h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

I remember quite clearly the axium about 2D vs 3D objects. On a plane the is only one line between any two given points while on three dimensional objects there could an infinite number as in a sphere.

It truly gets complex with N-dimensional objects. Physics have basically proven that space is not just 3 dimensional. Michio Kaku (MIT) wrote about Hyperdimensinality. Given the fact that we have no way of sensing anything above three dimensions and little ability to even visualize more than that where does it leave this nice theory of yours?

Well, in order to deal with higher-dimensional objects, I think you have to translate everything in abstract algebraic terms (like, an n-dimensional real vector space is just isomorphic to the set of n-tuples of real numbers). This will make objects which you cannot visualise accessible to calculations and analysis. Of course, if you need intuition, you can't but resort to 2 or 3-dimensional pictures .

In order to distinguish n-dimensional vector spaces up to homeomorphism, you use the same trick as the one explained in this entry (homotopy groups), only you have to use higher-dimensional spheres instead of a circle, which make things quite technical and unpleasant.