So I was thinking and I was wondering if we could have a set of vectors that spanned just the unit sphere, and nothing else beyond that. So, if we replace euclid's 5th postulate to give us spherical geometry, a line is a circle on the surface of some sphere. If we have two perpendicular vectors (or two linearly independent) vectors in our spherical geometry, then it would seem that the whole circle of radius r is spanned. If we combine this and do it for all real numbers x such that 0<x<r and add the orgin, we will get a basis for the whole sphere which is a subset of the euclidean plane. Alternatively, I can imagine we could've done the same thing with complex multiplication. I was wondering: 1) Does this actually work? 2) If it does work, what kind of geometrical shapes can you form a basis for?