Successive application of Archimedes' Recurrence Formula gives the Archimedes algorithm, which can be used to
provide successive approximations to
(Pi). The algorithm is also called the Borchardt-Pfaff
Algorithm. Archimedes obtained the first rigorous approximation of
by Circumscribing and Inscribing
-gons on a Circle. From Archimedes'
Recurrence Formula, the Circumferences
and
of the circumscribed and
inscribed Polygons are
where
![\begin{displaymath}
b(n)< C=2\pi r=2\pi\cdot 1=2\pi< a(n).
\end{displaymath}](a_1361.gif) |
(3) |
For a Hexagon,
and
where
. The first iteration of Archimedes' Recurrence Formula then gives
Additional iterations do not have simple closed forms, but the numerical approximations for
, 1, 2, 3, 4 (corresponding to
6-, 12-, 24-, 48-, and 96-gons) are
![\begin{displaymath}
3.00000 < \pi < 3.46410
\end{displaymath}](a_1371.gif) |
(8) |
![\begin{displaymath}
3.10583 < \pi < 3.21539
\end{displaymath}](a_1372.gif) |
(9) |
![\begin{displaymath}
3.13263 < \pi < 3.15966
\end{displaymath}](a_1373.gif) |
(10) |
![\begin{displaymath}
3.13935 < \pi < 3.14609
\end{displaymath}](a_1374.gif) |
(11) |
![\begin{displaymath}
3.14103 < \pi < 3.14271.
\end{displaymath}](a_1375.gif) |
(12) |
By taking
(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}](a_1377.gif) |
(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.
© 1996-9 Eric W. Weisstein
1999-05-25