Originally posted by ChronicLeakyAre you saying the angle-sum measurement, if I have it straight, is a series of triangles from the center to the edge and those interior angles sum up to 360? And if you were near a black hole, say, the angles would be maybe more than 360? Am I on the right track with your analogy?
This has me wondering, but here are some things, mostly in case I have time to think about this again and catch on to what you're saying:
Given a 2-dimensional Riemannian manifold M, let a "circle" in M be the set of points at a given distance (wrt the Riemannian metric) from a given point. It's almost 2 am and it's been some months since I messed w ...[text shortened]... ion is way better than the circumference/diameter definition, for sonhouse's purposes.
Originally posted by sonhouseBy "andgle sum measurement", I just mean the sum of the angles in a given triangles (try drawing different-sized triangles on a globe and you'll see how this changes from triangle to triangle, in general). I'm saying that, if you're trying to get a handle on pi in different geometric circumstances, perhaps it's best to define pi as the sum of the sngles in some given triangle; in Euclidean space this is equivalent to the circumference/diameter definition.
Are you saying the angle-sum measurement, if I have it straight, is a series of triangles from the center to the edge and those interior angles sum up to 360? And if you were near a black hole, say, the angles would be maybe more than 360? Am I on the right track with your analogy?
