Orbital Fractals

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Orbital Fractal Overview

The Fractal Science Kit fractal generator Orbital fractals collect statistics during the orbit of a fractal formula and use these to create the fractal image. In contrast to Mandelbrot Fractals where we generate an orbit for each pixel in the viewing window to produce a picture, Orbital fractals generate a single orbit and we keep track of all the points we visit during the orbit along with how many times we visit each point, what part of the fractal formula caused us to visit the point (the attractor index), the speed/acceleration of the orbit at that point, etc. These data values are used to color the fractal. This process is sometimes called the Chaos Game (Wikipedia).


Sierpinski Triangle

The classic example of an Orbital fractal is the Sierpinski Triangle (Wikipedia).

Sierpinski Triangle

A Fractal Science Kit fractal program to generate a Sierpinski Triangle fractal follows:

global:
 
  const Complex v[] = 1i, -0.866-0.5i, 0.866-0.5i
   
iterate:
 
  attractorIndex = Random.Integer(3)
  z = (z + v[attractorIndex])/2

The array v[] contains the vertices of an equilateral triangle on the complex plane. On each iteration, we select a vertex of the equilateral triangle at random and move the orbit point to the midpoint of the segment that connects the current point to the selected vertex. The Sierpinski Triangle is the result!

Sierpinski N-gons, IFS fractals, Strange Attractors, Dragon Flames, Rep-N Tiles, Apollonian Gasket fractals, Circle Inversion fractals, Kleinian Group fractals, Symmetric Icons, Symmetric Attractors, Mobius Dragon IFS, Mobius Patterns, Grand Julian IFS, Elliptic Splits IFS, Splits Ngon fractals, Frieze Group Attractors, Wallpaper Group Attractors, and Hyperbolic Attractors, are all examples of Orbital fractals.


Sierpinski N-gons

Fractal: Sierpinski N-gonSierpinski N-gons are a generalization of the Sierpinski Triangle (Wikipedia) attractor based on a polygon with N vertices. A Sierpinski N-gon is defined as a set of transformations about the vertices of a regular polygon. During the fractal iteration, one of the vertices is selected at random, and the current orbit point is passed through the associated transformations to obtain the next orbit point. Variations of this algorithm include points in addition to the polygon vertices and allow control of the individual transformations about each point.


IFS Fractals

Fractal: Dragon IFSIFS (Iterated Function System) fractals are defined as a set of affine transformations (usually), each assigned a probability value. During the fractal iteration, one of the transformations is selected at random based on the assigned probabilities, and the current orbit point is passed through the selected transformation to obtain the next orbit point. In general, in order to produce a fractal, the transformations should be contractions; i.e., when applied to any 2 points, the transformation should reduce the Euclidean distance between the points. Programs to display example IFS fractals and to search for different combinations of parameters that produce interesting results are provided. See Iterated function system (Wikipedia) for additional details.


Strange Attractors

Fracatal: Strange AttractorStrange Attractors are defined by an equation or system of equations. The orbit points are generated by passing the current orbit point through the equations to obtain the next orbit point. This process is repeated thousands (or millions) of times to produce the fractal data. Of course, most equations will not produce a fractal and the challenge is to find equations that do. Quadratic Attractors and Cubic Attractors are examples of these fractal types. Programs to display examples of these fractals and to search for different combinations of parameters that produce interesting results are provided. See Attractor (Wikipedia) for additional details.


Rep-N Tiles

Fractal: Rep-9 TileRep-N Tiles (replicating figures on the plane) are attractors based on a set of N affine transformations that generate a Rep-N Tile. Omitting 1 or more of the transformations and passing the orbit points through a symmetry transformation results in highly complex, and beautiful, symmetric designs. The term Rep-Tile was coined by Solomon W. Golomb in 1962. See Rep-Tile (MathWorld) for additional details.


Apollonian Gasket Fractals

Fractal: Apollonian Gasket AttractorApollonian Gasket fractals are fractals based on 2 pairs of Mobius transformations. The Apollonian Gasket, and the methods used to produce them, are described in the excellent book Indra's Pearls - The Vision of Felix Klein by David Mumford, Caroline Series, and David Wright. These fractals are usually displayed using a deterministic algorithm (see the Apollonian Gasket Orbit Trap) which produces the best results; however, the Orbital fractal images are still quite impressive. For additional details, see David Wright's Indra's Pearls site and Apollonian Gasket (Wikipedia).


Circle Inversion Fractals

Fractal: Circle InversionCircle Inversion fractals are defined by a set of mutually tangent circles. During the fractal iteration, one of the circles is selected at random, and the current orbit point is reflected in the circle to obtain the next orbit point. As usual, this process is repeated thousands of times to produce the fractal data. The process of reflecting a point in a circle is called inversion. See Circle Inversion Fractals (Yale) for additional details.


Kleinian Group Fractals

Fractal: Kleinian Group AttractorKleinian Group fractals are fractals based on 2 pairs of Mobius transformations and allow you to produce Quasifuchsian, Single Cusp, and Double Cusp, Two-Generator Group fractals described in the book Indra's Pearls - The Vision of Felix Klein by David Mumford, Caroline Series, and David Wright. These fractals are usually displayed using a deterministic algorithm (see the Kleinian Group Orbit Trap) which produces the best results; however, the Orbital fractal images are still quite impressive. For additional details, see David Wright's Indra's Pearls site.


