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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)
![{\displaystyle \prod _{k=3}^{\infty }\cos \left({\frac {\pi }{k}}\right)=0.1149420448\dots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/590ab6b16125657ca213121223cbc1ffa635f402)
- The natural logarithm of the Kepler-Bouwkamp constant is given by
![{\displaystyle -2\sum _{k=1}^{\infty }{\frac {2^{2k}-1}{2k}}\zeta (2k)\left(\zeta (2k)-1-{\frac {1}{2^{2k}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bffef434262f04de6fa9f5b4353c4caf8a9ac2f)
where
is the
Riemann zeta function.
If the product is taken over the odd primes, the constant
![{\displaystyle \prod _{k=3,5,7,11,13,17,\ldots }\cos \left({\frac {\pi }{k}}\right)=0.312832\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4751f2bf0718c4da0dc0007ca43db1582ff06e17)
is obtained (sequence
A131671 in the
OEIS).
References
Further reading
External links