New Orleans teenagers found a new proof of the Pythagorean Theorem | Hacker News Reader

New Orleans teenagers found a new proof of the Pythagorean Theorem

keith-mcnulty.medium.com

949 pointsmalshe251 comments

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.

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

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.

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?

> 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 imo the interesting part, and can stand on its own merits.

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.

Not at all - the hype part is that 2 teenagers derived a uniquely elegant proof that other highly-trained mathematicians had thus far failed to do so.

Any other claims seem to have been added by the media, not the teenagers themselves.

That is what I was referring to. I wasn’t accusing the teenagers of anything.

This theorem is essentially the Pythagorean theorem, so … a proof of one is a proof of the other.

Sine and cosine can take as their input any real number including negatives and including very large positive numbers. Their outputs can also be negative numbers between negative 1 and 1 if they have real inputs. None of this necessarily makes any sense if you're considering a purely geometric naive interpretation in terms only of ratios of lengths. You have to introduce concepts like modulo the angle in a circle and analytic coordinate system for it all to square with normal naive intuition.

In fact the sign and cosine can take as their inputs any and produce as their output any complex number. You have to come up with some very interesting triangles to make this makes sense. I'm sure it might be doable but they would potentially be four dimensional triangles and I haven't explored that concept very deeply.

4 dimensional triangles are the same as 3 dimensional lines. They don’t exist in the 4th dimension any more than they exist in any dimension >= 3. You would need a fourth side/point in the polygon in order for it to have any position in that dimension.

(It could be a triangle in dimensions 2-4 from our perspective but to the triangle it only has 3 dimensions any way you arrange it.)

Or you can bend a triangle in another dimension(s), but then it’s not a triangle by the commonly accepted definition. (E.g a 270° “triangle” on a sphere)

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?

I asked https://math.stackexchange.com/q/124887/6400

I wish I had a more modern summary of the papers mentioned in the linked paper

> Tannery, Fonctions d'une Variable, 1886, p. 147. Osgood, Lehrbueh der Funktionentheorie, 1912, p. 582. Van Vleck and H'Doubler, Transactions Amer. Math. Society, vol. 17 (1916), p. 30

because we spent an entire semester at the university in one class working on these two.

> because we spent an entire semester at the university in one class working on these two.

Yeah, the Math Overflow answers are a bit sparse, and I think the Euler formulation (while clever) is a bit of a red herring and might be circular. Trying to slowly go through that 1917 paper, thanks for linking!

I'm not familiar with this result, but this comment is phrased in the language of Ordinary Differential Equations, so I'd look for a textbook on solving systems of ODEs and expect to find a technique that can prove that this is the unique solution (at least assuming differentiability of S(x) and C(x)).

You can also consider the pair of functional equations as implicitly using a Taylor expansion

Also related: doi:10.1007/s00010-003-2700-z

f(x)f(y)f(z) = f(x) + f(y) + f(z) sort of implies f(x) is tan(x) -- it's actually tan(kx + (1-k) pi/3).

Sorry I left out the condition of x + y + z = π

I did not know that one! It's a more complex version of the well-known functional definition of the exponential function, i.e. the unique continuous function satisfying

  E(x) E(y) = E(x + y)

and a normalization, E(1) = (whatever).

> 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.

The Pythagorean theorem is the most frequently re-proved thing in the history of math, with hundreds of published proofs and entire books dedicated to collecting them. The reason to come up new proofs these days is solely for the novelty, not because we have a need for a simpler proof. The fact that it's such a popular subject for proofs is why novel proofs are inherently interesting, regardless of their complexity.

> The reason to come up new proofs these days is solely for the novelty, not because we have a need for a simpler proof.

On the contrary, every new way to prove a known theorem has the potential to be applicable in other areas of the same or related fields, extending the mathematicians' toolset with new instruments. These new methods often serve as a seed for new discoveries.

The mistake is seeing this as a constructed thing. Math is already there, we only uncover it. If they revealed a previously unseen chamber in the Great Pyramid or something, you wouldn't say "aw nuts, that overcomplicates our existing knowledge of the structure".

Hey, come to think of it, I'd be perfectly happy with it being the corner of a solid cube[*] of rock ploinked diagonally down into the sand; the whole idea of interior chambers and tunnels does kinda make it feel too fussy, not as clean and sleek as it could be...

___

[*]: Dangit, no, wrong shape.

For applied maths I agree with you. But when exploring novel and innovative approaches I disagree. There is so much we can learn from finding new ways to look at the world. The initial take may be complex or clever, but it can lead to a deeper understanding that allows for later simplification of entire subject areas.

