Optimal partitions of the flat torus into parts of smaller diameter
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909095432290304 |
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| author | Protasov, Dmitry Tolmachev, Alexander Voronov, Vsevolod |
| author_facet | Protasov, Dmitry Tolmachev, Alexander Voronov, Vsevolod |
| contents | We consider the problem of partitioning a two-dimensional flat torus $T^2$ into $m$ sets in order to minimize the maximal diameter of a part. For $m \leqslant 25$ we give numerical estimates for the maximal diameter $d_m(T^2)$ at which the partition exists. Several approaches are proposed to obtain such estimates. In particular, we use the search for mesh partitions via the SAT solver, the global optimization approach for polygonal partitions, and the optimization of periodic hexagonal tilings. For $m=3$, the exact estimate is proved using elementary topological reasoning. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_03997 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal partitions of the flat torus into parts of smaller diameter Protasov, Dmitry Tolmachev, Alexander Voronov, Vsevolod Metric Geometry Combinatorics We consider the problem of partitioning a two-dimensional flat torus $T^2$ into $m$ sets in order to minimize the maximal diameter of a part. For $m \leqslant 25$ we give numerical estimates for the maximal diameter $d_m(T^2)$ at which the partition exists. Several approaches are proposed to obtain such estimates. In particular, we use the search for mesh partitions via the SAT solver, the global optimization approach for polygonal partitions, and the optimization of periodic hexagonal tilings. For $m=3$, the exact estimate is proved using elementary topological reasoning. |
| title | Optimal partitions of the flat torus into parts of smaller diameter |
| topic | Metric Geometry Combinatorics |
| url | https://arxiv.org/abs/2402.03997 |