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 first and third pictures below show patterns on triply periodic polyhedra in Euclidean 3-space; the second picture shows a hyperbolic plane pattern that is the "universal covering pattern" of the pattern on the first polyhedron. The patterns are inspired by those of M.C. Escher, and like Escher's patterns, they 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 large scale geometry as their universal covering surface. The patterns of the first and second pictures have the same color symmetry.
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. The color group is the symmetric group S3.
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. The color group is the symmetric group S3.
This is a hyperbolic plane pattern of butterflies, eight of which meet at left front wing tips and three of which meet at their right rear wings. The pattern is inspired by M.C. Escher's Euclidean image Regular Division Drawing Number 70. Disregarding color, the symmetry group of this pattern is generated by 8-fold and 3-fold rotations about the respective meeting points of the left front and right rear wings, and is 832 in orbifold notation (or [3,8]+ in Coxeter notation). This pattern exhibits perfect 3-color symmetry and its color group is the symmetric 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 can be thought of as the "universal covering pattern" of the pattern on the {3,8} polyhedron. The covering map from this pattern to the polyhedron maps bufferflies to butterflies with the same coloring on the polyhedron.