Liu Hui's π Algorithm - Quick Method

Quick Method

Calculation of square roots of irrational numbers was not an easy task in the third century with counting rods. Liu Hui discovered a short cut by comparing the area differentials of polygons, and found that the proportion of the difference in area of successive order polygons was approximately 1/4.

Let DN denote the difference in areas of N-gon and (N/2)-gon

He found:

1

Hence:


begin{align}
D_{384} & {} approx tfrac{1}{4} D_{192} \
D_{768} & {} approx left(tfrac{1}{4}right)^2 D_{192} \
D_{1536} & {} approx left(tfrac{1}{4}right)^3 D_{192} \
D_{3072} & {} approx left(tfrac{1}{4}right)^4 D_{192} \
& {}   vdots
end{align}

Area of unit radius circle =

In which

That is all the subsequent excess areas add up amount to one third of the

area of unit circle2

Liu Hui was quite happy with this result because he had acquired the same result with the calculation for a 1536-gon, obtaining the area of a 3072-gon. This explains four questions:

  1. Why he stopped short at A192 in his presentation of his algorithm. Because he discovered a quick method of improving the accuracy of π, achieving same result of 1536-gon with only 96-gon. After all calculation of square roots was not a simple task with rod calculus. With the quick method, he only needed to perform one more subtraction, one more division (by 3) and one more addition, instead of four more square root extractions.
  2. Why he preferred to calculate π through calculation of areas instead of circumferences of successive polygons, because the quick method required information about the difference in areas of successive polygons.
  3. Who was the true author of the paragraph containing calculation of
  4. That famous paragraph began with "A Han dynasty bronze container in the military warehouse of Jin dynasty....". Many scholars, among them Yoshio Mikami and Joseph Needham, believed that the "Han dynasty bronze container" paragraph was the work of Liu Hui and not Zu Chongzhi as other believed, because of the strong correlation of the two methods through area calculation, and because there was not a single word mentioning Zu's 3.1415926 < π < 3.1415927 result obtained through 12288-gon.

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

Other articles related to "quick method, method":

Tea Egg - Preparation - Quick Method
... Another method of making tea eggs is to boil the eggs until fully cooked inside, then remove the hard boiled eggs from their shells and let them steep in the spiced tea mixture at low heat for a ... This method requires a shorter steeping time than the traditional method ... marbled effect from the traditional crack shell method of preparation ...

Famous quotes containing the words method and/or quick:

    The method of painting is the natural growth out of a need. I want to express my feelings rather than illustrate them. Technique is just a means of arriving at a statement.... I can control the flow of paint: there is no accident, just as there is no beginning and no end.
    Jackson Pollock (1912–1956)

    Where I would like to discover facts, I find fancy. Where I would like to learn what I did, I learn only what I was thinking. They are loaded with opinion, moral thoughts, quick evaluations, youthful hopes and cares and sorrows. Occasionally, they manage to report something in exquisite honesty and accuracy. That is why I have refrained from burning them.
    —E.B. (Elwyn Brooks)