[francoi8]

I don't understand why the 4 corners of the screen make up the single hole. You could scroll the screen by 1 "square" which would change the corners which makes me feel there is nothing special about the original four corners.

[feoren]

You're right that the original 4 corners are arbitrary. The hole isn't physically present in the actual space. In fact the word "hole" comes from visualizing the space embedded in a higher space, which we know is not necessary and invites misconceptions. So let's call it a discontinuity.

The discontinuity shows up when you try to continuously map the space to the surface of a sphere. You can almost do it, except for one point. Different nearly-continuous maps have a different point of discontinuity -- it's basically your choice when doing the mapping. I think the 4 corners feels like a natural place for the discontinuity when I visualize that mapping -- and scrolling feels like selecting a different mapping -- but indeed it could be any point in the space if you visualize that mapping differently.

[contravariant]

Yeah the corner can correspond to any one point on the torus. They are all the same point, but other than that there's nothing really interesting about the corner(s).

The edges are more interesting, two of them go around the 'hole' of the 'donut' and the other two wrap 'around' the 'donut' itself (i.e. around the dough if it was an actual american style donut). There's no way to tell which is which.

These edges have the interesting property that you can't shrink them to a point (compared to say a loop on a globe which you can make smaller until its a single point). Except when the donut is not hollow in that case one of the loops becomes contractible, turning the space into the equivalent of a circle.