Quote by Hurkyl
It's actually pretty easy to see that it does work. Pick any horizontal line: where does the geodesic pierce it?

Ahh yes so simple, thanks.
Now actually, thinking about it a bit, this is a very roundabout way of showing that the cardinality of RxR is equal to the cardinality of R. This torus construction is directly analogous to the diagonal counting used when you show ZxZ = Z. EDIT: Scratch that, I can only show it is dense...