On the number of facets of three-dimensional Dirichlet stereohedra IV: quarter cubic groups
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry - Tập 52 - Trang 237-263 - 2011
Tóm tắt
In this paper we finish the intensive study of three-dimensional Dirichlet stereohedra started by F. Santos and D. Bochiş, 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.
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