Crystallographic groups in space and time: I. General definitions and basic properties

Abstract

A generalization of the concept of n-dimensional magnetic group is considered which also admits discrete time translations. This leads to the study of crystallographic groups in n + 1-dimensional Euclidean, Minkowskian, Galilean and product spaces. Definitions are given for point groups, system groups, arithmetic and geometric crystal classes, Bravais classes, lattice systems and space(-time) groups in these spaces. As in the Euclidean case, space-time groups G n+1 appear in ( K, Z n+1 , φ)-extensions with K a crystallographic point group and φ a monomorphism φ: K → GL( n + 1, Z). As it is not yet known under which conditions groups appearing in such extensions may be interpreted as space-time groups, the classification of these groups is here restricted to the case of finite K. This classification arises by identifying space(-time) groups related by an isomorphism which takes into due account the various kinds of translation elements. For known geometric point groups a constructive method to derive all non-isomorphic space(-time) groups is given. The number of Bravais classes in Euclidean and Galilean space turns out to be finite. It is enumerably infinite in so-called product space and continuously infinite in Minkowskian space. The same is true for the number of non-isomorphic space(-time) groups.