Planar lattices and equilateral odd-gons

TL;DR

The paper addresses when a planar lattice $L$ contains an equilateral $n$-gon for odd $n$, reducing the problem to the arithmetic of the square-free part $\nu(L)$ of the area of a fundamental parallelogram. Using the rectangular lattice $\Lambda(m)$ and explicit edge-vector constructions, it derives that a necessary condition is $\nu(L) \equiv 3 \pmod 4$ and that the largest prime factor $p$ of $\nu(L)$ satisfies $p \le n$, with sufficiency established for $3 \le n < 17$ and computational evidence extending to $n < 29$, leading to an equivalence among existence of an equilateral $n$-gon, convex existence for all larger $k$, and similarity to a lattice with the stated $\nu$-condition. This yields a concrete, checkable criterion for small odd $n$ and answers Maehara's convexity question in these cases, while leaving the general case for future work. The results connect geometric polygon existence on planar lattices to number-theoretic properties of $D(L)$ and support algorithmic verification via $\nu(L)$ and $p$.

Abstract

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.

Theorems & Definitions (17)

• Theorem 1.1: Ball
• Theorem 1.2: Maehara
• Theorem 1.3: Maehara
• Theorem 1.4: Maehara
• Lemma 2.1: Educ
• Lemma 2.2: Maehara
• Lemma 2.3: Maehara
• Theorem 3.1
• proof
• Proposition 3.2