Quaternions and 4-Dimensional Groups

This chapter deals with quaternions and 4-dimensional groups. One of the tasks of the chapter is to establish that the general orthogonal map in 4 dimensions has the form x→l¯xrorl¯x¯r, where l and r are two unit quaternions. Coxeter’s notations (adapted from Schläfli) for regular polytopes and associated groups are widely used. The chapter extends his system slightly so as to obtain a complete set of notations for the “polyhedral” groups. When we call the symmetry group of an object “chiral,” what we really mean is that the object is chiral. In three dimensions, the finite subgroups of the orthogonal group, even the “chiral” ones, are all the same as their mirror images. However, among the 219 crystallographic space-groups in 3 dimensions are 11 that differ from their mirror images.