Geodesic
Periodic Fractal Patterns
Journal for geometry and graphics, Tome 21 (2017) no. 1, pp. 1-6
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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.
Classification : 28A80, 51F99
Mots-clés : Fractals, wallpaper groups, algorithm 
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D. Dunham; J. Shier. Periodic Fractal Patterns. Journal for geometry and graphics, Tome 21 (2017) no. 1, pp. 1-6. http://geodesic.mathdoc.fr/item/JGG_2017_21_1_JGG_2017_21_1_a0/