I disagree.

'Aesthetic' appearance in math is important in helping drive mathematical innovation and help new human beings derive pleasure from that wonderful field.

Programming is kind of an applied mathematics where efficiency does matter because it's a tool, a means to an end.

Not to say that people can't find aesthetics in programming, nor that they shouldn't, rather in math at least the pleasure of discovering a new way of doing/proving something is the end in of itself.

It's pleasure for me to see another way to do or prove something; I can only imagine the feelings this teenager got from actually making a discovery.

I find simplicity aesthetically appealing. Obviously some other people differ.

Well you're in luck since these girls' proof is pretty simple so should be pretty pleasing for you!

wouldnt downvote this because it makes sense in some cases.

while programming and math overlaps in so many places, i think mathemathicians are quite a separate species and KISS is often not on the menu.

a fun, non math heavy book with a window into this i enjoyed is https://en.wikipedia.org/wiki/Uncle_Petros_and_Goldbach%27s_...

really just an arts/hobby vs job/career disagreement. and gosh, they’re teenager students: we can’t suck all the joy out of school!

but related, i wonder what a mathematics demo party would look like.

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.

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.

For that matter the easiest proof of Pythag that I know of involves dropping an altitude:

Look at the right triangle the normal way up, clearly the area of the triangle is k c² (k = ½ sin α sin β if you like, but it just matters that it's the same nonzero k for all similar triangles).

Now roll it onto its hypotenuse, drop an altitude, and observe that both subtriangles are similar to the first one, kc² = ka² + kb².

The diagram that's a bit involved is the angle sum diagram, you start with a right triangle (a,b,c) with some angle α, extend it to a new triangle (a,b', c') with angle α+β, then make the new triangle with angle β that you stacked on top of the original triangle into a right triangle with angle β (c', d, e) by extending the hypotenuse of the (a,b,c) triangle to a point P, basically until the angle with the hypotenuse c' is 90°. Drop a dotted line to the x-axis from P and you can work out that the dotted line is at x=cos α cos β, and its distance to a is sin α sin β. Similarly the y-coordinates give sin α cos β + cos α sin β.

As you say, you can do all of this without angles except for defining the first triangle with angle α+β, which you might not even need... We just need it here for sin(2 α) which is something like reflecting the same triangle about its hypotenuse?

For this proof to be a proof, you first have to define what area is and why it should scale as kc² and why the sum of areas of the two the smaller triangles should equal the area of the larger triangle.

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.

,,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

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.

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

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

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/

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.

> [A]ll my best learning would be done outside school.

Choose one:

- Experience of discovery and survival of curiosity to adulthood;

- Set of job-relevant skills well defined by names of subjects;

- Standardized testing and easily comparable grades.

(In my admittedly limited teaching experience.)

I would guess that the last point will always get chosen, because it’s bureaucracy-friendly, and a bureaucracy makes the choice. But one of my most bizarre experiences is (some) HN readers being quite vocal about their support for it as well, where I haven’t seen it be anything but harmful. The bullshit admission process at US colleges might be to blame—I’m really not sure.

References: Lockhart’s “Lament”[1], of course, for describing the feelings that (good) teachers have on this subject; Quinn’s “Revolution in mathematics”[2], as a more clinical analysis of how the bureaucracy won and got to basically redefine what “mathematics” even means for the majority of the population (in a way that’s as hopelessly obsolete as it is intensely harmful to the subject proper). The point shouldn’t be specific to mathematics, but it’s what I have the references for.

[1] https://www.maa.org/external_archive/devlin/devlin_03_08.htm...

[2] http://www.ams.org/notices/201201/rtx120100031p.pdf

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.

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

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.

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.

> 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.)

So this statement is not true, you cannot prove its value. But I understand what you mean, let's just talk about provable statements.

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...

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.

> 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.)

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.

How much of ancient and modern math was developed by group B?

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.

well said

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.

That's a good point

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.

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.

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

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

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

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.

"...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?

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.

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.

I see your confusion. The thing is that “trigonometric proof” is not a well-defined mathematical object (even in this HN thread, you can see a lot of discussion about whether the trigonometry is really essential to the proof, ways to remove it, etc). So the “establishment” idea that “There are no trigonometric proofs” (Loomis, 1927) is not a mathematical conjecture, which can be demolished by the first counterexample (the first “trigonometric proof”), but a meta-mathematical statement or social convention, where minds change slowly. Note how he says “this point of view has been increasingly questioned in recent decades”: this is typical of the evolution of social consensus rather than of mathematical conjectures (which would quickly switch from false to true or vice-versa as soon as a proof or counterexample is found).

