Is there a regular hexagon with integral corners? I'm looking for a regular hexagon in $\mathbb{R}^2$, whose corners are integral, i.e. the coordinates are integers.
The hexagon cannot lie "flat" (with upper and lower line segments horizontal), since then $h = \frac{\sqrt{3}}{2} w$ with $w$ width and $h$ height of the hexagon, making the ratio $\frac{h}{w}$ irrational.

Yes we can have a regular hexagon with integral vettex coordinates ... in three dimensions.  Think of a face-centered cubic lattice.  If we set the length unit to half the edge of a cubic unit cell then all lattice points have integer coordinates -- including those forming a hexagonal close-packed plane perpendicular to any body diagonal of the cube.
