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New Orleans teenagers found a new proof of the Pythagorean Theorem
- malshe 4y agoThe 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.
- capableweb 4y 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 4y agoI 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.
- muststopmyths 4y agojust 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.
- varenc 4y agoThis 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.
- azinman2 4y agoIt also doesn’t tell you “how” either. Almost never does an article tell you “how” or “why” despite it being the first word.
- capableweb 4y ago> (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.
- anigbrowl 4y agoNo. 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 4y agoHN strips clickbait titles. Same thing for listicles.
- johnfn 4y agoHN has some sort of algorithm to rewrite post titles to be a bit less link-baity.
- eyelidlessness 4y agoI share your dislike for this title editing, and agree the original title is better.
- oceansea 4y ago[flagged]
- KRAKRISMOTT 4y agoExcellent points, I am not sure why you were flagged as dead.
- haskellandchill 4y agoBecause they are wrong? Read the article. > to make a proof of it would be to show why a^2 + b^2 = c^2 That's what the article shows as a final step. And it outlines what assumptions are made and how the final step is demonstrated. I have no idea what OP is rambling about.
- kurthr 4y agoYeah, they made a symmetry argument, a series expansion argument, and a law of sines argument. None of those require the Pythagorean Theorem and are all separately proven. There may be a reason that it's wrong (I'm not a mathematician but the article's author is). So, they showed why the sum of squares is true for the hypotenuse of a triangle. It's even an exact proof rather than a limit argument. That I could follow the steps, and other mathematicians seem unaware of this combination of tools makes it seem relatively novel.
- oceansea 4y ago[flagged]
- klyrs 4y agoSpeaking as a mathematician with a PhD, you're wrong about proofs. You're wrong about this site, which intends to promote a curious and celebratory attitude towards knowledge. Ironically, it is you who is displaying the hateful attitude that you bemoan. Please adjust your attitude, or see yourself out. And do re-read the article; the author is attempting to reconstruct their proof without seeing it. You're attacking the students without even seeing their work. Incomprehensible.
- HL33tibCe7 4y 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 4y agoI don’t think Privilege matters in all cases, look at the background of Ada Lovelace.
- HL33tibCe7 4y agoIt 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
- JKCalhoun 4y agoThat'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.
- pedrosorio 4y agoI would also take that interpretation if the author didn't explicitly mention "socio-demographic background". They should just leave that one out next time.
- eyelidlessness 4y ago
- zem 4y agoI'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.
- School-Cotton 4y agoIn 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 4y 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
- deleted 4y ago[deleted]
- n3uromancer 4y agoFrom cosβ = a/c and cosβ = third side/((2ac)/b)
- joe_the_user 4y agoI 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 4y agoInduction 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 4y agoThere is a very straightforward proof that involves dropping an altitude. https://www.mathopenref.com/lawofsinesproof.html https://www.mathopenref.com/lawofsinesproof.html You could use a similar technique to make this proof not reference trigonometric functions.
- crdrost 4y agoFor 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?
- pizza 4y agoIt 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)
- contravariant 4y agoYou'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.
- pizza 4y agoAh, 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?
- contravariant 4y agoThe 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 https://en.wikipedia.org/wiki/Bregman_divergence [2]: https://www.mathopenref.com/lawofsinesproof.html https://www.mathopenref.com/lawofsinesproof.html
- deleted 4y ago[deleted]
- Y_Y 4y agoI'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.
- Jarmsy 4y agoWhere is it you think you've seen it before?
- dendrite9 4y agoI 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/the-mathematical-mechanic 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 http://www.personal.psu.edu/mxl48/Welcome_files/Sample.pdf
- hypertexthero 4y agoThis 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/ https://jsomers.net/i-should-have-loved-biology/
- Waterluvian 4y agoI 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.
- mananaysiempre 4y ago> [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.html https://www.maa.org/external_archive/devlin/devlin_03_08.htm... [2] http://www.ams.org/notices/201201/rtx120100031p.pdf http://www.ams.org/notices/201201/rtx120100031p.pdf
- nolamark 4y agofor completeness, here is a reference to the AMS presentation. https://meetings.ams.org/math/spring2023se/meetingapp.cgi/Paper/23621 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.