The first few trigonometric proofs, being very complicated, might have just had a reaction (among the very few people who even care about this question) like “yeah ok, whatever, that's just too contrived, not very interested”, but when a proof like this comes along, being more beautiful and simpler, more people will change their minds — but even now it's not guaranteed, which is why the author says “might make a few established mathematicians eat their words”.

Ultimately, all proofs are just pushing around of axioms and implications; there's no clear separation of whether a proof is “different” from another or whether it's “trigonometric”, but in this case the authors say their proof “is based on a fundamental result in trigonometry—the Law of Sines” and it seems pretty easy to believe that that is how they came up with the proof (so it seems fair to call it a trigonometric proof even if that can be got rid of).

[PS: I just found that some scans of Loomis's book are online: https://personal.math.ubc.ca/~cass/Euclid/java/html/L.pdf (1927), https://files.eric.ed.gov/fulltext/ED037335.pdf / https://www.lapasserelle.com/documents/Pythagorean_Propositi... (1940 second edition) — see the foreword where he says “Fifth, that no trigonometric proof is possible”, elaborated on p. 193(1e)/244(2e) in section called “No trigonometric proofs”.]

Thanks, this has been very explanatory. It’s unusual that maths, the language known for precise definitions, has no unambiguous definition of trigonometry. It’s what it is though.

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?

I wonder, are there an infinite number of proofs of the theorem, each more complex than the last? Can I rephrase what they did, make it more convoluted, and call it a new proof?

You can always insert irrelevant details into a proof, so it's easy to construct an infinite number of proofs from a single proof.

If they are irrelevant details then that should not count as a new proof, in my opinion.

I guess it might be possible to have a "canonic form" of a proof so that if two proofs can be reduced to the same canonic proof then they are in fact the same proof.

> 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.

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-....

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.

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.

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.

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

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

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

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

It’s the infinite geometric series idea which makes it potentially novel.

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!

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.

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.

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.

How is it trivial?

Commented this on another thread:

Okay so take the triangle made by taking the diagonal of the unit square. This has side lengths 1, 1, and c and has area 1/2.

Now, take four of these and arrange them in a square with the side length being c. It would be easier to draw this... basically you stick the right angles in the center. If this isn't clear I can draw a diagram.

Anyway, you just made a square with side length c but since its made of four of those original triangles we know that the area of it is 4 * (1/2) = c^2 so c^2 = 2.

EDIT: made an excalidraw to explain this construction - maybe helpful https://excalidraw.com/#room=2298a8fd232d5f58e8ca,HmUwSqOt6J...

You're using geoemetrical construction not dissimilar from proving the theorem for a != b. So it's not in the spirit of this new method. No one disputes there are easier methods to prove the theorem.

Note that when a = b we have an equilateral triangle, with area a^2/2.

Draw a line from the 90 degree angle to side c, bisecting the 90 degree angle into two 45 degree angles. This divides the original triangle into two smaller triangles.

From the fact that the sum of the interior angles of a triangle is 180 degrees, it is not hard to see that the two smaller triangles are both equilateral, with sides of a, c/2, and c/2, and the angle between their two c/2 sides is 90 degrees.

That gives c^2/8 for the area of each of the smaller triangles, or c^2/4 for the area of the original triangle which we know to be a^2/2. So c^2/4 = a^2/2 or c^2 = 2 a^2 = a^ + a^2.

Ok, but those triangles aren't equilateral but isosceles :)

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

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.

On the partial sums, okay, I see.

Though, still for the 45-45-90 I don't think you can pick a different angle? At least for alpha > 45 (because it also doesn't work for this, the lines diverge), you can always swap it so beta is > 45 for those cases. If you pick something other than 90 to be 2 alpha, the reflection mentioned in step 1 can't be done

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.

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

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.

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.

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)

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).

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

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.

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.

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?

I can!

Okay so take the triangle made by taking the diagonal of the unit square. This has side lengths 1, 1, and c and has area 1/2.

Anyway, you just made a square with side length c but since its made of four of those original triangles we know that the area of it is 4 * (1/2) = c^2 so c^2 = 2.

EDIT: made an excalidraw to explain this construction - maybe helpful https://excalidraw.com/#room=2298a8fd232d5f58e8ca,HmUwSqOt6J...

Thanks. Not the same as the 1st proof in https://socratic.org/questions/what-is-one-method-for-provin..., but still reminds me of it.

