Tilings of the sphere by congruent pentagons V: Edge combination $a^{4}b$ with rational angles
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909681584177152 |
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| author | Liang, Jinjin Liao, Yixi Hu, Wenchuan Wang, Erxiao |
| author_facet | Liang, Jinjin Liao, Yixi Hu, Wenchuan Wang, Erxiao |
| contents | We classify edge-to-edge tilings of the sphere by congruent pentagons with the edge combination $a^4b$ and with rational angles in degree: they are a one-parameter family of symmetric $a^4b$-pentagonal subdivisions of the tetrahedron with $12$ tiles; a sequence of unique symmetric $a^4b$-pentagons admitting a symmetric $3$-layer earth map tiling by $4m$ tiles for any $m\ge4$, among which each odd $m$ case admits two standard flip modifications; and a unique non-symmetric and degenerate $a^4b$-pentagon admitting a non-symmetric $3$-layer earth map tiling and its standard flip modification with $20$ tiles. The full classification from this series and all induced non-edge-to-edge quadrilateral tilings from degenerate pentagons are summarized with their 3D pictures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_07038 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tilings of the sphere by congruent pentagons V: Edge combination $a^{4}b$ with rational angles Liang, Jinjin Liao, Yixi Hu, Wenchuan Wang, Erxiao Combinatorics 52C20, 05B45, 11R18, 11Y50, 14Q25 We classify edge-to-edge tilings of the sphere by congruent pentagons with the edge combination $a^4b$ and with rational angles in degree: they are a one-parameter family of symmetric $a^4b$-pentagonal subdivisions of the tetrahedron with $12$ tiles; a sequence of unique symmetric $a^4b$-pentagons admitting a symmetric $3$-layer earth map tiling by $4m$ tiles for any $m\ge4$, among which each odd $m$ case admits two standard flip modifications; and a unique non-symmetric and degenerate $a^4b$-pentagon admitting a non-symmetric $3$-layer earth map tiling and its standard flip modification with $20$ tiles. The full classification from this series and all induced non-edge-to-edge quadrilateral tilings from degenerate pentagons are summarized with their 3D pictures. |
| title | Tilings of the sphere by congruent pentagons V: Edge combination $a^{4}b$ with rational angles |
| topic | Combinatorics 52C20, 05B45, 11R18, 11Y50, 14Q25 |
| url | https://arxiv.org/abs/2507.07038 |