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,  ...
 * 5)  plane transformations
 * 6) Fractal software
 * 7) Fractal links

Fractals made by the iterations
Theory
 * 1) Definitions
 * 2) Iterations : forward and backward ( inverse ) and critical orbit
 * 3) Fractional iterations
 * 4) critical orbit
 * 5) Periodic points or cycle
 * 6) periodic points of complex quadratic map
 * 7) Period
 * 8) How to analyze map ?  How to read location from the image?
 * 9)  How to construct map with desired properities ?
 * 10) Algorithms ( graphical (coloring, transformations), numerical, symbolic, other)

Iterations of real numbers : 1D

 * (angle) doubling map
 * logistic map
 * real quadratic 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

 * 1) Analysis
 * 2) Herman rings

Complex quadratic polynomials
Dynamic plane: Julia and Fatou sets
 * 1) coloring the dynamic plane and the Julia and the Fatou sets
 * 2) Julia set
 * 3) with an non-empty interior ( connected )
 * 4) Hyperbolic Julia sets
 * 5) attracting : filled Julia set have attracting cycle  ( c is inside hyperbolic component )
 * 6) superattracting : filled Julia set have superattracting cycle( c is in the center of hyperbolic component ). Examples : Airplane Julia set, Douady's Rabbit, Basillica.
 * 7) Parabolic Julia set
 * 8) Elliptic Julia set: Siegel disc - a linearizable irrationaly indifferent fixed point
 * 9) with empty interior
 * 10) disconnected ( c is outside of Mandelbrot set )
 * 11) connected ( c is inside Mandelbrot set )
 * 12) Cremer Julia sets -a non-linearizable irrationaly indifferent fixed point
 * 13) dendrits or Dendrite Julia sets ( Julia set is connected and locally connected ). Examples :
 * 14) Misiurewicz Julia sets (c is a Misiurewicz point )
 * 15) 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 )
 * 16) others which have no description
 * 17) Fatou set
 * 18) exterior of all Julia sets = basin of attraction of superattracting fixed point (infinity)
 * 19) Escape time
 * 20) Boettcher coordinate
 * 21) Orbit portraits and lamination of dynamical plane
 * 22) Dynamic external rays
 * 23) Interior of Julia sets:
 * 24)  Basin of attraction of superattracting periodic/fixed point  - Boettchers coordinate, c is a center of period n component of Mandelbrot set
 * 25) Circle Julia set ( c = 0 is a center of period 1 component)
 * 26) Basilica Julia set ( c = -1 is a center of period 2 component)
 * 27)  Basin of attraction of attracting periodic/fixed point  - Koenigs coordinate
 * 28)  Local dynamics near indifferent fixed point/cycle
 * 29)  Local dynamics near rationally indifferent fixed point/cycle  ( parabolic ). Leau-Fatou flower theorem
 * 30) petal of the Leau-Fatou flower
 * 31) Repelling and attracting directions
 * 32) Rays landing on the parabolic fixed point
 * 33) parabolic checkerboard
 * 34) parabolic perturbation
 * 35) Fatou_coordinate
 * 36) Fatou_coordinate for f(z)=z/(1+z)
 * 37) Fatou_coordinate for f(z)=z+z^2
 * 38) Fatou_coordinate for f(z)=z^2 + c
 * 39)  Local dynamics near irrationally indifferent fixed point/cycle   ( elliptic ) - Siegel disc

Parameter plane and Mandelbrot set
 * 1) Topological model of Mandelbrot set : Lavaurs algorithm and lamination of parameter plane
 * 2) structure of Mandelbrot set and ordering of hyperbolic components
 * 3)  $$F_{1/2}$$ family:  real slice of Mandelbrot set.
 * 4) periodic part: period doubling cascade. Escape route 1/2
 * 5) the Myrberg-Feigenbaum point of $F_{1/2}$ family
 * 6) chaotic part main antenna is a shrub of $$F_{1/2}$$ family
 * 7) Transformations of parameter plane
 * 8) Sequences and orders on the parameter plane
 * 9) Parts of parameter plane
 * 10) exterior of the Mandelbrot set: escape time, Level Set Method ( LSM/M), Binary Decomposition Method (BDM/M)
 * 11) External (Parameter) Rays of:
 * 12) the wake ( root point)
 * 13) the principle Misiurewicz points for the wake k/r of main cardioid
 * 14) subwake (root points, tuning and internal address)
 * 15) branch tips of the shrub ( Misiurewicz points)
 * 16) islands ( root point, Douady tuning)
 * 17) Interior and the boundary : components
 * 18) Number of the Mandelbrot set's components
 * 19) Boundary of whole set and it's components
 * 20) parabolic points: root points and cusps
 * 21) unroll a closed curve and then stretch out into an infinite strip
 * 22) Misiurewicz points
 * 23) Devaney algorithm for principle Misiurewicz point
 * 24) interior of hyperbolic components
 * 25) centers of hyperbolic components = nuclesu of Mu-atoms
 * 26) Internal rays
 * 27) Islands
 * 28)  the biggest island of the wake
 * 29) distortion of mini Mandelbrot sets
 * 30) islands ( root point, Douady tuning)
 * 31) Points ( parameter of the iterated function)
 * 32) speed improvements
 * 33) coloring algorithm