Liu Hui's algorithm was invented by Liu Hui (fl. 3rd century), a mathematician of the state of Cao Wei. Before his time, the ratio of the circumference of a circle to its diameter was often taken experimentally as three in China, while Zhang Heng (78 - 139) rendered it as 3.1724 (from the proportion of the celestial circle to the diameter of the earth,) or as
\pi ≈ \sqrt{10} ≈ 3.162
Liu Hui remarked in his commentary to The Nine Chapters on the Mathematical Art,[2] that the ratio of the circumference of an inscribed hexagon to the diameter of the circle was three, hence must be greater than three. He went on to provide a detailed step-by-step description of an iterative algorithm to calculate to any required accuracy based on bisecting polygons; he calculated to between 3.141024 and 3.142708 with a 96-gon; he suggested that 3.14 was a good enough approximation, and expressed as 157/50; he admitted that this number was a bit small. Later he invented a quick method to improve on it, and obtained with only a 96-gon, a level of accuracy comparable to that from a 1536-gon. His most important contribution in this area was his simple iterative 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".
Apparently Liu Hui had already mastered the concept of the limit[3]
\limNareaofN-gon=areaofcircle.
Further, Liu Hui proved that the area of a circle is half of its circumference multiplied by its radius. He said:
"Between a polygon and a circle, there is excess radius. Multiply the excess radius by a side of the polygon. The resulting area exceeds the boundary of the circle".
In the diagram = excess radius. Multiplying by one side results in oblong which exceeds the boundary of the circle. If a side of the polygon is small (i.e. there is a very large number of sides), then the excess radius will be small, hence excess area will be small.
As in the diagram, when,, and .
"Multiply the side of a polygon by its radius, and the area doubles; hence multiply half the circumference by the radius to yield the area of circle".
When, half the circumference of the -gon approaches a semicircle, thus half a circumference of a circle multiplied by its radius equals the area of the circle. Liu Hui did not explain in detail this deduction. However, it is self-evident by using Liu Hui's "in-out complement principle" which he provided elsewhere in The Nine Chapters on the Mathematical Art: Cut up a geometric shape into parts, rearrange the parts to form another shape, the area of the two shapes will be identical.
Thus rearranging the six green triangles, three blue triangles and three red triangles into a rectangle with width = 3, and height shows that the area of the dodecagon = 3.
In general, multiplying half of the circumference of a -gon by its radius yields the area of a 2-gon. Liu Hui used this result repetitively in his algorithm.
Liu Hui proved an inequality involving by considering the area of inscribed polygons with and 2 sides.
In the diagram, the yellow area represents the area of an -gon, denoted by
AN
A2N
D2N=A2N-AN.
The red area is equal to the green area, and so is also
D2N
Yellow area + green area + red area =
A2N+D2N.
Let
AC
A2N<AC<A2N+D2N.
If the radius of the circle is taken to be 1, then we have Liu Hui's inequality:
A2N<\pi<A2N+D2N.
Liu Hui began with an inscribed hexagon. Let be the length of one side of hexagon, is the radius of circle.
Bisect with line, becomes one side of dodecagon (12-gon), let its length be . Let the length of be and the length of be .
, are two right angle triangles. Liu Hui used the Pythagorean theorem repetitively:
{}G2=r2-\left(\tfrac{M}{2}\right)2
{}G=\sqrt{r2-\tfrac{M2}{4}}
{}j=r-G=r-\sqrt{r2-\tfrac{M2}{4}}
{}m2=\left(\tfrac{M}{2}\right)2+j2
{}m=\sqrt{\left(\tfrac{M}{2}\right)2+j2}
{}m=\sqrt{\left(\tfrac{M}{2}\right)2+\left(r-G\right)2}
{}m=\sqrt{\left(\tfrac{M}{2}\right)2+\left(r-\sqrt{r2-\tfrac{M2}{4}}\right)2}
From here, there is now a technique to determine from, which gives the side length for a polygon with twice the number of edges. Starting with a hexagon, Liu Hui could determine the side length of a dodecagon using this formula. Then continue repetitively to determine the side length of an icositetragon given the side length of a dodecagon. He could do this recursively as many times as necessary. Knowing how to determine the area of these polygons, Liu Hui could then approximate .
