Why is the Asteroids world not a sphere ? Topology of a virtual surface : sphere vs torus

Question

I'm reading some basic topology and I have encountered several times examples involving video games. In some of these games -as Asteroids, often used as an example- a sprite can go through one of the four sides of the screen and appear on the opposed side, at the same speed, heading the same direction. I read then that, while you could think this world is a sphere, it is actually a torus. This statement is explained starting with a sheet of paper and (kind of) rolling it two ways. I certainly understand how it works on a torus but I can't figure out clearly why the Asteroids world can't be a sphere.

Answer 1

Here's a test: on a sphere, draw any closed curved. You can shrink the curve until it collapses to a point. There's no way for the curve to "snag" and be unable to shrink.

On the other hand, on the torus, if you draw a curve that circles around the torus (like a rope would be tied to a life preserver) and it's impossible to shrink it to a point: you can't make the curve any shorter than the radius of the circle making up the cross-sections of the torus.

Now consider asteroids world. If you draw a curve that stays within the screen, you can clearly shrink it to a point. But now draw a curve which starts in the center of the screen, hits the left side, teleports to the right side, and then closes up at the center. There's no way you can shrink it to a point: you can straighten it, so that it is a perfectly horizontal line, and this makes the curve as short as possible, but you can't make it any shorter. You can try to slide the curve up and down, and even drag it up past the top of the screen so that it teleports to the bottom, but no matter what you do you can't "unsnag" the curve from wrapping around the left and right sides of the screen. This tells you that the asteroid world cannot possibly have the same topology as the sphere, where shrinking is always possible.