On the Number of Facets of Three-Dimensional Dirichlet Stereohedra I: Groups with Reflections

Let G be a crystallographic group in I R n. A Dirichlet stereohedron for G is any region in the Voronoi diagram of any orbit of G. We prove that Dirichlet stereohedra for three-dimensional crystallographic groups containing reeexions in three, two or one independent directions cannot have more than eight, eighteen and fteen facets respectively. We show examples where the three bounds are attained. In the subsequent papers of this series ((Boc-San II] and Boc-San III]) we will study the Dirichlet stereohedra corresponding to groups without reeexions.