Embeddings of Equilateral Polygons in Unit Lattices

Abstract

It is well-known that a regular n-gon can be embedded in the unit lattice of ℝ^m if and only if m ≥ 2 and n = 4; or m ≥ 3 and n = 3 or 6. In this paper we consider equilateral polygons that can be embedded in the unit lattice of ℝ^k. These are called lattice equilateral polygons. We show that for any ε > 0, there exists a lattice equilateral 2n-gon in ℝ² such that the difference between the values of the maximum internal angle and the minimum internal angle is less than ε. We also show a similar result for lattice equilateral 3n-gons in ℝ³ and other related results.