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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.
ddunham@d.umn.edu

John Shier
6935 133rd Court, Apple Valley, MN 55124, U.S.A.
johnpf99@frontiernet.net



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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