Patterns on Triply Periodic Polyhedra

This paper discusses patterns on triply periodic polyhedra, infinite polyhedra that repeat in three independent directions in Euclidean 3-space. We further require that all the vertices be congruent by a symmetry of the polyhedron, i.e. that they be uniform, and also that each of the faces is a single regular polygon. We believe that we are the first to apply patterns to such polyhedra. The patterns we use are inspired by the Dutch artist M.C. Escher. The patterns are preserved, up to color symmetry, by the symmetries of the polyhedra.