10 - Three-Dimensional Apeirohedra

Summary

As an illustration of realization theory, the realization domain of each finite regular polygon is described, and that of the infinite apeirogon is commented on. In three dimensions, the regular polyhedra and apeirohedra of full rank are also classified. Thus the first non-trivial cases of nearly full rank are the apeirohedra (infinite polyhedra) in ordinary space. Since the blended apeirohedra have already been met, the core of the chapter is therefore the classification of the twelve pure 3-dimensional regular apeirohedra; here, the mirror vector plays an important part. The treatment in ‘Abstract Regular Polytopes’ is expanded on, by displaying new relationships among these apeirohedra; certain of these relationships are then used to describe the automorphism groups of the apeirohedra as abstract polytopes. Last, it is shown that the fine Schläfli symbols for nine of the twelve apeirohedra are rigid; the exceptions are the three apeirohedra with finite skew faces.