All Polyhedral Manifolds are Connected by a 2-Step Refolding
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arXiv
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2024
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| author | Chung, Lily Demaine, Erik D. Diomidova, Jenny Kamata, Tonan Lynch, Jayson Uehara, Ryuhei Zhang, Hanyu Alice |
| author_facet | Chung, Lily Demaine, Erik D. Diomidova, Jenny Kamata, Tonan Lynch, Jayson Uehara, Ryuhei Zhang, Hanyu Alice |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02174 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | All Polyhedral Manifolds are Connected by a 2-Step Refolding Chung, Lily Demaine, Erik D. Diomidova, Jenny Kamata, Tonan Lynch, Jayson Uehara, Ryuhei Zhang, Hanyu Alice Computational Geometry 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. |
| title | All Polyhedral Manifolds are Connected by a 2-Step Refolding |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2412.02174 |