Generalized Crystallography of Diamond-Like Structures. 1. Finite Projective Planes and Specific Clusters of Diamond-Like Structures Determined by These Planes

摘要

It is shown that the adequate mapping of the symmetry of determined crystalline and noncrystalline diamond-like structures requires a change in the Euclidean basis of the structural crystallography to a more general basis of projective (algebraic) geometry. It is established that the incidence graphs of specific subconfigurations of a finite projective plane PG(2, q) with q = 2, 3, 4 are isomorphic to the graphs of specific clusters of diamond-like structures. The clusters thus determined are invariant with respect to the groups of projective geometry that have orthogonal groups as subgroups and, in fact, are set by the subtables of the incidence tables PG(2, q) with q = 2, 3, 4. As an example, the a priori derivation of clusters, which are not traditional crystallogrpahic objects, is considered, namely, the parallelohedron of the diamond structure, the combination of two Bernal polyhedra Z8 and clusters containing icosahedra.