WDS>Noticing 1 and 11 form a multiplicative subgroup mod 120, I tried Ries on (11/120)! * (1/120)! = 0.95023643380439452766545957659649988150532 and (11/120)! / (1/120)! = 0.95931292205678601103043910565064773096559 but it failed to find anything I liked. A certain proportion of Ries formulas seem to be "tasteless" and the sort of thing that are never going to happen, like using (e^pi) or 1/ln(pi) or e or sin(1) in an exponent... ---------------------------------------- In:= FindRoot[EllipticTheta[2, 0, x] == Gamma[1/4]/(2*Pi)^(3/4), {x, 0},WorkingPrecision -> 22] Out= {x -> 0.04321391826377225235322} billgosper$ ./ries .04321391826377225235322 Your target value: T = 0.0432139182637723 www.mrob.com/ries ln(x) = -pi (exact match) {49} pi,/x = 1/e for x = T + 2.08167e-17 {59} (Stopping now because best match is within 1e-15 of target value.) e = base of natural logarithms, 2.71828... A,/B = Ath root of B ln(x) = natural logarithm or log base e pi = 3.14159... --LHS-- --RHS-- -Total- max complexity: 36 25 61 dead-ends: 1357 550 1907 CPU time: 0.000 expressions: 41 44 85 distinct: 39 39 78 Memory: 64KiB Total equations tested: 1521 (1521) Also, a rhumb line from the south pole to the north pole of the unit sphere at a constant bearing a east of north has parametric (x,y,z) = (sech(w) * cos(tan(a) * w), sech(w) * sin(tan(a) * w), tanh(w)) -oo<w<oo. As w->oo, the distance from the north pole reduces by a factor approaching e^(2 π/tan(a)) for each circuit of the pole (2π change in longitude). For due northeast, a=π/4. (Since e^(2π) > 535, when you try to plot this spiral, you just see a curve that appears to reach the pole in about one orbit.) --rwg But things like sin(1) and cos(2) are exceptionally rare, so sin and cos should maybe be replaced with sin pi # and cos pi #. Irony: tan(1) is a bit more interesting due its continued fraction, but we'd need (sinpi 1/pi)/(cospi 1/pi) ! Also there's that pi,/4 limit at the beginning of http://www.tweedledum.com/rwg/idents.htm