Mapping the Hypersphere

So I was at work, the other day, and randomly contemplating the fifth postulate and the ramifications that would have on the nature of space and FTL methods... yeah it was a weird day.

Anyway, so I was like, okay, what if space is a sphere, right? We're on the surface of a giant sphere. So maybe you could cut across it?... but that kind of stopped making sense, because clearly space is three-dimensional. So how could we be on the surface of a sphere?

And then I remembered the lessons learned from Flatland, and realized - aha, we are effectively a flat surface from a 4-dimensional perspective (and by "effectively," I mean we could be on the surface of a 4-dimensional sphere)! But then I had an "oh crap" moment when I realized that I was unaware of any useful ways of mapping a 4-dimensional space (because honestly, is that a skill that comes up all that often?).

Then I remembered some things that I read awhile ago about mapping, using charts and atlases and such. See an atlas is composed of a number of maps that overlap to display the entire area you're mapping, if the shape of the thing is not a thing you can map on a 2-d surface - much like the surface of the earth. I don't honestly remember why I came across it - probably something to do with Minkowski spaces, that's a thing that comes up a lot with my random studying - but it was mildly useful, so huzzah for that.

So I pondered - hmm. Could you do much the same with 4-dimensional spaces, by using 3-dimensional maps that overlap, that - when taken together - show you the entire thing?

Of course, it should be possible to map a 3-dimensional space (even when taking interior space into account, unlike with maps of the Earth) with an atlas of 2-dimensional maps.

Which thus leads me to the conclusion that it is entirely possible to map any n-dimensional object with an increasingly exponential (or maybe logarithmic, who knows) number of 2-dimensional maps.

Possibly even approaching infinite, since to map a 3-dimensional sphere, including its interior, you would theoretically have to take an infinite number of bisections. Well okay, you could theoretically take an infinite number of bisections, though the utility of that seems less than ideal to me, so it would probably be a finite number, but depending on the spaces (ha!) involved, might get unwieldy fast.

Not entirely sure how useful this thought is. But I think it serves to illustrate how my mental processes work when contemplating a problem.