A parametrization of equilateral triangles having integer coordinates

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 ${\mathbb{Z}}^3$: such a tetrahedron exists if and only if the sides are of the form $k\sqrt{2}$, for some $k\in \mathbb{N}$. The classification of all the equilateral triangles in ${\mathbb{Z}}^3$ 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.

3 fugures, 17 pages, submitted to Integers

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