Rationality of certain triangle tilings
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866912996423368704 |
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| author | Beeson, Michael Zhang, Yan X |
| author_facet | Beeson, Michael Zhang, Yan X |
| contents | We consider tilings of a triangle $ABC$ by congruent copies of a triangle that has one angle equal to $120^\circ$, has non-commensurable angles (that is, not all angles are rational multiples of $π$), and is not similar to $ABC$. We prove that any such tiling has commensurable sides, meaning that the side lengths can be taken to be integers after scaling. As a consequence, we show that outside of a couple of special cases, a triangle (allowing all angles) tiling must either have commensurable angles or commensurable sides (that is, all sides have rational ratios). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01314 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rationality of certain triangle tilings Beeson, Michael Zhang, Yan X Combinatorics 51M20, 51M04, 05E40 We consider tilings of a triangle $ABC$ by congruent copies of a triangle that has one angle equal to $120^\circ$, has non-commensurable angles (that is, not all angles are rational multiples of $π$), and is not similar to $ABC$. We prove that any such tiling has commensurable sides, meaning that the side lengths can be taken to be integers after scaling. As a consequence, we show that outside of a couple of special cases, a triangle (allowing all angles) tiling must either have commensurable angles or commensurable sides (that is, all sides have rational ratios). |
| title | Rationality of certain triangle tilings |
| topic | Combinatorics 51M20, 51M04, 05E40 |
| url | https://arxiv.org/abs/2604.01314 |