Embeddings of equilateral polygons in unit lattices

Abstract

It is well-known that a regular n-gon can be embedded inthe unit lattice of Rm if and only if m ≥ 2 and n = 4; or m ≥ 3 andn = 3 or 6. In this paper we consider equilateral polygons that can beembedded in the unit lattice of Rk. These are called lattice equilateralpolygons. We show that for any ε > 0, there exists a lattice equilateral2n-gon in R2 such that the difference between the values of the maximuminternal angle and the minimum internal angle is less than ε. We alsoshow a similar result for lattice equilateral 3n-gons in R3 and otherrelated results.