Appreciate all the thoughtful comments! If you're interested in pursuing more mathematics, I recommend checking out Brilliant.org. Brilliant's got an offer for my viewers right now. Head to brilliant.org/polymathematic/ to try everything Brilliant has to offer-free-for a full 30 days. Thanks for watching!
It's most simple to use the law of cosines (LoC) to prove the Pythagorean theorem, if someone wants a purely trigonometric way. LoC itself has a trigonometric proof itself, check the Wikipedia article for it: the proof does not use the sin²x+cos²x=1 identity anywhere, so it is not circular. Why they thought then trigonometric proofs were impossible? Or am I missing something? Someone please enlighten me!
So this still seems circular to me. Using the law of sines when the law of sines relies on the Pythagorean theorem and certainly wouldn’t be proven without it. If sine a equals b/c and sine b = a/c then a = c*sine b and b = c*sine a so a^2 + b^2 = c^2((sine a)^2 + (sine b)^2) so for a^2 + b^2 to be = c^2, that means (sine a)^2 + (sine b)^2 = 1. Instead of calculating it go back to the basics, opposite over hypotenuse. (b/c)^2 + (a/c)^2 = 1. OR multiply both sides by c^2 and you are back with the Pythagorean theorem. The calculus and hurdles just go in a big circle we just don’t see it when you forget how you calculate sine.
The real value that I see in what those students did is that it's a proof that's creative, resourceful, and possibly never thought of before; and that they tried to do something that they thought (mistakenly) mathematicians have considered impossible for 2000 years, and never gave up until they succeeded. Unfortunately, the reporting and public discussion has been almost exclusively around that false claim, instead of the real value of what they did.
@@polymathematic I would want those students to know that a person doesn't have to believe any false stories about what's new in their proof, to see value in it and in what they did.
@@polymathematic I'm going to try to come up with the same proof they did before watching this video or consuming any other media about them to see if I.can independently replicate their work..I mean if they did it, I should be able to without much difficulty right? Thanks for sharing .
@@marcomoreno6748 how did the news come to the assertion that mathematicians have thought it was impossible for 2k years when there's a paper from the early 2000s by a mathematician named Zimba that back then was already considered to be the first fully trigonometric proof of pythagoras? A lot of people reach some assertions that make no sense. It is good to question them. The problem is when you only question half of the nonsense.
Really cool proof. There's 2 general takeaways from this, 1) as scientists we should keep open minds to old problems and tackle them with ingenuity and 2) draw triangles
@@thatweakpowerlifter2515 I guess I will tell my mathematician friends that their B.S. and M.S. degrees are no longer "Bachelors of Science" or "Masters of Science"
@@slothbearanonymous when we say science we are referring to natural science. Mathematics is formal science, like computer science. It doesn't follow the scientific methods.
The infinite sum is typically taught in Calc II, which is a college-level math, especially with the geometric series convergence. But if your high-school offers AP Calc II then I guess it counts.
I'm guessing this "waffle cone" shape can be used to derive the Taylor series expansion for sin(x) or cos(x) using only geometry, too, which is neat. Props to Johnson & Jackson.
I can totally see why nobody had thought to prove it this way yet. This is really complicated, but still elegant. Major props to the students who discovered this.
@@telanis9 The idea that proofs should use weak techniques to prove strong results has been part of mathematics since Euclid, at least. So it's less of my opinion, and more of explaining what is valued in mathematical proofs over the past 2000 years.
@@JeremyOuelletteNH no comment about iyzie's claims in general, but you are misreading. Sledgehammer is presumably talking using calculus-ish tools to prove trig identities. Weak techniques is what SHOULD have been used, which the sledgehammer isn't.
@@ianjohnston379 why, not that hard probably if you are actually trying to solve it, most of the people just did not bother to since there was a good reason to believe that its not possible. proof itself is not that hard to come up with tho
@@ianjohnston379 We need ppl like you (cynics), who are impossible to convince no matter how much proof or evidence are presented. Unfortunately, while their distrust might serve a great purpose to the entire population (or subset thereof), they sometimes also tend to fall for the most common (and least sophisticated) scams at the same time. Quite puzzling.
Greαt insight by those students. Technical point: This proof doesn't work is α=β or a=b since those long lines added will be parallel. Not to worry though since 2α=90 degrees the we have sin(β)=a/c from original triangle and sin(β)=c/(2a) from larger right triangle. So 2a^2=c^2.
Very nice evaluation of the degenerate case! I dealt with it a little lazily: I just programmed it into Desmos to make that impossible :) Not sure how the girls dealt with it, but I'm eager to see once they release their final paper.
This is fantastic. I love any proofs that bring in a geometric series and these students did it twice. Very inspiring. I hope math instructors far and wide share this accomplishment with their students. Having great role models like these two could inspire many people to keep pushing themselves to achieve more than they had thought possible.
How about all the role models that have also reproven this or any other proof? Something that must be kept in mind with anyone doing a reproof is that xe knows what the outcome will be beforehand.
I love it when complex things are done with high school level math. That's one of the main reasons I love special relativity so much -- a kid can get a grip on it as a sophomore in high school. Congratulations to those kids!
Finally, some accurate description of the issue. Because the media grossly misinformed the people of what the students actually achieved and did. They were like "they solved a problem that mathematician couldn't do for 2000 years", etc. Or "First proof of Pythagoras Theorem", etc. Crazy!
Those students are doing really well! It's not until about Calc II that you start doing limits and integrals and only certain highschools will have lectures for that. Kudos to them for thinking of taking things to the limit to breathe life into the sciences.
@@theobserver314 yes Infinite series is Calculus II. You'd have to do algebra I/II/Trig by middle school so by about 10th grade you could complete Calc I/Calc II. Then 11th grade you could start applying the stuff from Calc that you've learned toward things that have not been proven.
I thought this would be fun, but my headache says otherwise. I am glad there are smarter people on this planet than myself, because this breaks my mind.
The biggest thing for looking at proofs, I've realized, is to slow down. Every step in this case is coming at a mile a minute. The girls who wrote the proof didn't do it this fast, and polymathematic didn't understand it and derive it as fast as he did in the video. If you slow each individual step down and don't move on until you full understand it, it makes a _lot_ more sense!
4:00 Consider extending the line with length b That creates a new right triangle with sides c, ca/b, and hypotenuse b+a/b = (a²+b²)/b Note that the hypotenuse also equals c²/b And we're done
The last step, "Note that the hypotenuse also equals c²/b"" comes from the similarity of the original triangle to the big triangle. Although the law of sines can be used, it is not essential; everything results from similar triangles. Nice job, ckq!
Excellent alternate proof!! See my other reply to Polymathmatic asking him to create video of your proof. And you inspired me to make a similar alternate proof (interior construction).
I’m not really skilled at math, but I understood the challenge was to prove the theorem using trigonometry, which had not been done for 2000 years until someone did in 2009 I think? And then these two young women each came up with a trigonometric solution, so now there are 3 after 2000 years of none. Yours seems like a geometric solution to me. Is that not the case?
@@apriljones1381 The other proof isn't trigonometric. It's basically an exact copy of a proof from 1896 by B. F. Yanney and J. A. Calderhead. Also this proof is a variation of a proof by a mathematician named John Arioni and you could take out all the trig and it would still work.
That's such a creative and beautiful proof. And thank you for the video. I was looking for the proof since the news came out, and looking at it, is certainly a great achievement.
I've been waiting for a video like this to drop. Every news outlet that I read had their brains frying over some high school math. Thanks for making the video
The key is that the mathematics involved are not difficult at all, even for me who left college decades ago and barely touched mathematics for years. I could see the ingenuity and elegance in the proof they discovered. The girls did a marvelous job.
@@Khemith_Demon_Hours Don't ignore the riddle here, is a woman with a penis a woman???.....since you're so smart with mathematical equations, why don't you solve this riddle?
Great work by the students! The part of the proof that you were saying was not purely trigonometric (infinite serries) can actually be proved by trigonometry. Consider the triangle made of all of the infinitely many triangles except the first two and apply Sine Law on this triangle. You will be able to find u and (v - c) in terms of a,b,c. The rest is the same.
They literally used an infinite number of non-trigonometric terms to prove a theorem by "only using trigonometry." I knew this story was complete nonsense before I did my five minutes of research to confirm it.
This is a wonderful argument for MOST right triangles where a is not equal to b. The technique of extending the triangle infinitely using a scale factor of a/b technically only works if b > a, as this ensures that the size of subsequent triangles are getting smaller and smaller. Thankfully, if a>b you would just extend in a different direction and use a scale factor of b/a, and you will end up with the same result, so therefore assuming b>a is fine in this regard. However, this technique falls apart if a=b, as a/b (or b/a) would be equal to 1, and the lines formed by the hypotenuses would in fact be parallel, never intersecting to form the larger triangle in question. This is further evidenced later in the proof where the construction has to deal with b^2 - a^2 in a denominator, except if a=b, then we've just divided by 0 and therefore the resulting expression is undefined. Hopefully when the full proof is released, we will see how they have (or haven't) accounted for this special case (specifically by way of the trigonometric ratios). Regardless, as a high school math teacher myself I still have to applaud these students' rare display of ingenuity in developing this approach to at least a partial proof.
