```Date: Dec 27, 2012 7:35 PM
Author: kirby urner
Subject: Re: A Point on Understanding

On Thu, Dec 27, 2012 at 3:37 PM, Robert Hansen <bob@rsccore.com> wrote:>> On Dec 27, 2012, at 6:24 PM, kirby urner <kirby.urner@gmail.com> 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, ifwrapping around and meeting up with itself, has one tetrahedron'sworth 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 shallowcurvature, but at the limit we lose curvature i.e. a curve is notflat.The contradiction is merely grammatical:  ever shallower curvatureapproaches zero curvature, but a curve never has zero curvature or itwouldn't be a curve.Just calling it a curve means it's not straight.  Just saying it's ageodesic sphere means 720 degrees have been subtracted already, andthat'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 perfectflatness.  At that point, it goes away.  But then so does the fact ofthis 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 ann-frequency icosa-sphere because our "surface" is all "windows" withedges for frames.  Per Cornwall, polyhedrons have become moreephemeral since the "marble bowling ball" days.The issue of limits is less all-important, whereas the calc heads areraving loons where limits are concerned (relatively speaking).Their assumptions allow for "perfect continuity" but we could simplyrule that out as an option, as an axiom.  "Perfect spheres" aredefined to not exist, period.  Don't need 'em, don't want 'em.That's the "ideal" one approaches in calculus, as edges e becomerelatively less (in length) compared to a sphere diameter.  Ultimatelythere's no room for edges as the points become adjacent in a perfectlycontinuous surface.  The points are no more "set apart by gaps" thanthose 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 weneed "real numbers" come to think of it.  We'll stick with whatevernumbers our physical computers manage to get by with.  Why not?Floats, extended precision, whatever types you like (and haveimplemented).> 3. Now state your contradiction.>The contradiction went away as soon as I let frequency be finite.  Thecontradiction:  how could a curve ever be flat, even at a limit?  It'snot a curve if it's flat.Likewise, how could a geodesic sphere approach perfect flatness as fincreases, given the subtracted tetrahedron is not coming back?Answer:  we don't need the concept of "limit" to shape the question inthe first place.  This isn't a math language where "perfectcontinuity" is even defined, let alone necessary.  This isn'tcalculus.  So what?  Most of math isn't.Kirby> Bob Hansen
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