12 - Four-Dimensional Apeirotopes

Summary

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.