On 8/14/09, Fred lunnon <fred.lunnon@gmail.com> wrote:
... What's more, using Gaussian curvature (in the form of a "2-D cosine") gives a far easier way to compute the face angle sums than employed by my existing kludgy program (or my dodgy sine formula). WFL
Well, no as it turns out --- computing face angles from polyhedron edges turns out less troublesome. But any algebraic proof of Fred's identity [with appropriate ambiguity concerning which Fred is thus immortalised], to avoid introducing arcsines, must deal with cosines and sines, which are not additive; and to avoid square roots, must deal with squares of sines. So for two angles [my earlier dodgy identity should have read] SAB^2 - 2*(SA-2*SA*SB+SB)*SAB + (SA-SB)^2 = 0, with SA -> sin^2(A), SB -> sin^2(B), SAB -> sin^2(A+B). But the polytore proof needs three angles, for which the equivalent identity (according to my Maple investigations) has degree 4 in SABC, 8796 terms in SA,SB,SC --- and anyway refuses to check out numerically, let alone formally. So where does one go from here? WFL