Regular polytopes can be discussed in a purely abstract setting. As we saw in Chapter 2, in that context, the theory of regular polytopes is equivalent to the theory of certain kinds of groups generated by involutions, namely, the string C-groups. However, much of the appeal of regular polytopes throughout their history has been their geometric symmetry. Even when the notion of symmetry was only hazily understood (at least from the present viewpoint of action of a group), it was very clearly appreciated that geometric regular polygons and polyhedra were (in some sense) as symmetric as they could possibly be.

It is this geometric picture of regular polytopes which we shall address in this chapter. More specifically, we shall define the idea of realizations of an abstract regular polytope, and discuss their various properties. In Section 5A, we shall consider the general theory of realizations, which is that part common to the whole subject. The realization theory for finite regular polytopes is fairly detailed, and we shall cover this in Section 5B. As yet, the theory of realizations of a regular apeirotope is a little sketchy; we shall describe what is known in Section 5C.

Realizations in General

There are many candidates for spaces in which regular polytopes might be realized geometrically, although the most natural are probably the spherical, euclidean and hyperbolic spaces.