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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.11281 |
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| _version_ | 1866917140312883200 |
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| author | Brander, David Gravesen, Jens |
| author_facet | Brander, David Gravesen, Jens |
| contents | We study $C^1$-regular surfaces in $R^3$ that admit tilings by a finite number of rigid motion congruence classes of tiles. We construct examples with various topologies and present a framework for a systematic study, mainly concentrating on monotilings. A finite edge prototile is a tile that has only a finite number of possible interfaces with adjacent copies of itself. We describe all monotilings by such tiles with three or less edges. We consider the question of whether a monohedral polyhedron can be smoothed to become a finite edge type tileable surface with the same graph structure, and we give an example where this is not possible. Finally we list some open problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11281 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tileable Surfaces Brander, David Gravesen, Jens Differential Geometry Computational Geometry Combinatorics Primary 53A05, Secondary 52C20, 52C22, 05B45, 51M20 We study $C^1$-regular surfaces in $R^3$ that admit tilings by a finite number of rigid motion congruence classes of tiles. We construct examples with various topologies and present a framework for a systematic study, mainly concentrating on monotilings. A finite edge prototile is a tile that has only a finite number of possible interfaces with adjacent copies of itself. We describe all monotilings by such tiles with three or less edges. We consider the question of whether a monohedral polyhedron can be smoothed to become a finite edge type tileable surface with the same graph structure, and we give an example where this is not possible. Finally we list some open problems. |
| title | Tileable Surfaces |
| topic | Differential Geometry Computational Geometry Combinatorics Primary 53A05, Secondary 52C20, 52C22, 05B45, 51M20 |
| url | https://arxiv.org/abs/2507.11281 |