Doug Dunham
My goal is to create aesthetically pleasing repeating patterns related to hyperbolic geometry. The first two pictures below show patterns on triply periodic polyhedra in Euclidean 3-space; the third is on the hyperbolic plane, and is the "universal covering pattern" for the first two. The patterns, being inspired by those of M.C. Escher, have no gaps or overlaps. The polyhedra are related to the hyperbolic plane in two steps. First, the polyhedra are approximations to triply periodic minimal surfaces (the vertices of the polyhedra all lie on the corresponding minimal surface). Second, since the minimal surfaces have negative curvature, the hyperbolic plane has the same geometry as their universal covering surface.
This hyperbolic pattern of butterflies, insprired by Escher's Regular Division Drawing 70, has eight butterflies meeting at left front wing tips and three meeting at right rear wings. The symmetry group of the uncolored pattern is generated by 8-fold and 3-fold rotations about the respective meeting points of the wings, and is 832 in orbifold notation. This pattern also exhibits perfect 3-color symmetry with color group S3. As in Escher's pattern, the wings of butterflies of two colors meeting at left front wing tips are decorated with small circles of the third color. This pattern is the "universal covering pattern" of the patterns on both {3,8} polyhedra above. The covering map maps bufferflies to butterflies with the same color.
This is a pattern of butterflies (inspired by M.C. Escher's Regular Division Drawing 70) on a triply periodic polyhedron composed of equilateral triangles meeting 8 at a vertex, thus denoted by the Schläfli symbol {3,8}. It is formed from octahedral hubs connected by octahedral struts. This polyhedron approximates Schwarz' D-Surface, the boundary between two congruent, complementary solids, both in the shape of a "thickened" diamond lattice (the hubs are the carbon atoms and the struts are the atomic bonds). This polyhedron has genus 3. As in Escher's drawing, fish of two colors meet at each vertex, with wing spots of the third color. In fact the polyhedron exhibits perfect 3-color symmetry, the color group being the symmetric group S3.