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Polygon Circumscribing

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PolygonCircumscribing

Polygon circumscribing is the iterative construction in which a regular polygon is circumscribed about a circle and a new circle is circumscribed about that regular polygon, with the number of sides increasing from 3 to 4, 5, ... at successive steps. The circumradius R and inradius r of a regular polygon with n sides are related by

r=Rcos(pi/n),
(1)

so the limiting ratio of the radii of the circles is

K=(r_(final circle))/(r_(initial circle))
(2)
=sec(pi/3)sec(pi/4)sec(pi/5)...
(3)
=product_(n=3)^(infty)sec(pi/n).
(4)

The approximate value K approx 12 reported by Kasner and Newman (1989) and Haber (1964) is incorrect (Pickover 1995, p. 148). The actual value is

K=8.700036625...
(5)

(OEIS A051762).

By writing

K=exp[sum_(n=3)^inftylnsec(pi/n)],
(6)

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

K=exp{sum_(k=1)^infty((4^k-1)zeta(2k)[4^k(zeta(2k)-1)-1])/(4^kk)},
(7)

where zeta(s) is the Riemann zeta function.

Bouwkamp (1965) produced the following infinite product formulas for the constant,

K=pi/2{product_(m=1)^(infty)product_(n=1)^(infty)[1-1/(m^2(n+1/2)^2)]}^(-1)
(8)
=1/2piproduct_(n=1)^(infty)[sinc((2pi)/(2n+1))]^(-1)
(9)
=6exp{sum_(k=1)^(infty)([lambda(2k)-1]2^(2k)[zeta(2k)-1-2^(-2k)])/k},
(10)

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

B=product_(n=3)^infty(1-(pi^2)/(2n^2)+(pi^4)/(24n^4))sec(pi/n)
(12)

(cited in Pickover 1995).

See also

Infinite Product, Nested Polygon, Polygon Inscribing, Whirl

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References

Bouwkamp, C. J. "An Infinite Product." Indag. Math. 27, 40-46, 1965. https://doi.org/10.1016/S1385-7258(65)50004-4.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."

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Polygon Circumscribing

Cite this as:

Weisstein, Eric W. "Polygon Circumscribing." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PolygonCircumscribing.html

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