Orbital Fractals 

Orbital Fractal OverviewThe 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 TriangleThe classic example of an Orbital fractal is the Sierpinski Triangle (Wikipedia). A Fractal Science Kit fractal program to generate a Sierpinski Triangle fractal follows: global: 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 Ngons, IFS fractals, Strange Attractors, Dragon Flames, RepN 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 NgonsSierpinski Ngons are a generalization of the Sierpinski Triangle (Wikipedia) attractor based on a polygon with N vertices. A Sierpinski Ngon 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 FractalsIFS (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 AttractorsStrange 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. RepN TilesRepN Tiles (replicating figures on the plane) are attractors based on a set of N affine transformations that generate a RepN 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 RepTile was coined by Solomon W. Golomb in 1962. See RepTile (MathWorld) for additional details. Apollonian Gasket FractalsApollonian 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 FractalsCircle 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 FractalsKleinian Group fractals are fractals based on 2 pairs of Mobius transformations and allow you to produce Quasifuchsian, Single Cusp, and Double Cusp, TwoGenerator 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 IconsSymmetric 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 AttractorsSymmetric 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 FractalsMobius 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 MobiusDragonScript and MobiusDesignPack. 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 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 FractalsGrand Julian IFS fractals are generated by an IFS using the the Orbital Equation Grand Julian IFS. Elliptic Splits FractalsElliptic 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 FractalsSplits 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 SplitsNgon 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 AttractorsFrieze Group and Wallpaper Group attractors are strange attractors that produce symmetric designs based on the 7 onedimensional frieze group patterns and the 17 twodimensional 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 AttractorsHyperbolic 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 FractalsHyperbolic 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. Builtin Orbital EquationsThe Fractal Science Kit fractal generator has over 50 builtin Orbital Equations including Sierpinski Triangle, Sierpinski Ngons, Koch Snowflake, Apollonian Gasket, Circle Inversion fractals, Kleinian Group fractals, Dragons, Dimers, RepN 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 Builtin Orbital Equations for a complete list. 
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