On 8/21/09, Fred lunnon <fred.lunnon@gmail.com> wrote:
A postscript --- it took some time to dawn on me that the constant coefficient of the polynomial in SABC is actually the square of 4*SA*SB*SC + SA^2 + SB^2 + SC^2 - 2*(SA*SB + SA*SC + SB*SC), and the vanishing of this polynomial is equivalent to A+B+C = k \pi.
This considerably reduces the computational cost of proving the h versus q relations for a (cuboid) polytore; the bad news is that it fatally entices one (me) into attempting to find a similar relation for polytores constructed from prisms based on regular even n-gons, rather than just square cuboids (n = 4).
And in a different --- doubtless equal fatal --- direction, into posing this generalisation: given 4 angles A,B,C,D, what polynomial in the squares of their sines gives a condition (necessary only, since the signs remain unspecified) that A+B+C+D = k \pi ? WFL