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) |

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

(13) |

(14) |

(15) |

(16) |

(17) |

(18) |

(19) |

(20) |

and

(21) |

(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) |

(26) |

**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.

© 1996-9

1999-05-25