Many years ago I posted this on sci.physics. It turns out that the value of the Inverse Fine Structure Constant (a dimensionless parameter which can be experimentally measured as about 137.036 but for which no theory of physics yet predicts a value) is remarkably close to (alpha^2)(pi^2)(pi^pi-1)/16 where alpha is the second Feigenbaum constant, about 2.502907875096. This formula gives a value for the IFSC of 137.0359996810.
After posting that, I got a message from James Gilson pointing out his work on the same subject - he has a different formula for the IFSC, pi/(29*cos(pi/137)*tan(pi/(137*29))), which is not quite so pretty but does have the advantage of having some geometrical justification (which I still don't completely understand). Gilson's formula gives a value for the IFSC as 137.0359997867.
Back in 2001 the most accurate measurement of the IFSC (CODATA 1999) gave a value of 137.03599976(50) (i.e. there is a 68% chance that the true value is between 137.03599926 and 137.03600026). Both the formula give answers in this range.
I thought I would revisit this briefly and see if the latest measurements were able to rule out one of both of these formulae. Work of G. Gabrielse et al in 2006 give the IFSC as 137.035999068(96), i.e. there is a 68% chance that the true value is between 137.035998972 and 137.035999164. This appears to rule out both formulae. An earlier version of the 2006 Harvard work (which was compatible with both formulae) was superceded by an erratum. I admit this post is a bit anticlimactic but when I started writing it I thought that the latest measurements ruled out Gilson's formula but not mine.