Oh dear. Musing on the curious and only almost obvious fact that in the sin^2 criterion f_{n+1} = 0 for n+1 angles the "constant coefficient" [terms not containing the final variable] is just (f_n)^2, I was crushed to discover the following trivial algorithm for generating these polynomials. [Note that they are symmetric in the a_i for n > 2.] Define f_n (a_1, ..., a_n) = 0 just when \sum (+/-) A_i = 0 mod \pi (for some combination of signs), where a_i == sin^2(A_i) ; then f_{n+1} is given in terms of f_n by f_2 (a_1, a_2) = a_1 - a_2 ; f_{n+1} (a_0, a_1, ..., a_n) = f_n (b+c, ..., a_n) f_n (b-c, ..., a_n) , where b = a_0 - 2 a_0 a_1 + a_1 , c^2 = a_0 a_1 (1-a_0) (1-a_1) . 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. Fred Lunnon [25/08/09]