[HN Gopher] New Orleans teenagers found a new proof of the Pytha...
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New Orleans teenagers found a new proof of the Pythagorean Theorem
Author : malshe
Score : 863 points
Date : 2023-04-08 21:13 UTC (1 days ago)
(HTM) web link (keith-mcnulty.medium.com)
(TXT) w3m dump (keith-mcnulty.medium.com)
| option wrote:
| how does one define sin without Pythagorean theorem? E. g. need
| to prove that it only depends on angle and not triangle size.
| istjohn wrote:
| [flagged]
| teo_zero wrote:
| You don't need to prove anything to _define_ sin.
| option wrote:
| the definition of _function_ in mathematics requires than
| 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.
| TheRealPomax wrote:
| "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.
| groffee wrote:
| [dead]
| FuriouslyAdrift wrote:
| Original post on Hacker News two weeks ago
| https://news.ycombinator.com/item?id=35311765
| HL33tibCe7 wrote:
| > 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.
| sbelskie wrote:
| 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!
| User23 wrote:
| 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.
| fancyham wrote:
| 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.
| sbelskie wrote:
| Are new articles about people's accomplishments "remiss"
| for not including statements about how much their parents
| cared about them?
| ummonk wrote:
| 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.
| sbelskie wrote:
| Ahh, the tuition was cut off on mobile and did not realize
| there was horizontal scroll.
| briandear wrote:
| 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."
| uberman wrote:
| 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?
| ThalesX wrote:
| > 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.
| threatofrain wrote:
| 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.
| 908B64B197 wrote:
| > 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...
| concordDance wrote:
| 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.
| secondcoming wrote:
| I don't think Privilege matters in all cases, look at the
| background of Ada Lovelace.
| HL33tibCe7 wrote:
| 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
| JKCalhoun wrote:
| 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.
| pedrosorio wrote:
| 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.
| eyelidlessness wrote:
| > > 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?
| pedrosorio wrote:
| Read the thread I am responding to.
| eyelidlessness wrote:
| I did.
| boomboomsubban wrote:
| ... The daughter of Lord Byron?
| foldr wrote:
| 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.
| boomboomsubban wrote:
| 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.
| foldr wrote:
| 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.
| [deleted]
| pelorat wrote:
| I think it's just fucked up these things needs to be mentioned
| at all.
|
| We're all fucking humans. Great job to these persons*
|
| * = a human being regarded as an individual
| lern_too_spel wrote:
| 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.
| [deleted]
| steve76 wrote:
| [dead]
| thrashh wrote:
| 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.
| lern_too_spel wrote:
| 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.
| throwayyy479087 wrote:
| [flagged]
| dang wrote:
| We've banned this account for using HN primarily for
| ideological battle and flamewar. That's not allowed, regardless
| of what you're battling or flaming for.
|
| If you don't want to be banned, you're welcome to email
| hn@ycombinator.com and give us reason to believe that you'll
| follow the rules in the future. They're here:
| https://news.ycombinator.com/newsguidelines.html.
| drno123 wrote:
| [flagged]
| eneim wrote:
| 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-....
| alex_smart wrote:
| 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.
|
| 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.
| gabereiser wrote:
| 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._
| pizza wrote:
| 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)
| contravariant wrote:
| 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.
| pizza wrote:
| 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?
| contravariant wrote:
| 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
| sedatk wrote:
| "...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?
| _Nat_ wrote:
| > 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/w
| iki/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.
| svat wrote:
| 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.
| sedatk wrote:
| 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.
| svat wrote:
| 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 / h
| ttps://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".]
| sedatk wrote:
| 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.
| galaxyLogic wrote:
| 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?
| qez wrote:
| 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?
| FartyMcFarter wrote:
| You can always insert irrelevant details into a proof, so
| it's easy to construct an infinite number of proofs from
| a single proof.
| galaxyLogic wrote:
| 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.
| zem wrote:
| 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.
| gerdesj wrote:
| [flagged]
| tomrod wrote:
| I have a few experts I work with I could run it by.
|
| Math isn't some esoteric thing, they teach to anyone who would
| like to learn it.
| fullsend wrote:
| An okay thought but to be honest one thing I like about HN is
| that some of the experts are here occasionally.
| gabereiser wrote:
| 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.