Good point, I would suggest that the case a=b is limiting. Using a sequence of triangles with fixed "a", and "bk" sequence such that "bk ->a" from above. This way the resulting sequence of hypothenuses "ck ->c" from above as well. Since for each term in the sequence we have "a^2 + bk^2 = ck^2" with "bk -> a" and "ck ->c", then by continuity of multiplication and addition. we must have "a^2 + b^2 = c^2" as well for the case a=b.
Might be a classic case of "if it doesn't hold at a single point (or a countable set of isolated points) then since everything's continuous it still holds for this case". Useful sometimes in physics, where you divide by a parameter that changes with time which might be zero at particular instants.
What intrigues me the most is the waffle cone, becuase it is a general idea. You can use it to prove the pythagorean theorem, but it might be useful for other things. It is a concept. You can generate waffle cones out of any straight line. Wherever there is a straight line, there is a waffle cone. But it is not just the waffle cone itself. It highlights how geometric problems can be solved using infinite converging series. Any shape that can generate an infinite number of shrinking (and possibly growing) copies of itself, can provide some insight to that shape. And you can do this for any symmetrical 2D object, using the actual waffle cone, i.e. use the side of a dodecagon to create a waffle cone, use the side of a smaller triangle to draw a new dodecagon. We can use the same method for some 3D objects, what about higher dimensions? Can we distinguish between waffleconeable shapes and non-waffleconeable objects, and is that distinction somehow useful?
Very nice proof. I am very impressed that high school students persisted with this multi-step proof. It's a level of maturity that you don't see in most college students these days. Congratulations to CJ and NJ!
Hi! I'm a 19-year-old student from Korea, and I’d like to share my experience. When I first heard about American students discovering a new proof of the Pythagorean theorem using trigonometry through a korean youtuber, I was immediately intrigued. I looked into it further, and that’s when I came across this video. At the time, I was also a high school student, and the idea of trying something similar excited me. I decided to give it a shot and at the end successfully proved the Pythagorean theorem using trigonometry myself. I shared my work with the Korean mathematician youtuber who initially introduced me to the topic, and he helped me refine and publish it. To my amazement, my proof was featured in a Korean math magazine! I want to thank you for uploading this video because it was such a life-changing moment for me
Great explanation of the proof. This proof is more about ratios and algebra. The trig notation, I admit is very useful to declare those ratios. Great job by the students.- very creative solution!
The proof has nothing to do with trigonometry. The trig. functions were just shorthand notions for ratios. That these young ladies did was to find yet another proof of the P.T.
Somebody shared an article about this on twitter and I can’t wrap my head around what I just read, so I came here to try to understand what a noble thing they contributed to humanity. I still don’t understand how any of that make sense, but great job to the two girls who discovered it.
It should be noted that mathematics often contributes nothing to society. It's great to see such things written in the news, because I think it's encouraging. Of course whenever such things are reported in the news, some percentage of the population blames the wrong people. The news is often full of hyperbole. I read a whole book on this Irish student who found a few way to encrypt, and the news was full of hyperbole. She herself, seemed very aware the news media wanted to make too much of it. I loved the book, it was down to earth, and a good story. Doesn't have to be earth shattering to be a good story.
@@michaelbauers8800 uh mathematics is the foundation of all engineering, physics, chemistry, and really everything at all relating to the construction of society at all. If you mean it doesn’t get spotlight, you would be correct. To mean it didn’t contribute anything to society? Look at your phone. Every pixel, code, and even the construction of the phone is all math.
@@harrys2331 and we had every single thing you just listed years and years and years before this “new proof”, supporting that it actually contributes nothing
@@brownie3454 I never even mentioned this new proof. I responded to Michael who stated that mathematics contributes nothing to society and I’m sorry but that’s completely not true.
The infinite series part is an elegant expansion, excellent thinking there. It's one step further than the typical "double angle" construction, well, I guess it's an infinite number of steps further!
The amount of creativity and intelligence needed to find a NEW proof for a theorem as old and well known as the Pythagorean theorem is immense. I hope those two young women get a full ride through college from this!
When I teach trig, we regularly use the theorem to derive a missing ratio, and we also use it in defining the law of cosines. So it is intriguing to see a student use a measure that is based on the theorem to prove the theorem.
@polymathematic I don't think those girls did this. I think someone else did and is letting the girls have the credit for woke reasons. This is why u ain't seeing these girls present. Also this proof is cool... but it ain't ground breaking lol. Yes I am a real mathematician lol. And real mathematicians secretly think something funny is going on
Fantastic proof! Congrats to the two girls. I wonder if Pythagoras himself would have actually liked this proof if you could present it to him? I know Archimedes used techniques that look a lot like summing infinite series over 2000 years ago, but even he came centuries after Pythagoras. I don't know if Pythagoras ever did anything like that, but considering the Pythagoreans were meant to have been horrified by discovering the irrationality of √2, they might have struggled with the idea of infinity and summing infinite series. Would he actually have been proud? He might have a lot of thinking to do before he became satisfied.
A most remarkable tour de force of patience and imagination by Calcea and Ne'Kiya (I would say patience was the larger part). I note that strictly speaking the steps leading to the two long sides as ratio'd over a^2-b^2 fails when a=b , still less of final cancellations of a^2-b^2, is strictly not justified when a=b , the 45 degree special case of the original triangle, for which the 'cone' goes to its u=v = \infty limit with parallel sides. Nevertheless, this is a 'trivial' lacuna because then it is certain that a^2+b^2=2a^2 and the equation on the right already says (for this triangle) that 2a^2=c^2 . Without having seen their proof, I will just assume they set aside this trivial case at the start. As a high-school math teacher myself, I say Kudos to these ladies !
My understanding is that this was only one of the proofs from one of the students, the other had a completely separate proof inscribing the triangle in a circle. Any chance you've gone through that one?? Would love to see it; I've wanted to see their proofs since I heard the news over a year ago, thanks so much for this!
Probably using Thales Theorm. It's interesting because the "waffle cone" is the same thing you get when you divide an obtuse isocles 108° 36° 36° (gnomon) via the acute 36° 72° 72° isoceles (golden) triangle. Idk... Whatever. The fact that Thales and Pythagoras theorms both prove right triangles. But Thales only isoscles and Pythagoras equilateral. Keeps me awake at night. On a side note Euclid uses two examples of Thales Theorm to make a equilateral in the first proof in Elements. In addition to that. Plato pulls this same S with the square grid divided by a diagonal in Meno. Analogy of the Divided Line in Republic VI 529d. And his "platonic solids" in Timæus. (lol "Timaeus" I just think it's funny to spell that blasted book with Æ. 🤭) But yeah ancient philosophy and geometry. Stuff is EXTREMELY interesting. Aristotle's square of opposition is based off of Empodcles's "transition of the elements" diagram. In sum. Philosophy makes Geometry & Geometry makes Philosophy. Seriously downright fascinating. Because before we had computers and graphics cards to do all the work. Philosophers & mathematicians used geometry and diagrams to achieve the same ends.
Awesome! I’m currently taking a Calculus 2 class, and found this proof fascinating. However, I wonder if this works for 45-45-90 triangles: because the infinite series couldn’t be assumed to be convergent if b=a, correct?
that's exactly right. you couldn't use the convergent series for any a greater than or equal to b. the a is greater than b case is easy enough to deal with, because you can always just flip the triangle around. but you have to deal with the a = b case differently. i'm not sure what the new orleans teens did to deal with that case, but i'm looking forward to finding out!
congrats to those 2 students! Love to see original thinking by kids, Id also like to say its great to see a teacher that RECOGNIZES this, actually bothered to understand the presented work and then promoted their students work. That I think is sadly the truly rare thing here. When I was at uni, my self and my lab partner came up with an algorithm for hit detection in 3d space that was an order faster (n squared + n) than the expected result presented in the course (n cubed). We thankfully had one professor who understood what we had done and pushed it out there, the professor who gave out the assignment simply told us we had 'done it wrong' - and we had to fight it because we knew it worked. So kudos to the young ladies! BUT also kudos to the teacher!
Actually, the limit part can be avoided. The shape (triangle) with side lengths u, 2a and v-c is similar to the one with side lengths v-c, 2a²/b and u-(2ac/b). Taking ratios, we get simultaneous linear equations involving u and v which can be easily solved. This was a really great proof!
My favourite proof is where you take 4 equal triangles with hypotenuse c (and sides a, b, a > b) and place them so that the 4 hypotenuses make a square with area c^2 with all triangles inside it, getting a smaller square in the middle, with sides a - b. Putting the area of the big square equal to the 4 triangles plus the small square you get immediately a^2 + b^2 = c^.
