September 24, 2004

That math problem

John Derbyshire has posted a solution to the Nested Polygons puzzle.

No real new news. As discussed before, there is no closed form. Derb sez:

So far as I am aware, it is known only as "the polygon-inscribing constant," and is unrelated to any other mathematical constants. Not that you can't get alternative expressions for it; but they are all worse than the one you started with. Take its log, for example, and apply the series for log(cos(z)) in Abramowitz and Stegun's Handbook of Mathematical Functions:

log(cos(z)) = (1/2) z2 + (1/12) z4 + (1/45) z6 + (17/2520) z8 + (31/14175) z10 +...

If you apply that to the successive angles and add up the results, you get an infinite series involving  yes!  zeta functions. It doesn't resolve to anything simple, though. Those coefficients are derived from the Bernoulli numbers; and zeta(N), for even values of N, resolves to an expression in Bernoulli numbers and powers of pi; so you end up with a mess of Bernoulli numbers and powers of pi.

Posted by jk at September 24, 2004 09:57 AM