The $d$-dimensional "integer lattice", $\mathbb{Z}^d$, is the set of points in $\mathbb{R}^d$ with integer coordinates.
First of all, for $n\neq 4$, a regular $n$-gon is impossible to place so that its vertices lie in $\mathbb{Z}^2$, the integer lattice in the plane. But what about the 3D space?
I have examples for regular triangle, hexagon, and square, but I think for the other $n$s, it is not possible. Why? Or are there examples for regular heptagons?