# 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 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 fractal pattern of circle motifs has p4 (or 442 in orbifold notation) as its symmetry group. This pattern shows four p4 translation units, each itself consisting of four copies of the fundamental region for p4. This pattern is thus an aesthetic combination of both the randomness of the placement of circles (which obey the area rule) within a fundamental region, and the symmetry of a p4 pattern. The color of each circle is found by looking up its color according to a simple 2D Fourier series evaluated at the circle's center. Compact motifs, such as circles, allow for relatively high values of c, in this case c = 1.44. Here N = 2.5, and 60 circles fill 78% of the fundamental region.

This fractal pattern of circle motifs has p4 (or 442 in orbifold notation) as its symmetry group. This pattern shows four p4 translation units, each itself consisting of four copies of the fundamental region for p4. This pattern is thus an aesthetic combination of both the randomness of the placement of circles (which obey the area rule) within a fundamental region, and the symmetry of a p4 pattern. The color of each circle is found by looking up its color according to a simple 2D Fourier series evaluated at the circle's center. Compact motifs, such as circles, allow for relatively high values of c, in this case c = 1.44. Here N = 2.5, and 60 circles fill 78% of the fundamental region.

This is a fractal pattern whose motifs are copies of the flag of Finland. The dark blue in each flag represents Finland's lakes and the sky; the white part represents snow. We placed flag motifs on a light blue background - a lighter version of the sky. The aspect ratio of the flag of Finland is 18:11, the same as for the enclosing rectangle of this pattern. 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 (in which N = 1; our algorithm starts with n = 0). For this pattern c = 1.25, N = 2, and 390 flags fill 75% of the bounding rectangle.