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!
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!
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.
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.
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.
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...
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.
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!
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.
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.
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!
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!
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 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!
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?
@@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 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.
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^.
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 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!
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).
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
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.
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.
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.
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.
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.
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 !
@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
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
Wow, this makes me think I should go back and look at how I found another way to solve matrix translations. It didn't work in specific circumstances (my hs precalc teacher made sure to put those circumstances on the test when I showed her my method) but maybe I could refine my rule on the exceptions and do something with it?
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.
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?
Meta comment: the chalk and talk technique used to guide us along the proof is awesomesauce (the proof clearly is too). Where can I read about what @polymathematic is using for a pen and tablet for the "chalking"?
you're very kind! if it moves, it's animated in Desmos. if it's my writing, it's in the Goodnotes app for iPad. I screen record myself writing and then I composite that behind me in Final Cut Pro.
@@polymathematic Ah, yes. You mention Desmos in the video, but I did not grok it to be a software package. Anyhow it just looks seamlessly well done, so I thought it would be one app for both (sans composition). Thank you for the insight here! The overall flow is what caught my attention, very well done.
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.
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.
Awesome explanation! Thanks a lot for taking the time to do this! I was wondering, what software are you using to write? it seems really easy to manipulate the texts and everything else. TIA
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.
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.
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.
As I understand it, the formula for the convergence of an infinite geometric series requires that the rate be between 0 and 1. This would mean that a/b is less than 1. This seems to be implied by the diagram, given that the infinite series of triangles keeps getting smaller. Is that sufficient?
yes, there's an assumption in this proof that ab, you can just switch the legs around, so it's symmetrically equivalent. for a=b, you can have to consider a separate case, but that's the case of the isosceles right triangle, and it's trivial to show pythagorean theorem using, if i remember correctly, a geometric mean.
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.
For perspective, we usually place the observer as point C. Drawing the peripheral vision as points A and B respectively. However, what if it’s actually the inverse? Bear with me. If you have 2 stakes in the ground, how far away (vertically or laterally) do you have to go until both points appear as 1? Are there any theories that describe this?
so their another solution for this form, and that is u can add the value of tan(a)+tan(b)=a/b+b/a as tan(a)=a/b,sin(a)=a/c,cos(a)=b/c similarly to tan(b) if u can find that next is we can add and convert the lhs to sin cos function sina.cob+sinb.cosa(whole divided by)cosa.cosb=a^2+b^2(whole divided by)ab here u have remember some situation related formulae is angle is lower than 90 degrees i.e sin(a+b)=sina.cosb+sinb.cosa and that angles a+b+90 degree=180 degree so a+b=90 and sin(a+b)=sin90=1 so u can hence write like 1/cosa.cosb=a^2+b^2(whole divided by)ab here u can do some mathematical operations, funny right we will multiply 2 to numerator and denominator respectively to lhs hers 2cosa.cosb can be written as 2sina.cosa why? cause u can find that cosb is equivalent ot sina so we have another formula that is sin2a=2sina.cosa we can replace it and we get 2/sin2a=a^2+b^2(whole divided by)ab here u can see 2ab/sin2a can be founded easy which is equivalent to a^2+b^2=c^2 thank u if u get it.
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.
Impressive, but I have a question regarding the validity of this prove: This prove involves that sin(alpha)/a = sin(beta)/b The prove for this is simple but it uses that sin(alpha) = opposite/hypothenuse AND sin(beta) = opposite/hypothenuse Now in trigonometrie we can define sin(alpha) = opposite / hypothenuse But if we use that well defined function sin then we have to prove that sin(beta) = opposite / hypothenuse As well. I don't know the prove for that and didn't find it online so maybe you could explain. Because if this prove involves the Pythagoras Theorem then this prove would be a circle proof. Non or less impreasive, but I really wanna know
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.
It seems the proof holds if aa when one draws the waffle the other way that is proof through sin(2beta)) for the series to converge but the proof for a=b does not lead to convergence.
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
at 4:00 the waffle triangle will be impossible to construct when alpha = beta because two of it's sides will be parallel. in other words, this proof is incomplete because it doesn't work for isosceles right triangles
for the proof i presented in this video, that's certainly true. i have no idea if the proof the high school students used didn't also consider the a = b case though.
This is definitely true, but since that special case is also very easy, it doesn't take much away from this proof. If a referee noticed this kind of omission, they would just say to include it, they wouldn't be likely to reject the paper...
