## Polygon Circumscribing Constant If a Triangle is Circumscribed about a Circle, another Circle around the Triangle, a Square outside the Circle, another Circle outside the Square, and so on. From Polygons, the Circumradius and Inradius for an -gon are   (1)   (2)

where is the side length. Therefore, (3)

and an infinitely nested set of circumscribed polygons and circles has (4)

Kasner and Newman (1989) and Haber (1964) state that , but this is incorrect. Write (5) (6)

Define (7)

Now define (8)

with (9) (10)

so (11)

and (12)

But for , so (13)       (14) (15)

If the next term is included, (16)

As before, (17) (18)

so (19)         (20)

and (21)

The process can be automated using computer algebra, and the first few bounds are 11.7485, 9.53528, 8.98034, 8.8016, 8.73832, 8.71483, 8.70585, 8.70235, 8.70097, and 8.70042. In order to obtain this accuracy by direct multiplication of the terms, more than 10,000 terms are needed. The limit is (22)

Bouwkamp (1965) produced the following Infinite Product formulas   (23)  (24)

where is the Riemann Zeta Function and is the Dirichlet Lambda Function. Bouwkamp (1965) also produced the formula with accelerated convergence  (25)
where (26)

(cited in Pickover 1995).

References

Bouwkamp, C. An Infinite Product.'' Indag. Math. 27, 40-46, 1965.

Finch, S. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/infprd/infprd.html

Haber, H. Das Mathematische Kabinett.'' Bild der Wissenschaft 2, 73, Apr. 1964.

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.

Plouffe, S. Product(cos(Pi/n),n=3..infinity).'' http://www.lacim.uqam.ca/piDATA/productcos.txt.