Five or ten new proofs of the Pythagorean Theorem
tandfonline.comI still maintain that this (originally from ancient China) is the clearest proof, and gives the best insight into why the Pythagorean Theorem holds.
https://cdn.britannica.com/43/70143-004-CCB17706/theorem-dem...
It is not immediately obvious why the area of the hypotenuse square should be equal to the sum of the areas of squares drawn on the other two sides of the triangle.
It is clear that the lengths of a, b and c are connected -- if we are given the length of any two of (a, b, c), and one angle, then the remaining side can only have one possible length.
So far, so simple; what is less clear is why the exact relationship for right triangles is c^2 = a^2 + b^2.
The other proofs demonstrate that the relationship holds, but give little insight.
The geometric proof linked above makes the relationship crystal-clear.
For any right triangle we can define a 'big square' with sides (a + b). The hypotenuse square is simply the area of the 'big square' with 4 copies of the original triangle removed.
Simple algebra then gives us the formula for the hypotenuse square:
The big square has area: (a+b)^2 = a^2 + 2ab + b^2
The original triangle has area: ab/2
1 big square minus four original triangles has area: (a+b)^2 - 4ab/2 = a^2 + b^2
Similarly, if you take the hypotenuse square, and subtract 4 copies of the original triangle, you get a square with sides (b - a). This is trivial to prove with algebra but the geometric visualisation is quite neat, and makes clear why the hypotenuse square must always equal the sum of the other two squares.
Tbh this is a bit over my head as my music degree only qualifies me to count to four. But all joking aside, I wonder how Pythagoras would feel if he knew that one day he would be better known for this theorem and not for music?
I’m amazed by how many people I meet who don’t know about his contribution to the discovery and development of tonality! You mean the triangle guy invented music???
Completely agree. One can be skeptical, but the actual man is likely greater than the legend.
I own a coin designed by Pythagoras. Well, it’s from 510 BC Croton, features the tripod from Delphi, and has little snakes at the bottom. Also 10 little dots. No tetractys, but that’d be a bit much. Also, the front is the opposite of the back (Aristotle describes the Pythagorean obsession with opposites).
I mean, maybe it wasn't Pythagoras — but his father was a gold smith and it is the most beautiful coin of the era, suggesting genius. But it might have been Hippasus, who was known for having conducted the first hypothesis driven experiment of all time: casting bronze chimes in musical proportions to see if the 1:2:3:4 intervals that make stringed music consonant apply with the thickness of chimes. They do. The mathematical model generalizes.
Currently, I’m working on a textbook callout that helps students learn about fractions using musical intervals — and introduces all the DEI glory of Pythagoras (multiethnic, gender-mixed community, credited his moral doctrines to a woman, Themistoclea of Delphi, etc). I’m leaving out the fact that he was kicked out of the boys Olympics when he was 16 for being too effeminate. He won the men’s Olympics in boxing, introducing some kind of new martial arts. Then he trained the most successful Olympic athlete of all time, Milo of Croton, who won 5 consecutive Olympics. No one has done that since.
Let me know if you need sources for any of these facts, I collect them all. Pythagoras is the bessst
There was some weirdness to him too. He basically founded a religion. There were some strange ideas about the hierarchies of food, etc.
Well, technically he founded “philosophy” more than a religion.
Polytheism was really old, of course. But intellectual polytheism (which requires an esoteric treatment of gods as metaphors) was basically new and then lasted for 1000 years. Pythagoras is also credited with coining the term “philosophy” and the term “Cosmos.”
In addition to his coinage, he also started the first communist society. So many opposites. Music/math, religion/science, communism/capitalism, harmony/war (he conquered the sybarites), etc. He was vegetarian but, when he discovered his eponymous theorem, he sacrificed a “Hecatomb” at Delphi — that’s 100 cattle.
Of course, all of this is disputed. (But happy to provide sources for any of it).
He is textually associated with the druids and the Jews! Nuts.
> Well, technically he founded “philosophy” more than a religion.