| TayaWayne wrote:
| [dead]
| [deleted]
| oceansea wrote:
| [flagged]
| johnfn wrote:
| The proof does indeed seem to demonstrate a^2 + b^2 = c^2 from
| first principles. If you have specific problems with
| assumptions the proof is making you should state them; as it is
| it's unclear what problem with the proof your post is
| responding to.
| civilized wrote:
| I have a PhD in math and I have no idea what you are talking
| about. This is a proof.
| joe_the_user wrote:
| I can't see how your claims relate to the article involved. The
| proof that's described uses a construction of infinitely many
| triangles and the formula for the sum of a geometric series, so
| it basically is a proof in it's form, whether it's correct or
| uses circular logic is another question but not it's merely a
| demonstration of properties (which is one way student proofs
| often fail but I don't see that here).
| silisili wrote:
| > This is the Louisiana education system in the USA, a state
| which is rather notorious for having sub-par education
|
| What does this have to do with anything? Exceptional people
| come from absolutely anywhere, and seek to learn on their own.
| It's as true in Louisiana as it is in Massachusetts.
|
| Ken Thompson was born and (at least partly, afaik) raised in
| Louisiana, for example. Are we all just dumb people using a
| dumb person's language and OS?
| selimthegrim wrote:
| There's even people from the state that read this forum now
| and then
| slightlytwisted wrote:
| Indeed. I'd like to add the Louisiana School for Math,
| Science, and the Arts as another example of a great high
| school. https://www.lsmsa.edu/
| selimthegrim wrote:
| I'm a grad student at Tulane in New Orleans I was being
| sarcastic
| slightlytwisted wrote:
| I understood that you were being sarcastic. I should have
| said I loved your comment and thanked you for it as
| someone who was born and raised in New Orleans.
| filoleg wrote:
| A lot of people also don't realize that, even though schools
| in states like Alabama and Louisiana on average perform
| poorly on primary education metrics compared to other states,
| each one of the states almost definitely has at least one
| absolutely great high school that is extremely competitive on
| a national level in academics.
|
| For example, look at LAMP High School[0] or Mountain Brook[1]
| in Alabama. One is listed as the 17th top school in the US,
| the other produced 3 Rhodes scholars. Both consistently rank
| very high on the lists of top schools in the US. And there
| are quite a few other pretty academically great schools in
| that state, just ranked a bit lower (but still as some of the
| best schools in the country).
|
| 0. https://en.wikipedia.org/wiki/LAMP_High_School
|
| 1. https://en.wikipedia.org/wiki/Mountain_Brook_High_School
| KRAKRISMOTT wrote:
| Excellent points, I am not sure why you were flagged as dead.
| dang wrote:
| It was an obvious flamewar comment, which is against the site
| rules, plus name-calling, which is also against the site
| rules, plus nitpicking in an aggressive way. That's not what
| HN is for.
|
| https://news.ycombinator.com/newsguidelines.html
| oceansea wrote:
| [flagged]
| beepbooptheory wrote:
| Perhaps this is a chance for some self-reflection on your
| part? This doesn't seem to be a mean pileon of people
| hating on you, and its not a particularly charged issue to
| begin with. You are simply not being clear at all in your
| argument and then otherwise using this whole thing as a
| platform to talk about your thoughts on the public
| education system in Louisiana, and most likely I think some
| broader thoughts of yours on how kids are just too damn
| coddled these days. (It should be noted that these students
| are not in the public education system anyway!)
|
| Maybe just try and reflect on whether you would take such
| an issue with this if the subjects of it didn't challenge
| your various implicit assumptions so much?
|
| Or at the very least, attempt to be clearer in your
| writing. and articulate your arguments. As a champion of
| life and knowledge, this should really be a priority for
| you.
| haskellandchill wrote:
| Because 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 wrote:
| Yeah, 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 wrote:
| [flagged]
| klyrs wrote:
| Speaking 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.
| dang wrote:
| Please don't respond to a bad comment by breaking the
| site guidelines yourself. That only makes things worse.
|
| https://news.ycombinator.com/newsguidelines.html
|
| Edit: just for clarity, your comment would be fine
| without the 3rd and 4th sentences and the final bit.
| dang wrote:
| You took this thread on an offtopic flamewar tangent, which
| breaks the site guidelines. We ban accounts that make a habit
| of this, so please don't do it again.
|
| You also threw in nasty regional flamebait and crude name-
| calling. Those things are also not allowed here.
