Automatic Generation of Iterated Function Systems

J. C. Sprott
Department of Physics, University of Wisconsin, Madison, WI 53706, USA


A set of affine mappings with randomly chosen coefficients is repeatedly iterated numerically using the random iteration algorithm to produce an attractor with fractal characteristics. The attractor is tested for boundedness, sensitivity to initial conditions, and correlation dimension. In this way, a computer can generate a large collection of fractal patterns that are all different and most of which have considerable aesthetic appeal. A simple computer program and examples of its output are provided. Many of the attractors have been systematically evaluated for visual appeal, and a correlation is found with the Lyapunov exponent and correlation dimension.

Ref: J. C. Sprott, Comput. & Graphics 18, 417-425 (1994)

The complete paper is available in PDF format.

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Fig. 1. Examples of iterated function systems produced by pairs of two-dimensional affine maps.
[Figure 1]

Fig. 2. Results of evaluating 7500 iterated function systems, showing that the most visually appealing cases are those with large negative Lyapunov exponents (L) and with correlation dimensions (F) greater than one.
[Figure 2]

Fig. 3. Examples of iterated function systems produced by pairs of three-dimensional affine maps in which the color is determined by one of the variables.
[Figure 3a]

[Figure 3b]

The computer source code ifs.bas from the article is available along with an executable version ifs.exe.