You don't actually have to sum the series, just show that it converges for all ab by swapping the letters around, or by using the incantation "wlog". We still have to do a special case for a=b, unless we are physicists when cancelling by infinity is called renormalization (which makes it OK -- see for example the renormalization of the mass of the electron in Quantum Electro Dynamics). For some reason mathematicians are doubtful of this procedure...
Thank you for the detailed explanation. Based on some of the news articles you would think they revolutionized maths as a whole and proven all mathematicians of the last 2000 years wrong so I was a bit sceptical. It's great to hear that maths still stand but they managed to do a really cool thing especially for their age, that deserves to be celebrated without the need for exaggeration😊!
I think the most important thing that Calcea and Ne’Kiya proved is that our high schools kids are capable of so much more that they are currently expecting of themselves.
This proof is certainly a beautiful and unique one, though the significance of it being one of the first "Trig" proofs is more up to debate. The same proof can essentially be completed without the need for trig (and keeping the same logic) by replacing all trig identities with side length ratios instead. After all, this is the original proof of the sin law. It's a few lines of algebra separated from a completely algebraic proof, so I'm not fully convinced it should be considered a trig proof. Jason Zimba's proof does use deeper trig realizations, but at the end of the day, the lines of what "method" a proof uses can become blurry.
hey i think i found a problem in this proof. in 8:20 you use the formula for infinite series, that only works when the common ratio's value is lower than 1. if this is proof for the pythagorean theorem then who says a^2/b^2 is less than 1? this then excludes the posibility of what happens when both sides are equal (if they are then the sum is undefined).
Sir, you gained a subscriber. Congrats to Mmes. Johnson and Jackson. A superb feat. The Greeks of Old would have adored this. Very geometric. Although Zeno might have had something to object For those who don't know: look up Zeno's paradox.
I admire the persistence of these students. I wish I'd have had students like this more often when I was active. However, this is just a very involved way of getting the theorem. It had to come out in the end, of course, simply because it is true. The problem with the approach is the following: If you use the definition of sin(\alpha) as a/c in any other triangle than the unit one (c=1), you are using similarity. I.e., triangles with equal angles have equal proportions between their sides. But there are very easy proofs of the theorem using similarity only. So the computation shown here is based on facts which yield the theorem in a much easier way.
I saw a similar explanation of the proof in another site. You have b^2 -a^2 in the denominator which will not work for an isosceles triangle. In that website, he made a separate case for this and proved it trivially using sin 45 degrees = 1/sqrt(2), which of course depends on Pythagoras in a circular fashion
It happened but I still find it hard to believe that no one "discovered" this before them. Others have definitely tried different methods so I don't see how no one uncovered this before the year 2023.
There is an entire book of proofs of the Pythagorean theorem, so if something like this was known it would likely be included for its novelty. That doesn't mean nobody ever thought of it without sharing, but we can say that about most mathematical results. The underlying reason this is new is probably that the use of an infinite series is not a very geometric type of argument, and most people who want to prove the Pythagorean theorem want to limit themselves to geometric arguments.
They did not say it had not been done before but there is no record of it since the book containing all Pythagorean proofs did not contain it and went further to say it would not be possible to use trigonometry to prove without recourse to cyclical basic trig identity for a right angle triangle which would be wrong. All they did is prove that notion wrong, and no more. I can see that a lot of derisive comments are directed at conclusions which are at variance with what was reported for reasons best known to them.
The construction of this proof is clever! Demonstrates how mathematics is done. You play around with an idea long enough and with a proper understanding of the fundamentals and some techniques, you can begin to make those leaps of insight and arrive a wonderful result -- or proof in this case.
And do a lot of borrowing like these girls did. Jackson was so smart her proof was merely an exact copy of proof 60 from cut-the-knot by B. F. Yanney and J. A. Calderhead. Genius.
This is a beautiful proof! mostly because it only uses what high-school students know and can play with. No "Higher math mind set" needed. It is direct and as far as I can see doesn't use any "dirty tricks". The building of infinite set of triangles in imaginative and ingenious, The use of sum of infinite series is great. Coming to think about it... when you PROVE the sum of those infinite series, don't you, somewhere, rely on Pythagoras theorem? that needs to be checked - or the proof is circular.
After seeing 60 minutes and other articles, I felt like taking a shower. Thank you for restoring the balance and giving credit where credit is due. These girls did good job.
Hi Polymathmatic. Please consider creating another video with 2 alternate proofs inspired by your video: 1) CK offered his alternate 7 days ago (search "4:00" that he referenced as starting point). His is much more elegant (simpler, direct and includes the problematic isosceles right triangle case) requiring only 2 external line segments. 2) And he inspired me to do a similar proof but "internal construction." Simply add a line segment in the abc right triangle connecting the right-angle-vertex perpendicular to the hypotenuse. This splits the right angle into alpha and beta angles; call the subsegment of c across from new alpha as x and the subsegment of c across from the new beta as y. Using only triangle proportionality: a/c = x/a thus x = (a^2)/c and b/c = y/b thus y = (b^2)/c. Then substitute into x + y = c to get [(a^2) + (b^2)]/c = c to directly prove a^2 + b^2 = c^2. Skipped 2 obvious steps to fit here. Kudos to the 2 students for their proof!!! Regards...Al Waller
UPDATE: these 2 alternate proofs are versions of well-known algebraic proofs. The high school students contend that their new proof is trigonometric but seems to me that it is also fundamentally algebraic (just 10 times harder than needed); we'll see what the AMS rules.
Very impressive to view things in such a dynamic way (never mind still in high school) that I despite my love of maths still wouldn't have came up with that with pointed questions. Didn't realise that there was a belief that trig couldn't simply be used to prove pythagora's theorem. I believe I have one (which I think is fairly simple) and sure avoided inherit cyclic proof.
The first few seconds finally explained what this was all about. Most of the mainstream media was saying a "Proof of the Pythagorean Theorem without using Trigonometry", which is basically all the proofs I know, so I was really confused about what was special.
thank you! If it moves, it's animated in an online graphing calculator called Desmos. If it's my writing, I screen-record my iPad while I'm using an app called Goodnotes with an Apple Pencil.
I am going to venture to say that the last page of the screenshot document shows a construction that will be used to determine the value of sin(beta - alpha), and possibly also sin(2*alpha).
i was a little curious about that slide myself, since I didn't end up needing that altitude. Their approach could turn out to be quite a bit different depending on what they were doing with that.
Thanks for breaking it down! These girls are awesome and their proof is so creative! I haven't really gotten their story yet, not that it's very important, but does anyone know if they developed the solution independently on the same bonus question or was it a pair-work situation wherein they created this new proof?
I like to think that despite involving an infinite series it’s still trig (you do get to see why Greece never arrived at this, which is cool. Newton would have been proud).
Wouldn't this proof technique fail for a 45-45-90 right triangle? For any other right triangle you can assign your alpha and beta, without loss of generality, such that (2 * alpha) is less than 90 degrees, leading to the convergent waffle cone shape. But in a 45-45-90 right triangle, (2 * alpha) will itself be 90 degrees, as well as the measure of (alpha + beta), so instead of a waffle cone, you'd get an infinitely-long rectangle
yes, you'd have to demonstrate the a = b case separately. i'm told they used four different proofs in their paper, so it's possible one of those other proofs addressed that case, but i don't know.
Wow! Amazing! I forgotten a how to simplify infinite series, but this is a amazing for high schools students to undertake such a task! The school must have a great math teacher !
I would have eaten this explanation back in high school when I was a math wiz, but now at 70 my brain hurts. Well done to the two students, this will look great on their résumé’s!
Wow, very interesting proof. I didn't follow the infinite series part because I forgot most of what I learned at school lol. But everything else surely makes sense. Good job to the students! 👏
When a=b, u and v are parallel and it break down, so it requires a separate approach. Also want to add that existence of the right triangle with sides u and v is non-trivial and requires a proof. The proof that I could come up relies on the continuity of the distance function and that relies on the triangle inequality. This is really tricky because you need to prove the triangle inequality without Pythagorean theorem. The standard proof relies on Cauchy-Schwarz which as far as I know relies on Pythagorean Theorem.
I have a question, the convergence of the geometric series is only possible if a^2/b^2 < 1, what happens when that ratio is equal to 1, that would be the case of alpha and beta equal to 45 degrees
It is not so much trigonometric since the sin and cos functions are relationships of the sides of the angles and applying them is also the same as applying the geometry rules for similar triangles. For example, we can use a similar proof that uses the sin(alpha) and sin(beta), by drawing the baseline from c to the right angled corner and work with the height h of that baseline, the height h can be derived using the ratio's of the sides of the triangles, or using the angles. You have the equivalent pairs sin(beta) = h/a = b/c and sin(alpha) = h/b = a/c, but we can also use the two little triangles created by the baseline that splits up the main triangle to get to c = h*(a/b)+h*(b/a). From those equations you can derive c^2 = a^2 + b^2. Or at least, if the proof in the video is trigonometric then the proof in the alinea above should be at least as much trigonometric.
