  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.
