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The missing radius in a Sangaku geometry: an old Japanese problem

Posted 10 months ago
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10 Replies

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Thank you for all your interesting posts. I always enjoy them. The Mathematics Certification Institute of Japan celebrates Jan 23rd every year as "the day to spread the Sangaku culture."

Mr Okazaki,

Thanks for the compliment and the link. I am working on using WL to demonstrate a large number of geometry problems and properties from this fantastic book It is a wonderful journey to work with ancient sangaku problems plus modern technology.

Interesting work. I have a question. How do I use this to find, either exactly or approximately, the value of the radius for the original problem (that is, with the equilateral triangle).

You can change a to {1,Sqrt[3]} to form a equilateral triangle and run the attached notebook. The radius is about 0.318.


Thanks Shenghui. I had actually tried that change but for some reason the code was not working for me. Possibly I missed an initialization snippet. Anyway, it's a nice problem and solution method.

From the video linked in the head post the symbolic value of the radius for an equilateral triangle is $\sqrt{3}-\sqrt{2}$ .

WL indeed handles the symbolic case for equilateral triangle very well.

Nice! I wonder, is there a symbolic expression for a general (not equilateral) triangle case?

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By the way, this is the book to check out: Sacred Mathematics: Japanese Temple Geometry by Fukagawa Hidetoshi and Tony Rothman. Quoting from the book description:

Between the seventeenth and nineteenth centuries Japan was totally isolated from the West by imperial decree. During that time, a unique brand of homegrown mathematics flourished, one that was completely uninfluenced by developments in Western mathematics. People from all walks of life--samurai, farmers, and merchants--inscribed a wide variety of geometry problems on wooden tablets called sangaku and hung them in Buddhist temples and Shinto shrines throughout Japan. Sacred Mathematics is the first book published in the West to fully examine this tantalizing--and incredibly beautiful--mathematical tradition.

Fukagawa Hidetoshi and Tony Rothman present for the first time in English excerpts from the travel diary of a nineteenth-century Japanese mathematician, Yamaguchi Kanzan, who journeyed on foot throughout Japan to collect temple geometry problems. The authors set this fascinating travel narrative--and almost everything else that is known about temple geometry--within the broader cultural and historical context of the period. They explain the sacred and devotional aspects of sangaku, and reveal how Japanese folk mathematicians discovered many well-known theorems independently of mathematicians in the West--and in some cases much earlier. The book is generously illustrated with photographs of the tablets and stunning artwork of the period. Then there are the geometry problems themselves, nearly two hundred of them, fully illustrated and ranging from the utterly simple to the virtually impossible. Solutions for most are provided.

A unique book in every respect, Sacred Mathematics demonstrates how mathematical thinking can vary by culture yet transcend cultural and geographic boundaries.

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