The sangaku are wooden tablets decorated with geometrical problems, suspended in Japanese temples during the Edo era. A fascinating testimony of the link between spirituality, knowledge and aesthetics, they form an unknown but captivating part of Japanese culture. This article proposes you to discover the history, meaning and forms of these mathematical objects both learned and devotional.
Origins and historical context of sangaku
The sangakus appeared in Japan during the seventeenth century, at a time marked by the isolation of the country (sakoku politics) and an internal cultural effervescence. The Edo period (1603-1868) saw the rise of many autonomous artistic and intellectual forms, including Japanese mathematics (wasan). The sangaku, literally “calculation tablets”, are one of the most amazing expressions.
Placed in Shinto shrines and Buddhist temples, these votive tablets were offered to the deities, either as a thank you or to manifest the intellectual virtuosity of their author. They were designed by scholars, teachers, traders and even samurai who were ferus of geometry. They combined spirituality, knowledge and formal beauty in a deeply original approach to mathematics.
Structure and contents of a sangaku
A classic sangaku is a rectangular wooden tablet, usually painted or engraved. Each side features a colourful and stylized geometric figure, accompanied by a statement, one or more formulas, sometimes a demonstration, as well as information such as the date, author and his school.
The majority of puzzles deal with flat geometry: tangential circles, regular polygons, ellipses, spheres and conicals. There are also chains of circles, embedded figures or complex volumes. The aesthetic counts as much as the mathematical difficulty: visual harmony is paramount. The challenge is to propose a beautiful, difficult problem, but the solution remains possible with the mathematical tools of Japan of the time.
Japanese mathematical approach (wasan)
In contrast to Western mathematics, which then progressed towards differential and integral calculus, the Wasan ignored the notions of infinitesimal or derivative. Japanese scholars developed algebraic, geometric and arithmetical methods that were often original in solving problems. To calculate areas or volumes, they used infinite series that they developed term in term.
This approach has given rise to a wide variety of methods and schools, some of which – such as Seki Kōwa or Fujita Kagen – have left major works. The sangaku also allowed a form of emulation: the authors responded from one temple to another, in a kind of silent and codified mathematical joute.
Examples of famous sangaku
Some sangakus have lived through centuries because of their complexity or elegance. The problem of Hotta Jinsuke (1788), staged an infinite series of tangent circles inserted into an initial disc. This chain, called Pappus, is associated with mathematical suites with remarkable properties.
Another, more classical example is the riddle of the three aligned circles: two large circles frame a small tangential circle at the first two. From their diameters, the challenge is to determine the one of the smallest circle, applying creative formulas.
These examples illustrate the richness of the forms proposed, while testifying to the mathematical refinement of the creators of these tablets.
Religious and symbolic function
The sangaku were offerings to the local deities, in the same way as the ema (votive plates illustrated with vows or thanks). By their mathematical content, they expressed gratitude for intellectual enlightenment or were used to solicit divine protection on the author’s school or family.
Some sangakus can be considered ex-votos: their resolution could meet a wish fulfilled, or their design could serve as a tribute after an intense period of study. Knowledge thus became an act of faith, and reasoning became a form of devotion.
Decline and rediscovery
The tradition of the sangaku has been reversed with the modernization of Japan, which began at the Meiji era (1868-1912), and the introduction of Western mathematics into the educational system. Many tablets were lost, but about 900 were preserved or reproduced in collections by the end of the 18th century.
In the 1980s, several Japanese and foreign researchers, including Hidetoshi Fukagawa, rediscovered and re-examined these objects. Some sangaku have even been published in scientific journals, drawing attention to the singularity of this mathematical tradition.
Today, some temples still keep these tablets, sometimes exposed to the public. Local initiatives attempt to improve this form of heritage, including through exhibitions, reconstructions or resolution workshops.
Seeing sangaku today
Although rare, some genuine or reconstituted tablets can be seen in regional museums, shrines or during temporary exhibitions. The Hōtoku Museum of the Samukawa-jinja Shrine (Kanagawa Prefecture) or the Mathematical Museum in Tokyo are among the most accessible places to discover these works.
For lovers of geometry or Japanese culture, the observation of a sangaku is a singular experience, where reasoning, art and spirituality meet.
The sangaku form a discreet but revealing part of Japanese intellectual history. By linking geometry, aesthetics and faith, these tablets embody a unique look at knowledge as an offering. A tradition that, although extinguished in practice, continues to fascinate with the subtlety of its puzzles and the beauty of its forms.
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