|
| If you'd please review
| https://news.ycombinator.com/newsguidelines.html and stick to
| the rules when posting to HN, we'd appreciate it.
| roywiggins wrote:
| There's no actual requirement for a proof to give you a "why."
| The steps just all have to be be logically correct and not
| accidentally, eg, assume what is to be proven.
|
| Lots of proofs are basically unhelpful at giving you a "why,"
| that doesn't disqualify them from being a proof.
| oceansea wrote:
| I was using Stanford's definition. https://web.stanford.edu/c
| lass/archive/cs/cs103/cs103.1202/l...
| mathteddybear wrote:
| "A mathematical proof is an argument that demonstrates why
| a mathematical statement is true, following the rules of
| mathematics."
|
| yes, and?
| roywiggins wrote:
| That definition (as interpreted by you) would appear to, at
| the very least, preemptively disqualify any proof by
| exhaustion ("it's true because we brute forced all the
| possibilities and didn't find any counterexamples"), which
| is a perfectly valid proof method.
|
| https://en.m.wikipedia.org/wiki/Proof_by_exhaustion
|
| "All the steps are valid but it doesn't explain anything
| and I'm dissatisfied" is not an argument against a proof's
| validity. Sometimes the argument is just "A implies B
| implies C implies D implies E implies P, QED" and as long
| as you are convinced that each step is valid, you've got to
| take the proof itself as valid, even if the gestalt isn't
| satisfying.
| avindroth wrote:
| ...I took several classes with Keith (who teaches this
| class), and he would definitely consider this a proof.
| Also, you should look to the math department, not the CS
| department if you want to appeal to authority.
| yCombLinks wrote:
| That's a misc slideshow from a class, not a formal
| definition.
| oceansea wrote:
| [flagged]
| threeseed wrote:
| Pretty relevant when you're criticising the work of these
| teenagers because it doesn't meet your definition of a
| proof.
| SalmoShalazar wrote:
| It was a useful comment and your odd defensive posture is
| the problem here.
| threeseed wrote:
| That is not Stanford's official definition. That is the
| definition from one of their Computer Science lecturers.
|
| I can't speak to it's accuracy but I would imagine better
| sources exist.
| uberman wrote:
| Apparently there are some university math professors such as
| Lozano-Robledo who disagree with you.
|
| I can't say if their proof will hold up under real peer review
| but I hope it does.
| pdpi wrote:
| I can't see it failing to pass muster.
|
| It's more complicated than most proofs of Pythagoras's
| theorem I know, but it's still simple enough that several
| people have reconstructed it from just a diagram and some
| basic notes on how it should work. For so many people to have
| now independently reconstructed it without anybody finding
| issues is a clear indication that, even if the way they wrote
| it up has mistakes, the core ideas are solid.
| ivan_gammel wrote:
| Can you elaborate on what is a proof in your opinion? Having a
| degree in applied physics and mathematics from MIPT I'm
| struggling to understand your point. It does look like a proof
| of the theorem to me.
| oceansea wrote:
| https://web.stanford.edu/class/archive/cs/cs103/cs103.1202/l.
| .. I am using Stanford University's definition.
| klyrs wrote:
| You're pointing at a document without specifying what
| precisely in that document would indicate that the argument
| (which is presented only as an outline and extrapolation of
| the students' work) does not constitute a proof. If you're
| going to attempt to prove something, _do_ try to use at
| least a modicum of rigor.
| kurthr wrote:
| Ok, I have read the first 20 slides for a CS class at an
| institution from which I have a degree. I feel no closer to
| understanding why it is relevant.
|
| Can you specify which step in the "new proof" is wrong,
| unproven, or tautological (itself requires the Pythagorean
| theorem)?
|
| The simplest reason why it is explanatory as defined in
| your cited reference is that the Euclidean distance is
| defined by the hypotenuse of the triangle, that triangle
| (and similar triangles by construction) have ratios which
| are also similar, the distance along a side is the sum of
| those distances in an infinite series, the series
| expansions of sin and cosine are known along with the law
| of sines and are independently (from the Pythagorean)
| proven.
| [deleted]
| yosito wrote:
| > 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 wrote:
| 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.
| yosito wrote:
| Maybe not "obliged", but it still seems a bit like stealing
| thunder to me.
| hgsgm wrote:
| 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).
