Several regular apeirotopes of nearly full rank in four dimensions have already been found. Unlike in the general case, various operations applied to these lead to many more apeirotopes. In addition, other symmetry groups give rise to families unrelated to these; all of them are described in this chapter. There are connexions that tie together two basic ways of constructing apeirotopes of nearly full rank from polytopes or apeirotopes of full rank which are not available in other dimensions. First to be considered are imprimitive symmetry groups; this is the only dimension in which they can make a contribution to apeirotopes of nearly full rank. The largest family of apeirotopes is derived from the infinite tilings related to the 24-cell; included here those derived from the cubic tiling, since these tilings are closely connected. The final family consists of the apeirotopes related to those with a non-string hyperplane reflexion group discussed in Chapter 9.

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).