Suppose that S is a discrete point set in Rⁿ. For any point p ∈ S, the Voronoi o region is the set of points in Rⁿ closer to p than to any other point of S. If the set S is the orbit of a crystallographic group that stabilizes p trivially, then the Voronoi o region is called a Dirichlet region. The decomposition of Rⁿ into Voronoi o regions is a Dirichlet tessellation and its tiles are Dirichlet stereohedra.

In two previous papers [1,2], Santos, the second author of this paper, and D. Bochiș search for the maximal number of facets Dirichlet stereohedra can have for three-dimensional crystallographic groups. In this paper, which can be considered the third paper in the series, the authors show that Dirichlet stereohedra for the 27 full three-dimensional crystallographic groups can have no more than 25 facets. The notion “full” refers to the recent classification of three-dimensional crystallographic groups. Apart from this upper bound on the number of facets, the authors construct stereohedra with 17 facets for one of these groups.

This paper is a nice contribution to the understanding of stereohedra related to crystallographic groups, which is still incomplete by comparison to the progress made in the classification of crystallographic groups in recent decades. The authors also announce a subsequent paper, in which they will deal with the number of facets of Dirichlet stereohedra for quarter cubic crystallographic groups.