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 .

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).

References

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

Kitson, Adrian R. (2006). "The prime analog of the Kepler–Bouwkamp constant". arXiv:math/0608186 [math.HO].
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.

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

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Kepler–Bouwkamp constant

Numerical value

See also

References