Thread: 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 z-axis 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 eye-opening example.

If you know the solution, please don't give it way.

.