THE SYMMETRY OPERATIONS FOR N-DIMENSIONAL PERIODIC AND QUASI-PERIODIC STRUCTURES

A discussion is given of the possible orthogonal and Euclidean transformations which are crystallographic. Especially the symmetry operations are considered which are used for the description of quasi-periodic systems. Since the latter are obtained as intersections of periodic n-dimensional structures with the physical d-dimensional space, the n-dimensional space in which the crystallographic operations act has a distinguished d-dimensional subspace. The consequences for the classification of symmetry operations are discussed.