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.[1] It is named after Johannes Kepler and Christoffel Bouwkamp [de], and is the inverse of the polygon circumscribing constant.

Numerical value

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

∏ k = 3 ∞ cos ⁡ ( π k ) = 0.1149420448 … . {\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
− 2 ∑ k = 1 ∞ 2 2 k − 1 2 k ζ ( 2 k ) ( ζ ( 2 k ) − 1 − 1 2 2 k ) {\displaystyle -2\sum _{k=1}^{\infty }{\frac {2^{2k}-1}{2k}}\zeta (2k)\left(\zeta (2k)-1-{\frac {1}{2^{2k}}}\right)}

where ζ ( s ) = ∑ n = 1 ∞ 1 n s {\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

∏ k = 3 , 5 , 7 , 11 , 13 , 17 , … cos ⁡ ( π k ) = 0.312832 … {\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).

References

  1. ^ Finch, S. R. (2003). Mathematical Constants. Cambridge University Press. ISBN 9780521818056. MR 2003519.

Further reading

  • Kitson, Adrian R. (2006). "The prime analog of the Kepler–Bouwkamp constant". arXiv:math/0608186.
  • Kitson, Adrian R. (2008). "The prime analogue of the Kepler-Bouwkamp constant". The Mathematical Gazette. 92: 293. doi:10.1017/S0025557200183214. S2CID 117950145.
  • Doslic, Tomislav (2014). "Kepler-Bouwkamp radius of combinatorial sequences". Journal of Integer Sequence. 17: 14.11.3.

External links

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