Some geometric characteristics of four-dimensional crystallographic point symmetry groups and their subgroups

Abstract

Using three-dimensional point and tablet antisymmetry groups and also rosette and zero-dimensional groups of generalized antisymmetry that model all possible subgroups of four-dimensional crystallographic point groups, we determined the number of various symmetry groups forming the four-dimensional crystallographic classes for each category of such subgroups. These results allowed us to establish (without the use of the complete catalogue of the groups themselves) that 168 four-dimensional crystallographic classes preserve not only the point but also some other geometrical objects. The remaining 103 classes preserve only one invariant point of the four-dimensional Euclidean space.