Convex polyhedron

Convex polyhedra

By forgetting the face structure, any polyhedron gives rise to a graph, called the skeleton of the polyhedron, with corresponding vertices and edges. Such figures have a long history: Leonardo da Vinci devised frame models of the regular solids, which he drew for Pacioli's book Divina Proportione, and similar wire-frame polyhedra appear in M.C. Escher's print Stars. One highlight of this approach is Steinitz's theorem, which gives a purely graph-theoretic characterization of the skeletons of convex polyhedra: it states that the skeleton of every convex polyhedron is a planar graph with three-connected, and every such graph is the skeleton of some convex polyhedron. In other words, these are the connected graphs that can be drawn in the plane without any edges crossing, and that stay connected after removing any two of its vertices.