Mathematics Colloquia and Seminars

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A Dirichlet stereohedron for a crystallographic group $G$ is the Voronoi region of any point with trivial stabiliser under $G$ in the Voronoi diagram of its orbit. Delone's {it fundamental theorem of the theory of stereohedra} (1961) implies that no three-dimensional Dirichlet stereohedron can have more than 390 faces. In contrast, the Dirichlet stereohedron with the maximum number of faces known so far has 38 faces (Engel, 1980).

We show new upper bounds on the number of faces of 3-dimensional Dirichlet stereohedra, dividing the 219 three-dimensional crystallographic groups in 3 blocks:

- For groups which contain reflections (100 groups) we prove an upper bound of 18 faces and, in fact, we explicitly construct a Dirichlet stereohedron with 18 faces for the group $Ifrac{4_1}{g}frac{2}{m}frac{2}{d}$.

- For non-cubic groups without reflections (97 groups) we prove a bound of 84 faces. Our bound is group-by-group. It is greater than 38 in only 22 out of the 97 groups and it is grater than 50 in only 10 of them. We construct a Dirichlet stereohedron with 32 faces for the group $P6_1 22$.

- For cubic groups without reflections (22 groups) we prove an upper bound of 162 faces. Engel's stereohedron for the group $I4_132$ belongs to this block.