Local approach and the theory of lovozerite structures

S. V. Krivovichev

Informatics and Automation, Geometry, topology, and applications, Tome 288 (2015), pp. 120-132

Résumé

The local theorem and local approach to regular point systems and tilings are considered as applied to lovozerite structures, which form isohedral tilings of the space $E^n$ into cubes with two $v$-octants situated along a solid diagonal of a cube. All lovozerite tilings that satisfy the following three basic conditions are derived: (1) the tilings are isohedral; (2) the cubes can be joined to share either entire faces or rectangular half-faces; (3) the $v$-octants of neighboring cubes share vertices but never share edges or faces. Local conditions of the regularity of tilings in terms of the first coronas and subcoronas are considered. With the use of the information entropy of structures corresponding to lovozerite tilings, it is shown that in nature one encounters, as a rule, the simplest structures (four of the ten possible tilings are realized in the crystal structures of minerals and inorganic compounds).