- hgsgm 4y agoNote that in math "flatly" means "asserted without proof", not "incontrovertible". "Trigonometry" is not formally defined, so Loomis's statement is merely tautological.
- dullcrisp 4y agoMaybe this is my mathematical background, but is that not the common meaning of “flatly” stating something?
- deleted 4y ago[deleted]
- alex_smart 4y agoThe 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 4y agoSorry but that’s not new
- deleted 4y ago[deleted]
- soVeryTired 4y agoMaybe you could provide us with a reference then?
- threatofrain 4y agoThe article itself says it's not new.
- Jarmsy 4y agoWhere does it say that?
- deleted 4y ago[deleted]
- Jun8 4y ago“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.
- gerdesj 4y ago[flagged]
- Xenya 4y ago[flagged]
- GreenWatermelon 4y agoWhat isn't this how it works + how exactly does it work?
- aaron695 4y ago[dead]
- maxk42 4y agoOn 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.
- nolamark 4y agoperhaps: https://forumgeom.fau.edu/FG2009volume9/FG200925.pdf 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.
- hgsgm 4y agoNo, that was a different proof using different trigonometry.
- ummonk 4y agoIf 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 4y agoI 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 https://en.wikipedia.org/wiki/Law_of_cosines
- deleted 4y ago[deleted]
- maxboone 4y agoThe concept isn't new or novel, the proof is (or might be).
- User23 4y agoDijkstra did something similar when he derived a generalization of the theorem[1]. [1] https://www.cs.utexas.edu/~EWD/transcriptions/EWD09xx/EWD975.html https://www.cs.utexas.edu/~EWD/transcriptions/EWD09xx/EWD975...
- lern_too_spel 4y agoIt's the Pythagorean theorem, a special case of the law of cosines. What's new is the proof method.
- starchild_3001 4y agoOk, 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 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 https://en.wikipedia.org/wiki/Law_of_sines
- hgsgm 4y agoIt'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)
- ummonk 4y agoIt'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 https://forumgeom.fau.edu/FG2009volume9/FG200925.pdf). This is definitely a much more elegant proof than the angle-sum proofs.
- starchild_3001 4y agoCorrectness 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 https://www.youtube.com/watch?v=p6j2nZKwf20 I would write a more modest abstract for this work. Just my $0.02.
- groffee 4y ago[dead]
- yosito 4y 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 4y agoThe 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.
- yosito 4y agoMaybe not "obliged", but it still seems a bit like stealing thunder to me.
- hgsgm 4y agoThey 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).
- agurjal 4y agoI 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 4y agoWell sure, there are other proofs of Pythagorean theorem. The interesting part is that this is a new proof.
- hgsgm 4y agoThe 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?
- agurjal 4y agoIt 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!
- no_butterscotch 4y ago[flagged]
- jackmott 4y ago[dead]
- deleted 4y ago[deleted]
- Mtinie 4y ago(Unsolicited comment and suggestion) I’ve had similar thoughts in the past but figured out I personally do better when I channel the doubt into excitement for anyone’s’ potential claim. I’m happy that a human—or humans, in this case—believe(s) they have discovered a novel way to do something and want to share it with the world. That doesn’t mean I take the claim at face value, I don’t, and want to wait for secondary confirmation. But it’s true that I no longer worry if I’m hearing about something because of an agenda…because I know I am, in all cases. So I skip that part and just stick with the hopeful awe.
- girvo 4y agoApproaching things in the way you’ve described is a lot more joyful and enjoyable, too, I find. The world has enough to be sad about.
- eyelidlessness 4y ago> When I read "New Orleans teenagers" I wanted to immediately give the benefit of the doubt, but a part of me suspected they may lead with identity. I didn’t have this reflexive reaction, but I can understand why one might if familiar with the way regional references can be coded language. That said, > It does a disservice to an achievement, if in fact there is one. A part of me is now sort of doubtful that this isn't a "cause celebre" sort of situation. I don’t think it does a disservice. The author isn’t noting the achievement because of the students’ identity, only noting that the objective achievement being achieved by students with systemic disadvantages might be inspiring to others facing similar disadvantages. As an educator, one of their responsibilities is to help students overcome arbitrary barriers to their education and enjoy the benefits of the same education as their peers. It’s one of zillions of implied responsibilities educators have beyond the material itself. But you can’t fault any educator for recognizing when one factor of that might be even a little less fraught by centuries of reinforced barriers to their students. The rest of the paragraph you quoted seems to reflect your own view: > 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 if you have enough joy and passion for what you do. The difference is that the author seems to recognize that wishing doesn’t make it so, that systemic and historical barriers don’t vanish if you don’t mention them. The rest of the post is focused on the objective details. I hope that won’t be lost because you or anyone is sensitive to acknowledgment that humans don’t exist in a vacuum even if you that for the math they’re reasoning about.
