PubMed · 6861360
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W Shaw. 1983. Previous work overlooked.. https://pubmed.ncbi.nlm.nih.gov/6861360/
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A new and relatively simple version of the quaternion calculus is offered which is especially suitable for applications in molecular symmetry and structure. After introducing the real quaternion algebra and its classical matrix representation in the group SO(4) the relations with vectors in 3-space and the connection with the rotation group SO(3) through automorphism properties of the algebra are discussed. The correlation of the unit quaternions with both the Cayley-Klein and the Euler parameters through the group SU(2) is presented. Besides rotations the extension of quaternions to other important symmetry operations, reflections and the spatial inversion, is given. Finally, the power of the quaternion calculus for molecular symmetry problems is revealed by treating some examples applied to icosahedral symmetry.
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.
The concept of stereographic projections of point groups is reviewed. If the focus of attention is moved from the symmetry-equivalence of points to the symmetry operations inter-relating points, it is possible to give simple diagrams of irreducible representations, at least for many axial groups. This approach may be elaborated by the use of trigonometric basis functions, when these may be shown on similar stereographic-related diagrams. These two simple steps enable some insights into direct products; in particular, it is possible to give a diagrammatic representation of an antisymmetric direct product.