Doug Dunham

Statement

Our goal is to design aesthetic patterns in the Poincaré hyperbolic circle model or on surfaces of polyhedra. One set of polyhedra that we have considered are triply periodic polyhedra in Euclidean 3-space. The most regular ones are transitive on vertices, edges, and faces, and are often called skew apeirohedra. H.S.M. Coxeter and John Flinders Petrie proved that there are exactly three of these: $\{4,6|4\}$, $\{6,4|4\}$, and $\{6,6|3\}$, where $\{p,q|r\}$ is composed of regular $p$-sided polygons meeting $q$ at a vertex and with regular $r$-sided polygonal holes. We use the $\{6,6|3\}$ for this patterned polyhedron. It is composed of invisible regular tetrahedral "hubs'' connected by "struts'' which are truncated tetrahedra.

Artworks

This is a fish pattern in the style of M.C. Escher on part of the regular triply periodic polyhedron $\{6,6|3\}$. That polyhedron has six families of embedded parallel lines that go through the centers of the hexagon faces. Our goal was to create a pattern of Escher-like fish with their backbones along those lines such that the fish on each family of parallel lines were all of the same color and also swam in the same direction along each line. The $\{4,6|4\}$ and the $\{6,4|4\}$ polyhedra also have six families of embedded parallel lines that go through the centers of their faces, but we proved that it was not possible to arrange a fish pattern on either polyhedron such that the fish swam in the same direction along those lines.