Polygon Circumscribing
Circumscribe a triangle about a circle, another circle around the triangle, a square outside the circle, another circle outside the square, and so on. The circumradius and inradius for an n-gon are then related by
| [画像: r=Rcos(pi/n), ] |
(1)
|
so an infinitely nested set of circumscribed polygons and circles has
Kasner and Newman (1989) and Haber (1964) state that K=12, but this is incorrect, and the actual answer is
| K=8.700036625... |
(5)
|
(OEIS A051762).
By writing
![画像: K=exp[sum_(n=3)^inftylnsec(pi/n)],](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/PolygonCircumscribing/NumberedEquation3.svg){kind=link}
it is possible to expand the series about infinity, change the order of summation, do the n sum symbolically, and obtain the quickly converging series
![画像: K=exp{sum_(k=1)^infty((4^k-1)zeta(2k)[4^k(zeta(2k)-1)-1])/(4^kk)},](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/PolygonCircumscribing/NumberedEquation4.svg){kind=link}
where zeta(s) is the Riemann zeta function.
Bouwkamp (1965) produced the following infinite product formulas for the constant,
![画像:pi/2{product_(m=1)^(infty)product_(n=1)^(infty)[1-1/(m^2(n+1/2)^2)]}^(-1)](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/PolygonCircumscribing/Inline16.svg){kind=link}
![画像:1/2piproduct_(n=1)^(infty)[sinc((2pi)/(2n+1))]^(-1)](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/PolygonCircumscribing/Inline19.svg){kind=link}
![画像:6exp{sum_(k=1)^(infty)([lambda(2k)-1]2^(2k)[zeta(2k)-1-2^(-2k)])/k},](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/PolygonCircumscribing/Inline22.svg){kind=link}
where sinc(x) is the sinc function (cf. Prudnikov et al. 1986, p. 757), zeta(x) is the Riemann zeta function, and lambda(x)=(1-2^(-x))zeta(x) is the Dirichlet lambda function. Bouwkamp (1965) also produced the formula with accelerated convergence
| K=1/(12)sqrt(6)pi^4(1-1/2pi^2+1/(24)pi^4)(1-1/8pi^2+1/(384)pi^4)csc((pi^2)/(sqrt(6+2sqrt(3))))csc((pi^2)/(sqrt(6-2sqrt(3))))B, |
(11)
|
where
(cited in Pickover 1995).
See also
Infinite Product, Nested Polygon, Polygon Inscribing, WhirlExplore with Wolfram|Alpha
More things to try:
References
Bouwkamp, C. "An Infinite Product." Indag. Math. 27, 40-46, 1965.Chatterji, M. "Product[Cos[Pi/n], n,3,infinity]." http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap102.html.Finch, S. R. "Kepler-Bouwkamp Constant." §6.3 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 428-429, 2003.Haber, H. "Das Mathematische Kabinett." Bild der Wissenschaft 2, 73, Apr. 1964.Hamming, R. W. Numerical Methods for Scientists and Engineers, 2nd ed. New York: Dover, pp. 193-194, 1986.Kasner, E. and Newman, J. R. Mathematics and the Imagination. Redmond, WA: Microsoft Press, pp. 311-312, 1989.Pappas, T. "Infinity & Limits." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 180, 1989.Pickover, C. A. "Infinitely Exploding Circles." Ch. 18 in Keys to Infinity. New York: W. H. Freeman, pp. 147-151, 1995.Pinkham, R. S. "Mathematics and Modern Technology." Amer. Math. Monthly 103, 539-545, 1996.Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 1: Elementary Functions. New York: Gordon & Breach, 1986.Sloane, N. J. A. Sequence A051762 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Polygon CircumscribingCite this as:
Weisstein, Eric W. "Polygon Circumscribing." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PolygonCircumscribing.html