New investigations of parallelohedra in \(R^{d}\). (English) Zbl 0955.51011

Engel, Peter

Engel, P. (ed.) et al., Voronoï’s impact on modern science. Book II. Transl. from the Ukrainian. Kyiv: Institute of Mathematics. Proc. Inst. Math. Natl. Acad. Sci. Ukr., Math. Appl. 21(2), 22-60 (1998).

Summary: General properties of Dirichlet-Voronoï parallelohedra are described. Results are given for dimensions 2, 3, 4, and 5. In a \(5\)-dimensional space, a complete description of 222 combinatorial types of primitive parallelohedra and of 84 types of totally zone-contracted parallelohedra is given. It could be shown that every type of totally zone-contracted parallelohedra has as a representative a Dirichlet-Voronoï parallelohedron. From this result it is concluded that in spaces of dimensions \(d \leq 5\) every parallelohedron is combinatorial equivalent to a Dirichlet-Voronoï parallelohedron.
For the entire collection see [Zbl 0928.00007].

MSC:

51M20	Polyhedra and polytopes; regular figures, division of spaces
20H15	Other geometric groups, including crystallographic groups
82D25	Statistical mechanics of crystals

Keywords:

Dirichlet-Voronoï parallelohedra; zone contraction; convex polytope