Sangaku
An excerpt from the article found here http://www.princeton.edu/main/news/archive/S15/04/04O77/index.xml?section=topstories
Rothman helps reveal intricacies of ancient math phenomenon
by Chad Boutin
This portion of a wooden sangaku — literally “mathematical tablet” — is from one of approximately 900 that survive in Japan from as far back as the 17th century. Sangaku illustrate solutions to puzzles using traditional Japanese geometrical techniques that developed independently of Western methods. Mathematicians frequently hung their often lavishly decorated tablets in temples and shrines as religious offerings.
The tablet, hung in Fukushima prefecture in 1885, measures 5.6 by 2.4 feet and includes a problem involving a folding fan, a popular item in the 19th century.
Across the sea
Sangaku became popular in Japan during the Edo period, which lasted from the early 17th century until 1857. Although the rulers had forcibly cut off the island nation from the West — allowing only one Dutch ship a year to make port in Nagasaki — the country experienced a rich cultural flowering. Intellectual pursuits such as mathematics were encouraged, but because of the country’s isolation, Japanese with a flair for the subject were forced to forge their own paths. What developed was a method of answering complex geometrical questions — such as how many circles of a certain size could fit into a particular triangle — without benefit of the techniques of calculus, which were relatively new in Europe at the time.
“Math aficionados of the period did not have access to the advances in math that Gottfried Wilhelm Leibniz and Isaac Newton had made, so they found other approaches,” Rothman said. “To solve problems like these, they used methods similar to those the ancient Greeks used, which were far more time-consuming. They worked fairly well, but you had to be willing to put in some effort.”
While modern mathematical methods, such as calculus, can sometimes simplify a sangaku problem that requires pages of calculation, Rothman said that the advantage to the math used in sangaku problems was that it was simple enough for young children to use, which opened the problems up to nearly anyone who wished to try them.
“Some of the tablets feature solutions provided by 12-year-olds,” he said. “But that doesn’t mean they were easy. Today’s high school geometry problems tend to require only five or six lines to solve, whereas the old problems often demand pages and pages of work. Sangaku were more like math Olympics problems, or the sort of thing your teacher might have put on the wall for extra credit.”
An additional complication is that the tablets were inscribed in Kanbun, a form of Chinese that, similar to Latin in the West, was the language of scholarship in Japan.
“Kanbun was not understandable to the average Japanese person even when the tablets were made,” Rothman said. “Even three centuries ago, they required footnote-like annotation to make them understandable. Now the tablets might be all but incomprehensible, were it not for the colorful accompanying illustrations of the geometry itself.”
Yet the images and math together made the tablets worthy to be given as offerings to Buddhist temples or Shinto shrines in lieu of money or animals, which were more common gifts.
“One sangaku hung in 1815 has a preface, in which a mathematician and his disciples introduce their work,” Rothman said. “Their inscription reads, ‘In this shrine, we ask god for progress in our mathematical ability and dedicate a sangaku.’ They took the religious aspect of their work seriously.”