With
r=10
{}A96=313{584\over625}
area of 192-gon
{}A192=314{64\over625}
Difference of 96-gon and 48-gon:
{}D192=314
| 64 | |
| 625 |
-313
| 584 | |
| 625 |
=
| 105 | |
| 625 |
from Liu Hui's inequality:
A2N<AC<A2N+D2N.
Since = 10,
AC=100 x \pi
therefore:
| {}314 | 64 |
| 625 |
<100 x \pi<314
| 64 | + | |
| 625 |
| 105 | |
| 625 |
| {}314 | 64 |
| 625 |
<100 x \pi<314
| 169 | |
| 625 |
{}3.141024<\pi<3.142704.
He never took as the average of the lower limit 3.141024 and upper limit 3.142704. Instead he suggested that 3.14 was a good enough approximation for, and expressed it as a fraction
\tfrac{157}{50}
Liu Hui carried out his calculation with rod calculus, and expressed his results with fractions. However, the iterative nature of Liu Hui's algorithm is quite clear:
2-m2=\sqrt{2+(2-M2)},
in which is the length of one side of the next–order polygon bisected from . The same calculation is done repeatedly, each step requiring only one addition and one square root extraction.
Calculation of square roots of irrational numbers was not an easy task in the third century withcounting rods. Liu Hui discovered a shortcut 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.[4]
Let denote the difference in areas of -gon and (/2)-gon
DN=AN-AN/2
He found:
D96 ≈ \tfrac{1}{4}D48
D192 ≈ \tfrac{1}{4}D96
Hence:
\begin{align} D384&{} ≈ \tfrac{1}{4}D192\\ D768&{} ≈ \left(\tfrac{1}{4}\right)2D192\\ D1536&{} ≈ \left(\tfrac{1}{4}\right)3D192\\ D3072&{} ≈ \left(\tfrac{1}{4}\right)4D192\\ &{} \vdots \end{align}
Area of unit radius circle =
{}\pi=A192+D384+D768+D1536+D3072+ … ≈ A192+F ⋅ D192.
In which
F=\tfrac{1}{4}+\left(\tfrac{1}{4}\right)2+\left(\tfrac{1}{4}\right)3+\left(\tfrac{1}{4}\right)4+ … =
| ||||
|
=\tfrac{1}{3}.
That is all the subsequent excess areas add up amount to one third of the
D192
area of unit circle
{}=\pi ≈ A192+\left(\tfrac{1}{3}\right)D192 ≈ {3927\over1250} ≈ 3.1416.
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:
\pi={3927\over1250}.
Liu Hui established a solid algorithm for calculation of to any accuracy.
From Liu Hui's formula for 2-gon:
A2N=mN x r
For 12288-gon inscribed in a unit radius circle:
A24576=3.14159261864<\pi
From Liu Hui's inequality:
A24576<\pi<A24576+D24576
In which
D24576=A24576-A12288=0.0000001021
A24576=3.14159261864<\pi<3.14159261864+0.0000001021
Therefore
3.14159261864<\pi<3.141592706934
3.1415926<\pi<3.1415927
Zu Chongzhi then used the interpolation formula by He Chengtian (何承天, 370-447) and obtained an approximating fraction:
\pi ≈ {355\over113}
However, this value disappeared in Chinese history for a long period of time (e.g. Song dynasty mathematician Qin Jiushao used =
{22\over7}
\pi=\sqrt{10})
\pi ≈ {355\over113}.
Liu Hui's algorithm was one of his most important contributions to ancient Chinese mathematics. It was based on calculation of -gon area, in contrast to the Archimedean algorithm based on polygon circumference. With this method Zu Chongzhi obtained the eight-digit result: 3.1415926 < < 3.1415927, which held the world record for the most accurate value of for centuries,[6] until Madhava of Sangamagrama calculated 11 digits in the 14th century or Jamshid al-Kashi calculated 16 digits in 1424; the best approximations for known in Europe were only accurate to 7 digits until Ludolph van Ceulen calculated 20 digits in 1596.
Correct value: 0.2502009052
Correct values:
A192=3.1410319509
D192=0.0016817478
\pi ≈ A192+
| 1 | |
| 3 |
D192 ≈ eq3.1410319509+0.0016817478/3
\pi ≈ 3.1410319509+0.0005605826
\pi ≈ 3.1415925335.