vsoy
Mar 24th, 2005, 08:49 PM
Cool! Blending art and math! Something to see in Nayoga, Japan next month. I think it's also neat that this high school teacher did all this research too.
Science, vol307p1715,2005
HISTORY OF MATHEMATICS:
"Amateur" Proofs Blend Religion and Scholarship in Ancient Japan
Dennis Normile
A 300-year-old Japanese art form presents some surprising mathematical discoveries on elegant wooden tablets
TOKYO--When Japan was isolated from the rest of the world, a unique brand of mathematics flourished in the country's shrines and temples. Amateur mathematicians crafted geometric theorems on elegant wooden tablets called sangaku (literally "mathematical tablets") and offered them to the gods. Remarkably, some of those theorems predate by more than a century the work of Western mathematicians.
Next month the Nagoya City Science Museum will present an exhibition of 130 sangaku from Japan's Edo Period (early 17th to mid-19th centuries). Assembling the show was a labor of love by Hidetoshi Fukagawa, a high school math teacher in central Japan, who has written the definitive texts on the unusual art form. "It's a really remarkable phenomenon, showing that ordinary people of that time studied mathematics purely for enjoyment," says Fukagawa about the sangaku, which were hung up at shrines and temples and often beautifully illustrated with miniatures of women in kimonos, teachers and pupils studying, and landscapes.
Their appeal crosses the oceans. The exhibition "is a unique occasion to see one of the great treasures of Japanese culture," says Freeman Dyson, a mathematician at the Institute for Advanced Study in Princeton, New Jersey. "I wish I could come to Japan."
The sangaku tradition flourished in an era when Japan was closed to outside influences and at peace both internally and with its neighbors. That calm meant that the samurai--traditionally schooled not only in swordsmanship but also literature, philosophy, sciences, and the arts--could turn their attention from martial to more intellectual matters. Adds Fukagawa, "There was no academia as we know it. So samurai, farmers, and merchants all felt free to study mathematics."
Figure 1 Artistic math. Illustrated mathematical tablets, or sangaku, include straightforward geometrical problems as well as suggestions for estimating the height of distant peaks (above). An exhibition opens next month in Nagoya, Japan.
CREDIT: HIDETOSHI FUKAGAWA
The amateur mathematicians built upon an existing tradition of hanging wooden tablets with poetry or paintings in Shinto shrines and Buddhist temples, painting or engraving sangaku that typically give the result of a problem but not the proof. "Ostensibly, the tablets were left as gifts to the gods," Fukagawa explains. "In reality, people were showing off and challenging others to work out the proof."
The vast majority of the problems involve plane geometry. But some involve calculating volumes of solids and others deal with algebra-like equations. The sangaku crafters typically included their names and the dates they hung the tablets.
Once Japan ended its isolation in the mid-1800s, the government encouraged the study of the European mathematical tradition as part of its push to catch up to the West technologically and economically. The archaic Chinese characters of Japanese mathematics fell into disuse, and the sangaku tradition disappeared. The rediscovery of sangaku is due in large part to 61-year-old Fukagawa, who holds a degree in mathematics and who has spent nearly 40 years teaching high school math in Aichi Prefecture. Looking for material to enliven his classes, he stumbled upon sangaku. "At the time, no Japanese mathematician had studied sangaku in any depth," he says.
His first step was to teach himself the archaic Chinese characters used on the tablets. The more sangaku Fukagawa deciphered, the more impressed he became with their sophistication. Japanese mathematicians were less enthralled, however, so Fukagawa started contacting geometers in other countries. His search led to a number of collaborations. In 1989 he and Daniel Pedoe of the University of Minnesota, Twin Cities, co-authored Japanese Temple Geometry Problems, which remains the most complete monograph on sangaku in any language. In 2002 he and John Rigby of Cardiff University in Wales published Traditional Japanese Mathematics Problems from the 18th and 19th Centuries.
The first book describes a number of Western geometrical theorems that were solved independently in Japan. One notable example is Soddy's hexlet, a theorem published in 1936 by Frederick Soddy, a British chemistry Nobel laureate, involving a complex construction of spheres within a sphere. Fukagawa and Pedoe found that the identical solution had been inscribed on a sangaku placed at a shrine in Kanagawa Prefecture in 1822. (The tablet is lost but is described in a written text.)
Even so, the mathematical significance of the sangaku tradition is an open question. Hikosaburo Komatsu, a mathematician at the Science University of Tokyo who studies Japan's indigenous math, agrees that their existence "shows that knowledge of math among ordinary citizens of that time was quite high." But the tablet format limits results so that "mathematically, sangaku are not very deep," he says. Serious Japanese mathematicians were producing much more significant theoretical work at the time, he notes. Still, Peter Wong, who grew up in Hong Kong and now teaches mathematics at Bates College in Lewiston, Maine, says the sangaku "open up all sorts of questions" about how laypeople developed sufficient mathematical skills to tackle nontrivial problems.
Fukagawa hopes further study will provide some answers. About 900 sangaku are known to remain, and dozens more that have been lost are known from written references. Only last year, during a visit to a shrine in Mie Prefecture, Wong used his knowledge of Chinese characters to point out a sangaku that Fukagawa had overlooked. Fukagawa also hopes the exhibition, which runs from 19 April to 26 June, will stimulate interest in the topic and yield additional sangaku.
