Liu Hui's π algorithm

Liu Hui's π algorithm was invented by <a href="/facts/Liu_Hui/cH5P5ahJ">Liu Hui</a> (fl. 3rd century), a mathematician of the state of <a href="/facts/Cao_Wei/MDpLYXyC">Cao Wei</a>. Before his time, the ratio of the circumference of a circle to its diameter was often taken experimentally as three in China, while <a href="/facts/Zhang_Heng/jBwoVGQ4">Zhang Heng</a> (78–139) rendered it as 3.1724 (from the proportion of the celestial circle to the diameter of the earth, 92/29) or as 
 
 
 
 π
 ≈
 
 
 10
 
 
 ≈
 3.162
 
 
 {\displaystyle \pi \approx {\sqrt {10}}\approx 3.162}
 
. Liu Hui was not satisfied with this value. He commented that it was too large and overshot the mark. Another mathematician <a href="/facts/Wang_Fan/a4pA9Xe1">Wang Fan</a> (219–257) provided π ≈ 142/45 ≈ 3.156. All these empirical π values were accurate to two digits (i.e. one decimal place). Liu Hui was the first Chinese mathematician to provide a rigorous algorithm for calculation of π to any accuracy. Liu Hui's own calculation with a 96-gon provided an accuracy of five digits ie π ≈ 3.1416.
Liu Hui remarked in his commentary to <a href="/facts/The_Nine_Chapters_on_the_Mathematical_Art/INGLo2Cm">The Nine Chapters on the Mathematical Art</a>, that the ratio of the circumference of an inscribed hexagon to the diameter of the circle was three, hence π must be greater than three. He went on to provide a detailed step-by-step description of an iterative algorithm to calculate π to any required accuracy based on bisecting polygons; he calculated π to between 3.141024 and 3.142708 with a 96-gon; he suggested that 3.14 was a good enough approximation, and expressed π as 157/50; he admitted that this number was a bit small. Later he invented a quick method to improve on it, and obtained π ≈ 3.1416 with only a 96-gon, a level of accuracy comparable to that from a 1536-gon. His most important contribution in this area was his simple iterative π algorithm.

Liu Hui's π algorithm open-in-new

Liu Hui's π algorithm