Über die Autorin bzw. den Autor

Fukagawa Hidetoshi is a retired high-school teacher in Japan, and one of the world's experts on sangaku. He is the coauthor of Japanese Temple Geometry Problems. Tony Rothman is a theoretical cosmologist who lectures in physics at Princeton University. His books include Everything's Relative and Other Fables from Science and Technology.

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"An enchanting history of Japanese geometry--of a time and place where 'geometers did not cede place to poets.' This intersection of science and culture, of the mathematical, the artistic, and the spiritual, is packed, like circles within circles, with rewarding Aha! epiphanies that drive a mathematician's curiosity."--Siobhan Roberts, author ofKing of Infinite Space

"Teachers will welcome this remarkable collection of mathematical problems, history, and art, which will enrich their curriculum and promote both logical thinking and critical evaluation. It is especially important that we maintain an interest in geometry, which needs, and for once gets, more than its share."--Richard Guy, coauthor ofThe Book of Numbers

"This remarkable book provides a novel insight into the Japanese mathematics of the past few hundred years. It is fascinating to see the difference in mathematical style from that which we are used to in the Western world, but the book also elegantly illustrates the cross-cultural Platonic nature and profound beauty of mathematics itself."--Roger Penrose, author ofThe Road to Reality

"A significant contribution to the history of mathematics. The wealth of mathematical problems--from the very simple to quite complex ones--will keep the interested reader busy for years. And the beautiful illustrations make this book a work of art as much as of science. Destined to become a classic!"--Eli Maor, author of The Pythagorean Theorem: A 4,000-Year History

"A pleasure to read. Sacred Mathematics brings to light the unique style and character of geometry in the traditional Japanese sources--in particular thesangaku problems. These problems range from trivial to utterly devilish. I found myself captivated by them, and regularly astounded by the ingenuity and sophistication of many of the traditional solutions."--Glen Van Brummelen, coeditor ofMathematics and the Historian's Craft

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"An enchanting history of Japanese geometry--of a time and place where 'geometers did not cede place to poets.' This intersection of science and culture, of the mathematical, the artistic, and the spiritual, is packed, like circles within circles, with rewarding Aha! epiphanies that drive a mathematician's curiosity."--Siobhan Roberts, author ofKing of Infinite Space

"Teachers will welcome this remarkable collection of mathematical problems, history, and art, which will enrich their curriculum and promote both logical thinking and critical evaluation. It is especially important that we maintain an interest in geometry, which needs, and for once gets, more than its share."--Richard Guy, coauthor ofThe Book of Numbers

"This remarkable book provides a novel insight into the Japanese mathematics of the past few hundred years. It is fascinating to see the difference in mathematical style from that which we are used to in the Western world, but the book also elegantly illustrates the cross-cultural Platonic nature and profound beauty of mathematics itself."--Roger Penrose, author ofThe Road to Reality

"A significant contribution to the history of mathematics. The wealth of mathematical problems--from the very simple to quite complex ones--will keep the interested reader busy for years. And the beautiful illustrations make this book a work of art as much as of science. Destined to become a classic!"--Eli Maor, author of The Pythagorean Theorem: A 4,000-Year History

"A pleasure to read. Sacred Mathematics brings to light the unique style and character of geometry in the traditional Japanese sources--in particular thesangaku problems. These problems range from trivial to utterly devilish. I found myself captivated by them, and regularly astounded by the ingenuity and sophistication of many of the traditional solutions."--Glen Van Brummelen, coeditor ofMathematics and the Historian's Craft

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Sacred Mathematics

Princeton University Press

Chapter One

Temple bells die out. The fragrant blossoms remain. A perfect eve ning! -Basho

Temples

No visitor to a foreign country has failed to experience the fascination and unease that accompanies an encounter with unknown traditions and customs. Some visitors attempt to overcome their fears, while the majority quickly retreats to familiar shores, and in this lies a distinction: Those who embrace culture shock are travelers; those who do not are tourists.

The most profound culture shock comes about when one is confronted by a different way of thinking. Most of us can hardly imagine walking into a Western church or cathedral to encounter stained glass windows covered by equations and geometrical figures. Even if we can conceive of it, the thought strikes us as alien, out of place, perhaps sacrilegious. Yet for well over two centuries, Japanese mathematicians-professionals, amateurs, women, children-created what was essentially mathematical stained glass, wooden tablets adorned with beautiful geometric problems that were simultaneously works of art, religious offerings, and a record of what we might call "folk mathematics." The creators of these sangaku-a word that literally means "mathematical tablet"-hung them by the thousands in Buddhist temples and Shinto shrines throughout Japan, and for that reason the entire collection of sangaku problems has come to be known as "temple geometry," sacred mathematics.

