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Commit1e2fdc6

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Refactor README.
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‎src/algorithms/graph/liu-hui-pi/README.md

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‎src/algorithms/graph/liu-hui-pi/__test__/liu-hui-pi.test.js

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‎src/algorithms/math/liu-hui/README.md

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#Liu Hui's π Algorithm
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Liu Hui remarked in his commentary to The Nine Chapters on the Mathematical Art,
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that the ratio of the circumference of an inscribed hexagon to the diameter of
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the circle was`three`, hence`π` must be greater than three. He went on to provide
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a detailed step-by-step description of an iterative algorithm to calculate`π` to
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any required accuracy based on bisecting polygons; he calculated`π` to
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between`3.141024` and`3.142708` with a 96-gon; he suggested that`3.14` was
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a good enough approximation, and expressed`π` as`157/50`; he admitted that
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this number was a bit small. Later he invented an ingenious quick method to
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improve on it, and obtained`π ≈ 3.1416` with only a 96-gon, with an accuracy
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comparable to that from a 1536-gon. His most important contribution in this
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area was his simple iterative`π` algorithm.
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##Area of a circle
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Liu Hui argued:
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>Multiply one side of a hexagon by the radius (of its
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circumcircle), then multiply this by three, to yield the
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area of a dodecagon; if we cut a hexagon into a
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dodecagon, multiply its side by its radius, then again
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multiply by six, we get the area of a 24-gon; the finer
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we cut, the smaller the loss with respect to the area
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of circle, thus with further cut after cut, the area of
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the resulting polygon will coincide and become one with
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the circle; there will be no loss
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![Liu Hui](https://upload.wikimedia.org/wikipedia/commons/6/69/Cutcircle2.svg)
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Liu Hui's method of calculating the area of a circle.
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Further, Liu Hui proved that the area of a circle is half of its circumference
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multiplied by its radius. He said:
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>Between a polygon and a circle, there is excess radius. Multiply the excess
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radius by a side of the polygon. The resulting area exceeds the boundary of
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the circle
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In the diagram`d = excess radius`. Multiplying`d` by one side results in
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oblong`ABCD` which exceeds the boundary of the circle. If a side of the polygon
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is small (i.e. there is a very large number of sides), then the excess radius
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will be small, hence excess area will be small.
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>Multiply the side of a polygon by its radius, and the area doubles;
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hence multiply half the circumference by the radius to yield the area of circle.
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![Liu Hui](https://upload.wikimedia.org/wikipedia/commons/9/95/Cutcircle.svg)
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The area within a circle is equal to the radius multiplied by half the
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circumference, or`A = r x C/2 = r x r x π`.
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##Iterative algorithm
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Liu Hui began with an inscribed hexagon. Let`M` be the length of one side`AB` of
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hexagon,`r` is the radius of circle.
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![Liu Hui](https://upload.wikimedia.org/wikipedia/commons/4/46/Liuhui_geyuanshu.svg)
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Bisect`AB` with line`OPC`,`AC` becomes one side of dodecagon (12-gon), let
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its length be`m`. Let the length of`PC` be`j` and the length of`OP` be`G`.
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`AOP`,`APC` are two right angle triangles. Liu Hui used
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the[Gou Gu](https://en.wikipedia.org/wiki/Pythagorean_theorem) (Pythagorean theorem)
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theorem repetitively:
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![](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbfc192c78539c3901c7bad470302ededb76f813)
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![](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccd12a402367c2d6614c88e75006d50bfc3a9929)
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![](https://wikimedia.org/api/rest_v1/media/math/render/svg/65d77869fc02c302d2d46d45f75ad7e79ae524fb)
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![](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a0d0d7f505a0f434e5dd80c2fef6d2b30d6100)
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![](https://wikimedia.org/api/rest_v1/media/math/render/svg/c31b9acf38f9d1a248d4023c3bf286bd03007f37)
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![](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dee798efb0b1e3e64d6b3542106cb3ecaa4a383)
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![](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ffeafe88d2983b364ad3442746063e3207fe842)
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From here, there is now a technique to determine`m` from`M`, which gives the
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side length for a polygon with twice the number of edges. Starting with a
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hexagon, Liu Hui could determine the side length of a dodecagon using this
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formula. Then continue repetitively to determine the side length of a
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24-gon given the side length of a dodecagon. He could do this recursively as
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many times as necessary. Knowing how to determine the area of these polygons,
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Liu Hui could then approximate`π`.
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##References
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-[Wikipedia](https://en.wikipedia.org/wiki/Liu_Hui%27s_%CF%80_algorithm)
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importliuHuifrom'../liuHui';
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describe('liHui',()=>{
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it('Dodecagon π',()=>{
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expect(liuHui(1)).toBe(3);
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});
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it('24-gon π',()=>{
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expect(liuHui(2)).toBe(3.105828541230249);
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});
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it('6144-gon π',()=>{
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expect(liuHui(10)).toBe(3.1415921059992717);
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});
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it('201326592-gon π',()=>{
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expect(liuHui(25)).toBe(3.141592653589793);
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});
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});

‎src/algorithms/graph/liu-hui-pi/liu-hui-pi.jsrenamed to‎src/algorithms/math/liu-hui/liuHui.js

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// Liu Hui used the Gou Gu theorem repetitively.
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constg=Math.sqrt((r**2)-(m**2));
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constj=r-g;
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returngetSideLength(Math.sqrt((j**2)+(m**2)),count-1);
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};
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* the smaller the loss with respect to the area of circle, thus with
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* further cut after cut, the area of the resulting polygon will coincide
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* and become one with the circle; there will be no loss
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*@param {Number} splitCount repeat times
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*@return {Number}
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*
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*@param {number} splitCount repeat times
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*@return {number}
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*/
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constpi=(splitCount=1)=>{
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exportdefaultfunctionliuHui(splitCount=1){
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constsideLength=getSideLength(r,splitCount-1);
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constsideCount=getSideCount(splitCount-1);
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constp=sideLength*sideCount;
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constarea=(p/2)*r;
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returnarea/(r**2);
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};
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// !test
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// for (let i = 1; i < 26; i += 1) {
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// const p = pi(i);
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// console.log(
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// 'split count: %f, side count: %f, π: %f, is Math.PI? %o',
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// i,
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// getSideCount(i),
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// p,
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// p === Math.PI,
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// );
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// }
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exportdefaultpi;
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returnarea/(r**2);
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}

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