Friday, September 21, 2012

Ancient proof of R/r=sqrt{5}

In the Meiji era, mathematicians of Japan didn't use trigonometry to prove R/r=sqrt{5} for the problem about a big ball covered by 30 small balls. I doubt that they knew the trigonometry as we know today. Instead, a regular pentagon as shown in the following figure was recognized. Then the answer follows naturally. It is a smart proof, isn't it?

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