In this book you will be invited not only to encounter temple geometry but to appreciate it. There is a bit of culture shock to be overcome. A single glance at a sangaku is enough for one to realize that they were created by a profoundly different esthetic than the Greek-inspired designs found in Western geometry books. On a deeper level, one learns that the methods Japanese geometers employed to solve such problems differed, sometimes significantly, from those of their Western counterparts. Ask any professional mathematician whether the laws of mathematics would be the same in another universe and he or she will reply, of course. Real mathematicians are Pythagoreans-they cannot doubt that mathematics exists independently of the human mind. At the same time, during their off hours, mathematicians frequently speculate about how different mathematics could look from the way it is taught in Western schools.

Temple geometry provides a partial answer to both questions. Yes, the rules of mathematics are the same in East and West, but yes again, the traditional Japanese geometers who created sangaku saw their mathematical world through different eyes and sometimes solved problems in distinctly non-Western ways. To learn traditional Japanese mathematics is to learn another way of thinking.

Traditional Japanese mathematics, and with it temple geometry, arose in the seventeenth century under a nearly unique set of circumstances. In 1603, three years after defeating his rival daimyo- warlords-at the battle of Sekigahara, Tokugawa Ieyasu became shogun of Japan, establishing the Tokugawa shogunate. (A contemporary depiction of the battle of Sekigahara can be seen in the color plate 1.) The Tokugawa family ruled Japan for the better part of three hundred years, until 1868, when a decade after Commodore Matthew C. Perry forcefully opened Japan to the West, the shogunate collapsed.

One of Ieyasu's first moves after Sekigahara was to establish his headquarters at a small fortress town in central Japan, a town that became known as Edo (pronounced "Yedo")-today's Tokyo. For that reason the rule of the Tokugawa is also known as the Edo period. During the first years of the Tokugawa shogunate, Ieyasu (who, although living until 1616, officially remained shogun only until 1605) consolidated power by confiscating the lands of other warlords, but nevertheless continued many of the foreign policies of his pre decessor, the great daimyo Toyotomi Hideyoshi (1537-1598). At the turn of the seventeenth century, Japan carried on substantial trade with foreign countries, both Eastern and Western. Nagasaki on the island of Kyushu had become the base for the "southern barbarians" to import their goods, as well as to print translations of Western literature, much of it religious.

Foreign missionaries had by then been in Japan for over fifty years. In autumn of 1543, three Portuguese were shipwrecked off Kyushu. The misfortune proved decisive in terms of Japan's relations with outsiders, for the men were carrying arquebuses, which were rapidly adopted by the Japanese warlords. Of equal or greater importance was that, within a few years of the fateful shipwreck, Portuguese merchants and Jesuit missionaries began to arrive, seeking both trade and converts. The Jesuits were particularly successful, converting as many as two hundred thousand Japanese over the next forty years and becoming de facto rulers of the Nagasaki region.

All of this alarmed the proponents of Buddhism and raised the distrust of Hideyoshi himself; he in 1587 took direct control of Nagasaki and issued two edicts designed to curb the spread of Christianity. But the Spanish soon arrived, with Spanish merchants vying with Portuguese for trade and Franciscans vying with Jesuits for converts. In 1596, after a Spaniard supposedly boasted that the missionaries were merely the vanguard of an Iberian conquest, Hideyoshi ordered the execution of twenty- six priests and converts. The warlord, though, had other affairs on his mind, in particular the conquest of China, and he failed to pursue a resolution of the growing tensions between the Japanese and Westerners.

The tensions were resolved, in a particularly decisive and brutal fashion, at the very end of Tokugawa Ieyasu's life and in the two decades that followed. In 1614 Ieyasu reissued an earlier edict with which he summarily ordered that all Christian missionaries leave the country, that places of worship be torn down, and that the practice of Christianity be outlawed. But other internal affairs intervened and Ieyasu died in 1616 without having taken much action. After his death, though, persecution of Christian converts began in earnest and by 1637, according to some estimates, three hundred thousand converts had apostasized or been killed. Throughout the 1630s Ieyasu's grandson, Togukawa Iemitsu, issued a series of decrees that offered rewards for the identification of kirishitan, forbade the sending of Japanese ships overseas, and forbade any Japanese from traveling abroad, on pain of death.

By 1641 the last of the Portuguese merchants had been expelled, leaving only the Dutch. The Dutch had arrived comparatively late to Japan, in 1609, and had shown more interest in trade than mission. For that reason they were allowed to remain after the expulsion of the Iberians. The Japanese, however, by now utterly suspicious of Westerners, put severe strictures on the Dutch presence: The representatives of the Dutch East India Company were forced to move onto a small, man-made island called Deshima in Nagasaki harbor (see color plate 2 and plate 1.2). The fan-shaped island, originally created for the Portuguese, measured only 200 by 70 meters. A wall surrounded Deshima, posted with signs warning the Japanese to keep away, and it was entirely cut off from the mainland except for a bamboo water pipe and a single, guarded bridge. On this oasis,...