Wonderful presentation. But I have a question. Law of Sines not depends from Pythagorean theorem? I'm brasilian professor and would like to use the same tools used in you presentation in my classes. What app/program have you played here (thanks in advance).
The Law of sine does not depend on the Pythagorean theorem. It is simply the definition of sin applied to the right-angle triangles built on the attitude of one of the vertices of the original triangle. For a triangle ABC, with side lengths AB=c, BC=a, and AC=b, draw the attitude from A: the line from A, perpendicular to BC. The attitude has length h. By definition, sin(C)=h/b. sin(B)=h/c. Consequently, b*sin(C)=c*sin(B). For visualization, draw the figure for an acute triangle. It's easy to extend to obtuse triangles using sin(Pi-B)=sin(B) when the attitude falls outside of the original triangle.
@@comeraczy2483 This demonstration makes me raise another question: Would it be correct to say that the demonstration presented by the two young girls (without any disrespect) are in fact highlighted because they are trigonometric consequences? Are not all demonstrations somehow related to such functions?
@@jaimeabs I have no idea why there is such a hype around this specific demonstration. I personally don't see anything intrinsically interesting in the proof. Considering that a recurring part of the story is that the proof was produces by two little girls, I suspect that this is an important factor in the mediatization. If that's the case, it might be appalling at many levels, including the possibility of implied gender bias in education, and perhaps levels of innumeracy substantially higher than humanity needs.
Wow! In hindsight, it really makes sense. A right triangle's area is always greater or equal to 2ab, so dividing that by a fraction sort of fits the topology of things. May I offer the observation of a dummy who just didn't get math in high school? My geometry and trig was taught via experimental education without any mention of the unit circle. When I found that on my own, I thought it was a beautiful thing. Identities like sin^2+cos^2 strike me as a little disembodied. I recently worked out a straightedge and compass proof of the law of cosines, and one thing that helped was to remember that identity is incomplete in the real world. I know, it's just me, but I always remember (sin(ϴ)c)^2 + (cos(ϴ)c)^2 = c^2. Only for the special case of c=1 is sin^2+cos^2=1, and at that the units of "1" don't show it's a squared quantity. I know, I'm not the brightest bulb. It's ok.
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😊!
Hello Polymathematic, my question might be dumb but how to demonstrate properly that a^2/b^2< 1 in your geometric serie. It seems obvious since it is a right triangle, but how to demonstrate that in a right triangle the hypothenuse is the longer side without actually using pythagorean theorem.
not dumb at all! so, this proof will only work if the geometric series converges, which as you say, requires that a² is less than b² (or simply that a is less than b). If a is greater than b, of course, you could just switch them around, so that case is no problem. where you do run into the problem is if a equals b. but in that case, the original right triangle can be turned into a self-similar isosceles right triangle, and you can get to 2a² = c² in the way that another commenter did below. hope that helps!
It is possible to prove that for any triangle, the side opposite the largest angle is the largest side. This can be done without pythagoras. Since the sum of angles in a triangle is 180 degrees (provable from the parallel postulate), this is all we need.
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!
i think as a concept, infinite series fit within calculus, but yes, students will encounter them in different classes. for example, i actually teach the formula for converging geometric series to my 8th graders as part of an algebra class. but they're not getting the rigorous background, just the formula.
WHen I was bored in year 10 math I stumbled upon a like 3 step proof of pythag with trig, but it wasn't really a proof since I used sin^2x+cos^2x=1, so I used pythag to prove pythag. But isn't this the fundamental basis of trigonometry? Are all pythag proofs based in trig just circular proofs or did these kids somehow find a way around this?
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.
2:44 If the first triangle is a right isosceles triangle then the second one is similar. Now that I watch farther I see that in this case the "waffle cone" forms a rectangle, and you'd end up with infinitely long u and v. I see other commenters have addressed this special case.
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!
