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# 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)

Pi Iterations

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## 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.

## Referenced on Wolfram|Alpha

Archimedes Algorithm

## Cite this as:

Weisstein, Eric W. "Archimedes Algorithm." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ArchimedesAlgorithm.html