The \(\{4,5\}\) isogonal sponges on the cubic lattice
type: Item
instance of: scholarly article
title: The \(\{4,5\}\) isogonal sponges on the cubic lattice
zbMATH Open document ID: 1160.52011
author: Steven B. Gillispie
author: Branko Grünbaum
published in: The Electronic Journal of Combinatorics
publication date: 2009-04-07T00:00:00Z
full work available at URL: https://eudml.org/doc/129862
full work available at URL: v16i1r22.html
review text: Summary: Isogonal polyhedra are those polyhedra having the property of being vertex-transitive. 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 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.
Mathematics Subject Classification ID: 52B70
Mathematics Subject Classification ID: 05B45
Mathematics Subject Classification ID: 51M20
zbMATH DE Number: 5541087
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5541087