ABSTRACT

This chapter aims to enumerate the finite subgroups of SO3 and GO3 and relate them to quaternions. In the standard language, the set of unit quaternions is a “double cover” of SO3 called the spin group Spin3. The chapter describes the finite subgroups of GO3 and the way they are represented by quaternions. The first three groups are collectively called the polyhedral rotation groups, since they consist of the rotations that fix the appropriate regular polyhedra, while the remaining two are the axial rotation groups, since they fix an axis. To work projectively is to regard elements as identical if they are scalar multiples of each other. For us, the only relevant scalars are ±1, so working projectively replaces +g and –g by a single element [g].