New Orleans teenagers found a new proof of the Pythagorean Theorem

https://keith-mcnulty.medium.com/heres-how-two-new-orleans-teenagers-found-a-new-proof-of-the-pythagorean-theorem-b4f6e7e9ea2d

by malshe • 3 years ago

949 209 3 years ago

keith-mcnulty.medium.com

malshe2• 3 years ago

The actual title is "Here’s How Two New Orleans Teenagers Found a New Proof of the Pythagorean Theorem" but I don't know why HN automatically converted it to this title.

capableweb2• 3 years ago

"Here's How" is usually used for clickbait stuff and adds nothing of substance to the title itself.

New Orleans Teenagers Found a New Proof of the Pythagorean Theorem

is both shorter and less editorial than

Here’s How Two New Orleans Teenagers Found a New Proof of the Pythagorean Theorem

majormajor• 3 years ago

I get why that's the default behavior but "adds nothing of substance" is a huge generalization that fails in this case.

An article with zero details about the proof could easily be titled "New Orleans Teenagers Found a New Proof of the Pythagorean Theorem" but couldn't accurately be titled "Here’s How Two New Orleans Teenagers Found a New Proof of the Pythagorean Theorem." Whereas here it says it'll have more details and it does. (although TBH this is less a "here's how they found" and more a "here's what they found", if I'm being extra pendantic).

The shorter title is less descriptive in this case.

anigbrowl• 3 years ago

No. The short version titles an assertion, the longer version titles an exposition.

HN automatically strips things like this on submission but you can edit the submission title after posting to put it back in when it's appropriate, as here.

satvikpendem• 3 years ago

HN strips clickbait titles. Same thing for listicles.

johnfn• 3 years ago

HN has some sort of algorithm to rewrite post titles to be a bit less link-baity.

eyelidlessness6• 3 years ago

I share your dislike for this title editing, and agree the original title is better.

HL33tibCe72• 3 years ago

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 excellence in your chosen field of study is always attainable

I think it’s remiss not to point out here that these students attend a private, fee-paying Catholic all-girls academy.

None of that detracts from the impressive achievement of discovering this elegant proof, of course.

secondcoming• 3 years ago

I don’t think Privilege matters in all cases, look at the background of Ada Lovelace.

HL33tibCe72• 3 years ago

It matters when you try and make the point that this case “should inspire anyone — no matter what their ethnic, gender or socio-demographic background“, as the article does

boomboomsubban2• 3 years ago

... The daughter of Lord Byron?

uberman• 3 years ago

How would it be remiss to omit these factors, several of which seen irrelevant.

Are you suggesting that their proof be discounted because it might be divinely inspired?

threatofrain2• 3 years ago

The article is highlighting the inspirational quality of two black women from an underachieving area; it's not just about the proof. The inspirational story is an important element.

Adding that these two youths came from a private school certainly relates to the "underachieving area" part of the story.

ThalesX• 3 years ago

Are you suggesting that their proof be discounted because it might be divinely inspired?

I read it in a totally different way, that the merit of such institutions should be acknowledged. More so because of the culture because of any divine inspiration.

concordDance• 3 years ago

I guess the main thing would be if people were trying to use this as an example of "poor kids from the hood achieving" then the fact they went to a private school makes it a less good example.

sbelskie3• 3 years ago

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!

User232• 3 years ago

Parents that care are a huge predictor of academic success. Putting your child in a private school, even an affordable one, is more effort than just going with the default option and thus serves as signal for parents caring.

ummonk3• 3 years ago

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.

pelorat• 3 years ago

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

lern_too_spel3• 3 years ago

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.

zem• 3 years ago

I'm impressed, I've been following the story in the news and various blogs and magazines, and this is probably the best write up I've seen of it.

umanwizard• 3 years ago

In the first paragraph under “Deriving lengths of the sides of the smaller triangle”, how do we know the length of the third side is 2a^2/b ?

mathteddybear• 3 years ago

"This" in "This leads to the third side of this triangle to be 2a²/b." means "the same line of reasoning".

it's not even trigonometry in that paragraph, just rudimentary proportions of similar triangles

n3uromancer• 3 years ago

From cosβ = a/c and cosβ = third side/((2ac)/b)

joe_the_user• 3 years ago

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 the sum of geometric series' to derive the length of the original triangle's hypotenuse. It certainly seems clever.