Science, vol307p1715,2005
HISTORY OF MATHEMATICS:
"Amateur" Proofs Blend Religion and Scholarship in Ancient Japan
Dennis Normile
A 300-year-old Japanese art form presents some surprising mathematical discoveries on elegant wooden tablets
TOKYO--When Japan was isolated from the rest of the world, a unique brand of mathematics flourished in the country's shrines and temples. Amateur mathematicians crafted geometric theorems on elegant wooden tablets called sangaku (literally "mathematical tablets") and offered them to the gods. Remarkably, some of those theorems predate by more than a century the work of Western mathematicians.
Next month the Nagoya City Science Museum will present an exhibition of 130 sangaku from Japan's Edo Period (early 17th to mid-19th centuries). Assembling the show was a labor of love by Hidetoshi Fukagawa, a high school math teacher in central Japan, who has written the definitive texts on the unusual art form. "It's a really remarkable phenomenon, showing that ordinary people of that time studied mathematics purely for enjoyment," says Fukagawa about the sangaku, which were hung up at shrines and temples and often beautifully illustrated with miniatures of women in kimonos, teachers and pupils studying, and landscapes.
Their appeal crosses the oceans. The exhibition "is a unique occasion to see one of the great treasures of Japanese culture," says Freeman Dyson, a mathematician at the Institute for Advanced Study in Princeton, New Jersey. "I wish I could come to Japan."
The sangaku tradition flourished in an era when Japan was closed to outside influences and at peace both internally and with its neighbors. That calm meant that the samurai--traditionally schooled not only in swordsmanship but also literature, philosophy, sciences, and the arts--could turn their attention from martial to more intellectual matters. Adds Fukagawa, "There was no academia as we know it. So samurai, farmers, and merchants all felt free to study mathematics."
Figure 1 Artistic math. Illustrated mathematical tablets, or sangaku, include straightforward geometrical problems as well as suggestions for estimating the height of distant peaks (above). An exhibition opens next month in Nagoya, Japan.
CREDIT: HIDETOSHI FUKAGAWA
The amateur mathematicians built upon an existing tradition of hanging wooden tablets with poetry or paintings in Shinto shrines and Buddhist temples, painting or engraving sangaku that typically give the result of a problem but not the proof. "Ostensibly, the tablets were left as gifts to the gods," Fukagawa explains. "In reality, people were showing off and challenging others to work out the proof."
The vast majority of the problems involve plane geometry. But some involve calculating volumes of solids and others deal with algebra-like equations. The sangaku crafters typically included their names and the dates they hung the tablets.
Once Japan ended its isolation in the mid-1800s, the government encouraged the study of the European mathematical tradition as part of its push to catch up to the West technologically and economically. The archaic Chinese characters of Japanese mathematics fell into disuse, and the sangaku tradition disappeared. The rediscovery of sangaku is due in large part to 61-year-old Fukagawa, who holds a degree in mathematics and who has spent nearly 40 years teaching high school math in Aichi Prefecture. Looking for material to enliven his classes, he stumbled upon sangaku. "At the time, no Japanese mathematician had studied sangaku in any depth," he says.
His first step was to teach himself the archaic Chinese characters used on the tablets. The more sangaku Fukagawa deciphered, the more impressed he became with their sophistication. Japanese mathematicians were less enthralled, however, so Fukagawa started contacting geometers in other countries. His search led to a number of collaborations. In 1989 he and Daniel Pedoe of the University of Minnesota, Twin Cities, co-authored Japanese Temple Geometry Problems, which remains the most complete monograph on sangaku in any language. In 2002 he and John Rigby of Cardiff University in Wales published Traditional Japanese Mathematics Problems from the 18th and 19th Centuries.
The first book describes a number of Western geometrical theorems that were solved independently in Japan. One notable example is Soddy's hexlet, a theorem published in 1936 by Frederick Soddy, a British chemistry Nobel laureate, involving a complex construction of spheres within a sphere. Fukagawa and Pedoe found that the identical solution had been inscribed on a sangaku placed at a shrine in Kanagawa Prefecture in 1822. (The tablet is lost but is described in a written text.)
Even so, the mathematical significance of the sangaku tradition is an open question. Hikosaburo Komatsu, a mathematician at the Science University of Tokyo who studies Japan's indigenous math, agrees that their existence "shows that knowledge of math among ordinary citizens of that time was quite high." But the tablet format limits results so that "mathematically, sangaku are not very deep," he says. Serious Japanese mathematicians were producing much more significant theoretical work at the time, he notes. Still, Peter Wong, who grew up in Hong Kong and now teaches mathematics at Bates College in Lewiston, Maine, says the sangaku "open up all sorts of questions" about how laypeople developed sufficient mathematical skills to tackle nontrivial problems.
Fukagawa hopes further study will provide some answers. About 900 sangaku are known to remain, and dozens more that have been lost are known from written references. Only last year, during a visit to a shrine in Mie Prefecture, Wong used his knowledge of Chinese characters to point out a sangaku that Fukagawa had overlooked. Fukagawa also hopes the exhibition, which runs from 19 April to 26 June, will stimulate interest in the topic and yield additional sangaku.