Can an equilateral triangle have vertices at integral lattice points?
Integral lattice points are such points as (101, 254) or (3453, 12), but not points such as (123.4, 1) or (√2, 5)
If you can't find a solution in the 2D Cartesian plane, can you find one in a 3 (or more) dimensional space?
As Richard said, (1,0,0), (0,1,0), and (0,0,1) are vertices of an equilateral triangle. If we add to this list (1,1,1), we get four vertices of a regular tetrahedron with integral lattice points as vertices!
Makes me wonder. What platonic solids can be placed in 3space so that their vertices are integral lattice points? Obviously the cube works, and, from the above, the tetrahedron works. I suspect that even other platonic solids also work.

Posted by McWorter
on 20050303 22:26:40 