Japanese Temple Geometry*

Tony Rothman

            With the American daily news having transformed itself into an incessant trumpeting of economic Armageddon, it might seem the height of escapism, not to mention irrelevance, to contemplate a vanished Japanese mathematical tradition known as temple geometry.  Escapism it surely is; irrelevant, maybe.

            In the year 1600 Tokugawa Ieyasu defeated his rival warlords at the battle of Sekigahara and three years later became shogun of Japan.  The Tokugawa family ruled Japan for the better part of three hundred years.  Immediately upon Ieyasu’s death in 1616, his successors, having had their fill of Jesuits and Franciscans, began the systematic expulsion of foreigners from the country.  By 1640, all Western missionaries and merchants had been forcibly evicted, and it became a crime punishable by death for Japanese to travel abroad.  For the next two centuries, Japan’s sole contact with the West was through the Dutch East India Company, which was permitted to occupy a small island called Deshima  in Nagasaki harbor.

            Drastic without a doubt, nonetheless the Tokugawa’s isolationist policies produced a brilliant cultural renaissance.  The late seventeenth century saw the flowering of many of the traditions for which Japan is famous: tea ceremonies, garden architecture, haiku, no and kabuki drama, the ukiyo-e or “floating world” prints.

            Histories of the Tokugawa shogunate do not mention that the seventeenth century also witnessed the rise of a homegrown mathematics, one largely uninfluenced by Western developments, such as the invention of calculus by Newton and Leibnitz.  The samurai warriors had been pacified, many became highly educated government functionaries and, to supplement their meager salaries, some moonlighted as math teachers.  They fanned out into the countryside to small private schools teaching reading, writing and ‘rithmatic—and geometry.

            It had long been a Japanese tradition to hang votive tablets in Buddhist temples and Shinto shrines.  Such tablets might display a picture of a horse, the image being a substitute for the offering of a genuine horse to the temple.  But around 1660—we don’t know the exact date—a strange and wonderful custom emerged.  Lovers of mathematics began to hang wooden tablets engraved with geometry problems under the eaves of religious buildings.  These sangaku, a Japanese word that literally means “mathematical tablet,” were often large—a meter wide and three or four long.  Typically a sangaku contained a dozen problems with descriptions of each, diagrams and answers—but rarely full solutions.  The tablets themselves are works of art.  One of the most beautiful is framed by a pair of carved dragons, another is gilded; all are adorned by attractive, brightly colored figures drawn to resemble fans or kites, or whatever everyday object inspired the particular problem.

During the Tokugawa period, thousands of tablets appeared over Japan.  Only 910 survive today.  We are certain that over 1700 have been lost or destroyed and from mentions of tablets in old books, the 910 could represent as few as two percent of the original number.  From the inscriptions, we know that people from all walks of life created sangaku—samurai mathematicians, merchants, farmers, women and children.  We have problems from twelve-year-old boys and sixteen-year-old girls.  Many sangaku exercises are elementary, similar to those we encounter in a high school geometry course.  Others are astonishingly difficult and a handful anticipate theorems discovered in the West.  Sometimes the methods the Japanese employed were unwieldy, especially in matters requiring calculus, but in other respects were simpler than those taught today.  In yet other cases we cannot be certain exactly how the folk mathematicians solved a given problem.

It is clear that sangaku were hung both as challenges to other devotees and as thanks to God for mathematical progress.  Westerners often refuse to concede that mathematics can constitute a form of worship, can be sacred.  These are merely math problems, nothing more.  But whether they are math problems or concert oratorios or church relics or rocks, the sacredness of objects is never intrinsic; their status is conferred by the believer.  Moreover, the motto of Zen Buddhism might be taken to be “discipline in the service of enlightenment.”  What could be more disciplining—and enlightening—than  mathematics? 

The practice of hanging sangaku gradually died out when the Japanese   adopted Western mathematics after the collapse of the Tokugawa shogunate in 1868, and today the custom is virtually unknown, even in Japan.  For the past twenty years, a high school geometry teacher Fukagawa Hidetoshi has been instrumental in bringing sangaku to a wider audience and recently I collaborated with him in writing a book on the subject.**

For me, a physicist, it is pleasing to realize that there have existed societies less math-phobic than our own.   Japan is not alone, although America might be.  At the Princeton physics department, where I teach, it has become a running joke that the best incoming students are foreigners and not long ago I quipped to a colleague that we should force anyone whose surname ends in “ovich,” “adzic” or “escu” to be in the honors section without discussion.  The Eastern Europeans’ sterling performance is due to the remnants of the high-powered, no-nonsense Soviet educational system.  Asian and Asian American students, no longer to anyone’s surprise, also outperform their “native” counterparts.  They are hardly genetically superior; they are imbued with the Buddhist work ethic.

Contrast this with the American position, whose schizophrenic attitude toward math is embodied in the catch phrase “Do the math,” which rarely refers to anything more advanced than counting.*  On the one hand, Americans acknowledge math’s utility, if not its beauty; on the other hand we refuse to adequately train our students, believing that one has to be a born “rocket scientist” to be competent.  A fourth-grade teacher recently told me that in her school district there is no science or math curriculum—it is every teacher for herself and most of them do not know how to plot a straight-line graph or compute the number of seconds in a year.  Even among my Princeton students, the mindset that one can do physics and engineering without calculus is present.  And how often have instructors heard non-majors complain, “I’m interested in the concepts, not the math.”  In science, unfortunately, the two are impossible to disentangle.

With their “show me” attitude, Americans are repeating the age-old question, “What is the point of higher mathematics?”  Mathematicians tend to reply by quoting Benjamin Franklin’s version of their perennial answer: “What science can there be more noble, more excellent, more useful for men, more admirably high and demonstrative than mathematics?”

Yet even in Franklin’s response, there is more than a hint of the utilitarian.  Certainly the technological civilization we inhabit would collapse were not a sufficient number of people versed in higher mathematics.  All modern physics, chemistry, electronics, engineering and computer security relies on more algebra, calculus, number theory and geometry than one could possibly imagine hearing “Do the math.”  The development of alternative energy strategies will not be carried out by people inadequately trained in mathematics.

Unfortunately, even from the utilitarian point of view the prospects do not appear promising.  In a recent review of our book, the critic lamented that the study of geometry has decayed to such a primitive state in Britain that students under forty would find the book a severe challenge.  The same might be said in this country.  But the reviewer was not bemoaning the lack of appreciation of geometry’s utility.  Real mathematicians tend to be insulted if you insinuate that their work has any practical value.  Neither are young people attracted to mathematics because of its practical applications.  They like math because it opens up an universe of infinite possibilities.  The reviewer was speaking of the loss of cultural heritage.

It is here that the feudal Japanese farmers can offer their services.  They did not solve their temple geometry problems because they were practical, but because they were beautiful.  It would be naïve to suppose that every student will become a competent mathematician, but emphasizing the utility of mathematics is perhaps the wrong strategy.  Emphasize its beauty, its pleasures, and the skills developed in the attainment of this beauty will follow.