- rvz 4y agoWell 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 4y agoWhy 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
- istjohn 4y agoProbably because it has nothing to do with the article. It's just a rant about something entirely unrelated.
- rvz 4y ago> ‘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.
- throwayyy479087 4y ago[flagged]
- dang 4y agoWe've banned this account for using HN primarily for ideological battle and flamewar. That's not allowed, regardless of what you're battling or flaming for. If you don't want to be banned, you're welcome to email hn@ycombinator.com and give us reason to believe that you'll follow the rules in the future. They're here: https://news.ycombinator.com/newsguidelines.html https://news.ycombinator.com/newsguidelines.html.
- malshe 4y agoThe author of this blog Keith McNulty is a mathematician by training and now a data scientist at McKinsey: https://keithmcnulty.org https://keithmcnulty.org
- hackernewds 4y agookay
- option 4y agohow does one define sin without Pythagorean theorem? E. g. need to prove that it only depends on angle and not triangle size.
- istjohn 4y ago[flagged]
- teo_zero 4y agoYou don't need to prove anything to define sin.
- option 4y agothe 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.
- sedatk 4y 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?
- svat 4y agoIt 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 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.
- sedatk 4y agoYour 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.
- marcusphi 4y agoI 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_zero 4y agoThat 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.
- GistNoesis 4y agoThe 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).
- GistNoesis 4y agoHere is a better picture https://github.com/unrealwill/PythagoreanProofImage https://github.com/unrealwill/PythagoreanProofImage [Drawn with https://www.geogebra.org/geometry?lang=en https://www.geogebra.org/geometry?lang=en]
- drno123 4y ago[flagged]
- ipince 4y agoCool 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.
- cobertos 4y agoGeometrically, 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 4y agoYou 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.
- cobertos 4y agoOn 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
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- camjw 4y agoWhen 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.
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- notorandit 4y agoNice 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 4y agoWho said it's nice? A proof is a proof, beautiful or not.
- slowmotiony 4y agoThe author of the article. You did read it didnt you? "most beautiful and simplest trigonometric proof we have seen to date"
- cobertos 4y agoDoesn't this proof not work when alpha or beta is 45°? 2 alpha would make the upper of the extension line in the diagram 90°, and alpha+beta for the bottom angle also 90°. The lines will never intersect, they're parallel
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- Someone 4y agoNice 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.
- camjw 4y agoI 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.
- Someone 4y agoI 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?
- camjw 4y agoI 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. 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,HmUwSqOt6JMXxyaJOOrXow https://excalidraw.com/#room=2298a8fd232d5f58e8ca,HmUwSqOt6J...
- Someone 4y agoThanks. Not the same as the 1st proof in https://socratic.org/questions/what-is-one-method-for-proving-the-pythagorean-theorem https://socratic.org/questions/what-is-one-method-for-provin..., but still reminds me of it.
- noobcoder 4y agoI 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 4y agoHow come the triangles in the picture are all congruent but they don't all look congruent.
- scatters 4y agoBecause 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 4y agoOriginal post on Hacker News two weeks ago https://news.ycombinator.com/item?id=35311765 https://news.ycombinator.com/item?id=35311765
- eneim 4y agoI 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-Theorem-45ef547badaa4ac2905a97bb95e8bed1 https://eneim.notion.site/On-a-new-proof-of-the-Pythagorean-....
- gabereiser 4y agoI 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_smart 4y agoThere 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. 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.
- yafbum 4y agoI 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 4y agoI 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 4y agoIt'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.
- TheRealPomax 4y 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.
- gabereiser 4y agoI'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 4y agoIt'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 4y agoThe 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.
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- TayaWayne 4y ago[dead]