| kzz102 wrote:
| It's a really nice idea. I wouldn't call it a proof in the
| rigorous sense, because you need to define trigonometric
| functions first and be careful you don't use Pythagorean theorem
| to avoid circular logic. It's however perfectly fine to call it a
| proof for students, which is about deducing logical relation
| between statements.
|
| I am however, against media hype of this type of student
| achievements. This is very nice for high school students, and
| they should be showered with praise and get some notoriety in
| their school. For societal validation, I think it's better to
| have objective standards. I am not talking about this particular
| proof, but all the news about various "inventions" and
| "discoveries" made by high school students that come up every
| year.
| SamBam wrote:
| 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.
| malshe wrote:
| The author of this blog Keith McNulty is a mathematician by
| training and now a data scientist at McKinsey:
| https://keithmcnulty.org
| hackernewds wrote:
| okay
| umanwizard wrote:
| 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 ?
| n3uromancer wrote:
| From cosb = a/c and cosb = third side/((2ac)/b)
| mathteddybear wrote:
| "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]
| cobertos wrote:
| Doesn't this proof not work when alpha or beta is 45deg? 2 alpha
| would make the upper of the extension line in the diagram 90deg,
| and alpha+beta for the bottom angle also 90deg. The lines will
| never intersect, they're parallel
| [deleted]
| GistNoesis wrote:
| 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+90deg =
| 180deg).
|
| Otherwise you can't understand why the ration A/C would be
| sin(2*alpha).
| GistNoesis wrote:
| Here is a better picture
| https://github.com/unrealwill/PythagoreanProofImage [Drawn with
| https://www.geogebra.org/geometry?lang=en]
| agurjal wrote:
| I think you can sidestep trigonometry (and the Law of Sines)
| completely. You can decompose any triangle A, B, C using their
| construction to create smaller triangle a, b, c where A =
| 2abc/(b2 - a2), C = c(b2 + a2)/(b2 - a2), and B = c. This can be
| shown with only similar triangles (it seems like they
| unnecessarily use sines in the article). It is then just algebra
| to show A2 + B2 = c2(b2 + a2)2/(b2 - a2)2 = C2.
|
| edit: any _right_ triangle A, B, C using their construction to
| create smaller _right_ triangle a, b, c
| erehweb wrote:
| Well sure, there are other proofs of Pythagorean theorem. The
| interesting part is that this is a new proof.
| agurjal wrote:
| I'm certainly no expert on proofs of the Pythagorean theorem,
| but if their construction is novel, then the above may be
| too.
| jonahx wrote:
| It's the infinite geometric series idea which makes it
| potentially novel.
| hgsgm wrote:
| 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?
| agurjal wrote:
| 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!
| amelius wrote:
| How come the triangles in the picture are all congruent but they
| don't all look congruent.
| scatters wrote:
| 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.
| ipince wrote:
| 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.
| camjw wrote:
| 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.
| chengiz wrote:
| How is it trivial?
| tzs wrote:
| 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.
| camjw wrote:
| 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=2298a8fd232d5f58
| e8ca,HmUwSqOt6J...
| chengiz wrote:
| 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.
| cobertos wrote:
| Geometrically, this happens when alpha is 45. The two lines in
| the diagram from the article will be parallel and never
| converge. 2*alpha = beta+alpha
|
| I was waiting for them to break this out into a special case or
| something but the article never did. Can't find any other
| material on this proof that mentions it
| scotty79 wrote:
| 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.
| cobertos wrote:
| 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
| [deleted]
| no_butterscotch wrote:
| [flagged]
| pugworthy wrote:
| I understand your comment, though I have say that if not for
| the hinted identity (even the use of the word "teenagers"),
| many would just pre-judge and assume "white guys" as the
| authors.
| jackmott wrote:
| [dead]
| [deleted]
| Mtinie wrote:
| (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 wrote:
| Approaching 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.
| krick wrote:
| > doubtful that this isn't a "cause celebre" sort of situation
|
| I mean, honestly... it just is. Is it a new proof? I believe
| that it is. Is it any good? Yeah, sure, it's pretty clever. Is
| it really an all-over-the-news special kind of achievement? No.
| It simply wouldn't be all over the news if not for... uh, non-
| mathematical reasons.
|
| Allegedly it's special, because it's trigonometric, and
| trigonometric proofs of it are indeed special. But this one
| is... let's say it's "trigonometric with an asterisk". Both
| because of the series, and because it is kinda analogous to a
| known geometric proof. And the first one truly trigonometric
| proof, which was really special, wasn't as much celebrated, as
| this one (but, of course, the internet was a bit quieter place
| back then).
