Liu Hui's π Algorithm

Liu Hui's π algorithm was invented by Liu Hui (fl. 3rd century), a mathematician of Wei Kingdom. Before his time, the ratio of the circumference of a circle to diameter was often taken experimentally as three in China, while Zhang Heng (78–139) rendered it as 3.1724 (from the proportion of the celestial circle to the diameter of the earth, 92/29) or as . Liu Hui was not satisfied with this value. He commented that it was too large and overshot the mark. Another mathematician Wan Fan (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: π ≈ 3.1416.

Liu Hui remarked in his commentary to the The Nine Chapters on the Mathematical Art, 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 an ingenious quick method to improve on it, and obtained π ≈ 3.1416 with only a 96-gon, with an accuracy comparable to that from a 1536-gon. His most important contribution in this area was his simple iterative π algorithm.

Read more about Liu Hui's π Algorithm:  Area of A Circle, Liu Hui's π Inequality, Iterative Algorithm, Quick Method, Later Developments, Significance of Liu Hui's Algorithm