Doug Dunham and John Shier
The goal of our art is to create aesthetically pleasing fractal patterns. We choose a motif and iteratively place smaller and smaller copies of it at random within a region. Each new motif copy must not overlap previously placed motifs or the region's boundary. If the sizes are chosen appropriately, the motifs will completely fill the region in the limit. For this to happen, the sizes 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 at least 1, and 1 < c < c_max, where c_max depends on the shapes of the motif and region and is typically less than 1.5. The fractal dimension of the motif pattern is 2/c.
This is a fractal pattern whose motifs are monarch butterflies. We modify our usual rule that motifs cannot overlap by allowing the antennas - but not the rest of the motif - to overlap another motif. Expanding on the area rule of the Goals statement, the area of the n-th motif is given by A/(zeta(c,N)(N+n)^c), where A is the area of the region, and zeta(c,N) is the Hurwitz zeta function, a generalization of the Riemann zeta function (for which N = 1; our algorithm starts with n = 0). For this pattern c = 1.26, N = 1.5, and 150 butterflies fill 72% of the bounding rectangle.
As an extension of our algorithm, we allow it to fill the fundamental region of one of the 17 wallpaper groups, p6 in this example. Then we apply transformations from that wallpaper group to the fractal pattern within the fundamental region, which can theoretically cover the plane. The algorithm is slightly modified to account for copies of the motif which fall on symmetry elements (rotation centers, reflection lines) of the wallpaper pattern. This technique produces patterns which are aesthetic combinations of random placement of the motifs, flowers here, and global wallpaper symmetry. Flowers are the same color if they are equivalent under the wallpaper group. Viewers are challenged to spot the centers of rotational symmetry.