|
| And, yes, I agree that it does a disservice to an achievement.
| And I'm completely positive that it _is_ an achievement no
| matter what -- every new proof is, and it 's especially true
| when the theorem is well-known (and there are very few more
| famous than this one). So, honestly, I'd like if we wouldn't
| have this discussion at all, and could just stick to the
| matter. But the problem is we probably wouldn't even hear of
| it, if not for those "non-mathematical" reasons. Because by
| itself it isn't _that_ huge. I would love if every new proof of
| a famous theorem would be highlighted and celebrated on the
| Internet as much, as this one, but it simply isn 't the case.
|
| It's understandable, so it's neither good nor bad IMO, but I'm
| just saying that your suspicion is definitely correct.
| briandear wrote:
| If it were two kids from New York, would the story be "New
| York teenagers do x?"
|
| The reason the framing rubs me the wrong way is because it
| feeds a narrative that people from New Orleans (regardless of
| race) are somehow novel for doing something fancy. I was born
| in New Orleans so I have some slight offense at the
| implication that the geography is somehow notable. It's like
| "oh wow, even people from some Southern flyover city can do
| some smart stuff too."
|
| However what is interesting about this story is the girls go
| to St Mary's which is a catholic school created for black
| people during the segregation era founded by an the second
| oldest order of American black nuns just after the Civil War.
| The history of the school is fascinating.
|
| Imagine if more kids had the opportunity to go to high
| quality private schools but can't due to financial
| constraints (St Mary's costs about $9k per year which is a
| lot of money for those in Louisiana.)
|
| These girls can write their own ticket now -- I hope they end
| up staying in math and do something extraordinary with their
| lives.
| eyelidlessness wrote:
| > 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.
| concordDance wrote:
| > 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.
|
| Also cultural. Don't underestimate how big a headwind an
| anti-education subculture can be to the people in it!
| Y_Y wrote:
| 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.
| Jarmsy wrote:
| Where is it you think you've seen it before?
| Xenya wrote:
| [flagged]
| GreenWatermelon wrote:
| What isn't this how it works + how exactly does it work?
| notorandit wrote:
| Nice proof? I understand that beauty is in the eyes of those who
| watch.
|
| Maybe i am blind, but i wouldn't call that a beautiful proof.
| sph wrote:
| Who said it's nice? A proof is a proof, beautiful or not.
| slowmotiony wrote:
| The author of the article. You did read it didnt you?
|
| "most beautiful and simplest trigonometric proof we have seen
| to date"
| [deleted]
| joe_the_user wrote:
| 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.
| jonahx wrote:
| 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
| [deleted]
| alex_smart wrote:
| 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.
| cabalamat wrote:
| > 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.
| Tao3300 wrote:
| 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".
| kibwen wrote:
| 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.
| TuringTest wrote:
| _> 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.
| iamerroragent wrote:
| 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.
| cabalamat wrote:
| I find simplicity aesthetically appealing. Obviously some
| other people differ.
| colinsane wrote:
| 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.
| aaronscott wrote:
| 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.
| minusf wrote:
| 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_G
| oldbach%27s_...
| fnordpiglet wrote:
| 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?
| jonahx wrote:
| > 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.
| hn_throwaway_99 wrote:
| 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.
| jonahx wrote:
| 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.
| boycott-israel wrote:
| 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.
| jonahx wrote:
| That is what I was referring to. I wasn't accusing the
| teenagers of anything.
| onos wrote:
| This theorem is essentially the Pythagorean theorem, so
| ... a proof of one is a proof of the other.
| pramodbiligiri wrote:
| 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.
| chx wrote:
| 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.
| [deleted]
| mturmon wrote:
| 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).
| dvt wrote:
| 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?
| SonOfLilit wrote:
| 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)).
| selimthegrim wrote:
| You can also consider the pair of functional equations as
| implicitly using a Taylor expansion
| chx wrote:
| 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.
| deathlight wrote:
| 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.
| doublerebel wrote:
| 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 270deg "triangle" on a sphere)
| silentguy wrote:
| I don't see any Cosine being mentioned in the proof. I don't
| see any circular logic here.
| [deleted]
| scythe wrote:
| 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.
| contravariant wrote:
| 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)
| chr1 wrote:
| 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.
