Non-side-to-side tilings of the sphere by congruent triangles with any irrational angle
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918030450098176 |
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| author | Chen, Wen Liang, Jinjin Wang, Erxiao |
| author_facet | Chen, Wen Liang, Jinjin Wang, Erxiao |
| contents | We develop the basic and new tools for classifying non-side-to-side tilings of the sphere by congruent triangles. Then we prove that, if the triangle has any irrational angle in degree, such tilings are: a sequence of 1-parameter families of triangles each admitting many 2-layer earth map tilings with $2n$($n\geq3$) tiles, together with rotational modifications for even $n$; a 1-parameter family of triangles each admitting a unique tiling with $8$ tiles; and a sporadic triangle admitting a unique tiling with $16$ tiles. Then a scheme is outlined to classify the case with all angles being rational in degree, justified by some known and new examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16629 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-side-to-side tilings of the sphere by congruent triangles with any irrational angle Chen, Wen Liang, Jinjin Wang, Erxiao Combinatorics 52C20, 05B45 We develop the basic and new tools for classifying non-side-to-side tilings of the sphere by congruent triangles. Then we prove that, if the triangle has any irrational angle in degree, such tilings are: a sequence of 1-parameter families of triangles each admitting many 2-layer earth map tilings with $2n$($n\geq3$) tiles, together with rotational modifications for even $n$; a 1-parameter family of triangles each admitting a unique tiling with $8$ tiles; and a sporadic triangle admitting a unique tiling with $16$ tiles. Then a scheme is outlined to classify the case with all angles being rational in degree, justified by some known and new examples. |
| title | Non-side-to-side tilings of the sphere by congruent triangles with any irrational angle |
| topic | Combinatorics 52C20, 05B45 |
| url | https://arxiv.org/abs/2505.16629 |