In this chapter and the next, we shall discuss some general techniques for constructing new regular polytopes from old ones, or, rather, for constructing groups of regular polytopes from other groups, which will be those of regular polytopes or those closely related to them. These techniques fall into two broad categories. First, we may select certain elements of such a group, which themselves generate a string C-group. Second, we may augment a given group by adjoining outer automorphisms. The two chapters will be devoted to these topics in turn.

The general name for the first kind of technique will be mixing, while that for the second will be twisting. However, the two techniques are by no means exclusive, and we shall see that some regular polytopes can arise in both ways.

The discussion of this chapter will be split into six sections. In Section 7A, we consider the general idea of mixing operations. We then concentrate in Section 7B on the special case of polytopes of rank 3, where mixing sometimes permits a deeper investigation of the combinatorial structure of a regular polyhedron. We next explore in Section 7C the notion of a cut of a regular polytope, which yields, as the name might suggest, a regular polytope of lower rank embedded in a natural way in the original.