Zhao Youqin's π algorithm


Zhao Youqin's algorithm was an algorithm devised by Yuan dynasty Chinese astronomer and mathematician Zhao Youqin to calculate the value of Pi| in his book Ge Xiang Xin Shu.

Algorithm

Zhao Youqin started with an inscribed square in a circle with radius r.
If denotes the length of a side of the square, draw a perpendicular line d from the center of the circle to side l. Let e denotes r − d. Then from the diagram:
Extend the perpendicular line d to dissect the circle into an octagon; denotes the length of one side of octagon.
Let denotes the length of a side of hexadecagon
similarly
Proceeding in this way, he at last calculated the side of a 16384-gon, multiplying it by 16384 to obtain 3141.592 for a circle with diameter = 1000 units, or
He multiplied this number by 113 and obtained 355. From this he deduced that of the traditional values of, that is 3, 3.14, and, the last is the most exact.