This tablet was hung in the Kinshouzan shrine in the Gifu Prefecture in 1865. It shows 12 different geometric problems. The third problem from the right was presented by a 16-year-old girl.
Fukagawa
Posted on 03/23/2008 8:33:40 PM PDT by neverdem
Hundreds of years ago in Japan, people offered thanks to the gods by sacrificing a horse or a pig. Horses and pigs, however, were valuable and expensive, so poor folks had a hard time expressing their gratitude. So they came up with a solution: Rather than sacrificing a horse, they would simply draw a painting of a horse on a wooden tablet and hang it in the temple.
Then someone, most likely an impoverished samurai, realized that horses and pigs were hardly the only thing that could be drawn on a tablet. He had the idea of painting something original, something beautiful, something creative. He offered mathematics.
Hundreds of beautifully painted, multi-colored wooden tablets showing problems and theorems of geometry have adorned Japanese temples. They are called "sangakus," which simply means mathematical tablets. The text on the tablets is written in an ancient form of Chinese, which was the language of scholars, much like Latin in the West. Only in the past couple of decades have these tablets been translated into modern languages in significant numbers.
A Japanese mathematics teacher, Hidetoshi Fukagawa, has been finding, translating, and researching the tablets. This spring, Fukagawa and Tony Rothman of Princeton University will publish a complete history of sangaku, including photographs of many sangakus that have never before been seen outside of Japan.
"Sangakus are exceptional," Rothman says. "They're not only exceptionally beautiful, but the problems are often exceptionally difficult. And the solutions can be very clever. Some of the things they do to solve these problems would never have occurred to me."
The sangakus were made during a period when Japan was mostly isolated from the outside world. The shogun leaders expelled all the foreign missionaries and forbade Japanese from leaving the country on pain of death in the early 1600s...
(Excerpt) Read more at sciencenews.org ...
Very interesting. Thanks for the post!
Me either. I never would have thought of painting a math problem.
The Japanese were believed to have some of the finest mathematicians in the world in WWII. Some of their top ones were involved in cryptanalysis, code making and breaking.
Well into the war, the Americans suddenly came up with a mathematical code that was amazing. Astounding. Totally unique with nothing to compare it to. It was indecipherable.
Despite their Herculean efforts, their mathematicians could not even slightly penetrate this code. Some were so overwrought that they began to suffer from emotional collapse. Even the end of the war brought no relief to their intellectual anguish.
It is rumored that the government of Japan begged the US to let them in on the secret, for fear of losing the best mathematicians of their generation.
We told them that there is no mathematical way possible to decipher the Navajo language, one of the most complex and subtle in the world.
With all respects to Japanese intelligence, even in war, they knew it was Navajo. What their linguists didn't know was that it was a synthetic slang invented and spoken by only a tiny handful of Navajos. It was a linguistic problem, not a mathematical one.
Still, their painted geometry seems like a wonderful combination of art and math. I definitely want to see more of it.
Let's see; "Susie has 10 apples, eats 1 and gives three to Mary..."
The Code Talkers.
God bless them.
Scientific American did an article about them a long time ago. Very nicely illustrated.
... ... to prove that the sum of the radii is constant for all such partitions of a given inscribed polygon.
The article states, "The most straightforward way to prove it relies on Carnot's Theorem, which wasn't proven in the West until 100 years after the sangaku was created."
Well, I think the diagram itself points to another tack. Note that the upper right partition shares a triangle with each of the others, even though they have none in common. This means that all three diagrams can be shown to have equal sums of radii by showing that the two pairs formed with the upper right partition have them, and each of these two equivalences only requires the proof for a pentagon.
Without belaboring it, we can see right away that the general case will collapse recursively to proving a "cyclic quadrilateral" has this property when its two possible partitions are compared.
This is still a problem, but obviously much reduced.
... and so too bed.
And the Code Talkers....didn't talk.
Fascinating! I never heard of this before. Following all the links in the references will occupy me for several enjoyable hours. Thanks.
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