2025/05/11 by Chung, Lily, Demaine, Erik D., Diomidova, Jenny +4
#Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2505.07147
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.