Mathematics in Japanese temples

Hundreds of years ago, people in Japan thanked gods by sacrificing a horse or pig. However, these are valuable assets, so it is difficult for poor people to express their sincerity. So they came up with a solution: instead of sacrificing horses, they just drew a horse on a wooden board and hung it in the temple.

Then one person, perhaps a deadly samurai, recognized that horses and pigs were not the only things that could be painted on a board. He came up with the idea of ​​drawing something really original, beautiful, something very creative. And he gave up the spirit of mathematics.

Hundreds of beautifully painted wooden boards on which there are mathematical problems and theorems used to decorate Japanese temples. They are called 'sangaku', simply meaning math tables. The characters on the boards are an ancient Chinese form, the language of scholars, similar to the Latin word in the West. These tables have only been translated into modern languages ​​in the last few decades.

This sangaku is hung in Kinshouzan Temple in Gifu Prefecture in 1865. There are 12 different geometric problems. The third problem from the right is given by a 16-year-old girl. (Photo: Fukagawa)

A Japanese math teacher, Mr. Hidetoshi Fukagawa, searched, translated and studied these boards. Next spring, Fukagawa and Tony Rothman of Princeton University will publish the entire history of sangaku, including photographs of many sangaku plates that have never been known outside of Japan.
Rothman said 'Sangaku is unique. They are not only particularly beautiful, but the problems on them are often particularly difficult. And the solution can be very smart. Some of the steps people used to solve those problems were never known. '

The sangaku panels were made during the period when Japan was almost isolated from the outside world. The shogun leaders (general) expelled foreign missionaries and banned Japanese people from leaving the country around the beginning of the 17th century. Unique cultural traditions such as tea ceremony, tangled stage and printing on wooden blocks.

It was created in 1814 but it was only discovered in 1994 when the temple stored it about to be destroyed. (Photo: Fukagawa)

At the same time, the shogun persuaded samurai warriors to lower their weapons and serve the government. However, the salary was so hard that the samurai had to find other jobs. One of those things is going to school math. Isolated from the development of mathematics in the West, these mathematicians and their students create a domestic geometry with unique Japanese characteristics. For example, many problems are solved based on the art of origami paper folding or paper fans .

This is an example of a sangaku problem. Draw a polygon in a circle with each of its corners lying on the circle. Select one of the vertices of the polygon and connect it to the other vertices, dividing this polygon into multiple triangles. In these triangles, draw a circle that just touches the sides of the triangle. The total radius of these circles will be constant, no matter which peak you choose.

A sangaku plate shows that the total number of radii of the small circles in each of these figures is equal. (Photo: Fukagawa)

Most sangaku simply give theorems and provide a chart, but they lack proof. The most direct proof is based on Carnot's theorem, this theorem is only proved in the West about 100 years after people make sangaku. Rothman believes that sangaku plates are not just religious sacrifices but 'play a role in challenging the courage of those who solve the problem.

Beginning around 1800, some collections of sangaku problems were turned into books, including the solution, so the researchers learned the original method of solving many problems. But the two sangaku plates still have no solution to this day. 'One of these problems has resulted in a quadratic equation of 1024. A mathematician later became famous because it went down to level 10, but it was still difficult ahead. We can't know how they do it. '

Temple of Kaizu Tenma, Shiga Prefecture has a sangaku plate containing up to 30 problems. It is about 25 cm wide and about 518 cm long. (Photo: Fukagawa)