Archimedes Algorithm

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

 (1) (2)

where
 (3)

For a Hexagon, and
 (4) (5)

where . The first iteration of Archimedes' Recurrence Formula then gives
 (6) (7)

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

 (8)

 (9)

 (10)

 (11)

 (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

 (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.