As in the case of the 3-dimensional regular apeirohedra described in the previous chapter, the mirror vector plays an important role in the classification of the 4-dimensional regular polyhedra. Thus the first task is to determine the possible mirror vectors of such polyhedra. The polyhedra with mirror vector (3,2,3) and their relatives under standard operations such as Petriality form a specially rich family. One particular family of these polyhedra is treated in detail, with a description of their realization domains. With the mirror vector (2,3,2), most of the standard operations lead to polyhedra in the same class. Though there is a close analogy between the infinite and finite cases, those with mirror vector (2,2,2) have symmetry groups that need not be related to reflexion groups; the treatment here employs quaternions. There are various connexions among these regular polyhedra, the most interesting being the way that the skewing operation takes certain polyhedra in class (3,2,3) into polyhedra of class (2,2,2).

Building on the previous chapter, this one moves on to the special case of polytopes. Wythoff’s construction is the basic way to obtain a polytope from a representation of an automorphism group, and leads to geometric analogues of some of the core operations on polytopes. Various connexions between rank and dimension of faithful realizations are then considered, in particular the concepts of full and nearly full rank. The important mirror vector lists the dimensions of the reflexion mirrors of a realization. Realizations that are degenerate in some respect also play a part; these are looked at next. Induced cosine vectors come from cuts such as vertex-figures and facets of polytopes; in a number of ways these provide additional tools for determining realization domains. A brief account of the alternating product of polytopes is then given, although these are rarely used. The theory of realizations of regular apeirotopes (infinite polytopes) is somewhat different; it is sketched here. For the most part, descriptions of realization spaces of particular polytopes are postponed until the polytopes themselves have been introduced; however, several basic examples are given including polygons, which are needed to formulate the notion of rigidity in Chapter 6.