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
where
 |
(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
 |
(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.
