Doug Dunham and Lisa Shier
Our goal is to design aesthetic 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.
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 fins of the other three colors. The patterned
polyhedron has perfect 6-color symmetry.