Actually we were all rejects from different clicks in high school and we sat in the middle table because we dominated it that shit no one wanted to sit with us so we were like fuck it let's sit in the middle most of us had TI calculators and so we would share like our different ways of expressing things in basic for the xylog z80 some of us had TI-85s and we would share you know strings you know machine code I had the little plug to plug thing so we can share stuff and I had the little USB thingy so I could download stuff and then share it away to other people which was a unique thing back in like the late '90s but yeah but each one of us was a reject from another group but we all found each other and we decided fuck it if we're going to be outcasts we're going to sit in the center of the goddamn lunch room no one's going to want to sit with us so we're going to have the whole middle to ourselves
thank you! i record myself on an iphone at the same time that i screen-record my ipad using goodnotes. then i composite the video together in final cut pro.
Another way to interpret this proof is that it gives a geometric interpretation of the construction of new pythagorean triples from old ones. The usual formula to generate Pythagorean triples over the integers is (k(m^2-n^2), 2kmn, k(m^2+n^2)), for arbitrary 0
This is an awesome proof, but I don’t think you need to use the law of sines at all to prove it. You can draw the altitude from the vertex of the angle beta to one of the sides of length c in the reflected triangle, which by definition is equal to c * sin 2alpha, which for brevity I will call h. But the area of that triangle can also be expressed two ways: either as (2ab)/2, or as hc/2. So h = 2ab/c as expected, and no law of sines required. Similar argument applies to the big triangle: its area is either cu/2, or hv/2, so h = cu/v. Of course one could argue that this is exactly how you would prove the law of sines in the first place, but I really think that the crux of this proof is in the use of the sum of the geometric series, not in the law of sines! The a=b case is trivial because it can be proved by arranging four of the same right triangles in a big square, with 2a being the diagonal. The big square’s are is either c^2, or 4(ab/2) = a^2 + b^2. So no infinite series needed.
Here, the result (identity) sin(2x)=2*sinx*cosx is used. But this result can be proved from the result of sin(x+y). Now sin(x+y) can be proved from cos(x+y). cos(x+y) can be proved from cos(x-y). & cos(x-y) can be proved from distance formula between two points in R². & Distance formula is derived from the Pythagoras theorem. So ultimately the result sin(2x)=2*sinx*cosx can be proved by using the Pythagoras theorem. So, according to me, this proof is invalid. Note that any trigonometric identity cannot be proved without using Pythagoras theorem. However the highschool students don't know the proof of the result sin(2x)=2*sinx*cosx. So, we should appreciate them & to motivate them for new result.
Here is the geometric series part of their "waffle cone" except it predates their proof: Do an internet search for: "math.stackexchange Is this series representation of the hypotenuse symmetric with respect to the sides of a right triangle?"
chapeau to the two young students, this was a beautiful journey. Isn't it mindblowing that thousands of years later we can still get interesting stuff from one of the most known cornerstones of elementary mathematics?
And for those of you that like geometric proofs, it is said an ancient Greek, whose name is somewhere in my hard drive, during the Hellenistic era actually derived the calculus, before Leibnitz (and Newton). A nice book along the lines of forgotten inventions is: Russo, "The Forgotten Revolution".
The interesting thing here is that law of sines doesn't really depends upon Pythagorean Theorem, which makes this proof technically unique in its own right.
For the sake of completeness, it should be mentioned somewhere that b>a, aka that by definition b is choosen to be the longest side length. This is necessary afterwards so that the series converge. From this side remark one can notice that there is a small issue with the case a=b (the waffle cone becomes an infinite strip). Of course it is quite trivial to complement with a proof for this special case.
These highstudents were insane for coming up with this thing. But my professor put this on my Calc 2 exam, and I instantly got like depression by looking at it.
It's a variation of John Arioni's proof at cut-the-knot. Also, I seriously doubt a girl who never had calculus before came up with a calculus proof all on her own without a lot of help. The trig part of their proof is completely unnecessary as the Law of Sines is just another way of stating the hypotenuse and can be used with any proof of the Pythagorean theorem.
Appreciate all the thoughtful comments! If you're interested in pursuing more mathematics, I recommend checking out Brilliant.org. Brilliant's got an offer for my viewers right now. Head to brilliant.org/polymathematic/ to try everything Brilliant has to offer-free-for a full 30 days. Thanks for watching!
It's most simple to use the law of cosines (LoC) to prove the Pythagorean theorem, if someone wants a purely trigonometric way. LoC itself has a trigonometric proof itself, check the Wikipedia article for it: the proof does not use the sin²x+cos²x=1 identity anywhere, so it is not circular.
Why they thought then trigonometric proofs were impossible? Or am I missing something? Someone please enlighten me!
ua-cam.com/video/ixgAQgmYoHI/v-deo.htmlsi=O3wU7qznu0AXP9Kl
Посмотрите тоже на это видео ролик
So this still seems circular to me. Using the law of sines when the law of sines relies on the Pythagorean theorem and certainly wouldn’t be proven without it. If sine a equals b/c and sine b = a/c then a = c*sine b and b = c*sine a so a^2 + b^2 = c^2((sine a)^2 + (sine b)^2) so for a^2 + b^2 to be = c^2, that means (sine a)^2 + (sine b)^2 = 1. Instead of calculating it go back to the basics, opposite over hypotenuse. (b/c)^2 + (a/c)^2 = 1. OR multiply both sides by c^2 and you are back with the Pythagorean theorem. The calculus and hurdles just go in a big circle we just don’t see it when you forget how you calculate sine.
The real value that I see in what those students did is that it's a proof that's creative, resourceful, and possibly never thought of before; and that they tried to do something that they thought (mistakenly) mathematicians have considered impossible for 2000 years, and never gave up until they succeeded. Unfortunately, the reporting and public discussion has been almost exclusively around that false claim, instead of the real value of what they did.
Totally agree. The news coverage definitely got out of hand. One story I read said mathematicians thought it was true, but it had never been proven!
@@polymathematic I would want those students to know that a person doesn't have to believe any false stories about what's new in their proof, to see value in it and in what they did.
@@polymathematic I'm going to try to come up with the same proof they did before watching this video or consuming any other media about them to see if I.can independently replicate their work..I mean if they did it, I should be able to without much difficulty right? Thanks for sharing .
@@leif1075"...i should be able to do it without MUCH difficulty right?"
How did you come to that assertion?
@@marcomoreno6748 how did the news come to the assertion that mathematicians have thought it was impossible for 2k years when there's a paper from the early 2000s by a mathematician named Zimba that back then was already considered to be the first fully trigonometric proof of pythagoras?
A lot of people reach some assertions that make no sense. It is good to question them. The problem is when you only question half of the nonsense.
Really cool proof. There's 2 general takeaways from this, 1) as scientists we should keep open minds to old problems and tackle them with ingenuity and 2) draw triangles
Scientists? Math is not science.
@@thatweakpowerlifter2515 I guess I will tell my mathematician friends that their B.S. and M.S. degrees are no longer "Bachelors of Science" or "Masters of Science"
Haha, how do we do this to come up with quantum gravity that fits with current observations of the universe?
@@thatweakpowerlifter2515 I literally have a Bachelor Degree of Science in Mathematics
@@slothbearanonymous when we say science we are referring to natural science.
Mathematics is formal science, like computer science.
It doesn't follow the scientific methods.
I love that while all the math here is high school level, it's still creative.
Bro i always double take seeing you in yt comments and not on disc 💀💀
hi mr arthur
We didn't do series in highschool
The infinite sum is typically taught in Calc II, which is a college-level math, especially with the geometric series convergence. But if your high-school offers AP Calc II then I guess it counts.
@@mitchratka3661 I mean I took it extensively starting from 10th grade, but I do believe they got help, with the “waffle cone” idea I presume
I'm guessing this "waffle cone" shape can be used to derive the Taylor series expansion for sin(x) or cos(x) using only geometry, too, which is neat. Props to Johnson & Jackson.
Was thinking just that. This infinite right triangle cascade can be a seed for series expansions of many trigonometry expression.
A series os even and odd powers speaks sin and cos!! I was thinking the same
Props to Calcea and Ne'Kiya. Use their first names, in order to hopefully show to young girls that they, too, can make history in maths.
Well...using geometry and a limit anyway. Calculus is such a handy thing.😊
I wonder if their proof can also lead to another proof of Fermat's theorem. It might reveal new insights to help find a proof.
I can totally see why nobody had thought to prove it this way yet. This is really complicated, but still elegant. Major props to the students who discovered this.
Nobody has thought to use a sledgehammer to kill a fly either, as effective as it would be
@@telanis9 The idea that proofs should use weak techniques to prove strong results has been part of mathematics since Euclid, at least. So it's less of my opinion, and more of explaining what is valued in mathematical proofs over the past 2000 years.
