On 8/25/09, Fred lunnon <fred.lunnon@gmail.com> wrote:
... Thus f_5 in 3 sec on my ancient Mac G4; though f_6 with an estimated 10^6 terms remains out of reach, leaving (you will be desolate to hear) the direct verification of planarity for the inner corners of any general polytore family likewise unattainable.
More twaddle from the apparently inexhaustible fountain of elementary incompetence which this fine problem has afforded me! The "W" corners of the polytore (i.e. not corners of the initial cuboid) are already feasible using f_4, since two pairs of the 6 angles there are equal; so we just substitute a_i -> 4 a_i (1-a_i), by the addition law for sines, and get on with it. After 5 minutes out pops a numerator of total degree 44 in q,h, which factors into --- among other things --- (oval branch) (cusp branch)^2 --- phew! Intriguingly, besides c^2 h^2 there are 3 other factors, of degree > 4, which of course must have no (new) real roots in 0 < q < 1, otherwise total angular defect from both U/V and W corners would be nonzero. Is this the first time the Gauss-Bonnet theorem has been employed to find roots of algebraic equations, I wonder? Fred Lunnon