feat: Add Liu Hui's π algorithm (trekhleb#61)

Author: mantou132

.gitignore

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 node_modules
 .idea
 coverage
+.vscode

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

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+# Liu Hui's π Algorithm
+
+Multiply one side of a hexagon by the radius (of its circumcircle), then multiply this by three, to yield the area of a dodecagon; if we cut a hexagon into a dodecagon, multiply its side by its radius, then again multiply by six, we get the area of a 24-gon; the finer we cut, the smaller the loss with respect to the area of circle, thus with further cut after cut, the area of the resulting polygon will coincide and become one with the circle; there will be no loss
+
+
+
+## References
+
+- [Liu Hui's π Algorithm on Wikipedia](https://en.wikipedia.org/wiki/Liu_Hui's_%CF%80_algorithm)

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

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+import pi from '../liu-hui-pi';
+
+describe('Liu Hui\'s π algorithm', () => {
+  it('Dodecagon π', () => {
+    expect(pi(1)).toBe(3);
+  });
+  it('24-gon π', () => {
+    expect(pi(2)).toBe(3.105828541230249);
+  });
+  it('6144-gon π', () => {
+    expect(pi(10)).toBe(3.1415921059992717);
+  });
+  it('201326592-gon π', () => {
+    expect(pi(25)).toBe(3.141592653589793);
+  });
+});

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

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+// Liu Hui began with an inscribed hexagon.
+// Let r is the radius of circle.
+// r is also the side length of the inscribed hexagon
+const c = 6;
+const r = 0.5;
+
+const getSideLength = (sideLength, count) => {
+  if (count <= 0) return sideLength;
+  const m = sideLength / 2;
+
+  // Liu Hui used the Gou Gu theorem repetitively.
+  const g = Math.sqrt((r ** 2) - (m ** 2));
+  const j = r - g;
+  return getSideLength(Math.sqrt((j ** 2) + (m ** 2)), count - 1);
+};
+
+const getSideCount = splitCount => c * (splitCount ? 2 ** splitCount : 1);
+
+/**
+ * Calculate the π value using Liu Hui's π algorithm
+ *
+ * Liu Hui argued:
+ * Multiply one side of a hexagon by the radius (of its circumcircle),
+ * then multiply this by three, to yield the area of a dodecagon; if we
+ * cut a hexagon into a dodecagon, multiply its side by its radius, then
+ * again multiply by six, we get the area of a 24-gon; the finer we cut,
+ * the smaller the loss with respect to the area of circle, thus with
+ * further cut after cut, the area of the resulting polygon will coincide
+ * and become one with the circle; there will be no loss
+ * @param {Number} splitCount repeat times
+ * @return {Number}
+ */
+const pi = (splitCount = 1) => {
+  const sideLength = getSideLength(r, splitCount - 1);
+  const sideCount = getSideCount(splitCount - 1);
+  const p = sideLength * sideCount;
+  const area = (p / 2) * r;
+  return area / (r ** 2);
+};
+
+// !test
+// for (let i = 1; i < 26; i += 1) {
+//   const p = pi(i);
+//   console.log(
+//     'split count: %f, side count: %f, π: %f, is Math.PI? %o',
+//     i,
+//     getSideCount(i),
+//     p,
+//     p === Math.PI,
+//   );
+// }
+
+export default pi;