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, transformations, curves, ...
5. Fractal software
6. Fractal links

Fractals made by the iterations

Theory

1. Definitions
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
5. How to analyze map ?
6. How to construct map with desired properities ?

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

1. escape and attracting time for (level sets method (LSM), level curves method (LCM)
   1. the Julia sets
      1. Target sets, trap and bailout tests
         1. Decomposition of the target set: Binary Decomposition Method ( BDM) which in parabolic case gives: zeros of Qn or parabolic checkerboard ( chessboard)
         2. Esher like tilings
         3. orbit trap
   2. the Mandelbrot set
2. Inverse iteration method ( IIM) for drawing:
   1. Julia set = IIM/J
3. atom domains
4. True shape
5. Discrete Langrangian Descriptors
6. curves
   1. boundary - scanning
   2. unroll a closed curve and then stretch out into an infinite strip
   3. equipotential curve
   4. external ray ( parameter and dynamic) trace
   5. internal ray
   6. path: escape route
7. DEM = Distance Estimation Method
   1. DEM/M- for Mandelbrot set
   2. DEM/J for Julia set
8. Maping component to the unit disk ( Riemann map ):
   1. Multiplier map and internal ray
      1. on the parameter plane
      2. on the dynamic plane
   2. Boettcher map, complex potential and external ray
      1. on the parameter plane
         1. parameter ray = field lines
         2. complex potential , external angle
      2. on the dynamic plane
9. histogram colorings
10. 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
    1. Triangle Inequality Average Coloring = TIA and curvature average algorithm ( CAA)
    2. Stripe Average Coloring = SAC
    3. Discrete Velocity of non-attracting Basins and Petals by Chris King
    4. Average distance
11. 2D to 3D : bump maping
    1. heightmap
    2. slope
    3. Embossing and Lighting
    4. lighting
12. Parameter plane: combinatorial algorithms
    1. wake : How to find angles of external rays that land on the root point of the wake ( = angles of the wake)?
    2. 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 ?
    3. branches and tips
    4. subwake, tuning and internal address
    5. 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 ?
    6. Period doubling cascade and the Myrberg-Feigenbaum point in the 1/2 family. Escape route 1/2
13. Zoom
    1. on the parameter plane
       1. Perturbation method
       2. Julia morphing - to sculpt shapes of Mandelbrot set parts ( zoom ) and Show Inflection

Iterations of real numbers : 1D

• 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

Dynamical plane Julia and Fatou set

1. 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 ( 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
2. Fatou set
   1. exterior of all Julia sets = basin of attraction of superattracting fixed point (infinity)
      1. Escape time
      2. Boettcher coordinate
      3. Orbit portraits and lamination of dynamical plane
   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. 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. 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. Islands
      1. the biggest island of the wake
      2. distortion of mini Mandelbrot sets
      3. islands ( root point, Douady tuning)
   3. 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
6. speed improvements