| SamReidHughes wrote:
| 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.
| crdrost wrote:
| 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 = 1/2 sin a sin b 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 a,
| extend it to a new triangle (a,b', c') with angle a+b, then
| make the new triangle with angle b that you stacked on top
| of the original triangle into a right triangle with angle b
| (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 90deg. Drop a dotted line to the x-axis
| from P and you can work out that the dotted line is at
| x=cos a cos b, and its distance to a is sin a sin b.
| Similarly the y-coordinates give sin a cos b + cos a sin b.
|
| As you say, you can do all of this without angles except
| for defining the first triangle with angle a+b, which you
| might not even need... We just need it here for sin(2 a)
| which is something like reflecting the same triangle about
| its hypotenuse?
| alex_smart wrote:
| 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.
| xiphias2 wrote:
| ,,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
| [deleted]
| soegaard wrote:
| 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.
| dendrite9 wrote:
| 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
| coliveira wrote:
| The nice thing about mathematics is that for every true
| statement there are infinitely many proofs. Granted, some are
| just variations of others, but there many ways to reach the
| same point.
| carapace wrote:
| > 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.)
| coliveira wrote:
| So this statement is not true, you cannot prove its value.
| But I understand what you mean, let's just talk about
| provable statements.
| steppi wrote:
| 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.
| hypertexthero wrote:
| 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/
| Tao3300 wrote:
| 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.
| Waterluvian wrote:
| 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.
| mananaysiempre wrote:
| > [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_0
| 8.htm...
|
| [2] http://www.ams.org/notices/201201/rtx120100031p.pdf
| Al0neStar wrote:
| 2023 edition:
| https://press.princeton.edu/books/paperback/9780691242057/th...
| Someone wrote:
| 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 a = b = p /4, and the triangle is isosceles and
| rectangular?
|
| I haven't seen the original text, but this proof may be
| incomplete.
| camjw wrote:
| 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.
| Someone wrote:
| 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?
| camjw wrote:
| 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.
|
| 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=2298a8fd232d5f58
| e8ca,HmUwSqOt6J...
| Someone wrote:
| 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.
| camjw wrote:
| Yes this is essentially a special case where the yellow
| square has side length 0.
| unnouinceput wrote:
| 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.
| Someone wrote:
| > I suggest you do see the original text.
|
| If I could find it, I would have. This discussion mentions ht
| tps://meetings.ams.org/math/spring2023se/meetingapp.cgi/Pa...
| , but I can't find the paper there. Do I overlook something
| on that page?
| yafbum wrote:
| 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...
| kmeisthax wrote:
| 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.
| yafbum wrote:
| That's a good point
| matthewdgreen wrote:
| I think it's excellent that you, a commenter on HN, understand
| that important contributions can come from all parts of
| society. I really wish that understanding was shared more
| broadly across this whole country. Unfortunately I've been
| alive long enough to see that it really is not.
|
| I suspect that more people felt as you do, we wouldn't have so
| many barriers to opportunity for kids. Moreover, the existence
| of those barriers wouldn't be so disproportionately correlated
| to race and place-of-birth.
| samgtx wrote:
| 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.
| yafbum wrote:
| > It's telling that your comment is currently 2nd ranked.
|
| Fwiw i think only because it's relatively recent. Not a lot
| of upvotes currently.
|
| > It comes across generous "there are bright math kids
| everywhere" but really boils down to "don't talk about how
| they're black" and "don't talk about how they're women." And
| finished with "my male friend was disadvantaged, the
| conversation should be about that."
|
| Wow now I think you're reading a lot more into it than what I
| wrote.
|
| > Obviously a large number of the HN crowd agrees with you
| because these types of comments always land at the top of any
| article praising a woman or underrepresented minority for
| their accomplishments. "Why does it matter? We are all
| people."
|
| That's not actually my claim. I do agree that representation
| matters. But I find it condescending when someone's
| accomplishments are only ever mentioned in the same sentence
| as some statistically surprising fact about their identity,
| as if what we were saying here is "not bad _for a X_ ". (And
| fwiw I do find it condescending when I'm a recipient of such
| praise in settings where I'm in the minority.)
| duneisagoodbook wrote:
| well said
| allenu wrote:
| 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.
| harvey9 wrote:
| [flagged]
| noobcoder wrote:
| 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.
