Classification, nomenclature and symbolism of 32 types of crystallography symmetries

Abstract

The classification of 32 types (or groups) of crystal symmetry, i.e., the basis for dividing them into systems, or systems, can be based on various principles. Of these principles, the following two are the main and most natural. It is possible to classify the types (or groups) of symmetry, i.e., certain spatial collections of symmetry elements as such, by themselves, without relation to the complexes of possible faces and edges of the crystal to which these types of symmetry are characteristic. Let us call such classifications “purely geometric”. It is possible to classify the types of symmetry, taking into account the properties of those complexes of possible faces and edges of the crystal, in other words, those spatial lattices to which these types (groups) of symmetry are characteristic. Let us call such classifications “crystallographic”. The proposed classification, nomenclature and symbolism are closely linked by a single principle and are entirely based on a genetic trait — on the generative elements of symmetry.