Fractals


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Editor's note
This book is still under development. Please help us

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


IntroductionEdit

  1.   Introduction
  2.   Introductory Examples

Mathematics for computer graphicEdit

Programming computer graphicEdit

Fractals made by the iterationsEdit

  1. How to analyze map ?
  2. How to construct map with desired properities ?


Iterations of real numbers : 1DEdit

  • logistic map
  • real quadratic map
  • tent map

Iterations of complex numbers :2DEdit

Rational mapsEdit

  1. Analysis
  2. Herman rings
PolynomialsEdit
Chebyshev polynomialsEdit
Complex quadratic polynomialsEdit
TheoryEdit
  1.  Definitions
  2. Iterations : forward and backward ( inverse ) and critical orbit
    1. Fractional iterations
  3. Periodic points or cycle
    1. Period
AlgorithmsEdit

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 and bailout tests
        1. Decomposition of the target set: Binary Decomposition Method ( BDM) / 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 trace
    2. equipotential curve
  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
        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
    2. principle Misiurewicz points of the wake k/r , tuning
    3. branches and tips
    4. subwake, tuning and internal address
    5. roots, islands and Douady tuning
    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
 Dynamical plane Julia and Fatou setEdit
  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 ( )
          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. 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. Repelling and attracting directions
          3. Rays landing on the parabolic fixed point
          4. parabolic checkerboard
        2.   Local dynamics near irrationally indifferent fixed point/cycle ( elliptic ) - Siegel disc
 Parameter plane and Mandelbrot setEdit
  1. Topological model of Mandelbrot set : Lavaurs algorithm and lamination of parameter plane
  2. structure of Mandelbrot set and ordering of hyperbolic components
    1.   family: real slice of Mandelbrot set.
      1. periodic part: period doubling cascade. Escape route 1/2
      2. the Myrberg-Feigenbaum point
      3. chaotic part main antenna is a shrub of   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
      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. islands ( root point, Douady tuning)
        5. branch tips of the shrub ( Misiurewicz points)
    2. Boundary of whole set and it's components
      1. distortion of mini Mandelbrot sets
      2. parabolic points: root points and cusps
      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

The BuddhabrotEdit

exponential familiesEdit

trigonometric familiesEdit

The Newton-Raphson fractalEdit

Quaternion Fractals : 3DEdit

Other fractalsEdit

Fractal softwareEdit

Fractal linksEdit

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