All Polyhedral Manifolds are Connected by a 2-Step Refolding

2024-12-03

arXiv.org

AI Summary

This paper investigates the two-step refolding problem between polyhedral manifolds: given any two polyhedral manifolds $P$ and $Q$, does there exist an intermediate manifold $I$ such that both $P$ and $I$, and $I$ and $Q$, share a common net (i.e., are mutually refoldable via a single unfolding–refolding operation)? Using techniques from computational geometry, polyhedral unfolding theory, gluing topology, and constructive existence proofs, the authors establish that the refolding graph over all polyhedral manifolds has connected diameter at most 2—a first such result. Specifically, when $P$ and $Q$ are boundaryless, embeddable 3-manifolds, $I$ can be explicitly constructed to preserve both boundarylessness and embeddability. The framework extends to $n$-manifold sequences and yields strong structure-preserving reconstructions—maintaining embeddability, planarity, and combinatorial structure—for classes including convex polygonal double covers and tree-like polycubes.

Technology Category

Machine Learning: Learning with ManifoldsKnowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSocial Networks and Social Media: Computational social scienceResponsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We prove that, for any two polyhedral manifolds P, Q, there is a polyhedral manifold I such that P, I share a common unfolding and I, Q share a common unfolding. In other words, we can unfold P, refold (glue) that unfolding into I, unfold I, and then refold into Q. Furthermore, if P, Q are embedded in 3D, then I can be embedded in 3D (without self-intersection). These results generalize to n given manifolds P_1, P_2, ..., P_n; they all have a common unfolding with an intermediate manifold 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 any two polyhedral manifolds via a 2-step unfolding and refolding
• Ensuring intermediate manifold preserves 3D embeddability for boundary-free cases
• Generalizing results to multiple manifolds with shared unfolding steps

Innovation

• Connects polyhedral manifolds via 2-step refolding
• Ensures intermediate manifold preserves 3D embeddability
• Extends to multiple manifolds with common unfolding