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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

#151

Post by chx »

pramodbiligiri wrote: post #144
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…

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.

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

#152

Post by ummonk »

starchild_3001 wrote: post #79

Ok, I read through the proof, and I think I understood it. Thanks for posting this! Until the authors' work is submitted to a journal and reviewed, it's hard to say everything claimed here is definitely correct & new. Update: Nice video on the proof: https://www.youtube.com/watch?v=nQD6lDwFmCc I like what they did :) Seems legit to me. Btw, law of sines can be proven independently of pythagoras. So using that as a st…

It's definitely correct, and quite trivial to verify. It's possible it has been discovered before, but none of the proof compilations I've seen (e.g. cut-the-knot) has it, and the trigonometric proofs I can find involve using angle-sums (https://forumgeom.fau.edu/FG2009volume9/FG200925.pdf).

This is definitely a much more elegant proof than the angle-sum proofs.

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

#153

Post by ummonk »

maxk42 wrote: post #75

On my phone at a restaurant right now so I'm not looking it up but this was previously in a published paper in 2009. Not to rain on their parade: this is a great start for a couple of math prodigies, but they didn't quite discover a new proof.

If you had bothered to read the article before commenting you'd see that the proof is indeed novel, and is unrelated to the previously found trigonometric proof.

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

#154

Post by GistNoesis »

The picture is drawn really poorly : The first composite angle : alpha+beta at the top of the A side, is a right angle and should be drawn as such (if comes from the hypothesis of the Pythagorean theorem that the two sides are orthogonal so alpha+beta+90° = 180°).

Otherwise you can't understand why the ration A/C would be sin(2*alpha).

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

#155

Post by ummonk »

sbelskie wrote: post #82
HL33tibCe7 wrote: post #9

> By all accounts, these two teenage math students are the exact opposite of the majority of the math establishment. They are female, they are African-American, and they come from an area which is not particularly renowned for producing high academic achievers. This is just an awesome turn of events and one which should inspire anyone — no matter what their ethnic, gender or socio-demographic background — that excell…

Perhaps the website for the school is incomplete and only listing registration fees and not tuition but the only figure I see is $750, which is not nothing for plenty of families but is hardly what I would call expensive when it comes to private education. Am I missing something? Very well could be!

I see tuition of several thousand (e.g. 7,800 for 8-11th grade): https://smaneworleans.com/tuition-fees

Still quite a steal as private schools go, and significantly less than one would have to pay in yearly rent / mortgage to get their kids into a good school district.

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

#156

Post by slightlytwisted »

selimthegrim wrote: post #149
slightlytwisted wrote: post #133

Earlier quoted context omitted.

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

I understood that you were being sarcastic. I should have said I loved your comment and thanked you for it as someone who was born and raised in New Orleans.

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

#157

Post by pedrosorio »

eyelidlessness wrote: post #126
pedrosorio wrote: post #37

Earlier quoted context omitted.

I would also take that interpretation if the author didn't explicitly mention "socio-demographic background". They should just leave that one out next time.

> > Your ethnicity and/or gender should not be seen as a barrier to advanced mathematics. > I would also take that interpretation if the author didn't explicitly mention "socio-demographic background". They should just leave that one out next time. Please help me understand. Are you saying that it’s okay to say that ethnicity and/or gender should not be seen as a barrier to advanced mathematics, so long as you don’t…

Read the thread I am responding to.

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

#158

Post by krick »

no_butterscotch wrote: post #90

[flagged]

> doubtful that this isn't a "cause celebre" sort of situation

I mean, honestly… it just is. Is it a new proof? I believe that it is. Is it any good? Yeah, sure, it's pretty clever. Is it really an all-over-the-news special kind of achievement? No. It simply wouldn't be all over the news if not for… uh, non-mathematical reasons.

Allegedly it's special, because it's trigonometric, and trigonometric proofs of it are indeed special. But this one is… let's say it's "trigonometric with an asterisk". Both because of the series, and because it is kinda analogous to a known geometric proof. And the first one truly trigonometric proof, which was really special, wasn't as much celebrated, as this one (but, of course, the internet was a bit quieter place back then).

And, yes, I agree that it does a disservice to an achievement. And I'm completely positive that it is an achievement no matter what — every new proof is, and it's especially true when the theorem is well-known (and there are very few more famous than this one). So, honestly, I'd like if we wouldn't have this discussion at all, and could just stick to the matter. But the problem is we probably wouldn't even hear of it, if not for those "non-mathematical" reasons. Because by itself it isn't that huge. I would love if every new proof of a famous theorem would be highlighted and celebrated on the Internet as much, as this one, but it simply isn't the case.

It's understandable, so it's neither good nor bad IMO, but I'm just saying that your suspicion is definitely correct.

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

#159

Post by svat »

sedatk wrote: post #135

"...a few trigonometric proofs of Pythagoras have made the rounds since then. Claims in the media that Johnson and Jackson’s proof is the first trigonometric proof of Pythagoras are overblown..." I don't like how the author drops the bomb "this has been done before" and tries to compensate that with the "simple and lively" language of the proof, or the background of the girls. Why didn't author reference any of the p…

It seems pretty clear to me:

- Trigonometric proofs of the Pythagorean theorem are rare, because many trigonometric identities depend on the Pythagorean theorem, so it's easy to end up with an argument that is in fact circular.

- Nevertheless, there have been a handful of trigonometric proofs in recent decades.

- These two students have come up with a new trigonometric proof, and it is a nice one too (“could well be the most beautiful and simplest trigonometric proof we have seen to date”). That is the novelty/notability.

Note that this is consistent with the talk abstract (quoted in https://news.ycombinator.com/item?id=35498423) — they don't claim to have the first trigonometric proof, just that “in our lecture we present a new proof […] based on a fundamental result in trigonometry—the Law of Sines—and we show that the proof is independent of the Pythagorean trig identity”. Pretty cool work IMO.

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