Where does Pac-Man live?

Here’s a question I’ve always liked: what kind of world does Pac-Man on?

We know we live on a sphere, or course, or something close to it. But we often imagine our world on a rectangular map. What are the rules? Well, if you go out the left side, you come back on the right side. If you go to the top or bottom, you arrive a single point (the north or south pole). So following the same logic, what are the rules in Pac-Man’s world? If he goes out the left side or his world, he comes back in the right side. But—if he goes out the top, he comes back in the bottom. Our map represents a sphere. What shape is he living on?

You could also think of it like this: take a stretchy square of paper, glue the left side to the right side and the top to the bottom. What do you get?

It turns out that this shape is of great mathematical interest. See if you can figure it out. If you get stuck, there’s a nice video here.