Mathematics – Combinatorics
Scientific paper
2007-08-15
Contributions to Algebra and Geometry 52:2 (2011), 237-263
Mathematics
Combinatorics
23 pages, 10 figures, 14 tables. Changes from v1: editing and typographic correction
Scientific paper
10.1007/s13366-011-0010-5
In this paper we finish the intensive study of three-dimensional Dirichlet stereohedra started by the second author and D. Bochis, who showed that they cannot have more than 80 facets, except perhaps for crystallographic space groups in the cubic system. Taking advantage of the recent, simpler classification of three-dimensional crystallographic groups by Conway, Delgado-Friedrichs, Huson and Thurston, in a previous paper we proved that Dirichlet stereohedra for any of the 27 "full" cubic groups cannot have more than 25 facets. Here we study the remaining "quarter" cubic groups. With a computer-assisted method, our main result is that Dirichlet stereohedra for the 8 quarter groups, hence for all three-dimensional crystallographic groups, cannot have more than 92 facets.
Sabariego Pilar
Santos Francisco
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