All Polyhedral Manifolds are Connected by a 2-Step Refolding

Abstract

We prove that, for any two polyhedral manifolds $\mathcal{P},\mathcal{Q}$, there is a polyhedral manifold $\mathcal{I}$ such that $\mathcal{P},\mathcal{I}$ share a common unfolding and $\mathcal{I},\mathcal{Q}$ share a common unfolding. In other words, we can unfold $\mathcal{P}$, refold (glue) that unfolding into $\mathcal{I}$, unfold $\mathcal{I}$, and then refold into $\mathcal{Q}$. Furthermore, if $\mathcal{P},\mathcal{Q}$ have no boundary and can be embedded in 3D (without self-intersection), then so does $\mathcal{I}$. These results generalize to $n$ given manifolds ${\mathcal{P}}_1,{\mathcal{P}}_2,\dots ,{\mathcal{P}}_n$; they all have a common unfolding with the same intermediate manifold $\mathcal{I}$. 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.

This work was intended as a replacement of arXiv:2412.02174 and any subsequent updates will appear there