If it executes a guy for revealing secrets and/or heretical discoveries, it's mroe of a religion than a philosophy.
> But intellectual polytheism (which requires an esoteric treatment of gods as metaphors)
This reminds me of a very funny parody of various theologies on the early Christian church:
If you have a list of links of stuff Pythagoras did (whether disputed or not), do let me know. I'd happily gobble them up.
https://en.wikipedia.org/wiki/Pythagoras#Attributed_discover... has a list. I think all of them are at least disputed if not outright considered untrue.
Fascinating. Provide sources please. I need to learn more about this man.
Haha, sources for which fact? I wrote a paper that deals with Pythagoras here.
https://www.sciencedirect.com/science/article/pii/S240587262...
"There were some strange ideas about the hierarchies of food, etc."
A food "pyramid", if you will.
/JK
Can you share a picture of the coin, that sounds like such an amazing historical artifact, regardless of whether it's actually by Pythagoras.
(also between this and Plato's failed Olympic career I feel like there's a lot more to the ancient Greek Olympic games than I'm aware of)
>Also, the front is the opposite of the back
Unrelated, but that's generally true.
Fascinating. Any recommended books / bios?
Most likely he didn’t come up with the theorem. He lead a cult, whose followers attributed their achievements to him and it is alleged that he himself had little interest in mathematics. I don’t know about music specifically, but it wouldn’t surprise me if the story was similar there. His core competency was religion.
Where did you come up with this ? This is just not true, that Pythagoras had little interest in math. He had a love of numbers and thought that math was a way to the divine or at least understanding the divine. His philosophy, not religion , but philosophy was a way of life that entangled mathematics profusely.
If I recall rightly, it's not even that clear that a single person named Pythagoras really existed. If he did then he never wrote anything down, there are few contemporary accounts of his life or work, and what details there are contradict each other. And, as you say, he was chiefly interested in things like life after death as opposed to mathematics.
This level of skepticism flies in the face of references to Pythagoras across multiple contemporary authors. Keep in mind that it is damn hard to prove something in the 6th century BC with the same level of evidence as today.
But Heraclitus, for instance, was contemporary and accused him of plagiarism and trickery. Why abuse a person that doesn’t exist? Ion of Chios was contemporary and said that he authored Orphic hymns.
The Orphic hymn to Apollo proclaims “your resonance attunes the whole globe”. Now, we don’t know that Pythagoras wrote that. We can’t prove that sort of thing. But it sure seems Pythagorean.
Did you know that Copernicus, Kepler, Galileo and Newton all claimed to be Pythagorean?
The man is a legend. Embrace the legend. He wrote lots of things down but he either destroyed it to avoid creating a doctrine or it was destroyed during one of the different massacres of Pythagoreans. Because that happened.
The evidence for figures such as Plato, Aristotle, Socrates and Heraclitus is far more compelling than it is for Pythagoras. There is a singular lack of evidence and sources for his existence.
No one is denying that a cult named the Pythagoreans existed, that many of their writings are preserved, and that they influenced Copernicus etc ...
Heraclitus was notorious for his cutting critiques of predecessors. There are a total of three very brief mentions of Pythagoras in fragments of his work (fragments 40, 81 and 129). This is not really enough to go on - it's not clear whether he's criticising someone he's met in person, some learnings that he's heard through a third party, or a cult figure that was invented and idolised by the Pythagoreans.
Is there any evidence supporting the hypothesis that Pythagoras was invented?
I'm just saying it's not very clear that the figure of Pythagoras existed. It's like Robin Hood or King Arthur - maybe there was a historical person on which the mythology is based, but at this point we just don't know with any certainty.
This just isn’t true either. There are contemporary sources who talk about him.
As far as we know, a developed idea of deductively proving theorems in the style of the Elements postdates Pythagoras by about a century.
Musicians know a whole lot more math than they are aware of. Music is math. Even if you don't read sheet music or study the intervals of scales and chords. Musicians that become programmers write some of the best code, same for lit majors.