I don't understand why you can't just use the double angle formula for the sine to work this out directly. After all, the big triangle is a right triangle (since α + β = π/2), so we have u/v = sin 2α = 2 sin α cos α = 2ab/c². Similarly, we have c/v = cos 2α = cos² α − sin² α = (b²−a²)/c². Solving these for u and v gives u = 2abc/(b²−a²) and v = c³/(b²−a²). (Note that a ≠ b because α ≠ β in the construction. If we had α = β, then they would both have to measure π/4, making the angle at the top π/2, a right angle. But the other angle in the big triangle is also right, which means the top and bottom lines would not intersect, so the diagram would be incorrect. This proof does not work for isosceles right triangles, but for any other right triangle, we simply take α < β wlog, and so a < b.) Then we use the law of sines on the "waffle cone" part of the diagram, i.e. the part of the triangle excluding the two original reflected triangles. This gives (sin α)/(v−c) = (cos 2α)/(2a), using cos x = sin(π/2−x). Substituting the above expressions in gives (a/c)/[c³/(b²−a²)−c] = [(b²−a²)/c²]/(2a), which simplifies to a² + b² = c². This relies only on the law of sines and the double angle formulas, neither of which depends on the Pythaogrean Theorem. In fact, they can all be demonstrated directly from the definitions of sine and cosine and some very elementary geometric theorems like the congruence of vertical angles and that acute angles in a right triangle are complementary. In particular, the proof in the video is _not valid_ over the constructible numbers or other geometries where limit points are not guaranteed to exist. It just seems like it requires a lot of extra machinery for what is ultimately a very simple derivation.
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! 👏
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).
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.
How does the math account for the absence of either one of the triangles of a pair, since each expansion requires 2 new triangles after the first, at the end of the cone? Fascinating concept, but ultimately this equation should be able to determine the smallest viable triangle with positive values, however there should be a final triangle that sits along both negative and positive values. Is that fact of non importance? Or perhaps is there more importance yet to be discovered in the shape resulting in that space?
I was paying attention at the crucial moments when you (should have) pointed out the exact novelty of the new proof. I'm sad to say that none of those statements made sense. What do you mean by a trigonometric proof EXACTLY? What are the axioms of trigonometry that I can use, and just as importantly, what am I not allowed to assume? You mentioned something like the "sum and difference identities" used by Zimba about a decade ago. Do you mean the identity sin(x+y)=sin(x)cos(y)+cos(x)sin(y)? If I"m allowed to use that, the Pythagorean theorem follows trivially in a line, by putting y=pi/2-x: 1=sin(pi/2)=sin(x)cos(pi/2-x)+cos(x)sin(pi/2-x)=sin^2(x)+cos^2(x). The proof you present uses similar triangles. Once again, if we are allowed to use that triangles with the same angles are similar, implying that the ratio of corresponding sides are equal, then there is a trivial proof of the Pythagorean theorem. Namely, draw the altitude corresponding to the hypotenuse. That divides the right triangle into two smaller right triangles, both similar to the original one, with similarity ratios a/c and b/c. Thus the two segments that appear on the hypotenuse are a^2/c and b^2/c, yielding a^2/c+b^2/c=c, that is, a^2+b^2=c^2. So what is the advantage of the new proof?
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.
Is there a reason to prefer your summations start at n=1 instead of n=0? When the n only appears as n-1 it seems legit to start at 0. Disclaimer: am a programmer 😅
Yo this is crazy. The waffle cone is a brilliant idea, and what seems to be the biggest insight here -- though there's still quite a bit of work even outside of that. Fucking brilliant work by those two on this. Wow!
That is a wonderful proof. But I'm a little confused on the algebra when you use the convergence test on sum from n = 1 to n of 2ac/b * (a^2/b^2)^n-1 turning into (2ac/b)/(1-a^2/b^2). I tried different things after doing algebra like bringing the (a^2/b^2)^-1 down to the denominator but I just can't figure out how it turned into a factor of 1-a^2/b^2. Anyone know what I mean?
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
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.
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.
This honestly might be one of my favorite proofs ever; it's just so clever and thoughtful. What a wonderful thing.
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.
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.
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 desmos link :) this was very nice to watch!!! Love the energy and love the math :)
glad you enjoyed it! thanks for watching :)
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.
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.
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.
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.
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.
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.
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 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!
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.
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 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?
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.
Thanks for a great explanation. The waffle cone was definitely extremely clever and an inspired path to take. Great job ladies, and congratulations !
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^.
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 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...
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).
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
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.
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.
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.
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.
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.
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 !
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!
This is gorgeous and creatively motivating! Thanks for explaining, and well done Johnson and Jackson!
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
Wow, this makes me think I should go back and look at how I found another way to solve matrix translations. It didn't work in specific circumstances (my hs precalc teacher made sure to put those circumstances on the test when I showed her my method) but maybe I could refine my rule on the exceptions and do something with it?
I like it when it is math that is elegant, creative, and I can follow the whole thing.
Thank you!
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.
