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{{Short description|Chinese mathematician and writer}}
{{other people}}
{{Family name hatnote|[[Liu]]|lang=Chinese}}
{{Infobox person
| name = Liu Hui
| native_name = 劉徽
| native_name_lang = zh
| image = Liu hui.jpg
| image_size = 
| caption = 
| birth_date = {{circa}} 225<ref name="lee">Lee & Tang.</ref>
| birth_place = [[Zibo]], [[Shandong]]
| death_date = {{circa}} 295<ref name="lee" />
| death_place = 
| occupation = Mathematician, writer
}}
{{Infobox Chinese
|t=劉徽
|p=Liú Huī
|mi={{IPAc-cmn|l|iu|2|-|h|wei|1}}
}}
'''Liu Hui''' ({{floruit| 3rd century CE}}) was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu ([[The Nine Chapters on the Mathematical Art]]).''<ref name=":1">{{Cite web |title=Liu Hui – Biography |url=https://mathshistory.st-andrews.ac.uk/Biographies/Liu_Hui/ |access-date=2022-04-17 |website=Maths History |language=en}}</ref> He was a descendant of the Marquis of Zixiang of the [[Eastern Han dynasty]] and lived in the state of [[Cao Wei]] during the [[Three Kingdoms]] period (220–280 CE) of China.<ref name=":0">{{Cite book |last=Stewart |first=Ian |title=Significant Figures: The Lives and Work of Great Mathematicians |date=2017 |publisher=Basic Books |isbn=978-0-465-09613-8 |edition=First US |location=New York |pages=40}}</ref> 

His major contributions as recorded in his commentary on ''The Nine Chapters on the Mathematical Art'' include a proof of the [[Pythagorean theorem]], theorems in solid [[geometry]], an improvement on [[Archimedes|Archimedes's]] [[Liu Hui's π algorithm|approximation]] of {{pi}}, and a systematic method of solving linear equations in several unknowns. In his other work, ''[[Haidao Suanjing]] (The Sea Island Mathematical Manual)'', he wrote about geometrical problems and their application to surveying. He probably visited [[Luoyang]], where he measured the sun's shadow.<ref name=":0" />

==Mathematical work==
Liu Hui expressed mathematical results in the form of [[decimal]] fractions that utilized [[Metrology|metrological]] units (i.e., related units of length with base 10 such as 1 ''[[Chi (unit)|chǐ]]'' = 10 ''[[Cun (unit)|cùn]]'', 1 ''cùn'' = 10 ''fēn'', 1 ''fēn'' = 10 ''lí'', etc.); this led Liu Hui to express a diameter of 1.355 feet as 1 ''chǐ'', 3 ''cùn'', 5 ''fēn'', 5 ''lí''.<ref>{{Cite book |last=Needham |first=Joseph |title=Science and Civilization in China, Volume 3, Mathematics and the Sciences of the Heavens and the Earth |date=1959 |publisher=Cambridge University Press |others=With the collaboration of Wang Ling |isbn=978-0521058018 |location= |pages=84-85 |oclc=}}</ref> Han Yen (fl. 780-804 CE) is thought to be the first mathematician that dropped the terms referring to the units of length and used a notation system akin to the modern decimal system  and [[Yang Hui]] (c. 1238–1298 CE) is considered to have introduced a unified decimal system.<ref>{{Cite book |last=Needham |first=Joseph |url= |title=Science and Civilisation in China, Volume 3, Mathematics and the Sciences of the Heavens and the Earth |date=1959 |publisher=Cambridge University Press |others=With the Collaboration of Wang Ling |isbn=978-0521058018 |location= |pages=86 |oclc=}}</ref><!-- is the difference that the first only has a limited number of digits, while the second is infinite about of decimal numbers after the decimal point?