> You mean the triangle guy invented music???
... You know that we've found flutes in perfect pentatonic tuning that date back at least 40,000 years right (in Germany, Slovenia, etc)?
Pythagoras certainly contributed but to say he 'invented music' you'd have to ignore tens of thousands of years of history.
People were also using 'his' theorem long before he was ever born. Not trynna diminish the guy but let's give the ancestors their due.
Tbh I feel like this is a ‘read the room’ situation. So if I’m on HN, providing thorough, cited sources, with as much nuance as possible is right. On the other hand, if I’m teaching a class of middle school theory students, giving them a fun digestible story about ‘the real thing about the triangle guy’ is more effective.
It’s impossible to know the true scope of how it was all made whether it’s Pythagoras and the origin of tonality or Bach and the birth of common practice. There should always be a ‘click here to go deeper down the rabbit hole’ option, but sometimes Pythagoras and Bach are easy focal points to begin delivering the concept that this all came from somewhere.
I really really recommend that people watch this 60 minutes interview with the authors' of these proofs.
https://www.youtube.com/watch?v=VHeWndnHuQs
What isn't stressed enough is that they both came up with their respective proofs independently.
>What isn't stressed enough is that they both came up with their respective proofs independently.
They just happened to have the same teacher...
Think of how it must feel to be that teacher.
Their school is phenomenal, https://www.cbsnews.com/news/the-inspiration-for-new-orleans...
> Rogers told Whitaker that Calcea and Ne'Kiya are not "unicorns." She said all the young ladies at St. Mary's are exceptional and are taught early that they can achieve great things. For the last 17 years, St. Mary's Academy has had a 100% graduation rate and a 100% college admission rate.
Yet another example of the power of Prizes....the authors mention that they were motivated by a $500 prize offered to students by a math volunteer at their high school.
What is so counter-intuitive to me is that if the authors had wanted to earn $500 (or $250 after splitting it) they could have just got a job at McDonalds. They would have earned that money with far less time and effort.
I'm kinda glad that nobody pointed that out to them though :-)
But Prize-awards seems to put us into an entirely different economic frame. You can't say they did it just for the recognition, because if the prize wasn't there they wouldn't have bothered. But you also can't say that they did it for the money, because the money was ludicrously low--even when valued at the rate of unskilled labor.
>They would have earned that money with far less time and effort.
Prize or not, time 'invested' in reasoning out an original solution will very likely 'pay off' in the future much better than investing in flipping burgers. In satisfaction and fulfillment for sure. What's life for? No doubt Erdos and Euler, and certainly van Gogh, might have made more at McDonalds as well.
If you haven't you should watch the video I linked. I think money did have something to do with, but their school is also extremely high performing. People tend to do better when better is the norm.
People want to do challenging things that are worthwhile. The $500 prize is necessary to prove that it's a worthwhile challenge.
It's easy for anybody to see that, say climbing a mountain with a death rate of >1% is challenging, or completing an ultramarathon; so no prizes need to be offered. Offering a monetary prize for illegible things like new math proofs creates common knowledge that those things are challenging and worthwhile.
> a job at McDonalds
> far less time and effort
Pick one
Another example: the X-Prize (now named something else, I think)
semi related: I found Norman J. Wildberger rational trigonometry work very interesting. He ditches trigonometry in order to work with rational quantities. There was also a playlist on youtube of his work but I'm unable to find it for some reason
https://en.wikipedia.org/wiki/Divine_Proportions:_Rational_T...
The conclusion of this paper was so beautiful. A real feel good story.
One of the things I love about hacker news is that there's no AI content. The other is that it's like reading a commentary on our encyclopedia. I get to read thought happening.
Apropos of nothing, just saying, and this thread is a great example.
I always want to read more books after a good dose of hacker news.
No AI content? Pretty much every other submission is about AI nowadays.
Tongue firmly planted in cheek. :)
One of the things I love about hacker news is that there's no AI content. The other is that it's like reading a commentary on our encyclopedia. I get to read thought happen.
