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 -gon are then related by
|
(1)
|
so an infinitely nested set of circumscribed polygons and circles has
Kasner and Newman (1989) and Haber (1964) state that , but this is incorrect, and the actual answer is
|
(5)
|
(OEIS A051762).
By writing
|
(6)
|
it is possible to expand the series about infinity, change the order of summation, do the
sum symbolically, and obtain the quickly converging series
|
(7)
|
where
is the Riemann zeta function.
Bouwkamp (1965) produced the following infinite product
formulas for the constant,
where
is the sinc function (cf. Prudnikov et al. 1986,
p. 757),
is the Riemann zeta function, and is the Dirichlet
lambda function. Bouwkamp (1965) also produced the formula with accelerated convergence
|
(11)
|
where
|
(12)
|
(cited in Pickover 1995).
See also
Infinite Product,
Nested
Polygon,
Polygon Inscribing,
Whirl
Explore with Wolfram|Alpha
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 Circumscribing
Cite this as:
Weisstein, Eric W. "Polygon Circumscribing."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PolygonCircumscribing.html
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