Significance of Liu Hui's Algorithm
Liu Hui's π algorithm was one of his most important contributions to ancient Chinese mathematics. It was based on calculation of N-gon area, in contrast to the Archimedean algorithm based on polygon circumference. Archimedes used a circumscribed 96-gon to obtain an upper limit, and an inscribed 96-gon to obtain the lower limit . Liu Hui was able to obtain both his upper limit 3.142704 and lower limit 3.141024 with only an inscribed 96-gon. Furthermore, both the Liu Hui limits were tighter than Achimedes's: 3.140845 < 3.141024 < π < 3.142704 < 3.142857. With his method Zu Chongzhi obtained the result: 3.1415926 < π < 3.1415927, which held the world record for the most accurate value of π for 1200 years, even by 1600 in Europe, mathematician Adriaen Anthoniszoom and his son obtained π value of 3.1415929, accurate only to 7 digit, still 3 digits short of Zu's result
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