Skip to main content
No. 3111:
Temple Geometry

by Andy Boyd

Today, a very different expression of religion. The University of Houston presents this series about the machines that make our civilization run, and the people whose ingenuity created them.

Imagine walking into a church, looking up at the walls, and seeing ... math problems. Not stained glass or statues, but wooden tablets inscribed with intricate geometry problems. That was the case for hundreds of years in Japan.

Temple Geometry arose in the seventeenth century during Japan's Edo period. It was a time of great peace and stability during which Japanese culture flourished.

example of temple geometry
Temple Geometry

But it was also a time when the country closed its borders to the outside world. For over two hundred years, Japan developed in seclusion. Calculus, the most transformative concept in the history of mathematics, was unknown to the Japanese. And that profoundly influenced mathematical development.

Prior to the seventeen hundreds, mathematics hadn't made much progress in Japan. It wasn't a priority. Doing arithmetic with the Chinese abacus was the state of the art. But that changed during the Edo period. A pay-as-you-go school system evolved, open to people from all social classes. People wanted to learn, and they were willing to pay for the luxury. And as education blossomed, so, too, did creativity.

Temple geometry grew out of a fascination with plane geometry; problems involving squares, circles, and other two dimensional shapes. Plane geometry is filled with delightful results. Many are relatively simple, but others are quite involved. And during the Edo period, when people discovered a new geometric truth, they'd engrave it on a tablet and use it to adorn a shrine or temple.

The tablets, called sangaku, bore geometrical figures, problem descriptions, and final answers -- but rarely an explanation of how the answers were derived. Instead, visitors to a house of worship were invited to ponder where the answers came from. Historians describe sangaku as being "simultaneously works of art, religious offerings, and a record of what we might call 'folk mathematics.'" 

example of sangaku
Sangaku of Soddy's hexlet in Samukawa Shrine. Photo Credit: Wikimedia

Parallels can be found in Western culture. Euclid's Elements, perhaps the most influential book in the history of mathematics, was a study of plane geometry. It arose from a long geometric tradition tracing back to Pythagoras. Geometric truths were fundamental to the Pythagoreans' religious mysticism, seen as more than mere facts, but a glimpse into realms largely hidden from mere mortals. Japanese temple geometry shared that religious quality but in a distinctly different way. Sangaku was a mathematical gift to the gods; a humble thank you for revealing resplendent truths. And a supplication to reveal even greater truths should the gods be so willing.

example of Pythagorean Sunrise
Pythagoreans Celebrate Sunrise by Fyodor Bronnikov. Photo Credit: Wikimedia

Today, we study plane geometry in high school. It's an ideal way to teach logical reasoning. And the problems still challenge and satisfy, just as they've done throughout human history. Though I must say, most students don't seem to find it a religious experience.

I'm Andy Boyd at the University of Houston, where we're interested in the way inventive minds work. 

(Theme music)

This episode is an updated version of episode 2441.

Fukagawa Hidetoshi and Tony Rothman. Sacred Mathematics. Princeton: Princeton University Press, 2008. Contains information on the history of temple geometry in addition to the solution of many temple geometry problems.

Sangaku: Reflections on the Phenomenon. Taken from the Cut the Knot website. Accessed February 21, 2017. Contains solutions to many temple geometry problems, some interactive.

The picture of the sangaku tablet is widely circulated and considered in the public domain. It was taken from the Princeton University website: Click here. Accessed February 21, 2017.


An example temple geometry problem. The three circles shown above are tangent to one another and to the horizontal line. Show that the three radii always satisfy the relationship

1/squareroot(R3) = 1/squareroot(R1) + 1/squareroot(R2)

Hint: Use the Pythagorean Theorem on appropriately chosen triangles.

For a solution, Click here.

This episode was first aired on February 23, 2017