On Thu, Dec 27, 2012 at 3:37 PM, Robert Hansen <email@example.com> wrote: > > On Dec 27, 2012, at 6:24 PM, kirby urner <firstname.lastname@example.org> wrote: > > The limit (360 - v) really could be 0 if perfect flatness is allowed, > but when we add the constraint of curvature, then even though > everything locally seems to stay the same -- given the epsilon / delta > treatment alone (Debater A) -- the added fact of curvature is new > information and tells us that as n --> infinity, there's a 720 > involved (Debater B). > > > 1. You don't have to allow flatness. The limit(360-v) is indeed 0. It is a > limit. The function doesn't have to actually get there.
In fact we can't allow it to get there, because if flatness obtains, then the 720, i.e. the subtracted tetrahedron, goes away.
By "subtracted tetrahedron" I mean that a network of triangles, if wrapping around and meeting up with itself, has one tetrahedron's worth of degrees missing from its vertexes i.e. each is 360 - an iotum (an epsilon) from each one.
We're like saying "flatness" is the limit of ever more shallow curvature, but at the limit we lose curvature i.e. a curve is not flat.
The contradiction is merely grammatical: ever shallower curvature approaches zero curvature, but a curve never has zero curvature or it wouldn't be a curve.
Just calling it a curve means it's not straight. Just saying it's a geodesic sphere means 720 degrees have been subtracted already, and that's never going to change.
> 2. The limit(n*(360-v)) is indeed 720, and again, allowing flatness is not > an issue.
Allowing flatness is an issue, as the moment the network becomes flat, we loose our 720. The 720 is there up to the instant of perfect flatness. At that point, it goes away. But then so does the fact of this being a geodesic sphere, which was stated as a given.
In a discrete mathematics, we let f (1,2,3...) have an upper bound, beyond which we detect no sensible difference.
I'd saydiscrete mathematics is a more comfortable setting for an n-frequency icosa-sphere because our "surface" is all "windows" with edges for frames. Per Cornwall, polyhedrons have become more ephemeral since the "marble bowling ball" days.
The issue of limits is less all-important, whereas the calc heads are raving loons where limits are concerned (relatively speaking).
Their assumptions allow for "perfect continuity" but we could simply rule that out as an option, as an axiom. "Perfect spheres" are defined to not exist, period. Don't need 'em, don't want 'em.
That's the "ideal" one approaches in calculus, as edges e become relatively less (in length) compared to a sphere diameter. Ultimately there's no room for edges as the points become adjacent in a perfectly continuous surface. The points are no more "set apart by gaps" than those on the real number line.
High frequency geodesics fit the bill in place of "perfect" spheres, i.e. they're close enough for whatever role (as in CG SFX). Nor do we need "real numbers" come to think of it. We'll stick with whatever numbers our physical computers manage to get by with. Why not? Floats, extended precision, whatever types you like (and have implemented).
> 3. Now state your contradiction. >
The contradiction went away as soon as I let frequency be finite. The contradiction: how could a curve ever be flat, even at a limit? It's not a curve if it's flat.
Likewise, how could a geodesic sphere approach perfect flatness as f increases, given the subtracted tetrahedron is not coming back?
Answer: we don't need the concept of "limit" to shape the question in the first place. This isn't a math language where "perfect continuity" is even defined, let alone necessary. This isn't calculus. So what? Most of math isn't.