I'm still confused what axioms they're effectively using relative to the usual Pythagorean theorem proofs - most of these use the formula for area of a right triangle and this seemingly doesn't. On the other hand, it seems an infinite construct would require things like the axiom of induction, which may or may not be included in axiom of axiomatic geometry.

scythe• 3 years ago

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.

SamReidHughes• 3 years ago

There is a very straightforward proof that involves dropping an altitude.

https://www.mathopenref.com/lawofsinesproof.html

You could use a similar technique to make this proof not reference trigonometric functions.

contravariant3• 3 years ago

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)

jonahx5• 3 years ago

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. The law of sines is required. But the proof of this law [1] also boils down to nothing more than the ratio definition and some algebra.

So afaict no circular logic is being used, but at the same time it doesn't seem to be doing anything previously thought to be impossible, unless there was a previous belief that the law of sines could not be used in a proof, which would be a strange belief to hold. I see it simply as a creative, unexpected proof.

[1] https://www.youtube.com/watch?v=4xO8xqLyEbA

fnordpiglet• 3 years ago

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 necessarily be replaceable by the ratio of angles and lengths of some triangle. What am I missing?

cabalamat2• 3 years ago

It's a very clever proof.

Which makes it a bad one, IMO. In maths, as in programming, one should go with the simplest way that works.

alex_smart4• 3 years ago

The trigonometry thing is simply a marketing gimmick for this proof. There is no more or less trigonometry in this proof than there is in Einstein's proof. In fact, you can just taken Einstein's construction and reformulated that proof in their language by using sine rule instead of similar triangles. But then the gimmick would be too obvious.

Somehow the second gimmick (the infinite series construction instead of Einstein's elegant and simple construction) makes our monkey brains not notice the first gimmick.

xiphias2• 3 years ago

,,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

silentguy• 3 years ago

I don't see any Cosine being mentioned in the proof. I don't see any circular logic here.

soegaard• 3 years ago

Here is a proof of Pythagoras using a geometric series:

https://www.cut-the-knot.org/pythagoras/Proof100.shtml

The proof is by John Arioni and also features an infinite number of similar triangles.

pizza2• 3 years ago

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)

contravariant3• 3 years ago

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.

pizza2• 3 years ago

Ah, was just a flicker in my mind. Thanks - though I will ask, is there anything like the law of sines/cosines (in high dimensional statistics) that could get you to at least the first step of the proof?

Y_Y• 3 years ago

I've seen this construction before. It's very cool, but I don't think it's novel. In any case it's a cool article and it must be great for a couple of young students to present to the AMS.

Jarmsy2• 3 years ago

Where is it you think you've seen it before?

dendrite9• 3 years ago

I remember reading a book in high school and realizing there could be other ways to prove things that I had been taught only one way. One that particularly stood out later was using a rotating fishtank to prove the pythagorean theorum. A good friend of mine was so delighted by the example I gave him a copy of the book I found it in. https://press.princeton.edu/books/paperback/9780691154565/th...)

The relevant section is available as a pdf here: http://www.personal.psu.edu/mxl48/Welcome_files/Sample.pdf

hypertexthero• 3 years ago

This brings to mind the visual solution to calculating triangle area in James Somers post “I should have loved biology”:

In his “Mathematician’s Lament,” Paul Lockhart describes how school cheapens mathematics by robbing us of the questions. We’re not just asked, hey, how much of the triangle takes up the box?
That’s a puzzle we might delight in. (If you drop a vertical from the top of the triangle, you end up with two rectangles cut in half; you discover that the area inside the triangle is equal to the area outside.)

—https://jsomers.net/i-should-have-loved-biology/

Waterluvian• 3 years ago

I vividly remember math class one year boring me to death so bad that I distracted myself with my own puzzles like this. It was when I discovered the Fibonacci sequence inside Pascal’s triangle. I didn’t think this was a new discovery but it was new to me and it felt like lightning.

