If we construct the [Riemann sphere](https://en.wikipedia.org/wiki/Riemann_sphere) by taking a single point at infinity and adding it to the complex plane, then if I start with the point
$$z = 1 + i$$
in the finite part of the complex plane, and then push that point off to the top right and to infinity on a straight line at a 45 degree angle, then which side of the finite part of the complex plane does that point come back from? Is it the bottom left at a 45 degree angle? I thought that was topologically the projective plane. Is there some kind of very subtle distinction here related to the metric, or do we need to use 8 arrows instead of 4 in the fundamental rectangle? Even so, I still can't see which side a point pushed off to the top right would come back from if its not the bottom left(except maybe it comes back from the top right going the other way, but I find that very confusing, although conceptually possible).

In order to glue together the sides of the complex plane into a projective plane, you glue the top to the bottom like a mobius strip, and the left and the right like a mobius strip.

![](https://upload.wikimedia.org/wikipedia/commons/thumb/9/9b/ProjectivePlaneAsSquare.svg/240px-ProjectivePlaneAsSquare.svg.png)
If the Riemann sphere is topologically distinct from a projective plane, then how can this distinction be represented in terms of gluing together the sides of a fundamental rectangle as in the picture above?

Thanks!

(NOTE: I recently asked this question [Is the Riemann sphere really just the entire complex plane where you add a "single point" at infinity?](https://math.stackexchange.com/questions/4057880/is-the-riemann-sphere-really-just-the-entire-complex-plane-where-you-add-a-sing?noredirect=1#comment8382762_4057880) and feel like the confusion I had was resolved by a comment, and now I have a different question.)