181–190 of 287 posts
Cool proof, though it doesn't consider the case where a=b. If so, the geometric series is non-converging since the ratio isn't less than 1. Geometrically, the construction wouldn't work because the sides A and C of the large "triangle" would be parallel to each other.
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Apparently (found this while reading an article on the girls' accomplishments) somebody proved sin^2x + cos^2x = 1 without using the Pythagorean theorem in 2009: https://forumgeom.fau.edu/FG2009volume9/FG200925index.html . I don't think the "uses trigonometry" part is hype. They do use the definitions and law of sines, they just cleverly avoid the parts of trigonometry that depend on the Pythagorean theorem.
I address this in my OP. If you watch the video of the proof, you will see that the "law of sines" is 1 step away from the ratio definition of sin. You just drop one altitude, apply the definition again to the similar triangles, and re-arrange. It is almost content free as a result -- I see no reason using this in a proof would have special significance. For example, the standard proof using similar triangles ( https…
Any other claims seem to have been added by the media, not the teenagers themselves.
Nice proof? I understand that beauty is in the eyes of those who watch. Maybe i am blind, but i wouldn't call that a beautiful proof.
Who said it's nice? A proof is a proof, beautiful or not.
"most beautiful and simplest trigonometric proof we have seen to date"
Cool proof, though it doesn't consider the case where a=b. If so, the geometric series is non-converging since the ratio isn't less than 1. Geometrically, the construction wouldn't work because the sides A and C of the large "triangle" would be parallel to each other.
Geometrically, this happens when alpha is 45. The two lines in the diagram from the article will be parallel and never converge. 2*alpha = beta+alpha I was waiting for them to break this out into a special case or something but the article never did. Can't find any other material on this proof that mentions it
Not sure how much that helps with "infinite triangle" with two 90deg and one 0deg angles.
That's btw how you get sin90=1 which doesn't have any geometrical sense when we consider finite triangles.
Or in case of triangle with 45, 45, 90 maybe you could just pick different angle than 90 to be 2alpha.
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Your explanation is very clear, but I still think that the article is confusing about that. Note how the author sets the expecatations at the beginning of the article: "...the proof these young trailblazers have proposed might make a few established mathematicians eat their words. This is because their proof uses trigonometry." This wording implies to me that Jackson and Johnson are actually the first who came up wit…
I think any new proof of PT is pretty cool. it doesn't even have to be "elegant" or better than previous proofs. Just a new one. Will there at some point be a proof that all possible proofs of Pythagorean Theorem have been found. Or for any theorem? Is it possible to prove that all possible proofs of something have been found?
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That's what centuries of oppression does. It makes this rare. Hopefully, elevating these achievements in the community can make a difference. Ignoring that this problem exists certainly won't. If you don't care about broadening the pool of mathematicians, simply ignore the part of the article about that subject.
It doesn’t need to be mentioned though. I’m Asian if I read an Asian name doing something that Asians don’t normally do (like the recent accolades for Everything Everywhere), it’s going to inspire me. You don’t have to point it out. The people who need to hear it are not dumb. Mentioning it just sounds pandering and cringy.
For people who care about speeding up progress in mathematics, the fact that this comes from a marginalized group is worthy of mention to understand how to repeat it. For everybody else, ranting about how it doesn't matter is simply off-topic flamebait utilized by culture warriors and their dupes.
It would be really interesting to see if this could be applied to other analogous scenarios that are sometimes called Pythagorean theorems, in particular I’m thinking of the Pythagorean Theorem of Information Geometry- p* = argmin p of P (a set of possible distributions) of D_KL (p||q) (where q is eg your model's distribution)
You're missing the actual theorem in your comment, but the one you are referring to is essentially the pythagorean theorem for Bregman divergences, which I think may be a bit too far removed from geometry to allow this proof to generalize.
I haven’t seen the original text, but this proof may be incomplete.
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> When I read "New Orleans teenagers" I wanted to immediately give the benefit of the doubt, but a part of me suspected they may lead with identity. I didn’t have this reflexive reaction, but I can understand why one might if familiar with the way regional references can be coded language. That said, > It does a disservice to an achievement, if in fact there is one. A part of me is now sort of doubtful that this isn'…
Also cultural. Don't underestimate how big a headwind an anti-education subculture can be to the people in it!