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
Meta comment: the chalk and talk technique used to guide us along the proof is awesomesauce (the proof clearly is too). Where can I read about what @polymathematic is using for a pen and tablet for the "chalking"?
you're very kind! if it moves, it's animated in Desmos. if it's my writing, it's in the Goodnotes app for iPad. I screen record myself writing and then I composite that behind me in Final Cut Pro.
@@polymathematic Ah, yes. You mention Desmos in the video, but I did not grok it to be a software package. Anyhow it just looks seamlessly well done, so I thought it would be one app for both (sans composition). Thank you for the insight here! The overall flow is what caught my attention, very well done.
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.
At 8:14 it is important to note a
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.
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 :)
Awesome explanation! Thanks a lot for taking the time to do this! I was wondering, what software are you using to write? it seems really easy to manipulate the texts and everything else. TIA
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.
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.
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.
As I understand it, the formula for the convergence of an infinite geometric series requires that the rate be between 0 and 1. This would mean that a/b is less than 1. This seems to be implied by the diagram, given that the infinite series of triangles keeps getting smaller. Is that sufficient?
yes, there's an assumption in this proof that ab, you can just switch the legs around, so it's symmetrically equivalent. for a=b, you can have to consider a separate case, but that's the case of the isosceles right triangle, and it's trivial to show pythagorean theorem using, if i remember correctly, a geometric mean.
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
For perspective, we usually place the observer as point C. Drawing the peripheral vision as points A and B respectively. However, what if it’s actually the inverse? Bear with me. If you have 2 stakes in the ground, how far away (vertically or laterally) do you have to go until both points appear as 1? Are there any theories that describe this?
I’m assuming such a theory would be the basis for calculations for microscopy. Useful for determining scale.
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.
so their another solution for this form, and that is u can add the value of
tan(a)+tan(b)=a/b+b/a
as tan(a)=a/b,sin(a)=a/c,cos(a)=b/c
similarly to tan(b) if u can find that
next is we can add and convert the lhs to sin cos function
sina.cob+sinb.cosa(whole divided by)cosa.cosb=a^2+b^2(whole divided by)ab
here u have remember some situation related formulae is angle is lower than 90 degrees
i.e
sin(a+b)=sina.cosb+sinb.cosa
and that angles a+b+90 degree=180 degree
so a+b=90
and sin(a+b)=sin90=1
so u can hence write like
1/cosa.cosb=a^2+b^2(whole divided by)ab
here u can do some mathematical operations, funny right
we will multiply 2 to numerator and denominator respectively to lhs
hers 2cosa.cosb can be written as 2sina.cosa why? cause u can find that cosb is equivalent ot sina
so we have another formula that is sin2a=2sina.cosa
we can replace it and we get
2/sin2a=a^2+b^2(whole divided by)ab
here u can see
2ab/sin2a can be founded easy which is equivalent to a^2+b^2=c^2
thank u if u get it.
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.
Impressive, but I have a question regarding the validity of this prove:
This prove involves that
sin(alpha)/a = sin(beta)/b
The prove for this is simple but it uses that
sin(alpha) = opposite/hypothenuse
AND
sin(beta) = opposite/hypothenuse
Now in trigonometrie we can define
sin(alpha) = opposite / hypothenuse
But if we use that well defined function sin then we have to prove that
sin(beta) = opposite / hypothenuse
As well.
I don't know the prove for that and didn't find it online so maybe you could explain. Because if this prove involves the Pythagoras Theorem then this prove would be a circle proof.
Non or less impreasive, but I really wanna know
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.
It seems the proof holds if aa when one draws the waffle the other way that is proof through sin(2beta)) for the series to converge but the proof for a=b does not lead to convergence.
Applying limits, since for ab result the formula a^2+b^2=c^2, then applying the limit a=b by the two cases, the formula should be the same.
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
at 4:00 the waffle triangle will be impossible to construct when alpha = beta because two of it's sides will be parallel.
in other words, this proof is incomplete because it doesn't work for isosceles right triangles
for the proof i presented in this video, that's certainly true. i have no idea if the proof the high school students used didn't also consider the a = b case though.
This is definitely true, but since that special case is also very easy, it doesn't take much away from this proof. If a referee noticed this kind of omission, they would just say to include it, they wouldn't be likely to reject the paper...
Wonderful presentation. But I have a question. Law of Sines not depends from Pythagorean theorem? I'm brasilian professor and would like to use the same tools used in you presentation in my classes. What app/program have you played here (thanks in advance).
