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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 (pi). The algorithm is also called the Borchardt-Pfaff algorithm. Archimedes obtained the first rigorous approximation of pi by circumscribing and inscribing n=6·2^k-gons on a circle. From Archimedes' recurrence formula, the circumferences a and b of the circumscribed and inscribed polygons are

a(n)=2ntan(pi/n)
(1)
b(n)=2nsin(pi/n),
(2)

where

 b(n)<C=2pir=2pi·1=2pi<a(n).
(3)

For a hexagon, n=6 and

a_0=a(6)=4sqrt(3)
(4)
b_0=b(6)=6,
(5)

where a_k=a(6·2^k). The first iteration of Archimedes' recurrence formula then gives

a_1=(2·6·4sqrt(3))/(6+4sqrt(3))=(24sqrt(3))/(3+2sqrt(3))=24(2-sqrt(3))
(6)
b_1=sqrt(24(2-sqrt(3))·6)=12sqrt(2-sqrt(3))
(7)
=6(sqrt(6)-sqrt(2)).
(8)

Additional iterations do not have simple closed forms, but the numerical approximations for k=0, 1, 2, 3, 4 (corresponding to 6-, 12-, 24-, 48-, and 96-gons) are

 3.00000<pi<3.46410
(9)
 3.10583<pi<3.21539
(10)
 3.13263<pi<3.15966
(11)
 3.13935<pi<3.14609
(12)
 3.14103<pi<3.14271.
(13)

By taking k=4 (a 96-gon) and using strict inequalities to convert irrational bounds to rational bounds at each step, Archimedes obtained the slightly looser result

 (223)/(71)=3.14084...<pi<(22)/7=3.14285....
(14)

See also

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

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