plane geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, the Kepler–Bouwkamp constant (or polygon inscribing constant) is obtained as a limit of the following

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...

. Take a

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

of radius 1. Inscribe a regular triangle in this circle. Inscribe a circle in this triangle. Inscribe a

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

in it. Inscribe a circle,

regular pentagon In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...

, circle, regular hexagon and so forth. The

radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...

of the limiting circle is called the Kepler–Bouwkamp constant. It is named after

Johannes Kepler Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best know ...

and , and is the inverse of the polygon circumscribing constant.

Numerical value

The decimal expansion of the Kepler–Bouwkamp constant is :

\prod_^\infty \cos\left(\frac\pi k\right) = 0.1149420448\dots.

: The natural logarithm of the Kepler-Bouwkamp constant is given by :

-2\sum_^\infty\frac\zeta(2k)\left(\zeta(2k)-1-\frac\right)

where

\zeta(s) = \sum_^ \frac

is the

Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...

. If the product is taken over the odd primes, the constant :

\prod_ \cos\left(\frac\pi k\right) = 
0.312832\ldots

is obtained .

References

External links

* {{DEFAULTSORT:Kepler-Bouwkamp constant Mathematical constants Infinite products

Numerical value

References

Further reading

External links