Gion Shrine Problem

This is probably the most notorious of the geometric sangaku. The tablet was hung at the Zenkoji temple by Saito Mitsukuni in 1815 [Fukagawa and Rothman, p. 250]. On the tablet Saito wrote:

This problem was first proposed by Tsuda Nobuhisa in 1749 on a sangaku of the Gion Shrine in Kyoto. Tsuda derived the answer with a high-degree equation, one of one thousand and twenty-four degrees. But the famous mathematician Ajima Naonobu showed how to solve it with an equation of only the tenth degree in the variable a. On this tablet, I will show Ajima's equation.

Ajima is on the record to have submitted his solution in 1774 which brought him great fame as a mathematician.

In a circular segment with base AB of length a and altitude m, there are a circle of radius r inscribed in one half of the segment and a square of side d inscribed in the other half, as shown.

Gion Shrine Problem

Form p = a + m + d + r and q = m/a + r/m + d/r. The task is to express a, m, d, and r in terms of p and q.

Following Fukagawa and Rothman I, too, admit to having no idea where or how approach that problem.

References

  1. H. Fukagawa, A. Rothman, Sacred Mathematics: Japanese Temple Geometry, Princeton University Press, 2008

Sangaku

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