Sangaku

Sangaku or San Gaku (算額; lit. mathematical tablet) are Japanese geometrical puzzles in Euclidean geometry on wooden tablets created during the Edo period (1603-1867) by members of all social classes. The Dutch Japanologist Isaac Titsingh first introduced "sangaku" to the West when he returned to Europe in the late 1790s after more than twenty years in the Far East. [Association of American Geographers. (1911). " Annals of the Association of American Geographers," (Vol. I) p. 35.] [http://books.google.com/books?id=GR4aAAAAMAAJ&q=isaac+titsingh&dq=isaac+titsingh&ie=ISO-8859-1&pgis=1 ]

During this period Japan was completely isolated from the rest of the world so the tablets were created using Japanese mathematics, ("wasan"), not influenced by western mathematical thought. For example, the fundamental connection between an integral and its derivative was unknown so Sangaku problems on areas and volumes were solved by expansions in infinite series and term-by-term calculation.

The Sangaku were painted in color on wooden tablets which were hung in the precincts of Buddhist temples and Shinto shrines as offerings to the gods or as challenges to the congregants. Many of these tablets were lost during the period of modernization that followed the Edo period, but around nine hundred are known to remain.

A typical problem, which is presented on an 1824 tablet in the Gunma Prefecture, covers the relationship of three touching circles with a common tangent. Given the size of the two outer large circles, what is the size of the small circle between then? "The answer is:"

$frac\{1\}\{sqrt\{r\_\{middle\}\; =\; frac\{1\}\{sqrt\{r\_\{left\}\; +\; frac\{1\}\{sqrt\{r\_\{right\}$

Fujita Kagen (1765-1821), 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".

In 1989, a Sangaku collection, "Japanese Temple Geometry Problems" was published by Hidetoshi Fukagawa and Daniel Pedoe, and in 2008 "Sacred Mathematics: Japanese Temple Geometry", was published by Hidetoshi Fukagawa and Tony Rothman.

ee also

* Seki Takakazu (Kowa Seki)
* Japanese theorem for concyclic polygons
* Japanese theorem for concyclic quadrilaterals
* Equal Incircles Theorem

Notes

References

* Association of American Geographers. "Annals of the Association of American Geographers," Vol. I, 1911.
* Fukagawa, Hidetoshi and Daniel Pedoe. "Japanese Temple Geometry Problems: Sangaku". Charles Babbage Research Centre, 1989. ISBN 0-919611-21-4.
* Rothman, Tony and Fugakawa, Hidetoshi. "Japanese Temple Geometry," "Scientific American", May 1998.
* Fukagawa, Hidetoshi and Tony Rothman, "Sacred Mathematics: Japanese Temple Geometry" (Princeton University Press, Princeton, 2008). ISBN 0-691127-45-X.
* Rehmeyer, Julie, [http://www.sciencenews.org/view/generic/id/9499/title/Math_Trek__Sacred_Geometry Sacred Geometry] , Science News, March 21, 2008.

External links

* [http://agutie.homestead.com/files/sangaku2.html Sangaku problem] by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
* http://www.sangaku.info/
* http://www.wasan.jp/english/
* http://www.loyola.edu/maru/sangaku.html
* [http://www.archimedes-lab.org/monthly_puzzles_66.html An interesting Sangaku problem] by Archimedes Laboratory
* http://www2.gol.com/users/coynerhm/0598rothman.html
* [http://www.cut-the-knot.org/Curriculum/Geometry/PythagorasWithVectenInJapan.shtml Pythagoras and Vecten Break Japan's Isolation]

* http://matcmadison.edu/is/as/math/kmirus/Reference/SanGaku.html
* [http://www.cut-the-knot.org/pythagoras/Sangaku.shtml Sangaku: Reflections on the Phenomenon]
* East Asia Institute, University of Cambridge: [http://www.oriental.cam.ac.uk/jbib/edoint11.html Further reading/bibliography]