@@iyziejane first its a "sledgehammer", now it's "weak". Sounds like someone's just feeling a bit jelly 🥹🤣
@@JeremyOuelletteNH I don't care about the girls, I just hate the media and the way they lie about everything.
@@JeremyOuelletteNH no comment about iyzie's claims in general, but you are misreading. Sledgehammer is presumably talking using calculus-ish tools to prove trig identities. Weak techniques is what SHOULD have been used, which the sledgehammer isn't.
Brilliant! I’m amazed that two high school students came up with something so creative! It’s a wonderful achievement!
Yeah, I don't buy it.
@@ianjohnston379 why, not that hard probably if you are actually trying to solve it, most of the people just did not bother to since there was a good reason to believe that its not possible. proof itself is not that hard to come up with tho
@@ianjohnston379 there’s always someone pathetic like you commenting
@@ianjohnston379 We need ppl like you (cynics), who are impossible to convince no matter how much proof or evidence are presented. Unfortunately, while their distrust might serve a great purpose to the entire population (or subset thereof), they sometimes also tend to fall for the most common (and least sophisticated) scams at the same time. Quite puzzling.
@@tinsalopek7740 I believe the “waffle cone” was an addition of a helping teacher
Greαt insight by those students. Technical point: This proof doesn't work is α=β or a=b since those long lines added will be parallel. Not to worry though since 2α=90 degrees the we have sin(β)=a/c from original triangle and sin(β)=c/(2a) from larger right triangle. So 2a^2=c^2.
Very nice evaluation of the degenerate case! I dealt with it a little lazily: I just programmed it into Desmos to make that impossible :) Not sure how the girls dealt with it, but I'm eager to see once they release their final paper.
Right angles with side a less than side b appear to be the preferred case in the proof(s).
Great observation, but probably the two stunning girls must have seen that.
@@jaimeabs I assume so, also.
@@polymathematic I don't think they did deal with it. It's not a proof.
This is fantastic. I love any proofs that bring in a geometric series and these students did it twice. Very inspiring. I hope math instructors far and wide share this accomplishment with their students. Having great role models like these two could inspire many people to keep pushing themselves to achieve more than they had thought possible.
Yeah, but how many of those will achieve enough to make the peer of this in whatever field they go at, even with the exact same diligence?
I’m gonna show my students today
How about all the role models that have also reproven this or any other proof? Something that must be kept in mind with anyone doing a reproof is that xe knows what the outcome will be beforehand.
Surprisingly, it was easier to solve the Pythagorean theorem problem than to prove that a woman with a penis is not a woman!!
I love it when complex things are done with high school level math. That's one of the main reasons I love special relativity so much -- a kid can get a grip on it as a sophomore in high school. Congratulations to those kids!
@@heisenberg4996 Calculus is taught at many high schools, albeit, as an AP class (Calc AB/BC).
President Grover Cleveland came up with a reproof, and I'm fairly certain that he only had a basic high-school mathematics background.
Surprisingly, it was easier to solve the Pythagorean theorem problem than to prove that a woman with a penis is not a woman!!
@@signumcrucis71 Some kids perhaps - not this one, still congratulations both the presentation and the authors of the proof.
This honestly might be one of my favorite proofs ever; it's just so clever and thoughtful. What a wonderful thing.
Finally, some accurate description of the issue. Because the media grossly misinformed the people of what the students actually achieved and did. They were like "they solved a problem that mathematician couldn't do for 2000 years", etc. Or "First proof of Pythagoras Theorem", etc. Crazy!
More likely it was solved long ago but the information was lost.
@@ramdom_assortment like the fast Fourier transform
Its frustrating, but what do you expect? Most folks have a hard enough time with sales tax.
@@ramdom_assortment What was solved exactly?
They still did more than you'll ever do. Maybe the media will get that part right.
Those students are doing really well! It's not until about Calc II that you start doing limits and integrals and only certain highschools will have lectures for that. Kudos to them for thinking of taking things to the limit to breathe life into the sciences.
Limits are taught in Calculus 1. Infinite series on the other hand is taught in Calculus 2. But I suppose it depends on the curriculum of the school.
@@theobserver314 yes Infinite series is Calculus II. You'd have to do algebra I/II/Trig by middle school so by about 10th grade you could complete Calc I/Calc II. Then 11th grade you could start applying the stuff from Calc that you've learned toward things that have not been proven.
They didn't, can't you tell? They were coached.
I thought this would be fun, but my headache says otherwise. I am glad there are smarter people on this planet than myself, because this breaks my mind.
ha! fair enough :)
The biggest thing for looking at proofs, I've realized, is to slow down. Every step in this case is coming at a mile a minute. The girls who wrote the proof didn't do it this fast, and polymathematic didn't understand it and derive it as fast as he did in the video. If you slow each individual step down and don't move on until you full understand it, it makes a _lot_ more sense!
@@polymathematic Nice video! I was wondering what tablet and software you are using for writing in this video?
@@regroff I'm also looking forward to hearing the answer to this important question of yours for those who teach.
I think all of us know about that headache. I worked on a problem for 7 days, about a year ago & I was exhausted for 2 months.
4:00
Consider extending the line with length b
That creates a new right triangle with sides
c, ca/b, and hypotenuse b+a/b = (a²+b²)/b
Note that the hypotenuse also equals c²/b
And we're done
The last step, "Note that the hypotenuse also equals c²/b"" comes from the similarity of the original triangle to the big triangle. Although the law of sines can be used, it is not essential; everything results from similar triangles. Nice job, ckq!
Excellent alternate proof!! See my other reply to Polymathmatic asking him to create video of your proof. And you inspired me to make a similar alternate proof (interior construction).
WOW, Fantastic Awesome Proof!!
I’m not really skilled at math, but I understood the challenge was to prove the theorem using trigonometry, which had not been done for 2000 years until someone did in 2009 I think? And then these two young women each came up with a trigonometric solution, so now there are 3 after 2000 years of none. Yours seems like a geometric solution to me. Is that not the case?
@@apriljones1381 The other proof isn't trigonometric. It's basically an exact copy of a proof from 1896 by B. F. Yanney and J. A. Calderhead. Also this proof is a variation of a proof by a mathematician named John Arioni and you could take out all the trig and it would still work.
Thank you for the desmos link :) this was very nice to watch!!! Love the energy and love the math :)
glad you enjoyed it! thanks for watching :)
That's such a creative and beautiful proof. And thank you for the video. I was looking for the proof since the news came out, and looking at it, is certainly a great achievement.
I've been waiting for a video like this to drop. Every news outlet that I read had their brains frying over some high school math. Thanks for making the video
The key is that the mathematics involved are not difficult at all, even for me who left college decades ago and barely touched mathematics for years. I could see the ingenuity and elegance in the proof they discovered. The girls did a marvelous job.
As a math teacher in the New Orleans area, super proud of these girls.
Surprisingly, it was easier to solve the Pythagorean theorem problem than to prove that a woman with a penis is not a woman!!
@@signumcrucis71 How do women come from Rib bones? explain it mathematically.
@@Khemith_Demon_Hours Don't ignore the riddle here, is a woman with a penis a woman???.....since you're so smart with mathematical equations, why don't you solve this riddle?
Great work by the students! The part of the proof that you were saying was not purely trigonometric (infinite serries) can actually be proved by trigonometry. Consider the triangle made of all of the infinitely many triangles except the first two and apply Sine Law on this triangle. You will be able to find u and (v - c) in terms of a,b,c. The rest is the same.
very cool!
They literally used an infinite number of non-trigonometric terms to prove a theorem by "only using trigonometry." I knew this story was complete nonsense before I did my five minutes of research to confirm it.
This is a wonderful argument for MOST right triangles where a is not equal to b. The technique of extending the triangle infinitely using a scale factor of a/b technically only works if b > a, as this ensures that the size of subsequent triangles are getting smaller and smaller. Thankfully, if a>b you would just extend in a different direction and use a scale factor of b/a, and you will end up with the same result, so therefore assuming b>a is fine in this regard. However, this technique falls apart if a=b, as a/b (or b/a) would be equal to 1, and the lines formed by the hypotenuses would in fact be parallel, never intersecting to form the larger triangle in question. This is further evidenced later in the proof where the construction has to deal with b^2 - a^2 in a denominator, except if a=b, then we've just divided by 0 and therefore the resulting expression is undefined.
Hopefully when the full proof is released, we will see how they have (or haven't) accounted for this special case (specifically by way of the trigonometric ratios). Regardless, as a high school math teacher myself I still have to applaud these students' rare display of ingenuity in developing this approach to at least a partial proof.
