Doug Dunham, Lisa M Shier

Statement

My goal is to design aesthetic tessellating patterns on hyperbolic surfaces such as the Poincaré 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}. Last year we designed a "proof of concept" hand-drawn version. The current polyhedron is a "production" version made with a computer-controlled cutter/plotter and solid color paper. The 463 pieces were hand-glued together. The polyhedron has six families of embedded parallel lines going through the centers of the hexagon faces. The Escher-like fish swim with their backbones along those lines such that the fish on each family of parallel lines are all of the same color. On each face there are "full fish" of three colors, with overlapping fins of the other three colors. The patterned polyhedron has perfect 6-color symmetry.