Kepler–Bouwkamp constant

A sequence of inscribed polygons and circles.

In plane geometry, Kepler–Bouwkamp 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. Radius of the limiting circle is called the Kepler–Bouwkamp constant.

Computing Kepler–Bouwkamp constant

The Kepler–Bouwkamp constant is equal to (sequence A085365 in the OEIS)

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

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

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

is defined (OEIS: A131671).

See also

• Johannes Kepler

References

• S. R. Finch, Mathematical Constants, Cambridge University Press, 2003

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

• Kitson, Adrian R. (2008). "The prime analogue of the Kepler-Bouwkamp constant". The Mathematical Gazette. 92: 293.