a) The 24-cell — a.k.a. {3,4,3} — is unique among all polytopes, and b) the 4-space it lives in is also unique among all Euclidean spaces in that it has more than 1 inequivalent differentiable structure. (But it's not content to just have a finite number of them (like all the n-spheres S^n for n >= 7) or even a countable number; it has *continuum many* — i.e., 2^aleph_0 — of them. And, the 4-sphere S^4 is the only sphere among all S^n for which it is not known whether it can have more than one differentiable structure.) So it's natural to ask if a) and b) are somehow directly related. I have no idea. But it sure would be cool if they were. Its 24 vertices can be thought of as the binary tetrahedral group 2T as a subgroup of the unit quaternion group S^3. The space of cosets S^3 / 2T is a 3-manifold that is the configuration space of a regular tetrahedron inscribed in a unit 2-sphere S^2 (or equivalently, centered at th origin of R^3). This is defined as the space of all possible rotational positions of such a tetrahedron, such that if two positions look the same (i.e., the 4 vertices are the same) then they are the same in the configuration space. --Dan