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)
			(8)

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

(9)

(10)

(11)

(12)

(13)

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

(14)

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.