I think that might have been an early glimpse of my later discovery that all my best learning would be done outside school.

Tao33002• 3 years ago

I hated math for most of my childhood. I tested into an advanced track and had to be sequestered into lower level courses in the next higher grade do to a lack of effort.

Then when I hit college and had Discrete and Calculus, I found out I loved it and wound up minoring in math. Though my arithmetic is still slow and my trig has major gaps in it due to school math just sucking in general.

coliveira2• 3 years ago

The nice thing about mathematics is that for every true statement there are infinitely many proofs. Granted, some are just variations of others, but there many ways to reach the same point.

carapace• 3 years ago

for every true statement there are infinitely many proofs

No. There are true statements which cannot be proven. For example: "This statement cannot be proven." (Technically it's truth value is neither true nor false. It is an imaginary Boolean value.)

steppi• 3 years ago

This is a really great book. It’s very accessible but the insights can also be appreciated by a mathematically sophisticated audience. I’m particularly fond of Chapter 11 on understanding complex analytic functions and the part in Chapter 2 that gives a very clear explanation why the determinant formula gives the (signed) volume of the parallelepiped determined by the column vectors of a matrix.

Al0neStar• 3 years ago

2023 edition: https://press.princeton.edu/books/paperback/9780691242057/th...

nolamark2• 3 years ago

for completeness, here is a reference to the AMS presentation. https://meetings.ams.org/math/spring2023se/meetingapp.cgi/Pa...

[my post updated to include abstract]

Abstract

In the 2000 years since trigonometry was discovered it's always been assumed that any alleged proof of Pythagoras’s Theorem based on trigonometry must be circular. In fact, in the book containing the largest known collection of proofs (The Pythagorean Proposition by Elisha Loomis) the author flatly states that “There are no trigonometric proofs, because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean Theorem.” But that isn’t quite true: in our lecture we present a new proof of Pythagoras’s Theorem which is based on a fundamental result in trigonometry—the Law of Sines—and we show that the proof is independent of the Pythagorean trig identity \sin^2x + \cos^2x = 1.

hgsgm5• 3 years ago

Note that in math "flatly" means "asserted without proof", not "incontrovertible".

"Trigonometry" is not formally defined, so Loomis's statement is merely tautological.

dullcrisp• 3 years ago

Maybe this is my mathematical background, but is that not the common meaning of “flatly” stating something?

alex_smart4• 3 years ago

The trigonometry thing is simply a marketing gimmick for this proof. There is no more or less trigonometry in this proof than there is in Einstein's proof. In fact, they could just as well have taken Einstein's construction and reformulated that proof in their language by using sine rule instead of similar triangles. But then the gimmick would be too obvious.

Somehow the second gimmick (the infinite series construction instead of Einstein's elegant and simple construction) makes our monkey brains not notice the first gimmick.

bblpeter• 3 years ago

Sorry but that’s not new

soVeryTired• 3 years ago

Maybe you could provide us with a reference then?

threatofrain2• 3 years ago

The article itself says it's not new.

maxk42• 3 years ago

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.

nolamark2• 3 years ago

perhaps: https://forumgeom.fau.edu/FG2009volume9/FG200925.pdf

On the Possibility of Trigonometric Proofs of the Pythagorean Theorem

Jason Zimba

Abstract. The identity cos2 x + sin2 x = 1 can be derived independently of the Pythagorean theorem, despite common beliefs to the contrary.

hgsgm5• 3 years ago

No, that was a different proof using different trigonometry.

ummonk3• 3 years ago

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.

privacyonsec• 3 years ago

I don't get how this is "new" or "novel" ? it seems to me the law of cosine (or al kashi's theorem) we study this in high school in France ...

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

maxboone• 3 years ago

The concept isn't new or novel, the proof is (or might be).

User232• 3 years ago

Dijkstra did something similar when he derived a generalization of the theorem[1].