Apropos of nothing, just saying, and this thread is a great example.
I always want to read more books after a good dose of hacker news.
On one hand, I was ready to be interested.
However, I just cannot get excited about an article with proofs that:
(1) give a different name for methods that use sin(90)=1 vs only working with sine of an acute angle ("cyclometric" vs "trigonometric", ugh)
(2) use "high-powered" methods like convergence of infinite geometric series to prove the Pythagorean theorem
(3) apply the law of sines several times to produce the Pythagorean theorem
I just couldn't give it a chance. Give me a good old fashioned proof by a dissection diagram any day.
No one is obligated to be interested in everything, but I don't understand why you are bragging about your lack of intellectual curiosity about precise mathematical thinking. The difference between triangle trigonometry and unit circle trigonometry is well known to mathematics and important for constructing correct proofs (see the OP's cited Zimba's paper for a recent explanation), and deserves a name for clarity in exposition.
If anything, "trigonometric" is the word they should have avoided, since, even though the word is etymologocally closely associated with triangles as they said, it is also commonly used to refer to exactly the thing they are trying to avoid -- dependency on the Pythagorian theorem, which was the spource of all the confusion and fuss and terrible media reporting when they first published their proof and referred to an ill-defined statement in a 100 year old textbook.
There are hundreds of old proofs of Pythagorean Theorem. I'm sure you can find one that satisfies you. For those of us who enjoy new ideas that push back the intellectual frontier, this paper is very nice.
Note that there were already hundreds of old fashioned proofs, the challenge was exactly to find a new one with the "high powered" methods without circularly referring to the PT, which these two kids achieved.
Tastes differ. Myself, I think it's fascinating that we can use the convergence of infinite geometric series to prove Pythagorean theorem. And particularly inspiring / interesting that two High School students did this.
> "cyclometric"
it's cyclotopic, a term they coined. I suggest the intro section juxtaposing trigonometry vs 'circular' approaches might best be read as guidance as to how interested high school students (their past selves?) might think about the topic rather than a necessary preface for their paper.
I am a college graduate and I found it quite interesting and enlightening. I don't understand the persistent effort some bystanders are making to discredit this work that has met the approval of mathematical professionals.
I also very much enjoyed: "In this section, we verify that our proofs aren’t circular."
It's called jealousy. They've got all these big brains and nuanced thoughts and somehow not a one of them accomplished something that got that kind of recognition at that age.
Didn't have the motivation? So what? They did.
The things they did seem obvious? Yes, I bet they do in hindsight. Touch screens on cell phones seem obvious today too.
>I don't understand the persistent effort some bystanders are making to discredit this work that has met the approval of mathematical professionals.
I don't try to discredit this stuff, and probably couldn't come up with similar proof myself since I'm not that interested in mathematics in general, but I am personally getting kinda tired of all these "child discovers x" stories where x turns out to either be something this is already well known or it turns out that they've just restated something that was well known in a somewhat trivial way. I'm not saying that's what happened in this case, but it is what happens in most of these sorts of stories that get published anytime it's a slow news day.
Please give them some slack, they were in high school when they wrote the proofs.
No slack needed. The authors did great work, and they have been assured of that, and PP is embarrassing themself with the kind of middlebrow dismissal that Hacker News guidelines discourage.
I'm a bit of two minds about this. On the one hand you're quite right, it's impressive work for high school students. What I don't like is how pointing that out feels like an insult, when what I really want to convey is that it's impressive but not entirely beyond ordinary high school students with an interest in mathematics.
And articles like this have been popping up for years (I think about the exact same two students even), and each time I have to decide whether to downplay the scale of their achievement so high school students don't lose hope about achieving something similar, or praise them with the qualifier for high school students because they couldn't be expected to have enough mathematical background to push the boundaries of one of the oldest and most extensively researched parts of modern mathematics.
I can't help but feel that each additional article is just further entrenching the stereotype that you're either a genius at mathematics or not, and is demotivating the students in question, because how on earth are they ever going to top this?