Speaking intuitively, I can visualize a lot of surfaces in my mind; but it seems that some of the ones I can imagine are not capable of being described by the 'usual suspects', i.e., elementary numerical expressions (roots, trigonometric functions, exponentials, etc.). Is there some way, or some branch of mathematics, that would allow one to characterize exactly which surfaces we can describe through formulas and which we cannot?
Addition: And more generally, can we decide exactly which figures (i.e., any type of geometrical entity) can be described in closed form? [This sort of reminds of the theory of which functions, when integrated, can be expressed in a closed elementary form.]