Liu Hui's π Algorithm - Later Developments

Later Developments

Liu Hui established a solid algorithm for calculation of π to any accuracy.

  • Zu Chongzhi was familiar with Liu Hui's work, and obtained greater accuracy by applying his algorithm to a 12288-gon.
From Liu Hui's formula for 2N-gon:
For 12288-gon inscribed in a unit radius circle:
.
From Liu Hui's π inequality:
In which
.
Therefore

Truncated to eight significant digits:

.

That was the famous Zu Chongzhi π inequality.

Zu Chongzhi then used the interpolation formula by He Chengtian (何承天, 370-447) and obtained an approximating fraction: .

  • Yuan dynasty mathematician Zhao Yu Xin worked on a variation of Liu Hui's π algorithm, by bisecting an inscribed square instead of a hexagon.

Read more about this topic:  Liu Hui's π Algorithm

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