~~~~ I believe the first method does not have defined terms for decimals after the fifth decimal point, however, the author notes that such decimals exist and continue but labels them "little nameless numbers" -->

Liu provided a proof of a theorem identical to the [[Pythagorean theorem]].<ref name=":0" /> Liu called the figure of the drawn diagram for the theorem the "diagram giving the relations between the hypotenuse and the sum and difference of the other two sides whereby one can find the unknown from the known."<ref>Needham, Volume 3, 95–96.</ref>

In the field of plane areas and solid figures, Liu Hui was one of the greatest contributors to [[empirical]] solid geometry. For example, he found that a [[Wedge (geometry)|wedge]] with rectangular base and both sides sloping could be broken down into a pyramid and a [[tetrahedral]] wedge.<ref name=":2">Needham, Volume 3, 98–99.</ref> He also found that a wedge with [[trapezoid]] base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid.<ref name=":2" /> He computed the volume of solid figures such as cone, cylinder, frustum of a cone, prism, pyramid, tetrahedron, and a wedge.<ref name=":1" /> However, he failed to compute the volume of a sphere and noted that he left it to a future mathematician to compute.<ref name=":1" />

In his commentaries on ''The Nine Chapters on the Mathematical Art'', he presented:

* An [[Liu Hui's π algorithm|algorithm for the approximation]] of [[pi]] ({{pi}}). While at the time, it was common practice to assume {{pi}} to equal 3,<ref>{{Cite book |last=Needham |first=Joseph |title=Science and Civilisation in China, Volume 3, Mathematics and the Sciences of the Heavens and the Earth |date=1959 |publisher=Cambridge University Press |others=With the Collaboration of Wang Ling |isbn=978-0521058018 |pages=99}}</ref> Liu utilized the method of inscribing a polygon within a circle to approximate {{pi}} to equal <math display="inline">\frac{157}{50}</math> on the basis of a 192-sided polygon.<ref>{{Cite book |last=Needham |first=Joseph |title=Science and Civilisation in China, Volume 3, Mathematics and the Sciences of the Heavens and the Earth |date=1959 |publisher=Cambridge University Press |others=With the Collaboration of Wang Ling |isbn=978-0521058018 |pages=100}}</ref> This method was similar to the one employed by Archimedes whereby one calculates the length of the perimeter of the inscribed polygon utilizing the properties of right-angled triangles formed by each half-segment. Liu subsequently utilized a 3072-sided polygon to approximate {{pi}} to equal 3.14159, which is a more accurate approximation than the one calculated by Archimedes or Ptolemy.<ref>{{Cite book |last=Needham |first=Joseph |title=Science and Civilisation in China, Volume 3, Mathematics and the Sciences of the Heavens and the Earth |date=1959 |publisher=Cambridge University Press |others=With the Collaboration of Wang Ling |isbn=978-0521058018 |pages=101}}</ref>

* [[Gaussian elimination]].
* [[Cavalieri's principle]] to find the volume of a cylinder and the intersection of two perpendicular cylinders<ref>Needham, Volume 3, 143.</ref><ref>Siu</ref> although this work was only finished by [[Zu Chongzhi]] and [[Zu Gengzhi]]. Liu's commentaries often include explanations why some methods work and why others do not. Although his commentary was a great contribution, some answers had slight errors which was later corrected by the [[Tang dynasty|Tang]] mathematician and Taoist believer [[Li Chunfeng]].
* Through his work in the [[The Nine Chapters on the Mathematical Art|Nine Chapters]], he could have been the first mathematician to discover and compute with negative numbers; definitely before Ancient Indian mathematician [[Brahmagupta]] started using negative numbers.

=== Surveying ===
[[File:Sea island survey.jpg|thumb|right|200px|Survey of sea island]]
Liu Hui also presented, in a separate appendix of 263 AD called ''[[Haidao Suanjing]]'' or ''The Sea Island Mathematical Manual'', several problems related to [[surveying]]. This book contained many practical problems of geometry, including the measurement of the heights of [[Chinese pagoda]] towers.<ref>Needham, Volume 3, 30.</ref> This smaller work outlined instructions on how to measure distances and heights with "tall surveyor's poles and horizontal bars fixed at right angles to them".<ref>Needham, Volume 3, 31.</ref> With this, the following cases are considered in his work:

* The measurement of the height of an island opposed to its [[sea level]] and viewed from the sea
* The height of a tree on a hill
* The size of a city wall viewed at a long distance
* The depth of a [[ravine]] (using hence-forward cross-bars)
* The height of a tower on a plain seen from a hill
* The breadth of a river-mouth seen from a distance on land
* The width of a valley seen from a cliff
* The depth of a [[transparency (optics)|transparent]] pool
* The width of a river as seen from a hill
* The size of a city seen from a mountain.