Yes this is essentially a special case where the yellow square has side length 0.

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.

> 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?

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.

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.

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

okay

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.

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?

The wikipedia lists a law of cosines [1] for the Bregman divergence and someone in this thread posted a proof that basically showed the law of sines follows directly from the definition of the sine. So neither form an obstruction per se, but the generalized pythagorean theorem kind of just follows directly from the definitions you need to make to even be able to say what it means for a triangle to be rectangular.

[1]: https://en.wikipedia.org/wiki/Bregman_divergence [2]: https://www.mathopenref.com/lawofsinesproof.html

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

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.

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

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.

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 ?

"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

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

> 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.

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.

It’s a “steal” if you make California wages. For Louisiana, that’s a lot of money. The median home price in New Orleans is roughly $230k — in Santa Clara county, CA, it’s $1.3 million. That makes the tuition comparable to $45k per year which is more expensive that the best private schools in Silicon Valley.

The math is sloppy, but the point is that the tutoring isn’t really a “steal.”

Ahh, the tuition was cut off on mobile and did not realize there was horizontal scroll.

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.

Are new articles about people’s accomplishments “remiss” for not including statements about how much their parents cared about them?

Wealth is a huge predictor of academic (and business) success. Let’s put family wealth at the top of any news article about achievements, please.

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?

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.

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

I don't want to take anything away from these two; the proof is novel and interesting and even if it wasn't novel after all it's still incredible to see high school students interested in math and capable of that level of reasoning.

But yeah, that the two researchers (they've earned the tittle by giving a conference talk!) are from a private, selective school undermines a lot the "bad area and underachieving" narrative. It reminds me of a tech company who had a panel about their black engineers, pointing to the gap between the proportion of black SWE in the bay area relative the the percentage of people who identify as black. Three out of the four panelists were Nigerian born. When discussing their path to tech, one of them explained it was hard for him to convince his parents that he was not going to study surgery like his father and uncle. I assumed it would have resonated with anyone from an "underachieving area" in the audience...

> 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.

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.

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.

Publicly stating that it doesn't matter sounds pandering and cringey to me. Pretending that you would be inspired by an Asian doing mathematics sounds even more pandering cringey. If the proof came from the adult sons of educated Asian immigrant parents in Boston, it would be far less newsworthy.

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.

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

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

That's not the way I took the article. Rather I took it to mean something slightly different:

Your ethnicity and/or gender should not be seen as a barrier to advanced mathematics.

There is a nuanced difference there.

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 explicitly acknowledge that ethnicity and/or gender may presently be a barrier to advanced mathematics?

How does that even work? We achieve a meritocracy inherently based on not mentioning that we don’t have anything resembling a meritocracy? Because identifying reality is what makes it real?

Read the thread I am responding to.

I did.

Cool. The summary then is:

Ethnicity and Gender and Socio-demographic background != Ethnicity and Gender

What you asked in your first comment seem to be orthogonal to the point I was making.

Ethnicity and gender are two socio-demographic factors of many, yeah. So if I’m understanding you, it’s okay for the educator to want students to feel encouraged to succeed regardless of their ethnicity and gender, but shouldn’t mention any other socio-demographic barriers they might overcome? Or should specify them all? I’m trying to understand why mentioning the term is objectionable.

Mentioning ethnicity, gender AND socio-demographic factors, implies this case is a good illustration of OTHER socio-demographic factors that have to be overcome in order to succeed.

In particular, with no further information, one might assume a lot more about the difficulties these two girls had to overcome being African American in Louisiana. The State's public schools famously rank very poorly, and there is a strong correlation between socio-economic reality and ethnicity.

That's why a comment upthread quoted that section of the article and mentioned it is remiss not to mention the two girls attended a private Catholic school with ~$10k/year tuition.

Further down the thread, someone says the message they got from the article was how ethnicity and/or gender should not be seen as a barrier to advanced mathematics.

I disagreed that is the only thing the article says. The article does mention this as an inspiring case for people who may see themselves excluded from advanced mathematics because of gender/ethnicity AND other socio-demographic factors. Some very important ones that hold back a lot of black girls (and boys) in the US do not apply in this case.

So your problem is that these individual students weren’t oppressed enough to warrant an educator’s broader generalization that all students shouldn’t be held back by their many and varied forms of barriers?

If you recognize that there’s a vast set of intersecting disadvantages, why would you want to turn it into Oppression Olympics? It’s fine to recognize that not all of those power dynamics might be in play in this particular case. But do they really have to all be in play to make a fairly bland statement that no one should be held back by any of them?

