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New Orleans teenagers found a new proof of the Pythagorean Theorem

keith-mcnulty.medium.com

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Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#141

Post by xiphias2 »

joe_the_user wrote: post #20

I have read about this proof for a bit and this is the first write-up that gives the slightest details. The phrase "using trigonometry" is confusing. What they do is assume functions sine and cosine exist, as normally defined, as ratios of triangle values, without assuming these have the various Pythagorean-theorem derived properties. They then construct an infinite series of nested triangles and use the formula for…

,,In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M.''

It would be interesting to see if the original proofs work with non-complete metric spaces or not, as probably this proof doesn't.

https://en.wikipedia.org/wiki/Complete_metric_space

https://sharegpt.com/c/qRsum0c

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#142

Post by istjohn »

rvz wrote: post #97

Well 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

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#143

Post by istjohn »

criloz2 wrote: post #125
rvz wrote: post #97

Well 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…

Why are being downvoted?, there have been crazy comment on this website, hyping to the max ChatGPT and how all the humans now are useless, I was just ignoring HN for the past weeks because it have been ridiculous

Probably because it has nothing to do with the article. It's just a rant about something entirely unrelated.

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#144

Post by pramodbiligiri »

fnordpiglet wrote: post #130
jonahx wrote: post #99

Earlier quoted context omitted.

It's a very clever proof. Agreed "using trigonometry" is potentially misleading. After reading the proof, the only 2 senses in which "trignometry" is being used are: 1. The term "sin a" is used to denote the ratio opposite / hypotenuse. But this can be considered a purely notational convenience. They could have called it "foo a" and nothing would change, or they could have inlined the referred-to ratio everywhere. 2.…

Is there more to trigonometry? I’m not a abstract math person, so forgive the ignorance, but my understanding was all trigonometric functions derive from ratios of angles and lengths of triangles so in the end each occurrence of a trigonometric function can be replaced by the corresponding ratios in some triangle. There are other ways to construct things, such as power series representations, etc, but even these must…

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.

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#145

Post by agurjal »

hgsgm wrote: post #121
agurjal wrote: post #89

I think you can sidestep trigonometry (and the Law of Sines) completely. You can decompose any triangle A, B, C using their construction to create smaller triangle a, b, c where A = 2abc/(b² - a²), C = c(b² + a²)/(b² - a²), and B = c. This can be shown with only similar triangles (it seems like they unnecessarily use sines in the article). It is then just algebra to show A² + B² = c²(b² + a²)²/(b² - a²)² = C². edit:…

The Law of Sines is proven using similar triangles. This is a convenient way to bookkeep similar triangles. Does your version using the infinite series, or only the waffle cone shape?

It uses the infinite geometric series which is how A and C are determined in terms of a, b, and c (as shown in the article). I think the infinite geometric series is definitely the coolest part of their proof!

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#146

Post by hn_throwaway_99 »

jonahx wrote: post #137
fnordpiglet wrote: post #130

Earlier quoted context omitted.

Is there more to trigonometry? I’m not a abstract math person, so forgive the ignorance, but my understanding was all trigonometric functions derive from ratios of angles and lengths of triangles so in the end each occurrence of a trigonometric function can be replaced by the corresponding ratios in some triangle. There are other ways to construct things, such as power series representations, etc, but even these must…

> What am I missing? Nearly all the nontrivial results of trigonometry do in fact rest on the pythagorean theorem. The trig identities you learned in high school, as well as more advanced results like power series, etc. These results would be inadmissible. So the “uses trigonometry” part of this story feels like an attempt to manufacture mystery and hype. Which is a shame, because the geometric series construction is…

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.

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#147

Post by thrashh »

lern_too_spel wrote: post #113
pelorat wrote: post #111

Earlier quoted context omitted.

I think it's just fucked up these things needs to be mentioned at all. We're all fucking humans. Great job to these persons* * = a human being regarded as an individual

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.

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#148

Post by dang »

throwayyy479087 wrote: post #100

[flagged]

We've banned this account for using HN primarily for ideological battle and flamewar. That's not allowed, regardless of what you're battling or flaming for.

If you don't want to be banned, you're welcome to email hn@ycombinator.com and give us reason to believe that you'll follow the rules in the future. They're here: https://news.ycombinator.com/newsguidelines.html.

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#149

Post by selimthegrim »

slightlytwisted wrote: post #133
selimthegrim wrote: post #61

Earlier quoted context omitted.

There’s even people from the state that read this forum now and then

Indeed. I’d like to add the Louisiana School for Math, Science, and the Arts as another example of a great high school. https://www.lsmsa.edu/

I’m a grad student at Tulane in New Orleans I was being sarcastic

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#150

Post by jonahx »

hn_throwaway_99 wrote: post #146
jonahx wrote: post #137

Earlier quoted context omitted.

> What am I missing? Nearly all the nontrivial results of trigonometry do in fact rest on the pythagorean theorem. The trig identities you learned in high school, as well as more advanced results like power series, etc. These results would be inadmissible. So the “uses trigonometry” part of this story feels like an attempt to manufacture mystery and hype. Which is a shame, because the geometric series construction is…

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://sumantmath.wordpress.com/2020/08/16/proof-of-pythago...) is implicitly using the law of sines.

The hype part is the implication that impossible trig barrier was shattered by their proof.

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