Liu Hui's π algorithm: Difference between revisions

Revision as of 21:19, 24 October 2016

Revision as of 02:24, 30 November 2016

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Liu Hui began with an inscribed hexagon. Let {{math|M}} be the length of one side {{math|AB}} of hexagon, {{math|''r''}} is the radius of circle.

Bisect {{math|AB}} with line {{math|OPC}}, {{math|AC}} becomes one side of [[dodecagon]] (12-gon), let its length be {{math|m}}. Let the length of {{math|PC}} be {{math|j}} and the length of {{math|OP}} be {{math|G}}.

{{math|AOP}}, {{math|APC}} are two right angle triangles. Liu Hui used [[Pythagorean theorem|Gou Gu theorem]] repetitively:

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: <math>{} j= r - G = r - \sqrt{r^2- \tfrac{M^2}{4}}</math>

: <math>{}m^2= \left(\tfrac{M}{2}\right)^2 + j^2</math>

: <math>{}m=\sqrt{m\left(\tfrac{M}{2}\right)^2 + j^2}.</math>

With {{math|''r''}} = 10 units, he obtained