All Polyhedral Manifolds are Connected by a 2-Step Refolding

TL;DR

The work resolves the connectivity of the unfold/fold graph for polyhedral manifolds by proving that any two such manifolds of equal surface area admit a common intermediate $\,\mathcal{I}$ with shared unfoldings, yielding a diameter of at most 2 in the unfolding graph. The construction relies on a general common dissection to pairwise align the inputs, followed by an abstract intermediate manifold built from reconciled edge gluings, and, when possible, an isometric embedding guaranteed by the Burago–Zalgaller theorem. The approach is extended to special cases, including planar intermediates for doubly covered convex polygons and tree-shaped polycubes, with explicit step bounds ($O(n)$ and $O(n^2)$ respectively). The results generalize to any number of inputs and open avenues for tighter bounds and broader classes of polyhedra, while highlighting the role of nonconvex intermediates in enabling two-step refoldings.

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.

Figures (9)

• Figure 1: Example 2-step refolding from the cube to the regular tetrahedron.
• Figure 3: Rearranging two pieces via a 2-step refolding.
• Figure 4: Refolding $\triangle ABC$ into a rectangle when $B$ is between $A$ and $C$.
• Figure 5: Refolding $\triangle ABC$ into a rectangle when $B$ is not between $A$ and $C$.
• Figure 6: Bounding the distance from $B$ to $Q_1, Q_2$.

Theorems & Definitions (16)

• Theorem 3.1
• Corollary 3.2
• Theorem 3.3: Burago-Zalgaller-1996
• Theorem 4.1
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Lemma 4.4
• proof