Symmetric Icons

Fractal: Symmetric IconSymmetric Icons are strange attractors that produce symmetric designs based on the equations given in the book Symmetry in Chaos by Michael Field and Martin Golubitsky. Programs to display example Symmetric Icons along with programs to search for interesting sets of parameters are included. See Images of Chaos and Symmetry for additional details.


Symmetric Attractors

Fractal: Symmetric AttractorSymmetric Attractors are strange attractors that produce symmetric designs based on equations given in the paper Chaotic attractors with cyclic symmetry revisited, by Kevin C. Jones and Clifford A. Reiter (Lafayette College). See References for details. Programs to display example Symmetric Attractors, along with programs to search for interesting sets of parameters are included. See Clifford A. Reiter's Gallery of Fractals, Chaos and Symmetry for additional details.


Mobius Dragon IFS Fractals

Fractal: Mobius Dragon IFS

Mobius Dragon IFS fractals are generated by an IFS formed from a set of Mobius Transformations using the Orbital Equation Mobius Dragon IFS. The Mobius Dragon IFS equation is based on information given by penny5775 on the pages Mobius-Dragon-Script and Mobius-Design-Pack. Note that the information found on these pages relates to the Apophysis fractal generator not the Fractal Science Kit but is included here for reference.


Mobius Patterns

Fractal: Mobius PatternsMobius Patterns are generated by an IFS formed from a set of Mobius Transformations using the Orbital Equation Mobius Patterns. The Mobius Patterns equation is based on information found in the excellent book Indra's Pearls - The Vision of Felix Klein by David Mumford, Caroline Series, and David Wright. For additional details, see David Wright's Indra's Pearls site.


Grand Julian IFS Fractals

Fractal: Grand Julian IFSGrand Julian IFS fractals are generated by an IFS using the the Orbital Equation Grand Julian IFS. The Grand Julian IFS equation is based on information given by Claire Jones on the page Apophysis Resource Center: Tutorials. Note that the information found on this page relates to the Apophysis fractal generator not the Fractal Science Kit but is included here for reference.


Elliptic Splits Fractals

Fractal: Elliptic Splits IFS

Elliptic Splits IFS fractals are based on the Orbital Equation Elliptic Splits IFS. The Elliptic Splits IFS equation is based on information given by SaTaNiA on the page Tutorial - Splits elliptic. Note that the information found on this page relates to the Apophysis fractal generator not the Fractal Science Kit but is included here for reference.

 


Splits Ngon Fractals

Fractal: Splits NgonSplits Ngon fractals are based on the Orbital Equation Flame (4 Transforms). The Splits Ngon fractals are based on information given by guagapunyaimel on the page Splits-Ngon Tutorial. Note that the information found on this page relates to the Apophysis fractal generator not the Fractal Science Kit but is included here for reference.


Frieze Group and Wallpaper Group Attractors

Fractal: Plane Symmetry GroupFrieze Group and Wallpaper Group attractors are strange attractors that produce symmetric designs based on the 7 one-dimensional frieze group patterns and the 17 two-dimensional wallpaper group patterns. A discussion of algorithms to produce these attractors is given in the paper Chaotic Attractors with Discrete Planar Symmetries, by Nathan C. Carter, Richard L. Eagles, Stephen M. Grimes, Andrew C. Hahn, and Clifford A. Reiter (Lafayette College). Programs to display example Frieze Group patterns and Wallpaper Group patterns, along with programs to search for interesting sets of parameters are included. See Clifford A. Reiter's Gallery of Fractals, Chaos and Symmetry for additional details.

These patterns can also be formed from any attractor using the Symmetry Transformations Plane Symmetry Groups - Square Lattice or Plane Symmetry Groups - Hexagonal Lattice.


Hyperbolic Attractors

Fractal: Hyperbolic AttractorHyperbolic Attractors are strange attractors that produce designs with hyperbolic symmetry. A discussion of algorithms to produce these attractors is given in the paper Iterated function systems with symmetry in the hyperbolic plane, by Bruce M. Adcock, Kevin C. Jones, Clifford A. Reiter (Lafayette College), and Lisa M. Vislocky. See References for details. See Clifford A. Reiter's Gallery of Fractals, Chaos and Symmetry for additional details.

These patterns can also be formed from any attractor using the Symmetry Transformation Plane Hyperbolic Tiling - Orbital.


Hyperbolic Mobius Fractals

Fractal: Hyperbolic MobiusHyperbolic Mobius fractals are generated by an IFS formed from a set of Mobius Transformations using the Orbital Equation Mobius Patterns and then passing the results through the Symmetry Transformation Plane Hyperbolic Tiling - Orbital. The Mobius Patterns equation is based on information found in the excellent book Indra's Pearls - The Vision of Felix Klein by David Mumford, Caroline Series, and David Wright. For additional details, see David Wright's Indra's Pearls site.


Built-in Orbital Equations

Fractal: Mobius Dragon IFSThe Fractal Science Kit fractal generator has over 50 built-in Orbital Equations including Sierpinski Triangle, Sierpinski N-gons, Koch Snowflake, Apollonian Gasket, Circle Inversion fractals, Kleinian Group fractals, Dragons, Dimers, Rep-N Tiles, IFS fractals, Quadratic Attractors, Cubic Attractors, Symmetric Icons, Symmetric Attractors, Mobius Dragon IFS, Mobius Patterns, Grand Julian IFS, Elliptic Splits IFS, Frieze Group, Wallpaper Group, Hyperbolic Attractors, and many more. Many of these programs define properties that can be used to produce countless different variations. Some of the programs search for interesting parameter settings based on user defined criteria and produce unique fractals every time they are run! See Built-in Orbital Equations for a complete list.

 

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