Periodic Fractal Patterns

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.