{4,5} cubic lattice sponges

Abstract

Isogonal polyhedra are those polyhedra having the property of being vertex-transitive. By this is meant that every vertex can be mapped to any other vertex via a symmetry of the whole polyhedron; in a sense, every vertex looks exactly like any other. The Platonic solids are examples, but these are bounded polyhedra and our focus here is on infinite polyhedra. When the polygons of an infinite isogonal polyhedron are all planar and regular the polyhedra are also known as sponges, pseudopolyhedra, or infinite skew polyhedra. These have been studied over the years but many have been missed by previous researchers. We first introduce a notation for labeling three-dimensional isogonal polyhedra and then show how this notation can be combinatorially used to find all of the isogonal polyhedra that can be created given a specific vertex star configuration. As an example, we apply our methods to the {4,5} vertex star of five squares aligned along the planes of a cubic lattice and prove that there are exactly 15 such unlabeled sponges and 35 labeled ones. Previous efforts had found only 8 of the 15 shapes.