That case can also be handled by using the Law of Sines. I show how to do it here in my video on the topic. ua-cam.com/video/wuyvdKxXwO8/v-deo.html
Good point, I would suggest that the case a=b is limiting. Using a sequence of triangles with fixed "a", and "bk" sequence such that "bk ->a" from above. This way the resulting sequence of hypothenuses "ck ->c" from above as well. Since for each term in the sequence we have "a^2 + bk^2 = ck^2" with "bk -> a" and "ck ->c", then by continuity of multiplication and addition. we must have "a^2 + b^2 = c^2" as well for the case a=b.
for the case a=b, see the answer from @dkuhlmann above.
Might be a classic case of "if it doesn't hold at a single point (or a countable set of isolated points) then since everything's continuous it still holds for this case". Useful sometimes in physics, where you divide by a parameter that changes with time which might be zero at particular instants.
What intrigues me the most is the waffle cone, becuase it is a general idea. You can use it to prove the pythagorean theorem, but it might be useful for other things. It is a concept.
You can generate waffle cones out of any straight line. Wherever there is a straight line, there is a waffle cone.
But it is not just the waffle cone itself. It highlights how geometric problems can be solved using infinite converging series. Any shape that can generate an infinite number of shrinking (and possibly growing) copies of itself, can provide some insight to that shape. And you can do this for any symmetrical 2D object, using the actual waffle cone, i.e. use the side of a dodecagon to create a waffle cone, use the side of a smaller triangle to draw a new dodecagon.
We can use the same method for some 3D objects, what about higher dimensions?
Can we distinguish between waffleconeable shapes and non-waffleconeable objects, and is that distinction somehow useful?
Surprisingly, it was easier to solve the Pythagorean theorem problem than to prove that a woman with a penis is not a woman!!
@@signumcrucis71 Please take your transphobia back to nazi Germany.
Very nice proof. I am very impressed that high school students persisted with this multi-step proof. It's a level of maturity that you don't see in most college students these days. Congratulations to CJ and NJ!
"these days"... nice. So what did your generation ever do for us?
@@arjankroonen4319 computers, quantum physics, etc
@@arjankroonen4319 A lot more than your sorry TikTok generation.
@@profd65 What about the ones born recently?
@@profd65 "Your sorry TikTok generation" Not sure how old you think I am but I guess you are a few years off...
Hi! I'm a 19-year-old student from Korea, and I’d like to share my experience.
When I first heard about American students discovering a new proof of the Pythagorean theorem using
trigonometry through a korean youtuber, I was immediately intrigued. I looked into it further,
and that’s when I came across this video.
At the time, I was also a high school student, and the idea of trying something similar excited me.
I decided to give it a shot and at the end successfully proved the Pythagorean theorem using trigonometry myself.
I shared my work with the Korean mathematician youtuber who initially introduced me to the topic,
and he helped me refine and publish it.
To my amazement, my proof was featured in a Korean math magazine!
I want to thank you for uploading this video because it was such a life-changing moment for me
Great explanation of the proof. This proof is more about ratios and algebra. The trig notation, I admit is very useful to declare those ratios. Great job by the students.- very creative solution!
I agree, the trig notation is obsolete. It would be more in line with Pythagorean approach not to use it.
The proof has nothing to do with trigonometry. The trig. functions were just shorthand notions for ratios. That these young ladies did was to find yet another proof of the P.T.
Somebody shared an article about this on twitter and I can’t wrap my head around what I just read, so I came here to try to understand what a noble thing they contributed to humanity. I still don’t understand how any of that make sense, but great job to the two girls who discovered it.
it doesn’t actually contribute to society, just more propaganda from Big Girl
It should be noted that mathematics often contributes nothing to society. It's great to see such things written in the news, because I think it's encouraging. Of course whenever such things are reported in the news, some percentage of the population blames the wrong people. The news is often full of hyperbole. I read a whole book on this Irish student who found a few way to encrypt, and the news was full of hyperbole. She herself, seemed very aware the news media wanted to make too much of it. I loved the book, it was down to earth, and a good story. Doesn't have to be earth shattering to be a good story.
@@michaelbauers8800 uh mathematics is the foundation of all engineering, physics, chemistry, and really everything at all relating to the construction of society at all. If you mean it doesn’t get spotlight, you would be correct. To mean it didn’t contribute anything to society? Look at your phone. Every pixel, code, and even the construction of the phone is all math.
@@harrys2331 and we had every single thing you just listed years and years and years before this “new proof”, supporting that it actually contributes nothing
@@brownie3454 I never even mentioned this new proof. I responded to Michael who stated that mathematics contributes nothing to society and I’m sorry but that’s completely not true.
The infinite series part is an elegant expansion, excellent thinking there.
It's one step further than the typical "double angle" construction, well, I guess it's an infinite number of steps further!
The amount of creativity and intelligence needed to find a NEW proof for a theorem as old and well known as the Pythagorean theorem is immense. I hope those two young women get a full ride through college from this!
Hi.
I wonder if they will go to the same college and collaborate
Thanks for a great explanation. The waffle cone was definitely extremely clever and an inspired path to take. Great job ladies, and congratulations !
When I teach trig, we regularly use the theorem to derive a missing ratio, and we also use it in defining the law of cosines. So it is intriguing to see a student use a measure that is based on the theorem to prove the theorem.
awesome proof! great accomplishment for the two high school students
yep! i hope they both had a great experience, and go on to more math conferences in the future :)
@polymathematic I don't think those girls did this. I think someone else did and is letting the girls have the credit for woke reasons. This is why u ain't seeing these girls present. Also this proof is cool... but it ain't ground breaking lol. Yes I am a real mathematician lol. And real mathematicians secretly think something funny is going on
@Zigest whatever you have to tell yourself!
Great proof! I'd also like to add that if we are using the infinite series sum formula "(first term)/1-(common ratio)" then that means (common ratio)
How about when a=b?
Fantastic proof! Congrats to the two girls. I wonder if Pythagoras himself would have actually liked this proof if you could present it to him? I know Archimedes used techniques that look a lot like summing infinite series over 2000 years ago, but even he came centuries after Pythagoras. I don't know if Pythagoras ever did anything like that, but considering the Pythagoreans were meant to have been horrified by discovering the irrationality of √2, they might have struggled with the idea of infinity and summing infinite series. Would he actually have been proud? He might have a lot of thinking to do before he became satisfied.
i presume the infinite series would definitely have been looked down on :)
The square root of two shows itself exactly when a=b which is a degenerate case. It is interesting.
Only if Pythagoras did exist. That is not really a given.
@@caspermadlener4191 true
@@imanplays89 Indeed. And the proof breaks down because of division by zero.
WELL, DONE GIRLS. CONGRATULATIONS TO YOU. I WISH BOTH OF YOU CAN GET THE FIELD MEDAL.
A most remarkable tour de force of patience and imagination by Calcea and Ne'Kiya (I would say patience was the larger part). I note that strictly speaking the steps leading to the two long sides as ratio'd over a^2-b^2 fails when a=b , still less of final cancellations of a^2-b^2, is strictly not justified when a=b , the 45 degree special case of the original triangle, for which the 'cone' goes to its u=v = \infty limit with parallel sides. Nevertheless, this is a 'trivial' lacuna because then it is certain that a^2+b^2=2a^2 and the equation on the right already says (for this triangle) that 2a^2=c^2 . Without having seen their proof, I will just assume they set aside this trivial case at the start. As a high-school math teacher myself, I say Kudos to these ladies !
The fact that highschoolers found this makes me wonder how many times someone did this without realizing that it was so special
Sadly, that's my take on it also.
My understanding is that this was only one of the proofs from one of the students, the other had a completely separate proof inscribing the triangle in a circle. Any chance you've gone through that one?? Would love to see it; I've wanted to see their proofs since I heard the news over a year ago, thanks so much for this!
i've had a number of people comment about that one, but i haven't been able to find it anywhere publicly. i would also love to see it!
Probably using Thales Theorm. It's interesting because the "waffle cone" is the same thing you get when you divide an obtuse isocles 108° 36° 36° (gnomon) via the acute 36° 72° 72° isoceles (golden) triangle.
Idk... Whatever. The fact that Thales and Pythagoras theorms both prove right triangles. But Thales only isoscles and Pythagoras equilateral. Keeps me awake at night.
On a side note Euclid uses two examples of Thales Theorm to make a equilateral in the first proof in Elements.
In addition to that. Plato pulls this same S with the square grid divided by a diagonal in Meno. Analogy of the Divided Line in Republic VI 529d. And his "platonic solids" in Timæus. (lol "Timaeus" I just think it's funny to spell that blasted book with Æ. 🤭)
But yeah ancient philosophy and geometry. Stuff is EXTREMELY interesting. Aristotle's square of opposition is based off of Empodcles's "transition of the elements" diagram.
In sum. Philosophy makes Geometry & Geometry makes Philosophy. Seriously downright fascinating. Because before we had computers and graphics cards to do all the work. Philosophers & mathematicians used geometry and diagrams to achieve the same ends.
