| On the number of facets of three-dimensional Dirichlet stereohedra I: Groups with reflexions (2007) | |||||||||||||
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| Let G be a crystallographic group in IR n . A Dirichlet stereohedron for G is any region in the Voronoi diagram of any orbit of G. We prove that Dirichlet stereohedra for three-dimensional crystallographic groups containing reflexions in three, two or one independent directions cannot have more than eight, eighteen and fifteen facets respectively. We show examples where the three bounds are attained. In the subsequent papers of this series ([Boc-San II] and [Boc-San III]) we will study the Dirichlet stereohedra corresponding to groups without reflexions. Introduction This paper deals with the combinatorial type of Dirichlet stereohedra in three dimensions. More precisely, we find the exact maximum number of facets of threedimensional Dirichlet stereohedra corresponding to groups which contain reflexions. Let Isom(IR d ) denote the group of all isometries (motions) of IR d . A subgroup G of Isom(IR d ) is called a discrete group of motions if all its orbits are discrete. If ... | |||||||||||||
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