[gerfowhn] Disc manifold

First, consider a 2D game character in a square universe.  The character goes out the top of the square and reappears at the bottom (similarly sides).  This is the manifold of a flat torus.  This idea can also easily be generalized to a rectangle.

It can also be generalized to any shape that tiles the plane, like the regular hexagon, though the shape is subject to some conditions.  What conditions?  Tiling by one translation only?  (Also consider higher dimensions and tiling non-Euclidean manifolds.)  We would like conservation of momentum, though momentum might not make sense when non-Euclidean.

I think it can be generalized to any even-sided polygon whose every pair of opposite edges are parallel and congruent.  A moving point goes through one edge and reappears through the opposite edge.  Velocity vector remains unchanged.  I'm not sure if a collection of points, e.g., a game character of positive area, will retain its shape if some of its points go through one edge but others go through other edges, i.e., traveling through a corner.  It might be interesting the way a compact collection of points initially moving together becomes shredded as they travel through different edges.

It can also then be generalized to a circle or disc.  A moving point exits the disc at one point on the circumference and reappears at the diametrically opposite point on the circumference.  This seems too good to be true.  A circle is a polygon with an even infinite number of sides.

It can also then be generalized to any shape with point symmetry.  Does it have to be convex?  Does it have to be connected?