| On the Number of Facets of Three-Dimensional Dirichlet Stereohedra IV: Quarter Cubic Groups (2007) | |||||||||
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| 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 finish the study of cubic groups, dealing with ''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 haver more than 92 facets.. Comment: 20 pages, 10 figures | |||||||||
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