Journal of Integer Sequences
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 p 2(m2 − mn + n2) 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 √ 2, for some k ∈ N. The classification of all the equilateral triangles in Z 3 contained in a given plane is studied and the beginning analysis for small side lengths is included. A more general parametrization is proven under special assumptions. Some related questions about the exceptional situation are formulated in the end.
Ionascu, Eugen J., "A parametrization of equilateral triangles having integer coordinates." (2007). Faculty Bibliography. 875.