Equilateral n-gons in planar integer lattices

TL;DR

This paper proves that for certain rectangular integer lattices, there exist equilateral polygons of all sufficiently large sizes, extending the classification of such polygons in planar lattices.

Contribution

It demonstrates that for lattices with specific parameters, equilateral polygons of all large enough sizes exist, building on prior partial results.

Findings

Existence of equilateral n-gons for large n in specific lattices

Extension of previous classifications of equilateral polygons

Identification of conditions on lattice parameters for polygon existence

Abstract

We study the existence of equilateral polygons in planar integer lattices. Maehara showed that it's sufficient to work with rectangular lattices $\Lambda(m) = L[(1,0),(0,\sqrt{m})]$ with $m \equiv 3 \pmod{4}$. Building on results of Maehara and of Iino and Sakiyama, we show that for every such $m$ there exists $N$ such that for all $n \geq N$, the lattice $\Lambda(m)$ contains an equilateral $n$-gon. This extends previous classifications of equilateral polygons in planar lattices.