
Lattice Problem!
Hey guys I also have this question that I have no idea on!! I am greatful for any help!
Lattice pie
Is it possible to have an equilateral triangle in the plane all of whose vertices lie on the integer lattice {(m,n)  m,n are integers }? If so, what is the shortest side length possible for such a triangle? What about other regular polygons?
Thanks

Dear jessismith,
Let one of the vertex in the origin, the other is (n, m). Rotate on zaxis with 60°, so we get:
n' = cos(60)*n  sin(60)*m
m' = sin(60)*n + cos(60)*m
Since sin(60) is rational and cos(60) is irrational so n' and m' cann't be rational.

My Calculus 3 professor posed this problem to us.
Most of us came up with a proof of impossibility
. . and were quite proud of ourselves.
Then he asked, "Is there a regular tetrahedron with vertices
. . having integer coordinates?"
We worked on it over a weekend and gave up.
On Monday, he gave us the answer (Yes)
. . and an eyeopening example.
If you know the solution, please don't give it way.
.