Paper by Erik D. Demaine

Reference:

Lily Chung, Erik D. Demaine, Jenny Diomidova, Tonan Kamata, Jayson Lynch, Ryuhei Uehara, and Hanyu Alice Zhang, “All Polyhedral Manifolds are Connected by a 2-Step Refolding”, Journal of Information Processing, to appear.

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 have no boundary and can be embedded in 3D (without self-intersection), then so does ℐ. These results generalize to n given manifolds 𝒫₁, 𝒫₂, …, 𝒫_n; they all have a common unfolding with the same 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.

Comments:

This paper is also available as arXiv:2412.02174.

Length:

The paper is 10 pages.

Availability:

The paper is available in PDF (2116k).

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