In this chapter we present three fundamental theorems on the structure of convex polyhedra that are essential to understanding folding and unfolding: Steinitz's theorem (below), Cauchy's rigidity theorem (Section 23.1), and Alexandrov's theorem (Section 23.3). It will be useful to view these results as members of a broad class of geometric “realization” problems: reconstructing or “realizing” a geometric object from partial information. Often the information is combinatorial, or combinatorics supplemented by some geometric (metric) information. A classic example here is “geometric tomography” (Gardner 1995b), one version of which is to reconstruct a 3D shape from several X-ray projections of it. When we fold a polygon to form a convex polyhedron, we have partial information from the polygon and seek to find a polyhedron that is compatible with, that is, which realizes, that information.

Steinitz's theorem. The constraints on realization available from purely combinatorial information are settled by Steinitz's theorem. The graph of edges and vertices of a convex polyhedron forms a graph (known as the 1-skeleton of the polyhedron).

Theorem 23.0.3. G is the graph of a convex polyhedron if and only if it is simple, planar, and 3-connected.

It is not so difficult to see that the three properties of G are necessary: G is simple because it has no loops or multiple edges; it is planar by stereographic projection, and it is at least plausible that the removal of no pair of vertices can disconnect G.