Mathologer | Math in the Simpsons: Homer's theorem @Mathologer | Uploaded 9 years ago | Updated 1 hour ago
After putting on some glasses he found in a toilet Homer feels very smart and declares: "The sum of the square root of any two sides of an iscosceles triangle is equal to the square root of the remaining side." Well, sounds like Pythagoras theorem but it's not. The Mathologer sets out to track down this mystery theorem to its lair and dissects the hell out of it.

Enjoy :)

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$Visualising irrationality with triangular squares NEW (Christmas 2019). Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details) Get ready for some brand new and very pretty visual proofs of the fact that root 2, root 3, root 5 and root 6 are irrational numbers. Root 2 being irrational also translates into the fact that the equation x^2+x^2=y^2 has no solutions in positive integers, root 3 being irrational translates into the fact that the equation x^2+x^2+x^2=y^2 has no solutions in positive integers, etc. What I find very attractive about these proofs is that the destructive core of these proofs by contradiction lead a second secret constructive life, giving birth to infinitely many nearest miss solutions of our impossible equations like for example 15^2+15^2+15^2=26^2-1. Here is the paper by Steven J. Miller and David Montague which features the basic root 3 and pentagonal root 5 choreographies. https://arxiv.org/abs/0909.4913 Footnotes: -our nearest miss solutions like, for example, 15^2+15^2+15^2=26^2-1 correspond to the solutions of the equation y^2 - n x^2 = 1 with n=2, 3, 5 and 6. This is the famous Pells equation, which happens to have solutions for all integers n that are not squares. -there is also a second type of nearest miss solutions like 4^2+4^2+4^2=7^2+1 (a plus instead of a minus at the end). Starting with one of these our choreographies also generate all other such nearest misses. -the original Tennenbaum square choreography and the first puzzle root three choreography generate both types of nearest misses from any nearest miss solution. -The close approximations to the various roots corresponding to our nearest miss solutions are partial fractions of the continued fraction expansion of the roots. -lots more things to be said here but we are getting close to the word limit for descriptions and so I better stop :) Thank you very much to Marty for all his nitpicking of the script for this video and Danil for his ongoing Russian support. Todays t-shirt is the amazing square root t-shirt (google square root tshirt). Note that the tree looks like a square root sign AND that the roots of the tree are really square. Enjoy!$

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Math in the Simpsons: Homer's theorem @Mathologer