[1] https://www.cs.utexas.edu/~EWD/transcriptions/EWD09xx/EWD975...

lern_too_spel3• 3 years ago

It's the Pythagorean theorem, a special case of the law of cosines. What's new is the proof method.

starchild_30012• 3 years ago

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 step is ok. https://en.wikipedia.org/wiki/Law_of_sines

hgsgm5• 3 years ago

It's obviously correct (it's basic high school math; as with most proofs, the cleverness is in the construction, not the computation) and nearly obviously new or at least newly published. (There are many easily searchable collections of proof, but not everyone published their proof of a theorem already proved and published over 400 different ways)

ummonk3• 3 years ago

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.

starchild_30012• 3 years ago

Correctness might be there. But there are also some grandiose claims in the abstract ("“There are no trigonometric proofs, because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean Theorem.”) and the argument being novel. I don't have a way to verify these two. That's what the peer review is for.

------------

update: This video refers to another trigonometric proof that doesn't rely on sin^2 + cos^1 = 1. https://www.youtube.com/watch?v=p6j2nZKwf20

I would write a more modest abstract for this work. Just my $0.02.

yosito2• 3 years ago

doing my best to explain how Johnson and Jackson proved it using simple trigonometry. Although their proof hasn’t been published

They didn't publish it, but this author is just going to take the liberty to publish their work himself? If I were one of these teenagers, this would make me angry.

robotresearcher• 3 years ago

The article's author is very clear whose ideas they are reporting. Neither do they aspire to be the definitive citation for the idea. They hope the girls publish, or attempt to publish, and in the process establish their priority by peer review.

This is perfectly legit in academic publishing. Credit where credit is due is the rule. You are not obliged to keep stuff secret until the originator has published, only that you attribute the idea properly.

This is good, because it means ideas can get out there and be useful without delay.

yosito2• 3 years ago

Maybe not "obliged", but it still seems a bit like stealing thunder to me.

hgsgm5• 3 years ago

They already presented it at conference. If they are angry (they aren't), it would be their own fault for rushing to the ignorant media for fake publicity before publishing to the math community. Already we see that their approach led to massive fake news reporting based on journalists who had no experts to explain what actually happened, and relied on unvetted claims by high school students (not totally their fault) and their school admins (their fault).

agurjal3• 3 years ago

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: any right triangle A, B, C using their construction to create smaller right triangle a, b, c

erehweb• 3 years ago

Well sure, there are other proofs of Pythagorean theorem. The interesting part is that this is a new proof.

agurjal3• 3 years ago

I'm certainly no expert on proofs of the Pythagorean theorem, but if their construction is novel, then the above may be too.

hgsgm5• 3 years ago

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?

agurjal3• 3 years ago

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!

rvz3• 3 years ago

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 'LLMs', 'AI', 'GPT', etc in this thread. I'll tell you why:

The difference is those teenagers were able to clearly explain the process of deriving this proof transparently with the proof itself being (and still is) subject to intense scrutiny even by experienced mathematicians, going against what was thought to have been 'impossible'. Unlike the finest of LLMs and AI models which just repeat the same nonsense it has been trained on and confidently outputs more nonsense, whilst many celebrate this sophistry as a so-called 'breakthrough' even when it cannot transparently reason with its own decisions.

It goes without saying that these teenagers are very intelligent in mathematics to create this proof as it is not straightforward to just 'generate' it, given that it requires an amount of creativity AND originality that not even ChatGPT or LLMs in general can bullshit it's way around and will still tell you that it is impossible.

That proof is the true breakthrough; not magic AI black-boxes that spit out nonsense.

criloz2• 3 years ago

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

istjohn3• 3 years ago

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

istjohn3• 3 years ago

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

rvz3• 3 years ago

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.

malshe2• 3 years ago

The author of this blog Keith McNulty is a mathematician by training and now a data scientist at McKinsey: https://keithmcnulty.org

hackernewds• 3 years ago

okay

option2• 3 years ago

how does one define sin without Pythagorean theorem? E. g. need to prove that it only depends on angle and not triangle size.

teo_zero2• 3 years ago

You don't need to prove anything to define sin.

option2• 3 years ago

the definition of function in mathematics requires that same input always gives same output for all valid inputs.

