Liu Hui's π algorithm

Liu Hui's π algorithm is an iterative method for computing the ratio of a circle's circumference to its diameter to any desired accuracy, devised by the third century mathematician Liu Hui of the state of Cao Wei and set out in his commentary to The Nine Chapters on the Mathematical Art. Earlier Chinese values for π were found by experiment and were accurate to about one decimal place. Liu Hui instead started from an inscribed regular hexagon, whose perimeter shows that π must exceed three, and repeatedly bisected the polygon, doubling the number of sides and using the Pythagorean theorem at each step to find the new side length. He also showed that the area of a polygon with N sides, together with a correction term taken from the difference between successive polygons, bounds the area of the circle from both sides, which gives his π inequality. With a 96-sided polygon he obtained bounds of 3.141024 and 3.142708 and a value of about 3.1416, an accuracy comparable to a 1536-sided polygon, and the article describes him as the first Chinese mathematician to give a rigorous algorithm for π.