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

$$a(n) = 2n\tan\left({\pi\over n}\right)$$ (1)

$$b(n) = 2n\sin\left({\pi\over n}\right),$$ (2)

where

$$b(n)< C=2\pi r=2\pi\cdot 1=2\pi< a(n).$$ (3)

For a Hexagon, $n=6$ and

$$a_0 \equiv a(6) = 4\sqrt{3}$$ (4)

$$b_0 \equiv b(6) = 6,$$ (5)

where $a_k\equiv a(6\cdot 2^k)$. The first iteration of Archimedes' Recurrence Formula then gives

$$a_1 = {2\cdot 6\cdot 4\sqrt{3}\over 6+4\sqrt{3}} = {24\sqrt{3}\over 3+2\sqrt{3}} = 24(2-\sqrt{3})$$ (6)

$$b_1 = \sqrt{24(2-\sqrt{3}\,)\cdot 6}=12\sqrt{2-\sqrt{3}}$$

$$= 6(\sqrt{6}-\sqrt{2}\,).$$ (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

$$3.00000 < \pi < 3.46410$$ (8)

$$3.10583 < \pi < 3.21539$$ (9)

$$3.13263 < \pi < 3.15966$$ (10)

$$3.13935 < \pi < 3.14609$$ (11)

$$3.14103 < \pi < 3.14271.$$ (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

$${\textstyle{223\over 71}} = 3.14084\ldots < \pi < {\textstyle{22\over 7}}=3.14285\ldots.$$ (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.