Looking for a coordinate system

I'm working with a cartestian system that has certain periodic properties I'd like to exploit with a new coordinate system, but I don't know one that would work. The trajectory of the state of the system is symmetric across non-adjacent squares (ie a checkerboard of sorts), so that [tex](x,y)[/tex] can always be contained in [tex][-a, a], [-b, b][/tex], if the following are true. Along y, the plane wraps up on itself at b, so that [tex](x,-b)=(x,b)[/tex]. For x, if the state travels beyond a, it goes back to -a, but y will also be shifted, so that [tex](a,y) = (-a,y+b)[/tex]. Note that this shift might also cause a jump in y from [tex](x,-b)=(x,b)[/tex].
So wrapping in y means I curl my cartesian into a cylinder, and the wrapping in x might change the cylinder into a torus, but it would have to be twisted somehow so that [tex](a,y) = (a,y+b) [/tex], which toroidal coordinates wouldn't allow(?). I'm not really sure what to search for. Suggestions?


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2018 Award
Coordinates with periodicity aren't unique anymore and so no longer coordinates. Mathematicians solve this problem by using local coordinates and patch them, i.e. consider your surface as a manifold with an atlas.

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