So yeah, to define sin as a function one absolute must prove that it only depends on angle and not triangle size.

This proof is done via similar triangle properties (same angles => same proptions of sides) btw.

sedatk3• 3 years ago

"...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 previous work? How do they differ from Jackson and Johnson's proof? What's the novelty here? How can we properly give credit to these young mathematicians?

svat2• 3 years ago

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.

sedatk3• 3 years ago

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 with a trigonometric proof because otherwise how an nth in a series make a few established mathematicians eat their words? Why would they eat their words now? Why not after the first trigonometric proof?

So, he raises the expectations then drops a "it's been done before" bomb, now what are we supposed to think about what's happened here? Is it what we read at the beginning? Or is it just the elegance of the proof that's cool? But, why does THIS one make the mathematicians eat their words then?

The whole "yeah it's awesome, amazing actually, but hey, forget what I said earlier, it's been done before" kind of composition confuses me.

_Nat_• 3 years ago

Why didn't author reference any of the previous work?

Doesn't seem like a serious article about math so much as a feel-good story.

I mean, Wikipedia shows a better proof in [this section titled "proof using similar triangles"](https://en.wikipedia.org/wiki/Pythagorean_theorem#Proof_usin...), which kinda starts the same way, but it's a lot shorter and doesn't require the Law-of-Sines nor infinite-series.

If anyone wants to design further "trigonometric" proofs, they can just start from what's on Wikipedia and add in steps that rely on trigonometry.

marcusphi• 3 years ago

I like the infinite triangles construct. Once A and C can be constructed from a,b,c, the theorem can be proven by proving A^2 +c^2 = C^2 with expanding/reducing the equation to 4a^2b^2c^2 - 2a^2b^2c^2 = 2a^2b^2c^2, which is true. No sin rule required.

teo_zero2• 3 years ago

That would only prove that, given a triangle abc, you can construct another, different triangle for which the PT holds, not that the PT is valid for the given triangle.

GistNoesis2• 3 years ago

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).

GistNoesis2• 3 years ago

Here is a better picture https://github.com/unrealwill/PythagoreanProofImage [Drawn with https://www.geogebra.org/geometry?lang=en]

ipince3• 3 years ago

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.

cobertos3• 3 years ago

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

scotty79• 3 years ago

You can divide partial sums first and then take the limit at infinity instead summing first and then dividing.

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.

camjw5• 3 years ago

When a = b then the proof of the Pythagorean theorem is trivial so this is the sort of nit-picking that actual mathematicians don't care about.

chengiz2• 3 years ago

How is it trivial?

ipince3• 3 years ago

As a mathematician myself, I beg to differ. Proofs should be rigorous, and at the very least it should warrant a mention of this case.

notorandit• 3 years ago

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.

sph• 3 years ago

Who said it's nice? A proof is a proof, beautiful or not.

slowmotiony• 3 years ago

The author of the article. You did read it didnt you?

"most beautiful and simplest trigonometric proof we have seen to date"

cobertos3• 3 years ago

Doesn'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

Someone4• 3 years ago

Nice trick, but this text only handles the case of “When our extended lines from steps 2 and 3 meet”. What if they don’t, that is, what if α = β = π/4, and the triangle is isosceles and rectangular?

I haven’t seen the original text, but this proof may be incomplete.

camjw5• 3 years ago

I mean this special case is also trivial, so it seems pretty reasonable to omit it. Feels quite uncharitable to describe this proof as a nice trick and then claim its incomplete because of such a simple special case. Mathematicians wouldn't consider this incomplete when the "missing" case can be solved almost by looking at it.

Someone4• 3 years ago

I don’t see it being trivial. Of course, ‘everybody’ knows the diagonal of the unit square has length √2, but don’t we know that because of the Pythagorean theorem?

Can you educate me?

unnouinceput• 3 years ago

Then you go to the opposite direction, and the math still holds, but now you have inverses. And those inverses, at proportionality final formulae in the text, still gives you the Pythagorean formulae. I suggest you do see the original text.

