Iterated function system
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Menger_sponge_(IFS).jpg
Iterated function systems or IFS, are a kind of fractal that was conceived in its present form by John Hutchinson in 1981 and popularized by Michael Barnsley's book Fractals Everywhere.
IFS fractals as they are normally called can be of any number of dimensions, but are commonly computed and drawn in 2D. An IFS fractal is a solution to a recursive set equation. The fractal is made up of the union of several copies of itself, each copy being transformed by a function (hence "function system"). The canonical example is Sierpinski gasket. The functions are normally "contractive" which means they bring points closer together and make shapes smaller. Hence the shape of an IFS fractal is made up of several possibly-overlapping smaller copies of itself, each of which is also made up of copies of itself, ad infinitum. This is the source of its self-similar fractal nature.
Formally, <math>S = \cup_i f_i(S)<math> where <math>S \in \mathbb{R}^2<math> and <math>f_i:\mathbb{R}^2\to\mathbb{R}^2.<math>
The most common algorithm to compute IFS fractals is called the chaos game. It consists of picking a random point in the plane, then iteratively applying one of the functions chosen at random from the function system and drawing the point.
Fractal flames are a generalization and refinement of IFS fractals.
Barnsley tried to use IFS to encode images and received a patent for his efforts. But his claims were exaggerated and the company failed.
Linear_and_nonlinear_IFS.png
Linear and nonlinear IFS
Random game IFS from a set using five linear and then two nonlinear, (reversed Julia set, C = [0, 0]), transformations in sequence.
External links
- The Collage Theorem (http://www.cut-the-knot.org/ctk/ifs.shtml)
- Interactive 3D fractal polyhedra in Java (http://ibiblio.org/e-notes/3Dapp/Sponge.htm)
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