Sangaku or san gaku () are Japanese

geometrical Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

problems or theorems on wooden tablets which were placed as offerings at

Shinto shrine A Stuart D. B. Picken, 1994. p. xxiii is a structure whose main purpose is to house ("enshrine") one or more kami, , the deities of the Shinto religion. The Also called the . is where a shrine's patron is or are enshrined.Iwanami Japanese dic ...

s or

Buddhist temples A Buddhist temple or Buddhist monastery is the place of worship for Buddhists, the followers of Buddhism. They include the structures called vihara, chaitya, stupa, wat, khurul and pagoda in different regions and languages. Temples in Buddhi ...

during the

Edo period The , also known as the , is the period between 1600 or 1603 and 1868 in the history of Japan, when the country was under the rule of the Tokugawa shogunate and some 300 regional ''daimyo'', or feudal lords. Emerging from the chaos of the Sengok ...

by members of all social classes.

History

The sangaku were painted in color on wooden tablets (

ema Ema or EMA may refer to: Biology and medicine * Anti-Endomysial Antibodies test * Epithelial membrane antigen * European Medicines Agency, a European Union agency for the evaluation of medicinal products * European Medical Association, associa ...

) 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 Modernization theory or modernisation theory holds that as societies become more economically modernized, wealthier and more educated, their political institutions become increasingly liberal democratic and rationalist. The "classical" theories ...

that followed the Edo period, but around nine hundred are known to remain. 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''. During this period

Japan Japan is an island country in East Asia. Located in the Pacific Ocean off the northeast coast of the Asia, Asian mainland, it is bordered on the west by the Sea of Japan and extends from the Sea of Okhotsk in the north to the East China Sea ...

applied strict regulations to commerce and foreign relations for western countries so the tablets were created using

Japanese mathematics denotes a distinct kind of mathematics which was developed in Japan during the Edo period (1603–1867). The term ''wasan'', from ''wa'' ("Japanese") and ''san'' ("calculation"), was coined in the 1870s and employed to distinguish native Japanese ...

, developed in parallel to western mathematics. For example, the connection between an integral and its derivative (the

fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of derivative, differentiating a function (mathematics), function (calculating its slopes, or rate of change at every point on its domain) with the concept of integral, inte ...

) was unknown, so sangaku problems on areas and volumes were solved by expansions in

infinite series In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...

and term-by-term calculation.

Select examples

* A typical problem, which is presented on an 1824 tablet in

Gunma Prefecture is a landlocked Prefectures of Japan, prefecture of Japan located in the Kantō region of Honshu. Gunma Prefecture has a population of 1,937,626 (1 October 2019) and has a geographic area of . Gunma Prefecture borders Niigata Prefecture and Fuk ...

, covers the relationship of three touching circles with a common

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

, a special case of

Descartes' theorem In geometry, Descartes' theorem states that for every four kissing, or mutually tangent circles, the radii of the circles satisfy a certain quadratic equation. By solving this equation, one can construct a fourth circle tangent to three given ...

. Given the size of the two outer large circles, what is the size of the small circle between them? ''The answer is:'' :

\frac = \frac + \frac.

(See also Ford circle.) *

Soddy's hexlet In geometry, Soddy's hexlet is a chain of six spheres (shown in grey in Figure 1), each of which is tangent to both of its neighbors and also to three mutually tangent given spheres. In Figure 1, the three spheres are the red inner sphere and tw ...

, 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.

Notes

References

* Fukagawa, Hidetoshi, and Dan Pedoe. (1989). ''Japanese temple geometry problems = Sangaku''. Winnipeg: Charles Babbage.
OCLC 474564475
* __________ and Dan Pedoe. (1991) Tōkyō : Mori Kitashuppan.
OCLC 47500620
* __________ and

Tony Rothman Tony Rothman (born 1953) is an American theoretical physicist, academic and writer. Early life Tony is the son of physicist and science fiction writer Milton A. Rothman and psychotherapist Doris W. Rothman. He holds a B.A. from Swarthmore Col ...

. (2008). '' Sacred Mathematics: Japanese Temple Geometry.'' Princeton:

Princeton University Press Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial ...

.
OCLC 181142099
* Huvent, Géry. (2008)
''Sangaku. Le mystère des énigmes géométriques japonaises.''
Paris: Dunod.
OCLC 470626755
* Rehmeyer, Julie,
"Sacred Geometry"
''Science News,'' March 21, 2008. *

External links

Sangaku (Japanese votive tablets featuring mathematical puzzles)Japanese Temple Geometry ProblemSangaku: Reflections on the PhenomenonSangaku Journal of Mathematics
{{Authority control Euclidean geometry Japanese mathematics Recreational mathematics