DATA STRUCTURES AND PROCEDURES FOR A POLYHEDRON ALGORITHM

Abstract

In this paper we describe the data structures and the procedures of a program, which is based on the algorithms of [5,6]. Knowing the incidence structure of a polyhedron, the program finds all the essentially different facet pairings. The transformations, pairing the facets generate a space group, for which the polyhedron is a fundamental domain. The program also creates the defining relations of the group. Thus, we obtain discrete groups of certain combinatorial spaces. We have still to examine which groups can be realised in spaces of constant curvature (or in other simply connected spaces). Finally, we mention some results: Examining the 4-simplex, our program disproves Zhuk's conjecture concerning the number of essentially different facet pairings of d-simplices [11]. The classification of 3-simplex tilings has also been completed [7]. We have found the fundamental tilings of the Euclidean space with marked cubes and the corresponding crystalIographic groups [8].