If you compare the area of the two squares shown in the figure, you can get a proof of the Pythagorean theorem.
For some reason this proof sticks in my mind and it is one of the first proofs in mathematics that I understood and remember. I believe I can give at least five proofs of this theorem of the top of my head.
In some sense this theorem is one of the cornerstones of modern mathematics. One can use it to build analytic geometry and the Cartesian coordinate system: it gives the definition of distance. One can also use it to define the notion of a metric in differential geometry: on small enough distances one has a distance function that satisfies the Pythagorean theorem. (The proper notion requires the tangent bundle to a manifold, but the meaning is clear intuitively.)
The reason I like this proof is that it suffices to translate the small triangles on the top to the configuration of the bottom. If one agrees that one can add and subtract areas (that area is an extensive concept as one would say in physics), then it is clear that the area of the smaller square on the top square is equal to the added area of the two smaller squares in the bottom. One does not even need the area law for triangles to make this one work and one can probably explain this to a very young kid/girl and he/she will still understand it: they can move the pieces on the square themselves.
“God is a geometer”, Plato – but is He another Thurston?
Hmm. If that were my coffee table, I could fold it up.
I saw this proof in the science museum in Paris. They had a video of someone moving around blocks. I was impressed that the museum was trying to explain mathematics to the public, it’s not so common, I think.
They also had soap bubbles for minimal surfaces and a magnetic on a pendulum displaying chaotic motion.
The first proof of Pythagoras’ theorem I can remember having seen and understood is very similar to this one – it was on TV, long ago, in Jacob Bronowski’s Ascent of Man. And this extremely fascinating demonstration is even on YouTube now!
Click this link and scroll to the bottom for a proof of the theorem discovered by Garfield.
Uhhh… the 19th century US President, not the cat.