I was lying awake the other night and thinking about Pi and flatlanders. I haven't done a lot of topology reading, so forgive my naivete. Pi on a flat surface is a number we know well, but what happens to the ratio of a circle's diameter to its circumference on curved surfaces? First question: it's not called Pi, is it? Pi is a constant. For the prupose of this discussion I'm going to call it rho, but pleasecorrect me if there is already a term for it. As a surface curves positively, rho will decrease. The diameter may be constant while the circumference contracts. If the surface contracts into a sphere, then rho becomes 2 (diameter = to half the circumference). It can never become less than two because you wouldn't be able to fit that fixed sized circle on the sphere. Of course, this is only true of the surface curved evenly in all directions. Presumably, there are shapes, such as an elliptoid where you could have a high curvature in one direction but low in the other. On a positively curved surface, I suspect that rho would have an upper limit of pi - the surface approaches a Euclidean plane. If the surface had a negative curvature, I think you wouldn't have such problems - you should be able to fit a circle on a negatively curved surface. I think. I imagine there would be surfaces where you could not make a continuous line with all points equidistant from a central point. What happens on a toroid? Is it possible that rho wold not exist i.e. no line could be drawn that is equidistant from a point? Ah. A toroid is an example of negatively-curved surface. Maybe I should do a little reading.