Equilateral n-gons in planar integer lattices

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 $m\equiv 3(\mathrm{mod}4)$. Building on results of Maehara and of Iino and Sakiyama, we show that for every such $m$ there exists $N$ such that for all $n\ge N$, the lattice $\Lambda (m)$ contains an equilateral $n$-gon. This extends previous classifications of equilateral polygons in planar lattices.

5 pages, 2 figures