All Polyhedral Manifolds are Connected by a 2-Step Refolding

TL;DR

This paper proves that any two polyhedral manifolds can be connected through a two-step process involving unfolding and refolding, with special cases maintaining planarity or tree shape throughout.

Contribution

It introduces a universal two-step refolding method connecting any two polyhedral manifolds via a common unfolding.

Findings

Any two polyhedral manifolds share a common unfolding and refolding pathway.

Special cases include maintaining planarity for convex polygons and tree shape for polycubes.

Results extend to multiple manifolds with a shared intermediate manifold.

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.