Refold rigidity of convex polyhedra

We show that every convex polyhedron may be unfolded to one planar piece, and then refolded to a different convex polyhedron. If the unfolding is restricted to cut only edges of the polyhedron, then we show that many regular and semi-regular polyhedra are “edge-unfold rigid ” in the sense that each of their unfoldings may only fold back to the original. For example, all of the 43,380 edge unfoldings of a dodecahedron may only fold back to the dodecahedron. We begin the exploration of which polyhedra are edge-unfold rigid, demonstrating infinite rigid classes through perturbations, and identifying one infinite nonrigid class: tetrahedra. (This is the full version of the 4-page abstract. [6Feb12]) 1