Wednesday, September 12, 2012

Sangaku problem and C20

In the book "算法助述" I got from Mr. Horibe, there is an interesting problem in pp. 51-52 as shown in the following picture. As I mentioned before, Mr. Horibe made two beautiful bead models with a large metal sphere inside. And more importantly, large and small balls have the correct ratio of radii, R/r=sqrt{5}. He told me that he gave one of these two models to Hidetoshi Fukagawa, the author of the book "Sacred Mathematica - Japanese Temple Geometry", because, through Dr. Fukagawa, he knew so much about these interesting Japanese sangaku problems.
This Sangaku problem was originally proposed by Ishikawa Nagamasa and written on a tablet hung in 1798 in Tokyo’s Gyuto Tennosha shrine. I was quite surprised by that I could understand this ancient problem without any translation. It was written in Chinese (or Kanji):



In English:

Thirty small balls cover one big ball where each small ball touches four other small balls and big balls.
The radius of small ball is 305 inches. Find the radius of large ball.

Answer: The radius of large ball is 682 inches.

In the page 52 of the book, there are some explanations on how to reach the answer by noting that thirty balls are located on the thirty vertices of an icosidodecahedron. It was written with some Japanese characters which I don't understand. But it is not hard to guess the rationale. One can see that if one starts from an arbitrarily chosen small ball, then moves to next ball along a fixed direction, and then the next one along the same direction. Eventually, one would trace over exactly 10 balls which are located on the large circle of a sphere with the radius equal to the sum of radii of small and large balls. Using some trigonometry, one can get the answer.

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