# 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

1.   Introduction
2.   Introductory Examples
3. Mathematics for computer graphic: numbers, sequences, functions, numerical methods, fields, ...
4. Programming computer graphic: files, plane, curves, ...
1. plane transformations
5. Fractal software

## Fractals made by the iterations

Theory

### Iterations of real numbers : 1D

• (angle) doubling map
• logistic map
• tent map

### Iterations of complex numbers :2D

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

#### Rational maps

##### Polynomials
###### Chebyshev polynomials

Dynamic plane: Julia and Fatou sets

1. coloring the dynamic plane and the Julia and the Fatou sets
2. Julia set
1. with an non-empty interior ( connected )
1. Hyperbolic Julia sets
1. attracting : filled Julia set have attracting cycle ( c is inside hyperbolic component )
2. superattracting : filled Julia set have superattracting cycle( c is in the center of hyperbolic component ). Examples : Airplane Julia set, Douady's Rabbit, Basillica.
2. Parabolic Julia set
3. Elliptic Julia set: Siegel disc - a linearizable irrationaly indifferent fixed point
2. with empty interior
1. disconnected ( c is outside of Mandelbrot set )
2. connected ( c is inside Mandelbrot set )
1. Cremer Julia sets -a non-linearizable irrationaly indifferent fixed point
2. dendrits or Dendrite Julia sets ( Julia set is connected and locally connected ). Examples :
1. Misiurewicz Julia sets (c is a Misiurewicz point )
2. 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 )
3. others which have no description
3. Fatou set
1. exterior of all Julia sets = basin of attraction of superattracting fixed point (infinity)
2. Interior of Julia sets:
1.   Basin of attraction of superattracting periodic/fixed point - Boettchers coordinate , c is a center of period n component of Mandelbrot set
1. Circle Julia set ( c = 0 is a center of period 1 component)
2. Basilica Julia set ( c = -1 is a center of period 2 component)
2.   Basin of attraction of attracting periodic/fixed point - Koenigs coordinate
3.   Local dynamics near indifferent fixed point/cycle
1. Topological model of Mandelbrot set : Lavaurs algorithm and lamination of parameter plane
2. structure of Mandelbrot set and ordering of hyperbolic components
1. ${\displaystyle F_{1/2}}$  family: real slice of Mandelbrot set.
1. periodic part: period doubling cascade. Escape route 1/2
2. the Myrberg-Feigenbaum point of ${\displaystyle F_{1/2}}$  family
3. chaotic part main antenna is a shrub of ${\displaystyle F_{1/2}}$  family
3. Transformations of parameter plane
4. Sequences and orders on the parameter plane
5. Parts of parameter plane
1. exterior of the Mandelbrot set: escape time, Level Set Method ( LSM/M), Binary Decomposition Method (BDM/M)
1. External (Parameter) Rays of:
2. Interior and the boundary : components
1. Number of the Mandelbrot set's components
2. Boundary of whole set and it's components
1. parabolic points: root points and cusps
2. unroll a closed curve and then stretch out into an infinite strip
3. Misiurewicz points
1. Devaney algorithm for principle Misiurewicz point
3. interior of hyperbolic components
4. Islands
3. Points ( parameter of the iterated function)
6. speed improvements
7. coloring algorithm

## Other fractals

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