Liu Hui's information about surveying was known to his contemporaries as well. The [[History of cartography|cartographer]] and state minister [[Pei Xiu]] (224–271) outlined the advancements of cartography, surveying, and mathematics up until his time. This included the first use of a [[Grid reference|rectangular grid and graduated scale]] for accurate measurement of distances on representative terrain maps.<ref>Hsu, 90–96.</ref> Liu Hui provided commentary on the Nine Chapter's problems involving building [[canal]] and river [[Dike (construction)|dykes]], giving results for total amount of materials used, the amount of labor needed, the amount of time needed for construction, etc.<ref>Needham, Volume 4, Part 3, 331.</ref>

Although translated into English long beforehand, Liu's work was translated into [[French language|French]] by Guo Shuchun, a professor from the [[Chinese Academy of Sciences]], who began in 1985 and took twenty years to complete his translation.

==See also==
*[[Chinese mathematics]]
*[[Fangcheng (mathematics)]]
*[[Lists of people of the Three Kingdoms]]
*[[Liu Hui's π algorithm]]
*[[Haidao Suanjing]]
*[[History of geometry]]

== Further reading ==
*Chen, Stephen. "Changing Faces:  Unveiling a Masterpiece of Ancient Logical Thinking." ''[[South China Morning Post]]'', Sunday, January 28, 2007.
*Crossley, J.M et al. The Logic of Liu Hui and Euclid, Philosophy and History of Science, vol 3, No 1, 1994
*Guo, Shuchun. [https://web.archive.org/web/20070929110532/http://203.72.198.245/web/Content.asp?ID=43261&Query=1 "Liu Hui"]. ''[[Encyclopedia of China]]'' (Mathematics Edition), 1st ed.
*Ho Peng Yoke. "Liu Hui." ''Dictionary of Scientific Biography'', vol. 8. Ed. Charles C. Gillipsie. New York: Scribners, 1973, 418–425.
*Hsu, Mei-ling. "The Qin Maps: A Clue to Later Chinese Cartographic Development." ''Imago Mundi'' (Volume 45, 1993): 90–100.
*Lee, Chun-yue & C. M.-Y. Tang (2012). [https://web.archive.org/web/20160306030624/http://hpm2012.org/proceeding/ot5/t5-01.pdf "A Comparative Study on Finding Volume of Spheres by Liu Hui (劉徽) and Archimedes: An Educational Perspective to Secondary School Students."]
*Mikami, Yoshio (1974). ''Development of Mathematics in China and Japan''.
*Siu, Man-Keung. Proof and Pedagogy in Ancient China: Examples from Liu Hui's Commentary On Jiu Zhang Suan Shu, 1993

==References==
{{Reflist}}

==External links==
{{commons category|Liu Hui (mathematician)}}
{{Wikisourcelang|1=zh|2=Author:劉徽|3=Liu Hui}}
* {{MacTutor Biography|id= Liu_Hui}}
*[https://www.jstor.org/pss/2691200  Liu Hui and the first Golden Age of Chinese Mathematics, by Philip D. Straffin Jr]
* {{Gutenberg author | id=32384| name=Hui Liu}}
* {{Internet Archive author |name=Hui Liu}}

{{People of Cao Wei}}

{{Authority control}}

{{DEFAULTSORT:Liu, Hui}}
[[Category:Cao Wei science writers]]
[[Category:Mathematicians from Shandong]]
[[Category:People from Zibo]]
[[Category:People of Cao Wei]]
[[Category:Writers from Zibo]]
[[Category:3rd-century Chinese mathematicians]]