Non-Euclidean Crystallographic Rigidity
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911379917635584 |
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| author | Esson, Jack Kastis, Eleftherios Schulze, Bernd |
| author_facet | Esson, Jack Kastis, Eleftherios Schulze, Bernd |
| contents | This paper establishes combinatorial characterisations of forced-symmetric and forced-periodic rigidity (under a fixed lattice) of bar-joint frameworks in non-Euclidean normed planes. In $\ell_q$-planes for $q\in(1,\infty)\backslash\{2\}$, we prove characterisations for forced-periodic rigidity and forced-reflectionally-symmetric rigidity. We also characterise forced-symmetric rigidity in this space with respect to the orientation-reversing wallpaper group $\mathbb{Z}^2\rtimes\mathcal{C}_s$, otherwise known as $pm$ in crystallography. In the $\ell_1$ and $\ell_\infty$-planes, we provide characterisations for forced-periodic rigidity and forced-$\mathbb{Z}^2\rtimes\mathcal{C}_s$-symmetric rigidity. All of these characterisations are proved by inductive constructions involving Henneberg-type graph operations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08128 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-Euclidean Crystallographic Rigidity Esson, Jack Kastis, Eleftherios Schulze, Bernd Combinatorics 52C25 (primary) 05C22, 52A21 (secondary) This paper establishes combinatorial characterisations of forced-symmetric and forced-periodic rigidity (under a fixed lattice) of bar-joint frameworks in non-Euclidean normed planes. In $\ell_q$-planes for $q\in(1,\infty)\backslash\{2\}$, we prove characterisations for forced-periodic rigidity and forced-reflectionally-symmetric rigidity. We also characterise forced-symmetric rigidity in this space with respect to the orientation-reversing wallpaper group $\mathbb{Z}^2\rtimes\mathcal{C}_s$, otherwise known as $pm$ in crystallography. In the $\ell_1$ and $\ell_\infty$-planes, we provide characterisations for forced-periodic rigidity and forced-$\mathbb{Z}^2\rtimes\mathcal{C}_s$-symmetric rigidity. All of these characterisations are proved by inductive constructions involving Henneberg-type graph operations. |
| title | Non-Euclidean Crystallographic Rigidity |
| topic | Combinatorics 52C25 (primary) 05C22, 52A21 (secondary) |
| url | https://arxiv.org/abs/2510.08128 |