Doug Dunham
The goal of my art is to create aesthetically pleasing repeating patterns related to hyperbolic geometry. In the past I have drawn patterns on the hyperbolic plane itself. But the pictures below are of patterns on triply periodic polyhedra in Euclidean 3-space, the last two of which are regular. The patterns are inspired by those of M.C. Escher, and like Escher's patterns, they have no gaps or overlaps. They share another characteristic of Escher's patterns - each one has color symmetry. These polyhedra are related to the hyperbolic plane in two steps. First, the polyhedra are approximations to three 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 large scale geometry as their universal covering surface.
This is a pattern of red, green, and blue angular fish on the regular triply periodic polyhedron composed of regular hexagons meeting six at each vertex, which is denoted by the modified Schläfli symbol {6,6|3} (the 3 indicates that there are equilateral triangular holes in it). This polyhedron can also be thought of as an approximation to Schwarz' D-Surface. The white backbones of the fish of any one color lie along (infinite) Euclidean lines that are embedded in the polyhedron and in Schwarz' D Surface . Three of these lines of fish, one of each color, pass through each vertex. The fish swim in one direction along the backbone lines, so the fish of one color enter a vertex from one hexagon and exit the vertex into the "opposite" hexagon. There are 3-fold color symmetries generated by 120-degree rotations about axes of symmetry that alternately go through vertices and centers of the hexagons; the axes are perpendicular to the hexagons.
This is a pattern of angular fish (inspired by M.C. Escher) on the regular triply periodic polyhedron composed of regular hexagons meeting four at each vertex, which is denoted by the modified Schläfli symbol {6,4| 4} (the last 4 indicates that the polyhedron has square holes). This polyhedron can be thought of as an approximation to Schwarz' P-Surface, a triply periodic minimal surface that has the same symmetries as the cubic lattice. This surface contains three families of embedded lines. The backbone lines of the fish lie along those embedded lines. The backbones of the red and yellow fish lie in parallel planes; those of the green and magenta fish lie in other parallel planes orthogonal to those of the red and yellow fish; the backbones of the blue and cyan fish lie in planes orthogonal to the other two sets of planes. This arrangement gives the patterned polyhedron perfect 6-color symmetry. In fact each symmetry of the polyhedron induces a non-trivial color permutation.
This is a pattern of fish (inspired by M.C. Escher's Circle Limit III) on the regular triply periodic polyhedron composed of equilateral triangles meeting 8 at each vertex, which can be denoted by the Schläfli symbol {3,8}. It is formed from octahedral hubs which have octahedral struts connecting the hubs; the struts are on alternate faces of the hubs. This polyhedron approximates Schwarz' D-Surface which is 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). There are fish of four colors. The blue fish all swim around the "waists" of the struts. The yellow, green, and red fish swim along lines that approximate the set of Euclidean lines that are embedded in Schwarz' D-Surface. In the image, the yellow fish swim right to left, the green fish swim from lower left to upper right, and the red fish swim from upper left to lower right.