 TOPICS # 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