Periodic double tilings of the plane
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916912599924736 |
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| author | Nobili, Francesco Novaga, Matteo Paolini, Emanuele |
| author_facet | Nobili, Francesco Novaga, Matteo Paolini, Emanuele |
| contents | We study tilings of the plane composed of two repeating tiles of different assigned areas relative to an arbitrary periodic lattice. We classify isoperimetric configurations (i.e., configurations with minimal length of the interfaces) both in the case of a fixed lattice or for an arbitrary periodic lattice. We find three different configurations depending on the ratio between the assigned areas of the two tiles and compute the isoperimetric profile. The three different configurations are composed of tiles with a different number of circular edges, moreover, different configurations exhibit a different optimal lattice. Finally, we raise some open problems related to our investigation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_08396 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Periodic double tilings of the plane Nobili, Francesco Novaga, Matteo Paolini, Emanuele Metric Geometry Analysis of PDEs Differential Geometry We study tilings of the plane composed of two repeating tiles of different assigned areas relative to an arbitrary periodic lattice. We classify isoperimetric configurations (i.e., configurations with minimal length of the interfaces) both in the case of a fixed lattice or for an arbitrary periodic lattice. We find three different configurations depending on the ratio between the assigned areas of the two tiles and compute the isoperimetric profile. The three different configurations are composed of tiles with a different number of circular edges, moreover, different configurations exhibit a different optimal lattice. Finally, we raise some open problems related to our investigation. |
| title | Periodic double tilings of the plane |
| topic | Metric Geometry Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2502.08396 |