Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The {4, 5} isogonal sponges on the cubic lattice. | The 4 5g isogonal sponges on the cubic lattice Steven B. Gillispie Department of Radiology Box 357987 University of Washington Seattle WA 98195-7987 USA gillisp@ Branko Griinbaum Department of Mathematics Box 354350 University of Washington Seattle WA 98195-4350 USA grunbaum@ Submitted Aug 28 2008 Accepted Feb 4 2009 Published Feb 13 2009 Mathematics Subject Classifications 52B70 05B45 51M20 Abstract 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 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 5g 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. 1 Introduction Convex polyhedra with regular polygons as faces and with all vertices alike have been known and studied since antiquity. The ones with all faces congruent are called regular or Platonic while allowing different kinds of polygons as faces leads to uniform THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R22 1 or Archimedean polyhedra. The aim of the present note is to study the analogues of these classical polyhedra obtained by replacing .
