Suppose one just wanted to draw a Moebius strip on a screen in 3D, and be able to rotate it, view it from any angle. Would this pose difficulties? (In OpenGL?) I don't know much about 3D representations of surfaces. There's something about breaking things into triangles, yes? But going with that isn't there also something in the data structure about specifying whether something is "inside" or "outside" having to do with orienting the data in a clockwise or counter-clockwise order? Would that be possible for a Moebius strip?
Clearly, 3D animation of a moebius strip has been done...https://www.youtube.com/watch?v=CMVNrxDQ5ZA
Also travelling on Klein bottles! (Note--I haven't heard the dialogue yet, my sound is effed up.)https://www.youtube.com/watch?v=sRTKSzAOBr4
Trying to think (naively, this is new to me) about the data structure for the Klien Bottle surface (to allow 3D representation and 'grid-oriented' movement). Suppose the surface was broken into unit-length "squares", and each unit has four triangles (left, forward, right, backward). For movement, the triangles would be treated "as if" each was identical in size and coplanar to interpolate the movement. However, the mapping of each quadrant-triangle to a 3D location could be more flexible to account for the various curvatures. Am assuming with the mapping that the triangles will not be the same in size nor coplanar.
Or maybe the unit would be a "diamond" with four square quadrants.
Any vehicles or objects moving on this surface would be 3D (have a top/bottom/sides), so that we can use both the transparency degradation (per surface) and the object's orientation to help determine visually where it is.
To get away from complications of 2-person games, maybe an "Asteroids" sort of game could be played on such a surface. Or such could be used as a stepping-stone project.