The Law of sine does not depend on the Pythagorean theorem. It is simply the definition of sin applied to the right-angle triangles built on the attitude of one of the vertices of the original triangle. For a triangle ABC, with side lengths AB=c, BC=a, and AC=b, draw the attitude from A: the line from A, perpendicular to BC. The attitude has length h. By definition, sin(C)=h/b. sin(B)=h/c. Consequently, b*sin(C)=c*sin(B). For visualization, draw the figure for an acute triangle. It's easy to extend to obtuse triangles using sin(Pi-B)=sin(B) when the attitude falls outside of the original triangle.
Thanks@@comeraczy2483 , you calculations are very accurate, I had forgotten about how proceed that simple geometric facts.
@@comeraczy2483 This demonstration makes me raise another question: Would it be correct to say that the demonstration presented by the two young girls (without any disrespect) are in fact highlighted because they are trigonometric consequences? Are not all demonstrations somehow related to such functions?
@@jaimeabs I have no idea why there is such a hype around this specific demonstration. I personally don't see anything intrinsically interesting in the proof. Considering that a recurring part of the story is that the proof was produces by two little girls, I suspect that this is an important factor in the mediatization. If that's the case, it might be appalling at many levels, including the possibility of implied gender bias in education, and perhaps levels of innumeracy substantially higher than humanity needs.
@@comeraczy2483 Most people don't publish mathematical proofs in High School, so it happening certainly will lead to media coverage.
Wow! In hindsight, it really makes sense. A right triangle's area is always greater or equal to 2ab, so dividing that by a fraction sort of fits the topology of things.
May I offer the observation of a dummy who just didn't get math in high school? My geometry and trig was taught via experimental education without any mention of the unit circle. When I found that on my own, I thought it was a beautiful thing.
Identities like sin^2+cos^2 strike me as a little disembodied. I recently worked out a straightedge and compass proof of the law of cosines, and one thing that helped was to remember that identity is incomplete in the real world.
I know, it's just me, but I always remember (sin(ϴ)c)^2 + (cos(ϴ)c)^2 = c^2. Only for the special case of c=1 is sin^2+cos^2=1, and at that the units of "1" don't show it's a squared quantity.
I know, I'm not the brightest bulb. It's ok.
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.
Was the 2ab/sin2a = c^2 identity known before this? It seems useful in its own right.
Hello Polymathematic,
my question might be dumb but how to demonstrate properly that a^2/b^2< 1 in your geometric serie.
It seems obvious since it is a right triangle, but how to demonstrate that in a right triangle the hypothenuse is the longer side without actually using pythagorean theorem.
not dumb at all! so, this proof will only work if the geometric series converges, which as you say, requires that a² is less than b² (or simply that a is less than b). If a is greater than b, of course, you could just switch them around, so that case is no problem. where you do run into the problem is if a equals b. but in that case, the original right triangle can be turned into a self-similar isosceles right triangle, and you can get to 2a² = c² in the way that another commenter did below. hope that helps!
It is possible to prove that for any triangle, the side opposite the largest angle is the largest side. This can be done without pythagoras.
Since the sum of angles in a triangle is 180 degrees (provable from the parallel postulate), this is all we need.
Can the Pythagoras triangle be proved by Fourier and/or Laplace transforms in the frequency domain?
i don't know, give it a shot and let me know how it goes!
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
Infinite series is calculus? I know in the US it's taught as part of one of the calculus courses, but I got taught these things as pre-calc
i think as a concept, infinite series fit within calculus, but yes, students will encounter them in different classes. for example, i actually teach the formula for converging geometric series to my 8th graders as part of an algebra class. but they're not getting the rigorous background, just the formula.
It's always neat to see how cool intersections of math branches can be - trigonometry and calculus in this case. Brilliant!
Wonderful. Congratulations to those two ladies. You have to wonder why this approach was never thought of or attempted before? Congratulations again.
WHen I was bored in year 10 math I stumbled upon a like 3 step proof of pythag with trig, but it wasn't really a proof since I used sin^2x+cos^2x=1, so I used pythag to prove pythag. But isn't this the fundamental basis of trigonometry? Are all pythag proofs based in trig just circular proofs or did these kids somehow find a way around this?
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
Absolutely brilliant work by the two students. I'm so impressed and excited that I can barely think.
Brilliant & humble young women from St. Mary’s high school in New Orleans! More blessings 🙏
2:44 If the first triangle is a right isosceles triangle then the second one is similar. Now that I watch farther I see that in this case the "waffle cone" forms a rectangle, and you'd end up with infinitely long u and v. I see other commenters have addressed this special case.
