All Polyhedral Manifolds are Connected by a 2-Step Refolding

Abstract

We prove that, for any two polyhedral manifolds P, Q, there is a polyhedral manifold I such that P, I share a common unfolding and I, Q share a common unfolding. In other words, we can unfold P, refold (glue) that unfolding into I, unfold I, and then refold into Q. Furthermore, if P, Q have no boundary and can be embedded in 3D (without self-intersection), then so does I. These results generalize to n given manifolds P₁, P₂,..., P_n; they all have a common unfolding with the same intermediate manifold I. Allowing more than two unfold/refold steps, we obtain stronger results for two special cases: for doubly covered convex planar polygons, we achieve that all intermediate polyhedra are planar; and for tree-shaped polycubes, we achieve that all intermediate polyhedra are tree-shaped polycubes.