Someone4• 3 years ago

I suggest you do see the original text.

If I could find it, I would have. This discussion mentions https://meetings.ams.org/math/spring2023se/meetingapp.cgi/Pa..., but I can’t find the paper there. Do I overlook something on that page?

noobcoder• 3 years ago

I think Pythagorean theorem can be seen as a foundational concept for trigonometry since it is equivalent to sin squared x + cos squared x = 1. Still its impressive that the students were able to do this, but it's important to keep in mind that mathematicians weren't completely stumped by this for 2000 years.

amelius• 3 years ago

How come the triangles in the picture are all congruent but they don't all look congruent.

scatters• 3 years ago

Because the first dropped vertical is too long. The left hand angle is beta plus alpha, so it should be ninety degrees, but instead it's an obtuse angle.

FuriouslyAdrift• 3 years ago

Original post on Hacker News two weeks ago https://news.ycombinator.com/item?id=35311765

eneim• 3 years ago

I think this approach briliant. However, I believe it can be simplified to avoid depending to trigonometry and geometry series, while keeping the original idea. I have no idea if it is similar to any existing proof of the theorom, but I put my thought into a post [1].

[1] https://eneim.notion.site/On-a-new-proof-of-the-Pythagorean-....

gabereiser2• 3 years ago

I like this as a proof better than the OP. Not only does it avoid the sin2(a)/cos2(a)=1 pitfall but shows clearly the math without use of trig. This part is the yummy bit: >Since a<b and a+b=90 degrees then a<45 degrees, and therefore 'BAD' < 90 degrees, which means that if we draw a line 'l' that is perpendicular to 'BA' at 'B', 'l' would never be parallel with the line 'AD' and they cut each other at a point 'E' like the Figure B.

alex_smart4• 3 years ago

There really isn't any trigonometry here apart from sine rule, and because sine rule is derived from similar triangles, anything you can say using sine rule, you can also say using similar triangles.

yafbum3• 3 years ago

I am puzzled by the article's spin on this, which really centers on the fact that this original proof was authored by two teenage African-American girls from the South, as if the interesting thing here was not so much the proof itself than the idea that there are gifted mathematicians from underrepresented backgrounds and skin colors. In my experience, pretty much in any place and social stratus you might visit, there are bright kids who love math. The challenge is more about what happens next in their career — can these kids get affordable higher education, and a career track that values their gift? My data point of one is a friend of mine, who was an extremely bright student of physics, but had to drop out of college early because he couldn't afford it and needed to start making money. That kind of thing could explain skewed representation in science more than lack of talented high schoolers...

matthewdgreen• 3 years ago

I think it's excellent that you, a commenter on HN, understand that important contributions can come from all parts of society. I really wish that understanding was shared more broadly across this whole country. Unfortunately I've been alive long enough to see that it really is not.

I suspect that more people felt as you do, we wouldn't have so many barriers to opportunity for kids. Moreover, the existence of those barriers wouldn't be so disproportionately correlated to race and place-of-birth.

samgtx• 3 years ago

It's telling that your comment is currently 2nd ranked. It comes across generous "there are bright math kids everywhere" but really boils down to "don't talk about how they're black" and "don't talk about how they're women." And finished with "my male friend was disadvantaged, the conversation should be about that."

Obviously a large number of the HN crowd agrees with you because these types of comments always land at the top of any article praising a woman or underrepresented minority for their accomplishments. "Why does it matter? We are all people." That's very easy to say when you are in the position of not having your accomplishments and intelligence questioned based on your race or gender. And it shows how homogeneous the HN community is that these types of comments continue to be upvoted to the top.

Representation matters. When you have no concept of what it is like to be black in the deep south. Or to be a woman in the deep south, much less both, you have no appreciation for why stories like this are so interesting and inspiring to the people who relate to them.

yafbum3• 3 years ago

It's telling that your comment is currently 2nd ranked.

Fwiw i think only because it's relatively recent. Not a lot of upvotes currently.