Awesome! I’m currently taking a Calculus 2 class, and found this proof fascinating. However, I wonder if this works for 45-45-90 triangles: because the infinite series couldn’t be assumed to be convergent if b=a, correct?
that's exactly right. you couldn't use the convergent series for any a greater than or equal to b. the a is greater than b case is easy enough to deal with, because you can always just flip the triangle around. but you have to deal with the a = b case differently. i'm not sure what the new orleans teens did to deal with that case, but i'm looking forward to finding out!
There is a comment from Dkuhlmann above, answering that exact question.
congrats to those 2 students! Love to see original thinking by kids, Id also like to say its great to see a teacher that RECOGNIZES this, actually bothered to understand the presented work and then promoted their students work. That I think is sadly the truly rare thing here.
When I was at uni, my self and my lab partner came up with an algorithm for hit detection in 3d space that was an order faster (n squared + n) than the expected result presented in the course (n cubed). We thankfully had one professor who understood what we had done and pushed it out there, the professor who gave out the assignment simply told us we had 'done it wrong' - and we had to fight it because we knew it worked.
So kudos to the young ladies!
BUT also kudos to the teacher!
Actually, the limit part can be avoided. The shape (triangle) with side lengths u, 2a and v-c is similar to the one with side lengths v-c, 2a²/b and u-(2ac/b).
Taking ratios, we get simultaneous linear equations involving u and v which can be easily solved.
This was a really great proof!
You can also use the law of sines.
My favourite proof is where you take 4 equal triangles with hypotenuse c (and sides a, b, a > b) and place them so that the 4 hypotenuses make a square with area c^2 with all triangles inside it, getting a smaller square in the middle, with sides a - b. Putting the area of the big square equal to the 4 triangles plus the small square you get immediately a^2 + b^2 = c^.
You don't actually have to sum the series, just show that it converges for all ab by swapping the letters around, or by using the incantation "wlog".
We still have to do a special case for a=b, unless we are physicists when cancelling by infinity is called renormalization (which makes it OK -- see for example the renormalization of the mass of the electron in Quantum Electro Dynamics). For some reason mathematicians are doubtful of this procedure...
Thank you for the detailed explanation. Based on some of the news articles you would think they revolutionized maths as a whole and proven all mathematicians of the last 2000 years wrong so I was a bit sceptical. It's great to hear that maths still stand but they managed to do a really cool thing especially for their age, that deserves to be celebrated without the need for exaggeration😊!
You got scared didn’t you? The relief that breathes through your typed words are telling.
This is gorgeous and creatively motivating! Thanks for explaining, and well done Johnson and Jackson!
I think the most important thing that Calcea and Ne’Kiya proved is that our high schools kids are capable of so much more that they are currently expecting of themselves.
Brilliant & humble young women from St. Mary’s high school in New Orleans! More blessings 🙏
1: Why did I click on this?
2: Why did I watch the whole thing?
3: Why was it so cool?
Elegant! My company logo is actually an implicit proof of the Pythagorean theorem :)
This proof is certainly a beautiful and unique one, though the significance of it being one of the first "Trig" proofs is more up to debate. The same proof can essentially be completed without the need for trig (and keeping the same logic) by replacing all trig identities with side length ratios instead. After all, this is the original proof of the sin law. It's a few lines of algebra separated from a completely algebraic proof, so I'm not fully convinced it should be considered a trig proof. Jason Zimba's proof does use deeper trig realizations, but at the end of the day, the lines of what "method" a proof uses can become blurry.
Thank you! A swarm of praising voices often makes it hard to express a cool headed angle even shroud the brain's excellent intuitions.
hey i think i found a problem in this proof. in 8:20 you use the formula for infinite series, that only works when the common ratio's value is lower than 1. if this is proof for the pythagorean theorem then who says a^2/b^2 is less than 1? this then excludes the posibility of what happens when both sides are equal (if they are then the sum is undefined).
Sir, you gained a subscriber. Congrats to Mmes. Johnson and Jackson. A superb feat. The Greeks of Old would have adored this. Very geometric. Although Zeno might have had something to object
For those who don't know: look up Zeno's paradox.
I don't think these girls came up with this. I think it's a media stunt
Mlles., I think you mean
I admire the persistence of these students. I wish I'd have had students like this more often when I was active. However, this is just a very involved way of getting the theorem. It had to come out in the end, of course, simply because it is true. The problem with the approach is the following: If you use the definition of sin(\alpha) as a/c in any other triangle than the unit one (c=1), you are using similarity. I.e., triangles with equal angles have equal proportions between their sides. But there are very easy proofs of the theorem using similarity only. So the computation shown here is based on facts which yield the theorem in a much easier way.
I like it when it is math that is elegant, creative, and I can follow the whole thing.
Thank you!
I saw a similar explanation of the proof in another site. You have b^2 -a^2 in the denominator which will not work for an isosceles triangle. In that website, he made a separate case for this and proved it trivially using sin 45 degrees = 1/sqrt(2), which of course depends on Pythagoras in a circular fashion
It happened but I still find it hard to believe that no one "discovered" this before them. Others have definitely tried different methods so I don't see how no one uncovered this before the year 2023.
I thought I am the only one asking this to themself. :D
There is an entire book of proofs of the Pythagorean theorem, so if something like this was known it would likely be included for its novelty. That doesn't mean nobody ever thought of it without sharing, but we can say that about most mathematical results. The underlying reason this is new is probably that the use of an infinite series is not a very geometric type of argument, and most people who want to prove the Pythagorean theorem want to limit themselves to geometric arguments.
They did not say it had not been done before but there is no record of it since the book containing all Pythagorean proofs did not contain it and went further to say it would not be possible to use trigonometry to prove without recourse to cyclical basic trig identity for a right angle triangle which would be wrong. All they did is prove that notion wrong, and no more.
I can see that a lot of derisive comments are directed at conclusions which are at variance with what was reported for reasons best known to them.
The construction of this proof is clever! Demonstrates how mathematics is done. You play around with an idea long enough and with a proper understanding of the fundamentals and some techniques, you can begin to make those leaps of insight and arrive a wonderful result -- or proof in this case.
And do a lot of borrowing like these girls did. Jackson was so smart her proof was merely an exact copy of proof 60 from cut-the-knot by B. F. Yanney and J. A. Calderhead. Genius.
This is a beautiful proof! mostly because it only uses what high-school students know and can play with. No "Higher math mind set" needed. It is direct and as far as I can see doesn't use any "dirty tricks". The building of infinite set of triangles in imaginative and ingenious, The use of sum of infinite series is great.
Coming to think about it... when you PROVE the sum of those infinite series, don't you, somewhere, rely on Pythagoras theorem? that needs to be checked - or the proof is circular.
After seeing 60 minutes and other articles, I felt like taking a shower. Thank you for restoring the balance and giving credit where credit is due. These girls did good job.
Hi Polymathmatic. Please consider creating another video with 2 alternate proofs inspired by your video: 1) CK offered his alternate 7 days ago (search "4:00" that he referenced as starting point). His is much more elegant (simpler, direct and includes the problematic isosceles right triangle case) requiring only 2 external line segments. 2) And he inspired me to do a similar proof but "internal construction." Simply add a line segment in the abc right triangle connecting the right-angle-vertex perpendicular to the hypotenuse. This splits the right angle into alpha and beta angles; call the subsegment of c across from new alpha as x and the subsegment of c across from the new beta as y. Using only triangle proportionality: a/c = x/a thus x = (a^2)/c and b/c = y/b thus y = (b^2)/c. Then substitute into x + y = c to get [(a^2) + (b^2)]/c = c to directly prove a^2 + b^2 = c^2. Skipped 2 obvious steps to fit here. Kudos to the 2 students for their proof!!! Regards...Al Waller
UPDATE: these 2 alternate proofs are versions of well-known algebraic proofs. The high school students contend that their new proof is trigonometric but seems to me that it is also fundamentally algebraic (just 10 times harder than needed); we'll see what the AMS rules.
The ladies each came up with their own proof. This was one of the two.
Very impressive to view things in such a dynamic way (never mind still in high school) that I despite my love of maths still wouldn't have came up with that with pointed questions. Didn't realise that there was a belief that trig couldn't simply be used to prove pythagora's theorem. I believe I have one (which I think is fairly simple) and sure avoided inherit cyclic proof.
The first few seconds finally explained what this was all about. Most of the mainstream media was saying a "Proof of the Pythagorean Theorem without using Trigonometry", which is basically all the proofs I know, so I was really confused about what was special.
The moment the waffle cone triangle popped onto the screen with it's series of like triangles; I was like "ooh that's clever." Props to those students
Same😎👍
amazing video....just wanna ask, what kind of tech(the tab and the softeware)were you using on the video to write ur math stuff?!?
thank you! If it moves, it's animated in an online graphing calculator called Desmos. If it's my writing, I screen-record my iPad while I'm using an app called Goodnotes with an Apple Pencil.
