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Journal for Geometry and Graphics 21 (2017), No. 1, 001--006
Copyright Heldermann Verlag 2017

Periodic Fractal Patterns

Douglas Dunham
Dept. of Computer Science, University of Minnesota, 1114 Kirby Drive, Duluth, MN 55812-3036, U.S.A.

John Shier
6935 133rd Court, Apple Valley, MN 55124, U.S.A.

We present an algorithm that can create patterns that are locally fractal in nature, but repeat in two independent directions in the Euclidean plane - in other word "wallpaper patterns". The goal of the algorithm is to randomly place progressively smaller copies of a basic sub-pattern or motif within a fundamental region for one of the 17 wallpaper groups. This is done in such a way as to completely fill the region in the limit of infinitely many motifs. This produces a fractal pattern of motifs within that region. Then the fundamental region is replicated by the defining relations of the wallpaper group to produce a repeating pattern. The result is a pattern that is locally fractal, but repeats globally a mixture of both randomness and regularity. We show several such patterns.

Keywords: Fractals, wallpaper groups, algorithm.

MSC: 28A80; 51F99

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