2024/12/03 by Lily Chung, Erik D. Demaine, Chung, Lily +11 · 2 voices
Computer Science · Engineering · Mathematics · #Advanced Materials and Mechanics #Computational Geometry and Mesh Generation #Geometric and Algebraic Topology #cs.CG
paper · pdf · doi:10.48550/arxiv.2412.02174
openalex publication_date 2024/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 P1,\mathcal P2, …, \mathcal Pn; 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.