关于三维Dirichlet立体面IV的刻面数：四分之一立方群
On the Number of Facets of Three-Dimensional Dirichlet Stereohedra IV: Quarter Cubic Groups
摘要：
本文完成了第二作者和D。Bochis指出，除了立方晶系中的晶体学空间群外，它们不可能有超过80个小平面。利用Conway、Delgado-Friedrichs、Huson和Thurston最近对三维晶体群进行的简单分类，在以前的一篇论文中，我们证明了27个“全”立方群中任何一个的狄利克雷立体面都不能超过25个刻面。在这里，我们研究剩余的“四分之一”立方群。用计算机辅助的方法，我们的主要结果是，狄利克雷立体面的8个四分之一组，因此所有的三维晶体群，不能有超过92个方面。
英文摘要：
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.
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