Stereographic projections and direct products

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.