On the Number of Facets of Three-Dimensional Dirichlet Stereohedra III: Full Cubic Groups

We are interested in the maximum possible number of facets that Dirichlet stereohedra for three-dimensional crystallographic groups can have. In two previous papers, D. Bochiş and the second author studied the problem for noncubic groups. This paper deals with “full” cubic groups, while “quarter” cubic groups are left for a subsequent paper. Here, “full” and “quarter” refers to the recent classification of three-dimensional crystallographic groups by Conway, Delgado-Friedrichs, Huson and Thurston. This paper’s main result is that Dirichlet stereohedra for any of the 27 full groups cannot have more than 25 facets. We also find stereohedra with 17 facets for one of these groups. Research partially supported by the Spanish Ministry of Education and Science, grant number MTM2005-08618-C02-02.