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Archimedes Algorithm

Successive application of Archimedes' Recurrence Formula gives the Archimedes algorithm, which can be used to provide successive approximations to $\pi$ (Pi). The algorithm is also called the Borchardt-Pfaff Algorithm. Archimedes obtained the first rigorous approximation of $\pi$ by Circumscribing and Inscribing $n=6\cdot 2^k$-gons on a Circle. From Archimedes' Recurrence Formula, the Circumferences $a$ and $b$ of the circumscribed and inscribed Polygons are

$\displaystyle a(n)$ $\textstyle =$ $\displaystyle 2n\tan\left({\pi\over n}\right)$ (1)
$\displaystyle b(n)$ $\textstyle =$ $\displaystyle 2n\sin\left({\pi\over n}\right),$ (2)

where
\begin{displaymath}
b(n)< C=2\pi r=2\pi\cdot 1=2\pi< a(n).
\end{displaymath} (3)

For a Hexagon, $n=6$ and
$\displaystyle a_0$ $\textstyle \equiv$ $\displaystyle a(6) = 4\sqrt{3}$ (4)
$\displaystyle b_0$ $\textstyle \equiv$ $\displaystyle b(6) = 6,$ (5)

where $a_k\equiv a(6\cdot 2^k)$. The first iteration of Archimedes' Recurrence Formula then gives
$\displaystyle a_1$ $\textstyle =$ $\displaystyle {2\cdot 6\cdot 4\sqrt{3}\over 6+4\sqrt{3}} = {24\sqrt{3}\over 3+2\sqrt{3}} = 24(2-\sqrt{3})$ (6)
$\displaystyle b_1$ $\textstyle =$ $\displaystyle \sqrt{24(2-\sqrt{3}\,)\cdot 6}=12\sqrt{2-\sqrt{3}}$  
  $\textstyle =$ $\displaystyle 6(\sqrt{6}-\sqrt{2}\,).$ (7)

Additional iterations do not have simple closed forms, but the numerical approximations for $k=0$, 1, 2, 3, 4 (corresponding to 6-, 12-, 24-, 48-, and 96-gons) are


\begin{displaymath}
3.00000 < \pi < 3.46410
\end{displaymath} (8)


\begin{displaymath}
3.10583 < \pi < 3.21539
\end{displaymath} (9)


\begin{displaymath}
3.13263 < \pi < 3.15966
\end{displaymath} (10)


\begin{displaymath}
3.13935 < \pi < 3.14609
\end{displaymath} (11)


\begin{displaymath}
3.14103 < \pi < 3.14271.
\end{displaymath} (12)

By taking $k=4$ (a 96-gon) and using strict inequalities to convert irrational bounds to rational bounds at each step, Archimedes obtained the slightly looser result

\begin{displaymath}
{\textstyle{223\over 71}} = 3.14084\ldots < \pi < {\textstyle{22\over 7}}=3.14285\ldots.
\end{displaymath} (13)


References

Miel, G. ``Of Calculations Past and Present: The Archimedean Algorithm.'' Amer. Math. Monthly 90, 17-35, 1983.

Phillips, G. M. ``Archimedes in the Complex Plane.'' Amer. Math. Monthly 91, 108-114, 1984.



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© 1996-9 Eric W. Weisstein
1999-05-25