TITLE: Regular $n$-gon in the plane with vertices on integers?
QUESTION [12 upvotes]: For which $n \geq 3$ is it possible to draw a regular $n$-gon in the plane ($\mathbb{R}^2$) such that all vertices have integer coordinates? I figured out that $n=3$ is not possible. Is $n=4$ the only possibility?

REPLY [2 votes]: For n>6 it's easy. Fix n>6, pick the smallest regular n-gon with integer coordinates. Move the edges without rotating so that they form a star in (0,0). The endpoints now are on integer coordinates and form a regular n-gon which is smaller than the smallest. That's impossible, so no such thing exists.  For 5 and 6 you need an additional argument with mirroring etc, not too difficult.