A parametrization of equilateral triangles having integer coordinates

TL;DR

This paper characterizes the side lengths of equilateral triangles and regular tetrahedra with integer coordinates in three-dimensional space, providing explicit formulas and classifications for their existence.

Contribution

It offers a new parametrization for equilateral triangles and tetrahedra with integer coordinates, extending previous understanding of such geometric figures in lattice points.

Findings

Equilateral triangles exist if and only if side lengths are of the form √(2(m^2 - mn + n^2)).

Regular tetrahedra exist if and only if sides are multiples of √2.

Classification of equilateral triangles in a given plane is initiated.

Abstract

We study the existence of equilateral triangles of given side lengths and with integer coordinates in dimension three. We show that such a triangle exists if and only if their side lengths are of the form $\sqrt{2(m^2-mn+n^2)}$ for some integers $m,n$. We also show a similar characterization for the sides of a regular tetrahedron in $\Z^3$: such a tetrahedron exists if and only if the sides are of the form $k\sqrt{2}$, for some $k\in\N$. The classification of all the equilateral triangles in $\Z^3$ contained in a given plane is studied and the beginning analysis is presented. A more general parametrization is proven under a special assumption. Some related questions are stated in the end.