Crystallography, geometry and physics in higher dimensions. I. Point-symmetry operations

Physical phenomena such as incommensurate phases or diffraction enhancement of symmetry are interpreted by using symmetry groups in four, five or six dimensions. This first paper concerns the point-symmetry operations (PSO) in these Euclidean superspaces. Elementary, non-elementary, degenerate and non-degenerate PSOs are defined and their geometrical supports and geometrical symbols are specified. A geometrical description is thus given of nineteen types of PSO which are either the crystallographic rotations of the four-dimensional space or the crystallographic rotations and improper rotations of the five-dimensional space or the improper crystallographic rotations of the six-dimensional space. These PSOs are elements of crystallographic point groups of these spaces and the physical application to polar point groups is given.