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{{short description|Wooden tablets inscribed with geometrical theorems in Edo Japan}}
{{For|the ancient Japanese performing art of sangaku (散楽)|Sarugaku}}
[[File:Sangaku of Konnoh Hachimangu 1859.jpg|thumb|A sangaku dedicated to Konnoh Hachimangu ([[Shibuya, Tokyo]]) in 1859.]]

'''Sangaku''' or '''san gaku''' ({{langx|ja|算額|lit=calculation tablet}}) are Japanese [[Euclidean geometry|geometrical]] problems or theorems on wooden tablets which were placed as offerings at [[Shinto shrine]]s or [[Buddhist temples in Japan|Buddhist temples]] during the [[Edo period]] by members of all social classes.

==History==
[[File:Sangaku at Enmanji.jpg|thumb|A sangaku dedicated at Emmanji Temple in [[Nara, Nara|Nara]]]]
The sangaku were painted in color on wooden tablets ([[Ema (Shinto)|ema]]) and hung in the precincts of Buddhist temples and Shinto shrines as offerings to the kami and buddhas, as challenges to the congregants, or as displays of the solutions to questions. Many of these tablets were lost during the period of [[modernization]] that followed the Edo period, but around nine hundred are known to remain.

[[Fujita Kagen]] (1765&ndash;1821), a Japanese mathematician of prominence, published the first collection of ''sangaku'' problems, his ''Shimpeki Sampo'' (Mathematical problems Suspended from the Temple) in 1790, and in 1806 a sequel, the ''Zoku Shimpeki Sampo''.

During this period [[Japan]] applied strict regulations  to commerce and foreign relations for western countries so the tablets were created using [[Japanese mathematics]], developed in parallel to western mathematics. For example, the connection between an integral and its derivative (the [[fundamental theorem of calculus]]) was unknown, so sangaku problems on areas and volumes were solved by expansions in [[infinite series]] and term-by-term calculation.

==Select examples==
[[File:Sangaku_three_circles.svg|200px|thumb|right|The smallest distinct integer solution to the sangaku puzzle in which three circles touch each other and share a tangent line.]]
{| class="wikitable infobox" style="width:1px;"
! ''r''<sub>middle</sub> !! ''r''<sub>left</sub> !! ''r''<sub>right</sub>
|-
| 1 || 4 || 4
|-
| 4 || 9 || 36
|-
| 9 || 16 || 144
|-
| 16 || 25 || 400
|-
| 72 || 200 || 450
|-
| 144 || 441 || 784
|-
| colspan="3"|The six primitive triplets of integer radii up to 1000
|}
* A typical problem, which is presented on an 1824 tablet in [[Gunma Prefecture]], covers the relationship of three touching circles with a common [[tangent]], a special case of [[Descartes' theorem]]. Given the size of the two outer large circles, what is the size of the small circle between them? ''The answer is:''<!-- Info from ScAm May98 article -->

:<math>\frac{1}{\sqrt{r_\text{middle}}} = \frac{1}{\sqrt{r_\text{left}}} + \frac{1}{\sqrt{r_\text{right}}}.</math>

(See also [[Ford circle]].)

* [[Soddy's hexlet]], thought previously to have been discovered in the west in 1937, had been discovered on a sangaku dating from 1822.
* One sangaku problem from Sawa Masayoshi and other from Jihei Morikawa were solved only recently.<ref>{{cite arXiv|last1=Holly|first1=Jan E.|last2=Krumm|first2=David|date=2020-07-25|title=Morikawa's Unsolved Problem|class=math.HO|eprint=2008.00922}}</ref><ref>{{cite journal|last=Kinoshita|first=Hiroshi|date=2018|title=An Unsolved Problem in the Yamaguchi's Travell Diary|url=http://www.sangaku-journal.eu/2018/SJM_2018_43-53_kinoshita.pdf|journal=Sangaku Journal of Mathematics |volume=2|pages=43–53}}</ref>

==See also==
* [[Equal incircles theorem]]
* [[Japanese theorem for concyclic polygons]]
* [[Japanese theorem for concyclic quadrilaterals]]
* [[Problem of Apollonius]]
* [[Recreational mathematics]]
* [[Seki Takakazu]]

==Notes==
{{reflist|1}}

==References==
* Fukagawa, Hidetoshi, and [[Daniel Pedoe|Dan Pedoe]]. (1989).  ''Japanese temple geometry problems = Sangaku''. Winnipeg: Charles Babbage. {{ISBN|9780919611214}}; [https://www.worldcat.org/oclc/474564475   OCLC 474564475]
* __________ and Dan Pedoe. (1991) {{nihongo|''How to resolve Japanese temple geometry problems?'' |日本の幾何ー何題解けますか?|Nihon no kika nan dai tokemasu ka}} Tōkyō : Mori Kitashuppan. {{ISBN|9784627015302}}; [https://www.worldcat.org/oclc/47500620   OCLC 47500620]
* __________ and [[Tony Rothman]]. (2008). ''[[Sacred Mathematics: Japanese Temple Geometry]].'' Princeton: [[Princeton University Press]]. {{ISBN|069112745X}}; [https://www.worldcat.org/oclc/181142099   OCLC 181142099]
* Huvent, Géry.  (2008). [https://books.google.com/books?id=WXMNPQAACAAJ&dq=Sangaku.+Le+myst%C3%A8re+des+%C3%A9nigmes+g%C3%A9om%C3%A9triques+japonaises&lr=&cd=1   ''Sangaku. Le mystère des énigmes géométriques japonaises.''] Paris: Dunod. {{ISBN|9782100520305}}; [https://www.worldcat.org/oclc/470626755   OCLC 470626755]
* Rehmeyer, Julie,  [https://web.archive.org/web/20120216201242/http://www.sciencenews.org/view/generic/id/9499/title/Math_Trek__Sacred_Geometry "Sacred Geometry"], ''Science News,'' March 21, 2008.
*{{cite magazine | last1 = Rothman | first1 = Tony| last2 = Fugakawa | first2 = Hidetoshi | title=Japanese Temple Geometry| magazine = [[Scientific American]]|date=May 1998|pages=84–91}}

==External links==
{{Commons category}}
*[http://www.sangaku.info/ Sangaku (Japanese votive tablets featuring mathematical puzzles)]
*[http://www.wasan.jp/english/ Japanese Temple Geometry Problem]
*[http://www.cut-the-knot.org/pythagoras/Sangaku.shtml Sangaku: Reflections on the Phenomenon]
*[https://web.archive.org/web/20230601100137/https://www.sangaku-journal.eu/ Sangaku Journal of Mathematics]

{{Authority control}}

[[Category:Euclidean geometry]]
[[Category:Japanese mathematics]]
[[Category:Recreational mathematics]]