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

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

limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

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 called ...

. Take a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...

of radius 1. Inscribe a

regular triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each ...

in this circle. Inscribe a circle in this triangle. Inscribe a

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

in it. Inscribe a circle, regular pentagon, circle,

regular hexagon In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexa ...

and so forth. The

radius In classical geometry, a radius (plural, : radii) 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 name comes from the latin ''radius'', ...

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 philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...

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 \operatorname(s) > ...

. 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