Polyhedral version of sphere eversion Is there a polyhedral version of sphere eversion (Smale's theorem), where you take a _polyhedron_ with triangular faces homeomorphic to the sphere, and _continuously move the vertices_ such that _no dihedral angle becomes zero_ (that is, no pair adjacent faces ever overlap) and such that it _ends inside-out_ of how it started? If so, how many vertices do we need? A simple argument shows the tetrahedron cannot be everted in that manner. I don't think the octahedron can, either. I imagine this probably needs a large polyhedron to work. (Alternate question: What if you require that the dihedral angles never leave $(\pi-\epsilon,\pi+\epsilon)$ for some $\epsilon$?)

Apery and Denner showed that the cuboctahedron is the minimum polyhedron that can be everted in the way you describe.