Theory of regular systems of points and partition of space. I. R properties of regular point system

The (r, R) properties of regular point systems (RPS) of the general positio ns are discussed. The theorem relating the R characteristics of the RPS of the general position in space group G to the diameter of the fundamental do main F-G of this group is proved. Using this theorem, the upper bounds of t he R characteristics of the RPS of the general positions in the space group s of cubic system are determined as functions of the lattice constant a. Co nsidering the space groups Pn (3) over bar n and Fd (3) over bar c, it was shown that, for asymmorphic and hemisymmorphic groups, these bounds can be reduced upon the detailed analysis of the geometries of the corresponding f undamental domains.