On a~convex polyhedron in a~regular point system

M. I. Shtogrin

Izvestiya. Mathematics , Tome 86 (2022) no. 3, pp. 586-619.

Résumé

Faceting with a ‘filling’. An ideal crystal structure consists of finitely many equal and parallel translational point lattices. In $\mathbb R^3$ it extends unboundedly in all directions. We distinguish in it a finite part situated in a closed convex polyhedron every face of which contains nodes of a translational point lattice involved in the structure not belonging to the same straight line. Such a polyhedron is called a possible faceting of the ideal crystal structure. There are 32 well-known crystal classes, or 32 crystallographic point groups. Among them is the symmetry group of the possible faceting calculated taking account of the nodes of the ideal crystal structure belonging to it. A cyclic subgroup $C_n$ of the symmetry group of any possible faceting has order $n\le 4$ or $n=6$. Faceting without ‘filling’. In this paper we construct two crystal structures in which there are crystal polyhedra whose symmetry groups, calculated without taking account of the nodes of the crystal structure belonging to it, have rotation axes of orders $n=8$ and $n=12$. In both cases, the crystal polyhedron is a right prism of finite height. Without taking account of the internal structure, a possible faceting of a crystal structure in three-dimensional Euclidean space cannot have an axes of rotation of order $n$ satisfying $6$. The proposed constructions are accompanied by a detailed analysis of ideal crystal structures, as well as Delone sets $S$ of type $(r, R)$ in $\mathbb R^2$ and $\mathbb R^3$. In particular, we produce an expanded proof of one of the theorems stated in 2010 at an international conference dedicated to the 120th anniversary of B. N. Delone.

Détail

Keywords: lattice, irrational coordinates, regular point system, polyhedron, faceting, order of an axis.
Mots-clés : rational direction, rotation, group

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