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 of the Kepler–Bouwkamp constant

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

See also

• Johannes Kepler
• Polygon circumscribing constant

References

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