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. Template:Stage short /Introduction/
2. Template:Stage short /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
6. Fractal links

Fractals made by the iterations

Theory

1. Template:Stage shortDefinitions
2. Iterations : forward and backward ( inverse ) and critical orbit
   1. Fractional iterations
3. critical orbit
4. Periodic points or cycle
   1. periodic points of complex quadratic map
   2. Period
5. How to analyze map ? How to read location from the image?
6. How to construct map with desired properities ?
7. 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

Polynomials

Complex quadratic 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)
      1. Escape time
      2. Template:Stage shortBoettcher coordinate
      3. Template:Stage shortOrbit portraits and lamination of dynamical plane
      4. Dynamic external rays
   2. Interior of Julia sets:
      1. Template:Stage short 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. Template:Stage short Basin of attraction of attracting periodic/fixed point - Koenigs coordinate
      3. Template:Stage short Local dynamics near indifferent fixed point/cycle
         1. Template:Stage short Local dynamics near rationally indifferent fixed point/cycle ( parabolic ). Leau-Fatou flower theorem
            1. petal of the Leau-Fatou flower
            2. Repelling and attracting directions
            3. Rays landing on the parabolic fixed point
            4. parabolic checkerboard
            5. parabolic perturbation
            6. Fatou_coordinate
               1. Fatou_coordinate for f(z)=z/(1+z)
               2. Fatou_coordinate for f(z)=z+z^2
               3. Fatou_coordinate for f(z)=z^2 + c
         2. Template:Stage short 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
   1. $F_{1/2}$ family: real slice of Mandelbrot set.
      1. periodic part: period doubling cascade. Escape route 1/2
      2. the Myrberg-Feigenbaum point of $F_{1/2}$ family
      3. chaotic part main antenna is a shrub of $F_{1/2}$ 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:
         1. the wake ( root point)
         2. the principle Misiurewicz points for the wake k/r of main cardioid
         3. subwake (root points, tuning and internal address)
         4. branch tips of the shrub ( Misiurewicz points)
         5. islands ( root point, Douady tuning)
   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
         1. centers of hyperbolic components = nuclesu of Mu-atoms
         2. Internal rays
      4. Islands
         1. the biggest island of the wake
         2. distortion of mini Mandelbrot sets
         3. islands ( root point, Douady tuning)
   3. Points ( parameter of the iterated function)
6. speed improvements
7. coloring algorithm