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!
I don't understand why you can't just use the double angle formula for the sine to work this out directly. After all, the big triangle is a right triangle (since α + β = π/2), so we have u/v = sin 2α = 2 sin α cos α = 2ab/c². Similarly, we have c/v = cos 2α = cos² α − sin² α = (b²−a²)/c². Solving these for u and v gives u = 2abc/(b²−a²) and v = c³/(b²−a²). (Note that a ≠ b because α ≠ β in the construction. If we had α = β, then they would both have to measure π/4, making the angle at the top π/2, a right angle. But the other angle in the big triangle is also right, which means the top and bottom lines would not intersect, so the diagram would be incorrect. This proof does not work for isosceles right triangles, but for any other right triangle, we simply take α < β wlog, and so a < b.)
Then we use the law of sines on the "waffle cone" part of the diagram, i.e. the part of the triangle excluding the two original reflected triangles. This gives (sin α)/(v−c) = (cos 2α)/(2a), using cos x = sin(π/2−x). Substituting the above expressions in gives (a/c)/[c³/(b²−a²)−c] = [(b²−a²)/c²]/(2a), which simplifies to a² + b² = c².
This relies only on the law of sines and the double angle formulas, neither of which depends on the Pythaogrean Theorem. In fact, they can all be demonstrated directly from the definitions of sine and cosine and some very elementary geometric theorems like the congruence of vertical angles and that acute angles in a right triangle are complementary.
In particular, the proof in the video is _not valid_ over the constructible numbers or other geometries where limit points are not guaranteed to exist. It just seems like it requires a lot of extra machinery for what is ultimately a very simple derivation.
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! 👏
Thanks for breaking this down into clear steps.
glad you liked it!
@5:31 where did you get the ratio 2a/b for c relative to b?
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).
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.
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😎👍
How does the math account for the absence of either one of the triangles of a pair, since each expansion requires 2 new triangles after the first, at the end of the cone? Fascinating concept, but ultimately this equation should be able to determine the smallest viable triangle with positive values, however there should be a final triangle that sits along both negative and positive values. Is that fact of non importance? Or perhaps is there more importance yet to be discovered in the shape resulting in that space?
I was paying attention at the crucial moments when you (should have) pointed out the exact novelty of the new proof. I'm sad to say that none of those statements made sense. What do you mean by a trigonometric proof EXACTLY? What are the axioms of trigonometry that I can use, and just as importantly, what am I not allowed to assume? You mentioned something like the "sum and difference identities" used by Zimba about a decade ago. Do you mean the identity sin(x+y)=sin(x)cos(y)+cos(x)sin(y)? If I"m allowed to use that, the Pythagorean theorem follows trivially in a line, by putting y=pi/2-x:
1=sin(pi/2)=sin(x)cos(pi/2-x)+cos(x)sin(pi/2-x)=sin^2(x)+cos^2(x).
The proof you present uses similar triangles. Once again, if we are allowed to use that triangles with the same angles are similar, implying that the ratio of corresponding sides are equal, then there is a trivial proof of the Pythagorean theorem. Namely, draw the altitude corresponding to the hypotenuse. That divides the right triangle into two smaller right triangles, both similar to the original one, with similarity ratios a/c and b/c. Thus the two segments that appear on the hypotenuse are a^2/c and b^2/c, yielding a^2/c+b^2/c=c, that is, a^2+b^2=c^2.
So what is the advantage of the new proof?
The United States has a history of slavery
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.
Beautiful proof, beautiful and clear presentation!
Is there a reason to prefer your summations start at n=1 instead of n=0? When the n only appears as n-1 it seems legit to start at 0. Disclaimer: am a programmer 😅
Yo this is crazy. The waffle cone is a brilliant idea, and what seems to be the biggest insight here -- though there's still quite a bit of work even outside of that. Fucking brilliant work by those two on this. Wow!
That is a wonderful proof. But I'm a little confused on the algebra when you use the convergence test on sum from n = 1 to n of 2ac/b * (a^2/b^2)^n-1 turning into (2ac/b)/(1-a^2/b^2). I tried different things after doing algebra like bringing the (a^2/b^2)^-1 down to the denominator but I just can't figure out how it turned into a factor of 1-a^2/b^2. Anyone know what I mean?
I wonder if there's an analogue of this proof in non Euclidean geometry where two similar triangles must be the same size.