It comes across generous "there are bright math kids everywhere" but really boils down to "don't talk about how they're black" and "don't talk about how they're women." And finished with "my male friend was disadvantaged, the conversation should be about that."

Wow now I think you're reading a lot more into it than what I wrote.

Obviously a large number of the HN crowd agrees with you because these types of comments always land at the top of any article praising a woman or underrepresented minority for their accomplishments. "Why does it matter? We are all people."

That's not actually my claim. I do agree that representation matters. But I find it condescending when someone's accomplishments are only ever mentioned in the same sentence as some statistically surprising fact about their identity, as if what we were saying here is "not bad for a X". (And fwiw I do find it condescending when I'm a recipient of such praise in settings where I'm in the minority.)

duneisagoodbook• 3 years ago

well said

allenu• 3 years ago

I read through the article as well and was put off mostly by how it was presented that the students are black females:

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 gender, ethnic or socio-demographic background — that excellence in your chosen field of study is always attainable if you have enough joy and passion for what you do.

I'm a person of color myself (not black) and seeing this statement (and the fact that the author is white) made it come across as "Look, even a black female can excel in math if they have enough joy and passion in what they do." On the surface, it seems like an innocuous statement, but what it really reads is "the only thing holding you back as an underrepresented person in society, especially being black and female, is your joy and passion, so keep working at it and you too can excel at math". It just reads as tone deaf to me.

My point is there's a way to present the fact that they're black and female, but you have to be careful how you word it because it can otherwise come across as almost condescending.

zelphirkalt• 3 years ago

I think many are simply tired of having scientific or mathematical or technological progress updates, that should be about the thing that is actually new, intermingled with political agenda or racism. Not everything needs to be turned into a discussion about race or the disadvantaged. It is not helpful and it is not on-topic. Many people don't like being off-topiced and would rather engage in on-topic discussion.

Articles which put much emphasis on these things are often coming across as racist themselves, because of the underlying meaning "Even a X can do it!" as if it was something unusual that an X actually manged to do whatever the article is about. Often this kind of expressed surprise hints at more racism of the authors of such an article, as it shows, that they still try to establish an associtation between race and ability.

In a way the "we are all people" mindset is less racist than all of the acting surprised about a X being able to do it and focussing on the "they are a X" aspect mindset.

kmeisthax• 3 years ago

Skewed representation is the end product of multiple filters working in tandem. You're correct that there are still further filters and that college is one of them. But the filters don't start in college, it just continues. The girls in question have already passed through several.

yafbum3• 3 years ago

That's a good point

TheRealPomax• 3 years ago

"OK, so here’s how I think it goes" what do you mean "you think"? It was presented to the American Mathematical Society, there is no guessing here, you show what they showed. Why laud their background and then immediately dismiss their finding by not showing their proof? And then you can't even be bothered to draw a right angled triangle as a right angled angle, something literally any drawing program with straight lines will let you do? Forget you, buddy.

gabereiser2• 3 years ago

I'm with the circular loop crowd. Pythagorean Theorem proofs must exist in geometry terms and not trigonometric terms. To use the dependent math to prove the math will always be true. This is like me saying "I made beer out of old yeast". Yeast being PT in this case. Beer being trig. The proof is using proof to provide proof. sin a. Invalid to me.

kzz102• 3 years ago

It's a really nice idea. I wouldn't call it a proof in the rigorous sense, because you need to define trigonometric functions first and be careful you don't use Pythagorean theorem to avoid circular logic. It's however perfectly fine to call it a proof for students, which is about deducing logical relation between statements.

I am however, against media hype of this type of student achievements. This is very nice for high school students, and they should be showered with praise and get some notoriety in their school. For societal validation, I think it's better to have objective standards. I am not talking about this particular proof, but all the news about various "inventions" and "discoveries" made by high school students that come up every year.

SamBam• 3 years ago

The only definition of the trigonometric function required is that sin (x) = opp/adj. It's just a ratio, nothing else. The students could have called this ratio "snoo" and the proof would have still worked.

The proof is rigorous. The original article explains this, and why it's not circular.

New Orleans teenagers found a new proof of the Pythagorean Theorem

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