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 for some integers . We also show a similar characterization for the sides of a regular tetrahedron in : such a tetrahedron exists if and only if the sides are of the form , for some . The classification of all the equilateral triangles in 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.