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 (or a relatively small number of orbits) 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.
Sierpinski Triangle
The classic example of an Orbital fractal is the Sierpinski Triangle.
A Fractal Science Kit fractal program to generate a Sierpinski Triangle fractal follows:
global:
const Complex point[] = 1i, -0.866-0.5i, 0.866-0.5i
iterate:
attractorIndex = Random.Integer(3)
z = (z + point[attractorIndex])/2
The array point[] 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, Rep-N Tiles, Circle Inversion fractals, Kleinian Group fractals, Symmetric Icons, Symmetric Attractors, Frieze Group Attractors, Wallpaper Group Attractors, and Hyperbolic Attractors, are all examples of Orbital fractals.
Sierpinski N-gons
Sierpinski N-gons are a generalization of the Sierpinski Triangle 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
IFS
(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.
Strange Attractors
Strange 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.
Rep-N Tiles
Rep-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.
Circle Inversion Fractals
Circle 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.
Kleinian Group Fractals
Kleinian 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.
Symmetric Icons
Symmetric 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.
Symmetric Attractors
Symmetric 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.
Frieze Group and Wallpaper Group Attractors
Frieze 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). See References for details. Programs to display example Frieze Group patterns and Wallpaper Group patterns, along with programs to search for interesting sets of parameters are included.
Hyperbolic Attractors
![Hyperbolic Attractor [4+,5]](../gallery/orbital/f1576.jpg)
Hyperbolic
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.
Built-in Orbital Equations
The
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, 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.