The Origami flip graph of the $2\times n$ Miura-ori
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| Format: | Preprint |
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2025
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| _version_ | 1866908419672244224 |
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| author | Christensen, Lumi Hull, Thomas C. O'Neil, Emma Pappano, Valentina Ter-Saakov, Natalya Yang, Kacey |
| author_facet | Christensen, Lumi Hull, Thomas C. O'Neil, Emma Pappano, Valentina Ter-Saakov, Natalya Yang, Kacey |
| contents | Given an origami crease pattern $C=(V,E)$, a straight-line planar graph embedded in a region of $\mathbb{R}^2$, we assign each crease to be either a mountain crease (which bends convexly) or a valley crease (which bends concavely), creating a mountain-valley (MV) assignment $μ:E\to\{-1,1\}$. An MV assignment $μ$ is locally valid if the faces around each vertex in $C$ can be folded flat under $μ$. In this paper, we investigate locally valid MV assignments of the Miura-ori, $M_{m,n}$, an $m\times n$ parallelogram tessellation used in numerous engineering applications. The origami flip graph $OFG(C)$ of $C$ is a graph whose vertices are locally valid MV assignments of $C$, and two vertices are adjacent if they differ by a face flip, an operation that swaps the MV-parity of every crease bordering a given face of $C$. We enumerate the number of vertices and edges in $OFG(M_{2,n})$ and prove several facts about the degrees of vertices in $OFG(M_{2,n})$. By finding recurrence relations, we show that the number of vertices of degree $d$ and $2n-a$ (for $0\leq a$) are both described by polynomials of particular degrees. We then prove that the diameter of $OFG(M_{2,n})$ is $\lceil \frac{n^2}{2}\rceil$ using techniques from 3-coloring reconfiguration graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_19700 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Origami flip graph of the $2\times n$ Miura-ori Christensen, Lumi Hull, Thomas C. O'Neil, Emma Pappano, Valentina Ter-Saakov, Natalya Yang, Kacey Combinatorics Computational Geometry 05C90, 68U05 Given an origami crease pattern $C=(V,E)$, a straight-line planar graph embedded in a region of $\mathbb{R}^2$, we assign each crease to be either a mountain crease (which bends convexly) or a valley crease (which bends concavely), creating a mountain-valley (MV) assignment $μ:E\to\{-1,1\}$. An MV assignment $μ$ is locally valid if the faces around each vertex in $C$ can be folded flat under $μ$. In this paper, we investigate locally valid MV assignments of the Miura-ori, $M_{m,n}$, an $m\times n$ parallelogram tessellation used in numerous engineering applications. The origami flip graph $OFG(C)$ of $C$ is a graph whose vertices are locally valid MV assignments of $C$, and two vertices are adjacent if they differ by a face flip, an operation that swaps the MV-parity of every crease bordering a given face of $C$. We enumerate the number of vertices and edges in $OFG(M_{2,n})$ and prove several facts about the degrees of vertices in $OFG(M_{2,n})$. By finding recurrence relations, we show that the number of vertices of degree $d$ and $2n-a$ (for $0\leq a$) are both described by polynomials of particular degrees. We then prove that the diameter of $OFG(M_{2,n})$ is $\lceil \frac{n^2}{2}\rceil$ using techniques from 3-coloring reconfiguration graphs. |
| title | The Origami flip graph of the $2\times n$ Miura-ori |
| topic | Combinatorics Computational Geometry 05C90, 68U05 |
| url | https://arxiv.org/abs/2506.19700 |