We now discuss a general method for constructing regular polytopes P from certain groups W by what are called twisting operations. These operations provide an important tool in the classification of the locally toroidal regular polytopes in later chapters. In our applications, W will usually be a Coxeter group or a complex reflexion group with a diagram admitting certain symmetries. In this chapter, we shall mainly study twisting operations on Coxeter groups; we shall leave the case of complex reflexion groups to Chapter 11.

Our use of the term “twisting operation” is motivated by the analogy with the importance of diagram symmetries for the twisted simple groups (see [75]).

The present chapter is organized as follows. In Section 8A we explain the basic concept of twisting. Then in Section 8B we construct the regular polytopes LK,G and describe their basic properties. This construction is very general and leads to a remarkable class of polytopes. In Sections 8C and 8D we shall treat two particularly interesting special cases of this construction, namely, the polytopes 2K and 2K,G(s). Then in Section 8E we prove a universality property of LK,G, and apply it to identify certain universal regular polytopes. Finally, in Section 8F we describe a method for finding polytopes with small faces as quotients of polytopes constructed in earlier sections.