LATEST DIALOGUES Art & Spirit in Mathematics: The Lessons of Japanese Temple Geometry (part I)

Temple Geometry faded from memory—a shameful episode best forgotten—but in recent years it has been rehabilitated. The surviving sangaku panels, about 900 of them, are being conserved, and the amazing story of a culture profoundly in love with geometry is being unearthed. This movement has been led Fukagawa Hidetoshi, a high school mathematics teacher, who wrote the first modern history of Temple Geometry and then collaborated with Tony Rothman in translating it into Sacred Mathematics. Rothman, a physicist who loves geometry, added a wider perspective and much new material to this elegant and groundbreaking volume.

The story of JTG that we are telling, taken largely from Fukagawa and Rothman, has its critics. Alexander Bogomolny, an ardent lover of the mathematics of Temple Geometry, is skeptical of its broader social and spiritual significance (see his wonderful geometry website In judging this issue on the side of Fukagawa and Rothman, I am reminded that monks in medieval Europe studied Euclid as a way of putting their minds into harmony with the mind of God, where mathematical truth was believed to reside. During that time, little new geometry was created, as everything was referred back to the authority of Euclid. The Temple Geometers faced no such constraints. They were inspired by Euclid, but not bound by him. The growth of geometrical knowledge in Japan was explosive. And the centers of that growth were Zen and Shinto temples.

Fukagawa suggests the magnitude of his own achievement in his Preface: “When I became a high school mathematics teacher forty years ago, I studied the history of Western mathematics and would present some of this historical material to my students. In those days, it was said that traditional Japanese mathematics had no material of any value for high school students.”

By auspicious coincidence he was given an obscure text that he managed to decipher, from which, “I found that traditional Japanese mathematics of the seventeenth, eighteenth, and nineteenth centuries had much good material for high school students.”

Our interest is not just in the quality of the mathematics judged by our standards, as in the special qualities of the endeavor: “We call this mathematical world “Japanese temple geometry.” The mathematics lovers who formed this world enjoyed solving geometry problems.”

This world of mathematics lovers assembled in a temple is pictured in this unusual sangaku, which also poses three problems.

The group on the left includes two women and a child who are learning to use a Japanese abacus. On the right the teacher is leading a discussion of geometric calculations that involve solving higher degree equations. (Plate 4 in Sacred Mathematics)

When Temple Geometry met Western mathematics the latter won, because math-based technology gave Commodore Perry bigger guns and faster ships. Math continues to be a vital weapon for our military, and also contributes to many aspects of the unsustainable destruction of our living environment. In stark contrast, Temple Geometry, which was cultivated for peaceful purposes, for personal development and pleasure of the Temple Geometers, was part of a more stable and harmonious culture.

From the viewpoint of the Temple Geometers, it is our relationship with mathematics they appears badly out of balance. They may have been at one extreme, where mathematics was widely enjoyed and understood, but rarely applied; we have gone the other way, with mathematics central not just to technology and war, but also to communication, medicine, economics, education, etc. Yet all too many of us suffer negative emotions around math, from fear and loathing to boredom and indifference. Our view of mathematics is profoundly influenced by the dominant beliefs of the guild of academic mathematicians, which place mathematics above culture and beyond us. We are likely to feel powerless when decisions are made on mathematical grounds. Perhaps we could learn from the example of Temple Geometry how to reclaim our mathematics for peaceful purposes, to see in it the potential for pleasure and growth rather than domination, to experience it as “our most fascinating and imaginative art form”, rather than as a foundation for commerce and war (quote from the subtitle of Paul Lockhart’s lovely little book A Mathematician’s Lament). We would need to take it seriously as an integral part of our culture.

The widespread enjoyment of contemplating such problems on multiple levels points to a culture where math anxiety was not rampant, where people were at home with mathematics and did not regard it as alien. They were free to experience it as play, but play with serious intellectual, aesthetic, and even spiritual dimensions. The pleasures can be greatly magnified by exploring problems with a dynamic geometry program like the wonderful (and free!) GeoGebra which I have used to create the geometric figures in Japanese Temple Geometry on Beautiful Trees. The Temple Geometers would have loved GeoGebra.

After 1854, Japan came to devote a national effort to catching up with the West, and with the USA in particular. The STEM disciplines of science, technology, engineering, and mathematics were intensely cultivated. After 87 years, Japan paid Commodore Perry back at Pearl Harbor. Math played an even more direct role in the end of WW II, since the weapons used at Hiroshima and Nagasaki were unlike any that had come before, both in their destructive force and in their essentially mathematical nature. Our mathematics-based weapons technology had triumphed again. And again Japan tried to catch up, this time by embracing atomic power, a choice starkly illuminated by the Fukushima meltdown.