Kepler–Bouwkamp constant

A sequence of inscribed circles and regular polygons with increasing numbers of sides, which converge towards a circle at the limit.

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. It is named after Johannes Kepler and Template:Interlanguage link, and is the inverse of the polygon circumscribing constant.

Numerical value

Template:Unsolved The decimal expansion of the Kepler–Bouwkamp constant is Template:OEIS $$\prod _{k=3}^{\mathrm{\infty }}\cos \frac{\pi }{k}=0.1149420448\dots .$$ The natural logarithm of the Kepler-Bouwkamp constant is given by $$-2\sum _{k=1}^{\mathrm{\infty }}\frac{2^{2k}-1}{2k}\zeta (2k)(\zeta (2k)-1-\frac{1}{2^{2k}})$$

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

If the product is taken over the odd primes, the constant $$\prod _{\genfrac{}{}{0ex}{}{k=3,5,7,}{11,13,17,\dots }}\cos \frac{\pi }{k}=0.312832\dots$$ is obtained Template:OEIS.

Further reading

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External links

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