Kepler–Bouwkamp constant explained

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, and is the inverse of the polygon circumscribing constant.

Numerical value

The decimal expansion of the Kepler–Bouwkamp constant is

\prod ^{inf ty} _{k =3} \cos\left( (\pi)/(k\right)) = 0 . 1149420448... .

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

-2 \sum ^{inf ty (2 ^{2k} -1)/(2 k)} _{k =1} \zeta( 2 k)\left(\zeta( 2 k) -1- (1)/(2 ^{2k}) \right)

where

\zeta(s) = \sum ^{inf ty} _{n =1} (1)/(n ^{s}) is the Riemann zeta function.

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

\prod _{k=3,5,7,11,13,17,\ldots} \cos\left( (\pi)/(k\right)) = 0 . 312832 \ldots is obtained .

Further reading

• Kitson . Adrian R. . 2006 . The prime analog of the Kepler–Bouwkamp constant . math/0608186.
• Kitson . Adrian R. . 2008 . The prime analogue of the Kepler-Bouwkamp constant . The Mathematical Gazette . 92 . 293 . 10.1017/S0025557200183214. 117950145 .
• Doslic . Tomislav . 2014 . Kepler-Bouwkamp radius of combinatorial sequences . Journal of Integer Sequence . 17 . 14.11.3.

Notes and References

1. Book: S. R. . Finch . Mathematical Constants . registration . Cambridge University Press . 2003. 9780521818056 . 2003519.