I have solidarity with these students just as I have with others who have different socio-demographic factors to contend with. They’re not and shouldn’t be in competition for access to education or achievement in life. I don’t understand why that should be controversial even if each and every student has a different configuration of advantages and disadvantages. And I think it does a disservice to the students who have different disadvantages to say no one should speak about them in a unifying way unless they meet an exacting criteria of total disadvantage.

The articles’s author does not say that “students shouldn’t be held back”.

He says:

“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”

“Inspire”. A child who is abused or malnourished or whose caregivers care little or nothing about their education or whose classroom is constantly disrupted by other children living in terrible situations and whose teachers are overworked with large class sizes and incapable of providing them with a proper education (“no matter their socio-demographic background”) will find little to be inspired in this story (if they know the full story that is), regardless of their ethnicity or gender.

Read the comment I initially replied to and see how it interprets the article’s quote as a message of empowerment about “gender/ethnicity” only. It could have been. All the author had to do was remove the “and regardless of socio-demographic background”. As I stated in my initial comment.

I’ve been a child in several of the situations you describe, so I have little reason to speculate about how I would react, and I think you’re engaging this whole discussion in the worst faith possible to find much greater fault in a statement, which itself is a pretty milquetoast blanket wish for students to succeed without artificial barriers. You’re parsing a thing so far out of its intended context that it can’t help but be absurd, because it doesn’t have any tether to the idea the person expressed at all.

Thanks for the feedback.

... The daughter of Lord Byron?

Their point is that Ada Lovelace is still a salient example of a woman who made significant discoveries in computer science, even though she was also privileged in other ways. Indeed, in general, it's often otherwise-privileged members of disadvantaged groups who are the first to make advances.

The equivalent in Lovelace's case would be saying she's a woman from an area where most women are illiterate and work sixteen hour days in death trap factories. It's true, but ignoring that she was part of the nobility adds a disingenuous element to the statement.

And saying that privilege doesn't matter is nonsense, without it she's unlikely to have had the opportunity to do what she did.

I don’t think OP was suggesting that Ada Lovelace wasn’t aided by her privileged position. Rather, they were saying that even though she was privileged, that doesn’t take away from her significance as a woman who was a pioneer in computer science.

I doubt that is a controversial point. I think all three of us probably agree, and you may just have misinterpreted the original post.

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

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

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

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

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

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.

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.

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

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.

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

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

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.

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.

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.

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.

No, that was a different proof using different trigonometry.

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

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

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.

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.

"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

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.

This is getting off topic, but IMHO the HN default behavior here is reasonable.

There's certainly situations where removing the leading "Here's How" makes the title worse, but I think those instances are rare and in general this rule leads to better titles much more often than worse titles. Manual human review would of course be better, but dang only runs in O(n) time. Basically, it's not perfect but I think it does much more good than harm.

It also doesn’t tell you “how” either. Almost never does an article tell you “how” or “why” despite it being the first word.

> (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)

Yeah, I guess I'm a bit extra pedantic when it comes to the titles, because I agree the most with this, the article doesn't seem to actually go into how they found it out, meaning the original title was misleading after all.

just taking out the "here's" and leaving "How New Orleans Teenagers Found a New Proof of the Pythagorean" would have preserved the meaning and also made it seem less clickbaity.

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.

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

HN strips clickbait titles. Same thing for listicles.

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

> 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.

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.

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

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).

Sorry but that’s not new

Maybe you could provide us with a reference then?

The article itself says it's not new.

Where does it say that?

“However, this point of view has been increasingly questioned in recent decades, and 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, but their proof could well be the most beautiful and simplest trigonometric proof we have seen to date”

my understanding from this part is that their general approach is not new (though certainly astonishing for HS students) but their proof is novel.

"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.

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.

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.

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.

> ‘Probably because it has nothing to do with the article.’

Assuming you have read the comment, it is totally relevant to the article with AI (in this case LLMs or GPTs like ChatGPT) still not being able to transparently prove unsolved mathematical problems or even generate such solutions with and without supervision, since even if it was supervised, it will still generate it incorrectly and as with its black-box nature, it cannot reason or explain transparently.

The fact the those teenagers were able to create and derive this proof without regurgitation and withstood the scrutiny of experienced mathematicians tells us that it requires creative thought with transparent reasoning in the field to go against the books written by experts that once said it was ‘impossible’ until proven otherwise.

Reply on news.ycombinator.com