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If I tell you the radii of the spheres 𝑎 and 𝑏 in this picture, can you figure out the radii 𝑟₁,...,𝑟₆ of the six spheres that touch them and snugly fit inside the big sphere? Can you at least do it if you know 𝑟₁?

Irisawa Shintarō Hiroatsu did it in 1822! He was a merchant who sold tea, textiles and ingredients for traditional Chinese medicine - and he had a hobby of solving math puzzles.

In 1932 his technique was rediscovered by a Nobel-prize-winning chemist, so it's often called Soddy’s Hexlet Theorem. But Hiroatsu did it earlier as part of a Japanese mathematical tradition called 𝑤𝑎𝑠𝑎𝑛 - and as part of this tradition, he donated a plaque containing this result to a shrine!

He wasn't the only one who did this sort of thing. This kind of plaque is called a 𝑠𝑎𝑛𝑔𝑎𝑘𝑢. These plaques were used to commemorate newly discovered solutions to hard math problems during the Edo Period from 1603 to 1868. There's a lot of interesting math in these 𝑠𝑎𝑛𝑔𝑎𝑘𝑢, and you can see some of them here:

• Abe Haruki, Japan’s “𝑊𝑎𝑠𝑎𝑛” mathematical tradition: surprising discoveries in an age of seclusion, https://www.nippon.com/en/japan-topics/c12801/

You can also learn more about the solution to the puzzle I gave! The most surprising thing is that the reciprocals of the opposite pairs of spheres in the "hexlet" of 6 spheres add up to the same number:

1/𝑟₁+1/𝑟₄ = 1/𝑟₂+1/𝑟₅ = 1/𝑟₃+1/𝑟₆

See also:

• Wikipedia, Soddy's hexlet, https://en.wikipedia.org/wiki/Soddy%27s_hexlet

This math is secretly all about conformal transformations, which map spheres to spheres... or planes!

Thanks to @highergeometer for pointing this out!