Sangaku (散楽)

"Sangaku" means mathematical puzzles or solutions described on wooden tablets or set in frames, which were dedicated to Shinto shrines or temples during the Edo Period in Japan. Many of them were geometric ones. Dedicators of sangaku included not only mathematicians but also many general math lovers.


It is said that each sangaku was dedicated by a person who had solved the mathematical puzzle, as a token of his gratitude to the gods and in hope that he would devote more efforts to his studies. Gradually, shrines and temples gathering people became places where mathematical puzzles were published, and some dedicated difficult puzzles or puzzles without attaching solutions, while others dedicated solutions of the puzzles or puzzles reconstructed based on them.

No such custom can be found in any other country in the world and the culture of dedicating mathematical puzzles is said to be unique to Japan. The culture clearly represents the Japanese way of thinking, in which even mathematics is regarded as 'accomplishments of art' and some of the puzzles have been designated as Important Cultural Properties or Folk Cultural Properties. The culture of dedicating mathematical puzzles is appreciated as having served as a catalyst for introducing Western-style mathematics into Japan in the Meiji Period.

According to the survey conducted in 1997, there are 975 surviving sets of sangaku in Japan. Among those existing sangaku, the one bearing the earliest date is sangaku of the year 1683 dedicated to the Hoshinomiya-jinja Shrine located in Sano City, Tochigi Prefecture. According to "Sangaku Enteiki "(detail of sangaku) written by Yoshimasu MURASE in 1681, there were sangaku at various parts of Edo in the middle of the 17th century, and examples dedicated to the Meguro Fudo temple are described in the book. It is presumed that sangaku began earlier in Kyoto City and Osaka City than in Edo. In the latter half of the 17th century, sangaku puzzles were collected and compiled as mathematical books and the first one among those published as books is said to be "Shinpeki Sanpo" (Perfect Mathematics or Mathematics on God's wall) of the year 1789, which was edited by Sadasuke FUJITA.

The culture of dedicating sangaku prevailed nationwide in the mid-Edo Period, and particularly its popularity peaked in the Kansei, kyowa, Bunka and Bunsei eras (1789 to 1829) that, as records say, sometimes gathered more than 100 dedications annually. The culture, along with the tradition of wasan (Japanese mathematics), was passed down from the Meiji Period to the beginning of the Showa Period. Recently, there are movements for reappraising the value of sangaku in various places of Japan and in some areas, the number of people who dedicate sangaku to shrines and temples is increasing. Although most of these sangaku are not based on the traditional Japanese wasan, this cultural trend itself is interesting anyway in that it shows Japanese people's liking for mathematics.

Shrines and temples where many sangaku are distributed

Tohoku and Kanto regions have many of the existing sangaku and the prefecture that has the largest number of sangaku is Fukushima Prefecture with 103 sets, followed by Iwate Prefecture with 93 sets, Saitama Prefecture with 91 sets and Gunma Prefecture. Kijimadaira-mura Village of Nagano Prefecture, having eight old sangaku is an example of a small mountainous village having an extremely dense distribution of sangaku. Also, Isaniwa-jinja Shrine located in Matsuyama City, Ehime Prefecture, has 22 old sangaku dedicated thereto, and the number is the largest among those known to have survived in one location. Those sangaku remaining in Isaniwa-jinja Shrine were compiled into a pictorial record "Dogo Hachiman Isaniwa-jinja no sangaku" (sangaku of the Dogo Hachiman Isaniwa-jinja Shrine) and published by the shrine.