All Polyhedral Manifolds are Connected by a 2-Step Refolding

2026-04-11

AI Summary

This paper resolves the refolding connectivity problem for polyhedral manifolds: it proves that any two closed embeddable polyhedral manifolds $P$ and $Q$ admit an intermediate polyhedral manifold $I$ such that $P o I o Q$ forms a two-step, intersection-free refolding (i.e., both share a common net), and $I$ is embeddable in $mathbb{R}^3$. Methodologically, the work integrates computational geometry, topological deformation analysis, and boundary-gluing matching techniques to constructively define $I$ and rigorously establish its embeddability. Key contributions include: (1) the first general proof of two-step refolding connectivity between arbitrary closed embeddable polyhedral manifolds; (2) an extension to $n$ input manifolds sharing a single common intermediate manifold; and (3) strengthened results—preserving planarity or combinatorial structure—for double-covered convex polygons and tree-like polycubes.

Technology Category

Machine Learning: Learning with ManifoldsPlanning, Routing, and Scheduling: Replanning and Plan RepairKnowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsResponsible Web: Human-perceived consequences of algorithmic deployment on the webSystems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Connecting polyhedral manifolds through unfolding and refolding
• Establishing a common intermediate manifold for multiple shapes
• Ensuring intermediate shapes preserve embedding and boundary properties

Innovation

• Two-step refolding connects polyhedral manifolds via unfolding
• Intermediate manifold enables common unfolding between given shapes
• Generalizes to multiple manifolds using shared intermediate steps