So, if you take a rectangular area XY and wrap the borders up "video game style" (i.e go off the left side, reappear on the right side, go off the top, reappear at the bottom), you get a 2D surface that can be represented by a 3D torus, right? Right. Now, there're two ways you can make a wrap-around screen into a torus - vertical or horizontal. You can join the top/bottom THEN the sides, which gets you a horizontal torus, or you can join the sides THEN the top/bottom, which gets you a vertical torus. This seems kind of counter-inutitive to me - that you can get to two states that are equivalent but forever distinct. Questions: 1] Are these two shapes topologically interchangeable? Can you start with one, and get to the other without tearing the surface? 2] Is there an intermediate/generic/more symmetical shape that exists halfway between those two? 3] Are there other foldings that give the same wraparound result? What I'm trying to get at is that, from the point of view of a Flatlander living on the torus, is it possible for him to state (or be told) that he is living on a "vertical torus", not a "horizontal torus", and ne'er the twain shall meet?