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)

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

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

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

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

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

\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).

See also

Johannes Kepler

References

Finch, S. R. (2003). Mathematical Constants. Cambridge University Press. MR 2003519.

Further reading

Kitson, Adrian R. (2006). "The prime analog of the Kepler–Bouwkamp constant". arXiv:math/0608186 |class= ignored (help).
Kitson, Adrian R. (2008). "The prime analogue of the Kepler-Bouwkamp constant". The Mathematical Gazette. 92: 293.
Mathar, Richard J. "Tightly circumscribed regular polygons". arXiv:1301.6293.

External links

Doslic, Tomislav (2014). "Kepler-Bouwkamp radius of combinatorial sequences". J. Int. Seq. 17 (14.11.3).

Weisstein, Eric W. "Polygon Inscribing". MathWorld.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.