Liu Hui's π algorithm: Difference between revisions

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:"''Multiply one side of a hexagon by the radius (of its circumcircle), then multiply this by three, to yield the area of a dodecagon; if we cut a hexagon into a dodecagon, multiply its side by its radius, then again multiply by six, we get the area of a 24-gon; the finer we cut, the smaller the loss with respect to the area of circle, thus with further cut after cut, the area of the resulting polygon will coincide and become one with the circle; there will be no loss''".

This is essentially equivalent to:

Apparently Liu Hui had already mastered the concept of the limit<ref>First noted by Japanese mathematician [[Yoshio Mikami]]</ref>

: <math>\lim_{N \to \infty}\text{area of }N\text{-gon} = \text{area of circle}. \, </math>

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Liu Hui was quite happy with this result because he had acquired the same result with the calculation for a 1536-gon, obtaining the area of a 3072-gon. This explains four questions:

# Why he stopped short at {{math|''A''}}₁₉₂ in his presentation of his algorithm. Because he discovered a quick method of improving the accuracy of {{pi}}, achieving same result of 1536-gon with only 96-gon. After all calculation of square roots was not a simple task with [[rod calculus]]. With the quick method, he only needed to perform one more [[subtraction]], one more division (by 3) and one more addition, instead of four more square root extractions.

# Why he preferred to calculate {{pi}} through calculation of areas instead of circumferences of successive polygons, because the quick method required information about the difference in '''areas''' of successive polygons.

# Who was the true author of the paragraph containing calculation of <math>\pi = {3927 \over 1250}.</math>

# That famous paragraph began with "A [[Han dynasty]] bronze container in the military warehouse of [[Jin Dynasty (265–420)|Jin dynasty]]....". Many scholars, among them [[Yoshio Mikami]] and [[Joseph Needham]], believed that the "Han dynasty bronze container" paragraph was the work of Liu Hui and not Zu Chongzhi as other believed, because of the strong correlation of the two methods through area calculation, and because there was not a single word mentioning Zu's 3.1415926 < {{pi}} < 3.1415927 result obtained through 12288-gon.

==Later developments==

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==Significance of Liu Hui's algorithm==

Liu Hui's {{pi}} algorithm was one of his most important contributions to ancient [[Chinese mathematics]]. It was based on calculation of {{math|N}}-gon area, in contrast to the Archimedean algorithm based on polygon circumference. With this method Zu Chongzhi obtained the eight-digit result: 3.1415926 < {{pi}} < 3.1415927, which held the world record for the most accurate value of {{pi}} for centuries,<ref>Robert Temple, The Genius of China, a refined value of pi, p144-145, {{isbn|1-85375-292-4}}</ref> until [[Madhava of Sangamagrama]] calculated 11 digits in the 14th century or [[Jamshid al-Kashi]] calculated 16 digits in 1424; the best approximations for {{pi}} known in Europe were only accurate to 7 digits until [[Ludolph van Ceulen]] calculated 20 digits in 1596.

== See also ==

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==Notes==

{{Original research|section|date=March 2009}}

:{{note|1|1}} Correct value: 0.2502009052

:{{note|2|2}} Correct values:

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{{DEFAULTSORT:Liu Hui's Pi Algorithm}}

[[Category:Pi algorithms]]

[[Category:Chinese mathematicsmathematical discoveries]]

[[Category:Cao Wei]]