Kepler–Bouwkamp constant

In plane geometry, the Kepler–Bouwkamp constant (or polygon inscribing constant) is obtained as a limit of the following sequence. Take a circle of radius 1. Inscribe a regular triangle in this circle. Inscribe a circle in this triangle. Inscribe a square in it. Inscribe a circle, regular pentagon, circle, regular hexagon and so forth. The radius of the limiting circle is called the Kepler–Bouwkamp constant.^[1] It is named after Johannes Kepler and Christoffel Bouwkamp [de], and is the inverse of the polygon circumscribing constant.

A sequence of inscribed polygons and circles

Numerical value

The decimal expansion of the Kepler–Bouwkamp constant is (sequence A085365 in the OEIS)

${\displaystyle \prod _{k=3}^{\infty }\cos \left({\frac {\pi }{k}}\right)=0.1149420448\dots .}$

The natural logarithm of the Kepler-Bouwkamp constant is given by

${\displaystyle -2\sum _{k=1}^{\infty }{\frac {2^{2k}-1}{2k}}\zeta (2k)\left(\zeta (2k)-1-{\frac {1}{2^{2k}}}\right)}$

where ${\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}}$ is the Riemann zeta function.

If the product is taken over the odd primes, the constant

${\displaystyle \prod _{k=3,5,7,11,13,17,\ldots }\cos \left({\frac {\pi }{k}}\right)=0.312832\ldots }$

is obtained (sequence A131671 in the OEIS).