Tiling Triangles with $2π/3$ Angles
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | |
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| _version_ | 1866911566391148544 |
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| author | Zhang, Yan X |
| author_facet | Zhang, Yan X |
| contents | Motivated by a question of Erdös and inquiries by Beeson and Laczkovich, we explore the possible $N$ for which a triangle $T$ can tile into $N$ congruent copies of a triangle $R$. The \emph{reptile} cases (where $T$ is similar to $R$) and the \emph{commensurable-angles} cases (where all angles of $R$ are rational multiples of $π$) are well-understood. We tackle the most interesting remaining case, which is when $R$ contains an angle of $2π/3$ and when $T$ is one of $6$ ``sporadic'' specific triangles, of which only $2$ were known to have constructions. For each of these, we create a family of constructions and conjecture that they are the only possible $N$ that occur for these triangles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22696 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tiling Triangles with $2π/3$ Angles Zhang, Yan X Combinatorics 52C20, 05B45 Motivated by a question of Erdös and inquiries by Beeson and Laczkovich, we explore the possible $N$ for which a triangle $T$ can tile into $N$ congruent copies of a triangle $R$. The \emph{reptile} cases (where $T$ is similar to $R$) and the \emph{commensurable-angles} cases (where all angles of $R$ are rational multiples of $π$) are well-understood. We tackle the most interesting remaining case, which is when $R$ contains an angle of $2π/3$ and when $T$ is one of $6$ ``sporadic'' specific triangles, of which only $2$ were known to have constructions. For each of these, we create a family of constructions and conjecture that they are the only possible $N$ that occur for these triangles. |
| title | Tiling Triangles with $2π/3$ Angles |
| topic | Combinatorics 52C20, 05B45 |
| url | https://arxiv.org/abs/2512.22696 |