On the generation of point groups in spaces of various dimensions

In this paper the use of Clifford algebra in the parametrization of point groups in spaces of various dimensions is shown. Higher-dimensional spaces are of great interest especially when modulated crystals or quasicrystals are studied. While the quaternion units, which are useful to parametrize rotations in 3 dimensions, can be identified with rotations, the basic Clifford units may be regarded as mirrors from which all proper and improper symmetry operations can be generated. The practical implementation of this method of parametrization is demonstrated for the group of the hypercube in the 4-dimensional space, and generalisations to spaces of dimensions higher than 4 are suggested.