Regular polygons in non-euclidean geometries

Question

How is a regular hexagon described in elliptic and hyperbolic geometries in $\mathbb R ^3$?

Answer 1

The same way it is described in $\mathbb{R}^2$: take 6 rays in a plane, with the same base point and angles $60^\circ$ between consecutive rays; pick $r>0$ and take the 6 points that are at distance $r$ from the base point in each ray; connect up those points with consecutive segments. In elliptic geometry this will work for sufficiently small $r>0$; in euclidean and hyperbolic geometry this will work for all $r>0$.