Creating Self-Similar Fractals with Hutchinson Operators

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A map is a contraction mapping if for all points , , , where . A similitude is a contraction mapping that is a composition of dilations, rotations, translations, and reflections. A two-dimensional Hutchinson operator maps a plane figure to the union of its images under a finite collection of similitudes. The orbit of a plane figure under such an operator can form a self-similar fractal. In this Demonstration you can vary three similitudes (without reflection) to see what self-similar fractals are possible.

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As long as the initial subset of the plane is compact, iterations of the Hutchinson operator converge to the same fractal, yet the convergence is faster for some subsets than others; in particular, the set of the three fixed points of the three similitudes gives fast convergence.

To better see what a Hutchinson operator does, use "constant points" to start with the same three points rather than .

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Contributed by: Garrett Nelson (July 2014)
Suggested by: Branko Curgus
Open content licensed under CC BY-NC-SA


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Reference

[1] J. E. Hutchinson, "Fractals and Self-Similarity," Indiana University Mathematics Journal, 30(5), 1981 pp. 713–747. doi:10.1512/iumj.1981.30.30055.



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