I am going to venture to say that the last page of the screenshot document shows a construction that will be used to determine the value of sin(beta - alpha), and possibly also sin(2*alpha).
i was a little curious about that slide myself, since I didn't end up needing that altitude. Their approach could turn out to be quite a bit different depending on what they were doing with that.
@@polymathematic One news story mentioned that the students gave four proofs in their presentation.
Absolutely brilliant work by the two students. I'm so impressed and excited that I can barely think.
Thanks for breaking it down! These girls are awesome and their proof is so creative! I haven't really gotten their story yet, not that it's very important, but does anyone know if they developed the solution independently on the same bonus question or was it a pair-work situation wherein they created this new proof?
actually the other student created another independent proof. watch 60 minutes on their theorems
Wonderful. Congratulations to those two ladies. You have to wonder why this approach was never thought of or attempted before? Congratulations again.
I like to think that despite involving an infinite series it’s still trig (you do get to see why Greece never arrived at this, which is cool. Newton would have been proud).
Kudos to the teacher for recognising what was going on and not just dismiss it out of hand.
Thanks for breaking this down into clear steps.
glad you liked it!
It’s amazing that two high school students came up with that demonstration.
Wouldn't this proof technique fail for a 45-45-90 right triangle?
For any other right triangle you can assign your alpha and beta, without loss of generality, such that (2 * alpha) is less than 90 degrees, leading to the convergent waffle cone shape. But in a 45-45-90 right triangle, (2 * alpha) will itself be 90 degrees, as well as the measure of (alpha + beta), so instead of a waffle cone, you'd get an infinitely-long rectangle
yes, you'd have to demonstrate the a = b case separately. i'm told they used four different proofs in their paper, so it's possible one of those other proofs addressed that case, but i don't know.
Wow! Amazing! I forgotten a how to simplify infinite series, but this is a amazing for high schools students to undertake such a task!
The school must have a great math teacher !
I would have eaten this explanation back in high school when I was a math wiz, but now at 70 my brain hurts. Well done to the two students, this will look great on their résumé’s!
The straightforwardness of the solution procedures is just so wonderful
Wow, very interesting proof. I didn't follow the infinite series part because I forgot most of what I learned at school lol. But everything else surely makes sense. Good job to the students! 👏
When a=b, u and v are parallel and it break down, so it requires a separate approach.
Also want to add that existence of the right triangle with sides u and v is non-trivial and requires a proof. The proof that I could come up relies on the continuity of the distance function and that relies on the triangle inequality. This is really tricky because you need to prove the triangle inequality without Pythagorean theorem. The standard proof relies on Cauchy-Schwarz which as far as I know relies on Pythagorean Theorem.
I have a question, the convergence of the geometric series is only possible if a^2/b^2 < 1, what happens when that ratio is equal to 1, that would be the case of alpha and beta equal to 45 degrees
yes, you have to presume that a
@@polymathematic thanks I had not seen that comment
That was awesome, thank you so much for your explanation, so much better than just reading the news article
It is not so much trigonometric since the sin and cos functions are relationships of the sides of the angles and applying them is also the same as applying the geometry rules for similar triangles.
For example, we can use a similar proof that uses the sin(alpha) and sin(beta), by drawing the baseline from c to the right angled corner and work with the height h of that baseline, the height h can be derived using the ratio's of the sides of the triangles, or using the angles. You have the equivalent pairs sin(beta) = h/a = b/c and sin(alpha) = h/b = a/c, but we can also use the two little triangles created by the baseline that splits up the main triangle to get to c = h*(a/b)+h*(b/a). From those equations you can derive c^2 = a^2 + b^2.
Or at least, if the proof in the video is trigonometric then the proof in the alinea above should be at least as much trigonometric.
See also: en.wikipedia.org/wiki/Pythagorean_theorem#Trigonometric_proof_using_Einstein's_construction
The true Value is the friends made a long the way
These highschoolers must be in some insane math club, I only understood maybe 60% of the entire proof.
Actually we were all rejects from different clicks in high school and we sat in the middle table because we dominated it that shit no one wanted to sit with us so we were like fuck it let's sit in the middle most of us had TI calculators and so we would share like our different ways of expressing things in basic for the xylog z80 some of us had TI-85s and we would share you know strings you know machine code I had the little plug to plug thing so we can share stuff and I had the little USB thingy so I could download stuff and then share it away to other people which was a unique thing back in like the late '90s but yeah but each one of us was a reject from another group but we all found each other and we decided fuck it if we're going to be outcasts we're going to sit in the center of the goddamn lunch room no one's going to want to sit with us so we're going to have the whole middle to ourselves
This is awesome, thanks for the explanation - also I'd love to know what hardware and whiteboard software you use to make your videos?
thank you! i record myself on an iphone at the same time that i screen-record my ipad using goodnotes. then i composite the video together in final cut pro.
Another way to interpret this proof is that it gives a geometric interpretation of the construction of new pythagorean triples from old ones.
The usual formula to generate Pythagorean triples over the integers is (k(m^2-n^2), 2kmn, k(m^2+n^2)), for arbitrary 0
This was on the news. I didn't understand how they did it until now. Thanks for the explanation.
This is an awesome proof, but I don’t think you need to use the law of sines at all to prove it. You can draw the altitude from the vertex of the angle beta to one of the sides of length c in the reflected triangle, which by definition is equal to c * sin 2alpha, which for brevity I will call h. But the area of that triangle can also be expressed two ways: either as (2ab)/2, or as hc/2. So h = 2ab/c as expected, and no law of sines required. Similar argument applies to the big triangle: its area is either cu/2, or hv/2, so h = cu/v. Of course one could argue that this is exactly how you would prove the law of sines in the first place, but I really think that the crux of this proof is in the use of the sum of the geometric series, not in the law of sines!
The a=b case is trivial because it can be proved by arranging four of the same right triangles in a big square, with 2a being the diagonal. The big square’s are is either c^2, or 4(ab/2) = a^2 + b^2. So no infinite series needed.
Here, the result (identity)
sin(2x)=2*sinx*cosx
is used.
But this result can be proved from the result of sin(x+y).
Now sin(x+y) can be proved from cos(x+y).
cos(x+y) can be proved from cos(x-y).
& cos(x-y) can be proved from distance formula between two points in R².
& Distance formula is derived from the Pythagoras theorem.
So ultimately the result
sin(2x)=2*sinx*cosx can be proved by using the Pythagoras theorem.
So, according to me, this proof is invalid.
Note that any trigonometric identity cannot be proved without using Pythagoras theorem.
However the highschool students don't know the proof of the result
sin(2x)=2*sinx*cosx.
So, we should appreciate them & to motivate them for new result.
Just as I thought... this doesn't require ANY trigonometry. The trigonometry can be reduced out of it. This is Geometry and a bit of calculus.
It's just a slightly modified version of John Arioni's proof "Pythagorean Theorem via Geometric Progression"
@@kirkb2665
Would you look at that... you're right
Here is the geometric series part of their "waffle cone" except it predates their proof:
Do an internet search for:
"math.stackexchange Is this series representation of the hypotenuse symmetric with respect to the sides of a right triangle?"
chapeau to the two young students, this was a beautiful journey. Isn't it mindblowing that thousands of years later we can still get interesting stuff from one of the most known cornerstones of elementary mathematics?
It's always neat to see how cool intersections of math branches can be - trigonometry and calculus in this case. Brilliant!
And for those of you that like geometric proofs, it is said an ancient Greek, whose name is somewhere in my hard drive, during the Hellenistic era actually derived the calculus, before Leibnitz (and Newton). A nice book along the lines of forgotten inventions is: Russo, "The Forgotten Revolution".
The Egyptians taught the Greeks it.
@@calicoesblue4703 Greeks invented everything...it's in the movie, My Big Fat Greek Wedding.
@@raylopez99 lol🤣🤣🤣
Ooohhh I love how the girls used different mathematical tools to find another proof of Pythagorean’s theorem❤❤❤❤❤ So much math skills😊😊😊😊
We must be insanely dum to not have found this and now I am thinking what are we missing
The interesting thing here is that law of sines doesn't really depends upon Pythagorean Theorem, which makes this proof technically unique in its own right.
For the sake of completeness, it should be mentioned somewhere that b>a, aka that by definition b is choosen to be the longest side length. This is necessary afterwards so that the series converge. From this side remark one can notice that there is a small issue with the case a=b (the waffle cone becomes an infinite strip). Of course it is quite trivial to complement with a proof for this special case.
Beautiful proof, beautiful and clear presentation!
These highstudents were insane for coming up with this thing. But my professor put this on my Calc 2 exam, and I instantly got like depression by looking at it.
It's a variation of John Arioni's proof at cut-the-knot. Also, I seriously doubt a girl who never had calculus before came up with a calculus proof all on her own without a lot of help. The trig part of their proof is completely unnecessary as the Law of Sines is just another way of stating the hypotenuse and can be used with any proof of the Pythagorean theorem.
Happy to see an explanation. When the story broke awhile back, they never went into the math.