the wake k/r , tuning: How to find angles of external rays landing on the principle Misurewicz point of the wake ?
- branches and tips
- subwake, tuning and internal address
- roots, islands and Douady tuning - How to find the angles of external rays that land on the root point of any Mandelbrot set's component which is not accesible from main cardioid ( M0) by a finite number of boundary crossing ?
- Period doubling cascade and the Myrberg-Feigenbaum point in the 1/2 family. Escape route 1/2

- Zoom

### Iterations of real numbers : 1D[edit | edit source]

- logistic map
- real quadratic map
- tent map

### Iterations of complex numbers :2D[edit | edit source]

- complex-analytic formulas (like Mandelbrot set and Julia set)
- non-complex-analytic formulas (like Mandelbar and Burning Ship)

#### Rational maps[edit | edit source]

##### Polynomials[edit | edit source]

###### Chebyshev polynomials[edit | edit source]

###### Complex quadratic polynomials[edit | edit source]

**Dynamical plane Julia and Fatou set**

**Julia set**- with an non-empty interior ( connected )
- Hyperbolic Julia sets
- attracting : filled Julia set have attracting cycle ( c is inside hyperbolic component )
- superattracting : filled Julia set have superattracting cycle( c is in the center of hyperbolic component ). Examples : Airplane Julia set, Douady's Rabbit, Basillica.

- Parabolic Julia set
- Elliptic Julia set: Siegel disc - a linearizable irrationaly indifferent fixed point

- Hyperbolic Julia sets
- with empty interior
- disconnected ( c is outside of Mandelbrot set )
- connected ( c is inside Mandelbrot set )
- Cremer Julia sets -a non-linearizable irrationaly indifferent fixed point
- dendrits ( Julia set is connected and locally connected ). Examples :
- Misiurewicz Julia sets (c is a Misiurewicz point )
- Feigenbaum Julia sets ( c is Generalized Feigenbaum point: the limit of the period-q cascade of bifurcations and landing points of parameter ray or rays with irrational angles )
- others which have no description

- with an non-empty interior ( 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 set**

- Topological model of Mandelbrot set : Lavaurs algorithm and lamination of parameter plane
- structure of Mandelbrot set and ordering of hyperbolic components
- family: real slice of Mandelbrot set.
- periodic part: period doubling cascade. Escape route 1/2
- the Myrberg-Feigenbaum point of family
- chaotic part main antenna is a shrub of family

- family: real slice of Mandelbrot set.
- Transformations of parameter plane
- Sequences and orders on the parameter plane
- Parts of parameter plane
- exterior of the Mandelbrot set: escape time, Level Set Method ( LSM/M), Binary Decomposition Method (BDM/M)
- Islands
- Components
- Number of the Mandelbrot set's components
- Boundary of whole set and it's components
- parabolic points: root points and cusps
- unroll a closed curve and then stretch out into an infinite strip
- Misiurewicz points

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

- speed improvements