Doug Dunham
The goal of this work is to create aesthetically pleasing patterns that can fill a region "in the limit" with a sequence of ever smaller non-overlapping motifs which are placed by random search. The robust nature of this algorithm allows broad scope for the artist, as is demonstrated here using stars and the Texas flag. For the algorithm to work, the sizes of the motifs must obey an inverse power law: the area of the n-th motif is proportional to 1/(N+n)^c, where N and c are parameters with N > N_min > 0 and 1 < c < c_max, and c_max depends on the shapes of the motif and region and is typically less than 1.5 for 2D patterns. The "gasket" region between motifs is a single connected set if the motif is not a hollow one.
The inspiration for this work is the Texas state flag. The challenge here was to give the impression of the flag by filling the different regions with appropriately colored stars as motifs. Because of their "pointy" shape, many random positions must be tried in order to fill a region with stars. There are three kinds of regions to be filled: the horizontal red and white stripes, the large white star, and the vertical blue rectangle with a star hole in it. The red and white horizontal stripe regions are filled with the same pattern of stars, so that a translation of the white stripe down by a stripe-width, or translating the red stripe up by a stripe-width interchanges red and white stars, and is thus a kind of color symmetry.