Enantiomorphism of crystallographic groups in higher dimensions with results in dimensions up to 6

Abstract

This paper gives classification results for crystallographic groups in dimensions up to 6 which refine earlier enumeration results. Based on the classification data, the asymptotic growth of the number of space-group types is discussed. The classification scheme for crystallographic groups is revisited and a new classification level in between that of geometric and arithmetic crystal classes is introduced and denoted as harmonic crystal classes. Enantiomorphic pairs are determined on all classification levels from space-group types to crystal families and the enantiomorphic pairs of fixed-point-free space groups are given. A general algorithm to compute enantiomorphic pairs is described.