| marcusphi wrote:
| I like the infinite triangles construct. Once A and C can be
| constructed from a,b,c, the theorem can be proven by proving A^2
| +c^2 = C^2 with expanding/reducing the equation to 4a^2b^2c^2 -
| 2a^2b^2c^2 = 2a^2b^2c^2, which is true. No sin rule required.
| teo_zero wrote:
| 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.
| nolamark wrote:
| 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.
| [deleted]
| alex_smart wrote:
| 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.
| hgsgm wrote:
| Note that in math "flatly" means "asserted without proof", not
| "incontrovertible".
|
| "Trigonometry" is not formally defined, so Loomis's statement
| is merely tautological.
| dullcrisp wrote:
| Maybe this is my mathematical background, but is that not the
| common meaning of "flatly" stating something?
| aaron695 wrote:
| [dead]
| rvz wrote:
| Well deserved and well done to them on this proof. It is quite
| interesting to see that the AI bros continue to hype and worship
| hallucinating sophists like ChatGPT and GPT 4.
|
| By now we should have already expected that AIs like LLMs are
| able to create unique proofs and new solutions to existing
| unsolved mathematical problems. They still haven't after years of
| hype and not even one single mention of buzzwords like 'LLMs',
| 'AI', 'GPT', etc in this thread. I'll tell you why:
|
| The difference is those teenagers were able to clearly explain
| the process of deriving this proof transparently with the proof
| itself being (and still is) subject to intense scrutiny even by
| experienced mathematicians, going against what was thought to
| have been 'impossible'. Unlike the finest of LLMs and AI models
| which just repeat the same nonsense it has been trained on and
| confidently outputs more nonsense, whilst many celebrate this
| sophistry as a so-called 'breakthrough' even when it cannot
| transparently reason with its own decisions.
|
| It goes without saying that these teenagers are very intelligent
| in mathematics to create this proof as it is not straightforward
| to just 'generate' it, given that it requires an amount of
| creativity AND originality that not even ChatGPT or LLMs in
| general can bullshit it's way around and will still tell you that
| it is impossible.
|
| That proof is the true breakthrough; not magic AI black-boxes
| that spit out nonsense.
| criloz2 wrote:
| 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
| istjohn wrote:
| Probably because it has nothing to do with the article. It's
| just a rant about something entirely unrelated.
| rvz wrote:
| > '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.
| istjohn wrote:
| 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
| rvz wrote:
| 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.
| istjohn wrote:
| 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.
| maxk42 wrote:
| 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.
| ummonk wrote:
| 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.
| nolamark wrote:
| 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.
| hgsgm wrote:
| No, that was a different proof using different trigonometry.
| malshe wrote:
| 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.
| capableweb wrote:
| "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
| anigbrowl wrote:
| 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.
| majormajor wrote:
| 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.
| muststopmyths wrote:
| 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.
| varenc wrote:
| 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.
| azinman2 wrote:
| It also doesn't tell you "how" either. Almost never does
| an article tell you "how" or "why" despite it being the
| first word.
| capableweb wrote:
| > (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.
| satvikpendem wrote:
| HN strips clickbait titles. Same thing for listicles.
| johnfn wrote:
| HN has some sort of algorithm to rewrite post titles to be a
| bit less link-baity.
| eyelidlessness wrote:
| I share your dislike for this title editing, and agree the
| original title is better.
| bblpeter wrote:
| Sorry but that's not new
| [deleted]
| soVeryTired wrote:
| Maybe you could provide us with a reference then?
| threatofrain wrote:
| The article itself says it's not new.
| Jarmsy wrote:
| Where does it say that?
| [deleted]
| Jun8 wrote:
| "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.
| privacyonsec wrote:
| I don't get how this is "new" or "novel" ? it seems to me the law
| of cosine (or al kashi's theorem) we study this in high school in
| France ...
|
| https://en.wikipedia.org/wiki/Law_of_cosines
| maxboone wrote:
| The concept isn't new or novel, the proof is (or might be).
| User23 wrote:
| Dijkstra did something similar when he derived a generalization
| of the theorem[1].
|
| [1]
| https://www.cs.utexas.edu/~EWD/transcriptions/EWD09xx/EWD975...
| lern_too_spel wrote:
| It's the Pythagorean theorem, a special case of the law of
| cosines. What's new is the proof method.
| [deleted]
| starchild_3001 wrote:
| 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
| ummonk wrote:
| 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.
| hgsgm wrote:
| 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)
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