# Kepler–Bouwkamp constant

A sequence of inscribed polygons and circles.

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 (Finch, 2003), it is the inverse of the polygon circumscribing constant.

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

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).

• Kitson, Adrian R. (2006). "The prime analog of the Kepler–Bouwkamp constant". arXiv:math/0608186 |class= ignored (help).