Mathematics – Combinatorics
Scientific paper
2006-08-01
Discrete Comput. Geom., 40(2) (2008), 159-189.
Mathematics
Combinatorics
28 pages, 12 figures. Changes from v1: apart of some editing (mostly at the end of the introduction) and addition of reference
Scientific paper
10.1007/s00454-008-9063-0
We are interested in the maximum possible number of facets that Dirichlet stereohedra for three-dimensional crystallographic groups can have. The problem for non-cubic groups was studied in previous papers by D. Bochis and the second author (Discrete Comput. Geom. 25:3 (2001), 419-444, and Beitr. Algebra Geom., 47:1 (2006), 89-120). 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 (math.MG/9911185, Beitr. Algebra Geom. 42.2 (2001), 475-507). Our main result in this paper 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.
Sabariego Pilar
Santos Francisco
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