I know from this answer that hexagonal space is topologically equivalent to a torus (and therefore to square space): Wrapped hexagon topology
My question is, if we allow a regular hexagon to exist in four-dimensional space, can we fold it up so opposite sides touch, and angles, distances, and areas are not distorted on the hexagon?
No. For instance, if the opposite sides touch, then distance between a point close to the midpoint of one side and a point close to the midpoint of the other side collapses.