Isogonal polyhedra are those polyhedra having the property of being vertextransitive. By this we mean 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 innite polyhedra. When the polygons of an innite isogonal polyhedron are all planar and regular, the polyhedra are also known as sponges, pseudopolyhedra, or innite skew polyhedra. These have been studied over the years, but many have been missed by previous researchers. We rst introduce a notation for labeling three-dimensional isogonal polyhedra and then show how this notation can be combinatorially used to nd all of the isogonal polyhedra that can be created given a specic vertex star conguration. As an example, we apply our methods to the f4; 5g vertex star of v e squares aligned along the planes of a cubic lattice and prove that there are exactly 15 such unlabeled sponges and 35 labeled ones. Previous eorts had found only 8 of the 15 shapes.
['Steven Gillispie', 'Branko Grünbaum']
The $\{4, 5\}$ Isogonal Sponges on the Cubic Lattice