This is a kind of silly-sounding question I never realized puzzled me until moments ago, when I looked up the algorithm for spherical coordinates in n dimensions. In two dimensions, we have polar coordinates, consisting of r from 0 to ∞, and θ from 0 to 2π. In spherical coordinates, we have a third angle, from 0 to π, measured from the z-axis. Intuitively the reason for this is clear. So I wondered about n dimensions - is it the case that a 4-sphere requires an angle running only from 0 to π/2? Apparently not - all angles invoked for more than two dimensions run fron 0 to π. What in a mathematical sense, is special about the two-dimensional case that does not hold for higher dimensions? I'm a final-year physics undergrad; I've taken courses in multilinear algebra, multivariable calculus, and complex caluculus, and I have a casual interest in topology and advanced geometry, but basically I do not have a good grasp of anything beyond the level of, say, Spivak's "Calculus on Manifolds". EDIT: I realize this is somehow related to there being no straightforward 1d equivalent because you cannot account for parity in a continuous manner unlike higher dimensions, but that's as far as I get.