Spatio-temporal symmetry – crystallographic point groups with time translations and time inversion

The crystallographic symmetry of time-periodic phenomena has been extended to include time inversion. The properties of such spatio-temporal crystallographic point groups with time translations and time inversion are derived and one representative group from each of the 343 types has been tabulated. In addition, stereographic symmetry and general-position diagrams are given for each representative group. These groups are also given a notation consisting of a short Hermann–Mauguin magnetic point-group symbol with each spatial operation coupled with its associated time translation.