2D grid which is topologically equivalent to a sphere? I admit that my knowledge of topology is limited to the idea that a mug and doughnut are homomorphic since you can morph one into the other with a continuous deformation. I am a game dev working on a game which is played on a 2D grid. We were talking about how to wrap the edges of the map such that the grid is topologically equivalent to a sphere. The first idea we had was to wrap the horizontal edges so you warp from the left edge to the right edge, and you warp from the top most to the bottom edge. However, after some hand waving and a few badly folded pieces of paper, we determined that this was actually equivalent to a torus. The next idea was just to wrap the horizontal edges, but this is also a problem since there are two points on that shape which cannot be traversed at the north and south poles. The third idea was to wrap the horizontal edges as before, but have the edges at the top and bottom wrap so that if you went off the top edge, you would be warped horizontally half the grid width (modulo the grid width or something), but still be on the top edge. Same thing for the bottom edge. Now the bottom and top edges of the grid do not touch. You could travel in one direction vertically and end up exactly where you started. My intuition gathered from folding a piece of paper in half and looking at where the edges touch says this is correct, but I was wondering if there was a more rigorous way to think about this problem.
 A: Depending on your criteria, this can either be done or not done. If the edges of the grid are labelled clockwise with $a,b,c,d$ then by gluing $a$ to $b$ and $c$ to $d$, you are left with something which is topologically a sphere, but its metric properties are not those of the usual sphere (constant positive curvature). It has points which act like 'cone points' where you only have to turn $90$ degrees in order to end up facing the same way you were facing before.
If you want the metric properties of the sphere to be preserved, then this is impossible due to the grid being flat (it's a portion of Euclidean space) which has $0$ curvature, and the sphere having strictly positive (so non-zero) curvature.
A: Your intuition is wrong. Putting in a half-cycle shift doesn't change the result topologically -- you still get something that's a torus. Imagine a dog's large soft rubber chew-toy in the shape of a torus. Let it lie on a table, and slice it vertically, but only through one side, so you get something that's like a letter "C" with the top and bottom ends right next to each other. Now grab the two sides of the cut, one in each hand, and twist them while keeping their faces adjacent, adn then glue them back together with some hot-glue. Whether you twisted 5 degees, 20 degrees, or 180 degrees, you've still got a torus. That's what you've done in your description above: you've applied a twist along the glue-line. 
