[kennon]

Very cool, but I'm still a little confused. The earth's surface more or less just extends out in two dimensions, but doesn't space extend out in three? How does all of this work when you have take into account the z-axis as well as x and y? How would a three-dimensional object "close" in around itself?

[InclinedPlane]

There are different kinds of dimensions. There are simple, Euclidian dimensions where each dimension is perfectly orthogonal to the others, where parallel lines never intersect, etc. But there are other possibilities, non-Euclidean spaces and volumes. This is where the example of the sphere comes in. If you were a 2-dimensional being you could be on a plane or you could be on the surface of a sphere. Similarly, if you are a 3-dimensional being, such as us, you could be within a simple R^3 volume, or you could be on a hyper-sphere or another non-euclidian volume. The math is a little more complex, but similar.

[jmmcd]

To imagine a 2d surface closing in on itself, you imagine it in 3d. To imagine a 3d object closing in on itself, you unfortunately have to try to imagine it in 4d.

[Splines]

I read Flatland a long time ago as a student and found it enlightening when thinking about 4d.

https://en.wikipedia.org/wiki/Flatland