Sangaku, sometimes written san gaku, are a unique product of Japan's sakoku policy- wooden tablets bearing mathematical theorems explored during this period of national isolation. As with other disciplines that experienced a renaissance at this time, such as Haiku, tea-ceremonies and flower arranging, the creation of sangaku was not simply a practical exercise but an aesthetic one. Given that they were hung in shrines, it is likey that they also draw upon an earlier (pre 15th century) Shinto tradition of hanging tablets depicting horses in place of carrying out an actual sacrifice. However, examples can more rarely be found in Buddhist temples, and since the mathematical truths they offered were often supplied without proof, they can be interpreted as much as a challenge to other mortals as a tribute to the kami.
Sangaku flourished at around the same time as Kowa Seki (1642-1708), a Japanese mathematician who was, by popular accounts, of a similar calibre to his contemporary, Newton. Much of the work attributed to Seki is lost, but it is known that his theory of determinants both predated and outpowered that developed in Germany by Leibniz.
Of course, the reason Leibniz pursued his own theory rather than working with Seki's was the total isolation of Japan from the rest of the world under the dictates of the Tokugawa shogunate. This isolation cut both ways- and so Japanese mathematics, uninfluenced by the calculus being developed in the Western world, became a distinct entity, wasan (traditional Japanese mathematics, to be contrasted with yosan, western mathematics).
Although founded on arithmetic manipulation of the soroban (Japanese abacus) and including work on algebraic puzzles, the aspect of wasan most commonly featured on sangaku were geometric thoerems. Whilst these were Euclidean in nature, there was a particular emphasis on circles and ellipses (western work often being more concerned with triangles and other polygons). Some sangaku pose problems that a modern schoolchild could solve in a few lines, whilst others would today be tackled by advanced calculus techniques then unknown in either the east or west. Some results on sangaku precede their discovery in their west (such as the Malfatti problem); many more are mathematically uninteresting today yet retain their stunningly elegant artistic merit.
There is also debate as to precisely who created the sangaku. On the one hand, they were written in Kambun, which is to Japanese as Latin is to English, suggesting that the practitioners were highly educated. Yet the simplicity of the results, and the attribution of some tablets to children, suggest that they have been a more universal pleasure. There is evidence to suggest that everyone from merchants to farmers to samurai engaged in wasan:
It is pleasant to realize that some sangaku were the works of ordinary mathematics devotees, carried away by the beauty of geometry. Perhaps a village teacher, after spending the day with students, or a samurai warrior, after sharpening his sword, would retire to his study, light an oil lamp and lose the world to an intricate problem involving spheres and ellipsoids. Perhaps he would spend days working on it in peaceful contemplation. After finally arriving at a solution, he might allow himself a short rest to savor the result of his hard labor. Convinced the proof was a worthy offering to his guiding spirits, he would have the theorem inscribed in wood, hang it in his local temple and begin to consider the next challenge. Visitors would notice the colorful tablet and admire its beauty. Many people would leave wondering how the author arrived at such a miraculous solution. Some might decide to give the problem a try or to study geometry so that the attempt could be made. A few might leave asking, "What if the problem were changed just so...."1
Today, less than a thousand sangaku survive, although collections of the problems, which began to be published in 1789, describe many more. Whilst tablets continued to be created well into the 20th century, wasan fell from favour in the 19th as yosan at first supplemented then eventually replaced the old ways with the fall of the Tokugawa shogunate in 1867. Many tablets were lost during modernisation. The oldest surviving tablet dates from 1683, although other historical documents refer to examples from as early as 1668. Much of the research into sangaku was carried out by Hidetoshi Fukagawa, a high school teacher and holder of a doctorate in mathematics, one of a handful of people still able to understand Kambun. In 1989, he co-published an English collection of these puzzles2, but even amongst Japanese mathematicians the beautiful tablets are a mostly forgotten piece of history.
References
1 T. Rothman "Japanese Temple Geometry" Scientific American, May 1998 (accessed by EBSCO archives, with thanks to my University library).
2 H. Fukagawa & D. Pedoe "Japanese Temple Geometry Problems Sangaku" 1989, ISBN: 0919611214. Not even Amazon seems to have a copy, though.
- http://mathworld.wolfram.com/SangakuProblem.html
- http://www.sangaku.info/
- http://www.sunnyblue.net/tp/sangaku/san_info.html
- http://www.wasan.jp/english/
Recently, I have been working on Triangle Geometry, and in my research I encountered the phrase 'Japanese Temple geometry problem'. Node what intrigues you.