> > On Dec 16, 2012, at 11:22 PM, kirby urner > <email@example.com> wrote: > > > Descartes' proved that adding the angular deficits > of all such vertexes, no matter their number, yields > a constant number, 720 degrees. Ergo Sigma (360 - v) > over all N = 720. This proves the limit at each > vertex is never zero, as every vertex contributes > some tiny "tax" or "tithe" to the invariant constant > 720. 720/N > 0. |360 - v| > 0 even as N -> infinity. > > > > > Contradiction? > > A contradiction would involve two statements, but > here there is only one, that the deficit (360 - v) > approaches zero as N increases without bound. That a > function has a limit at a point doesn't mean that the > function exists at that point. The limit of 1/x as x > increases without bound is 0, but 0 isn't in the > range of 1/x, nor is infinity in the domain, nor can > it even be in any domain. > > Bob Hansen
The two statements would be:
(a) the limit as N->infinity (where N is the number of vertexes on the sphere), is 0 (where 0 is the diff between 360 and the number of degrees surrounding a vertex). The sphere approaches a limit of perfect flatness at each point.
(b) the sum of the non-zero difference at every vertex adds to a constant 720
Whether the function is defined at the limit doesn't matter to the contradiction's being intelligible: the epsilon / delta approach suggests a real limit of 0, i.e. I can always get within your epsilon with the right delta, yet those vertexes, not just at the limit but all along the way, have a computable non-zero difference, i.e | 360 - v | can get as close to 0 as we like, but could never really be 0, even at a limit, as the constant 720 is always the sum of the differences from 0.
It would seem as if one line of reasoning is saying "at the limit, it's 0" while another is saying "zero could never be the limit as each vertex contributes a positive amount to a constant total and +n > 0."