The Ontario Science Centre once had a "proof" that was made using wooden strips that were about an inch wide and perhaps a quarter of an inch deep. There was a right angled triangle with squares on each side. It looked just like typical diagrams except that (a) it was mounted on the wall, (b) the triangle and the squares were covered with glass, (c) there was exactly enough coloured liquid inside to fill the square on the hypotenuse. Every minute or so, the triangle rotated in such a way as to allow all the water from the square on the hypotenuse to flow into the squares on the other two sides. Of course, the fit was exact. I remember watching the exhibit, thinking about it, and suddenly feeling a greater depth of understanding, not of geometry, but of trigonometry! Of course, not a proof, but a brilliant exhibit. (Here is a similar demo on youtube: <a href="https://www.youtube.com/watch?v=CAkMUdeB06o" rel="nofollow">https://www.youtube.com/watch?v=CAkMUdeB06o</a>)
Since we are on HN and talking of the Pythagorean theorem, I must mention E. W. Dijkstra's proof. <a href="https://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/EWD975.html" rel="nofollow">https://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/E...</a> he proves the generalization sgn(α + β - γ) = sgn(a² + b² - c²) also by using similar triangles. Incredibly neat.
This is the most succcinct proof, it's an ancient Chinese visual proof:<p><a href="https://www.dbai.tuwien.ac.at/proj/pf2html/proofs/pythagoras/pythagoras/pythagoras7.gif" rel="nofollow">https://www.dbai.tuwien.ac.at/proj/pf2html/proofs/pythagoras...</a><p>It's not labelled this way but notice that the BIG squares are identical, having sides of a+b. Only the four triangles are rearranged. Just subtract the four identical triangles.<p>In the second arrangement, a c^2 area is present in addition to the same four triangles. In the first arrangement this had been rearranged to an a^2 and a b^2 area.<p>It's important to assure yourself the triangles are the same and the big square is the same - there is no tiny hidden sliver or something, and right angles are preserved, there are no shenanigans.<p>second explanation of same:<p><a href="http://www.math.toronto.edu/colliand/notes/pythagoras.html" rel="nofollow">http://www.math.toronto.edu/colliand/notes/pythagoras.html</a>
Einstein’s 1949 article in the Saturday Review is “Notes for an Autobiography” [0]. It’s worth reading.<p>0. <a href="https://archive.org/details/EinsteinAutobiography" rel="nofollow">https://archive.org/details/EinsteinAutobiography</a>
> Einstein’s predictions, during a solar eclipse in 1919—was asked if it was really true that only three people in the world understood the theory, he said nothing. “Don’t be so modest, Eddington!” his questioner said. “On the contrary,” Eddington replied. “I’m just wondering who the third might be.”<p>for another side of eddington, see 'empire of the stars' [<a href="https://www.amazon.com/Empire-Stars-Obsession-Friendship-Betrayal/dp/061834151X" rel="nofollow">https://www.amazon.com/Empire-Stars-Obsession-Friendship-Bet...</a>] quite a fun read, i picked it up on a whim, and could not just put it down over the course of a 8hr train ride :)
Every time Pythagoras is brought up, I can't help but think of the cult of Pythagoras as well.<p>One of the tenants is the refutation of beans. For what reason, I have no idea. I've always found it strange that a mind so capable was also equally capable of such folly.
Trying to figure out the proof of a² + b² = c² by myself, without looking up the solution, was somehow exiting. Being exposed to a riddle and trying to find the solution is kinda cool.<p>However, not solving it after 10 minutes left me feeling a bit dumb... :)
Had to stop reading long enough to see if I could prove it myself using similar triangles. This is definitely cleaner than what I came up with though, which involved creating a similar triangle with side b1 set equal to side c of the original.
Can anyone point to a proof that for similar non-isosceles right triangles, the area is proportional to the square of a side? Can that be generalized to other shapes?
I remember watching a documentary once, and it kind of made the case that Einstein wasn't an isolated genius, but more like Mark Zuckerberg. Basically organizing and putting together other scientists work,then unfairly getting all the credit. Anyone seen this documentary before? Its been years since I watched it, so memory is vague.
I find this style a bit frustrating to read. I just want to know what his proof is, but there's so much fluff in the article that the actual meat is kind of hidden. So anyway, "his" proof is to drop a height to the hypotenuse and use similar triangles. Ah, right, I know this proof and I remember it as the canonical proof my grade 9 geometry textbook presented.