On the number of facets of three-dimensional Dirichlet stereohedra II: Non-cubic Groups

Details On the number of facets of three-dimensional Dirichlet stereohedra II: Non-cubic Groups On the number of facets of three-dimensional Dirichlet stereohedra II: Non-cubic Groups

2002-04-18

arxiv.org/abs/math/0204231v1

Contributions to Algebra and Geometry, Vol. 47, No. 1, pp. 89-120 (2006)

Mathematics

Combinatorics

27 pages, 18 figures, 7 tables

Scientific paper

We prove that Dirichlet stereohedra for non-cubic crystallographic groups in dimension 3 cannot have more than 80 facets. The bound depends on the particular crystallographic group considered and is above 50 only on 9 of the 97 affine conjugacy classes of them. We also construct Dirichlet stereohedra with 32 and 29 facets for a hexagonal and a tetragonal group, respectively.

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