On Laves' Graph Of Girth Ten

Abstract

1. Introduction. This note shows how a certain infinite graph of degree three, discovered by Laves in connection with crystal structure, can be inscribed (in sixteen ways, all alike) in an infinite regular skew polyhedron which has square faces, six at each vertex. One-eighth of the vertices of the polyhedron are vertices of the graph, and the three edges of the graph that meet at such a vertex are diagonals of alternate squares. Thus either diagonal of any face of the polyhedron can serve as an edge, and the whole graph can then be completed in a unique manner.