Rigidity of polytopes with edge length and coplanarity constraints
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912956563849216 |
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| author | Himmelmann, Matthias Schulze, Bernd Winter, Martin |
| author_facet | Himmelmann, Matthias Schulze, Bernd Winter, Martin |
| contents | We investigate a novel setting for polytope rigidity, where a flex must preserve edge lengths and the planarity of faces, but is allowed to change the shapes of faces. For instance, the regular cube is flexible in this notion. We present techniques for constructing flexible polytopes and find that flexibility seems to be an exceptional property. Based on this observation, we introduce a notion of generic realizations for polytopes and conjecture that convex polytopes are generically rigid in dimension $d\geq 3$. We prove this conjecture in dimension $d=3$. Motivated by our findings we also pose several questions that are intended to inspire future research into this notion of polytope rigidity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00874 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rigidity of polytopes with edge length and coplanarity constraints Himmelmann, Matthias Schulze, Bernd Winter, Martin Combinatorics Discrete Mathematics Metric Geometry 51M20, 52C25, 52B11 We investigate a novel setting for polytope rigidity, where a flex must preserve edge lengths and the planarity of faces, but is allowed to change the shapes of faces. For instance, the regular cube is flexible in this notion. We present techniques for constructing flexible polytopes and find that flexibility seems to be an exceptional property. Based on this observation, we introduce a notion of generic realizations for polytopes and conjecture that convex polytopes are generically rigid in dimension $d\geq 3$. We prove this conjecture in dimension $d=3$. Motivated by our findings we also pose several questions that are intended to inspire future research into this notion of polytope rigidity. |
| title | Rigidity of polytopes with edge length and coplanarity constraints |
| topic | Combinatorics Discrete Mathematics Metric Geometry 51M20, 52C25, 52B11 |
| url | https://arxiv.org/abs/2505.00874 |