lived in the Kingdom of Wei so it is likely that he worked in what is now the Shansi province in...
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Transcript of lived in the Kingdom of Wei so it is likely that he worked in what is now the Shansi province in...
lived in the Kingdom of Wei so it is likely that he worked in what is now the Shansi province in north-central China. The Kingdom of Wei had
come about after the Han Empire, which lasted from around 200 BC to 220 AD, collapsed. However, the collapse of the Han Empire led to three Kingdoms coming into existence for, in addition to the Kingdom of Wei,
two former Han generals set up Kingdoms, one to the south of the Yangtze and one in the west of China in the present Szechwan
Province. This situation lasted for about sixty years, from 220 to 280, which must have been almost exactly the period of Liu Hui's life.
The period of the Three Kingdoms was one of almost constant warfare and political intrigue. However this fascinating period is now thought of
as the most romantic in all of Chinese history. What influence the events of the period had on Liu Hui is unknown, for nothing is known of his life
except that he wrote two works. One was an extremely important commentary on the Jiuzhang suanshu or, as it more commonly called
Nine Chapters on the Mathematical Art, and the other was a much shorter work called Haidao suanjing or Sea Island Mathematical Manual. That no record of Liu Hui's life was written, or at least if it was it was not
considered worth preserving, does not mean that he was particularly obscure during his lifetime. Although mathematics was an important
topic in China, nevertheless being a mathematician seems to have been considered an occupation of minor importance. As a consequence many
Chinese mathematical works are anonymous.
In the diagram we have a circle of radius r with centre O. We know AB, it is pn-1 , the length of the side of a regular polygon with 3 2n-1 sides,
so AY has length pn-1/2. Thus OY has length
√(r2 - (pn-1/2)2).
Then YX has length r - √[r2 - (pn-1/2)2].
But now we know AY and YX so we can compute AX using the Gougu theorem (Pythagoras) to be √{r[2r - √(4r - pn-1
2)]}.
Then pn= AX is the length of a side of a regular polygon with N = 3
2n sides. Putting r = 1 and taking n = 1 gives a regular hexagon of side p6= 1.
Then the perimeter of the hexagon is 6p1= 6 giving an approximate
value of π as 6p1/2 = 3 (assuming the circumference of the circle is
approximately the perimeter of the hexagon and using π = circumference/diameter). In general we obtain an approximate value of π as N pn/2. Larger
values of n give more accurate values of π. Liu Hui used the approximation 3.14 which he obtained from taking n = 5, in other words using a regular polygon of 96 sides. He did not, like Archimedes, find bounds by using an inscribed as well as a circumscribed circle. We iterate Liu Hui's procedure using a modern computer algebra program to obtain:
was a Chinese mathematician who wrote a commentary of the Nine Chapters on the Mathematical Art
