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 extensions to multiple manifolds and special cases like convex polygons and tree-shaped polycubes.

Contribution

It introduces a universal two-step refolding method connecting any two polyhedral manifolds via a common unfolding, extending to multiple manifolds and special geometric cases.

Findings

Any two polyhedral manifolds can be connected through a common unfolding and refolding.

The method preserves embedding in 3D for boundaryless manifolds.

Stronger results for special cases with planar intermediate shapes.

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.