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
![$\sqrt{2(m^2-mn+n^2)}$](abs/img1.gif)
for some integers
![$m,n$](abs/img2.gif)
. We also
show a similar characterization for the sides of a regular
tetrahedron in
![$\mathbb Z^3$](abs/img3.gif)
: such a tetrahedron exists if and only
if the sides are of the form
![$k\sqrt{2}$](abs/img4.gif)
, for some
![$k\in\mathbb N$](abs/img5.gif)
.
The classification of all the equilateral triangles in
![$\mathbb Z^3$](abs/img3.gif)
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.