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)
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(2)
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where
(3)
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For a hexagon, and
(4)
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(5)
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where . The first iteration of Archimedes' recurrence formula then gives
(6)
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(7)
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(8)
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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)
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(10)
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(11)
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(12)
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(13)
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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)
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