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.
New Orleans teenagers found a new proof of the Pythagorean Theorem
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Re: New Orleans teenagers found a new proof of the Pythagorean Theorem
#172Nice 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.
Re: New Orleans teenagers found a new proof of the Pythagorean Theorem
#173Well deserved and well done to them on this proof. It is quite interesting to see that the AI bros continue to hype and worship hallucinating sophists like ChatGPT and GPT 4. By now we should have already expected that AIs like LLMs are able to create unique proofs and new solutions to existing unsolved mathematical problems. They still haven't after years of hype and not even one single mention of buzzwords like 'LL…
I used GPT-4 in a reply in this thread to rewrite my explanation to make it as easy as possible to follow[1]. 1. https://news.ycombinator.com/item?id=35499894
That doesn’t counter what I have said. The AI *did not* come up with the proof. It just summarized your own explanation based on the text you have given it.
Summarization of existing text is not the same thing as creating a proof from scratch.
Re: New Orleans teenagers found a new proof of the Pythagorean Theorem
#174Earlier quoted context omitted.
I used GPT-4 in a reply in this thread to rewrite my explanation to make it as easy as possible to follow[1]. 1. https://news.ycombinator.com/item?id=35499894
That doesn’t counter what I have said. The AI *did not* come up with the proof. It just summarized your own explanation based on the text you have given it. Summarization of existing text is not the same thing as creating a proof from scratch.
It actually rewrote my explanation and made a substantial change to part of it, using a completely different chain of logic than I did. But you're right, it struggles to do original mathematics. When I asked it to write an explanation from scratch, it made a mistake in the proof.
I just wanted to point out that GPT 4 can be quite useful despite its shortcomings.
Re: New Orleans teenagers found a new proof of the Pythagorean Theorem
#175Doesn't this proof not work when alpha or beta is 45°? 2 alpha would make the upper of the extension line in the diagram 90°, and alpha+beta for the bottom angle also 90°. The lines will never intersect, they're parallel
Re: New Orleans teenagers found a new proof of the Pythagorean Theorem
#176Cool 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
Re: New Orleans teenagers found a new proof of the Pythagorean Theorem
#177Earlier quoted context omitted.
Induction isn't so bad. I'm not sure how they're getting away with the Law of Sines, though. The usual proof of LoS that I know is dependent on the existence of the circumcircle. But I don't know how to prove the existence of the circumcircle without dragging in a lot of geometry. Or you can use the area formula, which makes the proof similar to other arguments that use the area formula.
Is there not a proof using similar triangles somehow? Because the whole 'sine' part seems like a bit of a red herring, they're basically just considering a couple of ratios between different lengths, they do not use any properties of the sine function as such (in particular it does not look like they're using sin(x)^2 + cos(x)^2 = 1, which would make the proof trivial)
Yes, if they droped height h to the side of isoscales triangle, then from similar triangles they would have h/A=c/C and h/b=2a/c and would get the result without mention of sine rule or areas.
Re: New Orleans teenagers found a new proof of the Pythagorean Theorem
#178The author of this blog Keith McNulty is a mathematician by training and now a data scientist at McKinsey: https://keithmcnulty.org
okay
Re: New Orleans teenagers found a new proof of the Pythagorean Theorem
#179Earlier quoted context omitted.
The sine function can be defined in terms of its own behavior, using its first-order differentiation and no reference to triangles. See this detailed article on sine. https://betterexplained.com/articles/intuitive-understanding... There’s section there titled Part 2: Understanding the definitions of sine.
You can define sine and cosine together using the functional equations S(X)C(Y)+C(X)S(Y)=S(X+Y) C(X)C(Y)−S(X)S(Y)=C(X+Y) The only solutions to this are the constant 0 functions and the sine-cosine pair.
This is super cool, I've never seen it before! Do you know what this is called so I can look up a proof/theorem on it?
Re: New Orleans teenagers found a new proof of the Pythagorean Theorem
#180Doesn't this proof not work when alpha or beta is 45°? 2 alpha would make the upper of the extension line in the diagram 90°, and alpha+beta for the bottom angle also 90°. The lines will never intersect, they're parallel
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