Planar lattices and equilateral odd-gons
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912976838066176 |
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| author | Iino, Akira Sakiyama, Masashi |
| author_facet | Iino, Akira Sakiyama, Masashi |
| contents | For a planar integral lattice $L$, let $ν(L)$ denote the square-free part of the integer $D(L)^2$, where $D(L)$ stands for the area of a fundamental parallelogram of $L$. For each odd integer $n$ with $3 \leq n<29$, a planar lattice $L$ contains an equilateral $n$-gon if and only if $L$ is similar to an integral lattice $L'$ such that $ν(L')\equiv 3 \pmod 4$ and the largest prime factor $p$ of $ν(L')$ satisfies $p \leq n$. Moreover, such $L$ contains a convex equilateral $n$-gon, which answers a problem posed by Maehara. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_01911 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Planar lattices and equilateral odd-gons Iino, Akira Sakiyama, Masashi Combinatorics Metric Geometry Number Theory 52C05, 11H06, 51M04 For a planar integral lattice $L$, let $ν(L)$ denote the square-free part of the integer $D(L)^2$, where $D(L)$ stands for the area of a fundamental parallelogram of $L$. For each odd integer $n$ with $3 \leq n<29$, a planar lattice $L$ contains an equilateral $n$-gon if and only if $L$ is similar to an integral lattice $L'$ such that $ν(L')\equiv 3 \pmod 4$ and the largest prime factor $p$ of $ν(L')$ satisfies $p \leq n$. Moreover, such $L$ contains a convex equilateral $n$-gon, which answers a problem posed by Maehara. |
| title | Planar lattices and equilateral odd-gons |
| topic | Combinatorics Metric Geometry Number Theory 52C05, 11H06, 51M04 |
| url | https://arxiv.org/abs/2503.01911 |