On Sat, Dec 29, 2012 at 9:55 PM, Louis Talman <email@example.com> wrote: > On Sat, 29 Dec 2012 19:52:40 -0700, kirby urner <firstname.lastname@example.org> > wrote: > >> ...the sphere-at-the-limit >> concept does not apply. > > > The issue isn't whether the concept applies or not. > > It's "What do you mean by 'sphere-at-the-limit'?" >
I maybe should have said "perfect sphere as the limiting case" (for an algorithm that never really gets there, because it's always in terms of a network -- that sets up a tension).
I also left open the option of defining away "perfect spheres" entirely if we don't care about perfect continuity i.e. nothing really touches anything else in our preferred set of definitions. So no "limit case" to worry about. That wouldn't be calculus though so epsilon / delta need not apply.
> And, remember, you'll have to deal with the notion of vertex of a > sphere---not to mention edge and face, before you convince anyone that > you're doing anything but demonstrating that you don't have a good, > mathematical, handle on these notions. >
I see myself more in the role of a Berkeley, fostering free-wheeling / open debate so students might better appreciate the need for formalisms ala those of Cauchy and Weierstrass (credited for really getting The Calculus on a secure footing).
Those with the strong grip (secure handle) will complement my loosening with their contrastingly tight treatments.
In the meantime, lots of important concepts get introduced and we do some proofs of:
(a) Descartes' Deficit (b) V + F == E + 2 (the proof in Cromwell, as he presents it, attributed to von Staudt, is a fave) (c) V = 10 * f * f + 2 (I've done one of these -- as Coxeter exulted, it only takes high school level math)
It's a recognized pedagogical / andragogical technique: to re-create the confusions so as to re-live the satisfaction of having them dispelled.