# 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

## Introduction edit

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

## Fractals made by the iterations edit

**Theory**

- Definitions
- Iterations : forward and backward ( inverse ) and critical orbit
- critical orbit
- Periodic points or cycle
- How to analyze map ? How to read location from the image?
- How to construct map with desired properities ?
- Algorithms ( graphical (coloring, transformations), numerical, symbolic, other)

### Iterations of real numbers : 1D edit

- (angle) doubling map
- logistic map
- real quadratic map
- tent map

### Iterations of complex numbers :2D edit

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

#### Rational maps edit

##### Polynomials edit

###### Chebyshev polynomials edit

###### Complex quadratic polynomials edit

**Dynamic plane: Julia and Fatou sets**

- coloring the dynamic plane and the Julia and the Fatou sets
**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 or Dendrite Julia sets ( 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)
- Interior and the boundary : 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

- Islands

- Points ( parameter of the iterated function)

- speed improvements
- coloring algorithm