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
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.
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.
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.
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
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.
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.
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.
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.
IsaacL|1 year ago
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.
tonystride|1 year ago
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???
dr_dshiv|1 year ago
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
bjordnoygbi|1 year ago
sitkack|1 year ago
mandmandam|1 year ago
... 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.
sitkack|1 year ago
https://www.youtube.com/watch?v=VHeWndnHuQs
What isn't stressed enough is that they both came up with their respective proofs independently.
spidersenses|1 year ago
They just happened to have the same teacher...
rhelz|1 year ago
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.
8bitsrule|1 year ago
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.
sitkack|1 year ago
khafra|1 year ago
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.
thewarpaint|1 year ago
> far less time and effort
Pick one
1024core|1 year ago
user070223|1 year ago
https://en.wikipedia.org/wiki/Divine_Proportions:_Rational_T...
dr_dshiv|1 year ago
eointierney|1 year ago
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.
imp0cat|1 year ago
Tongue firmly planted in cheek. :)
eointierney|1 year ago
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.
fn-mote|1 year ago
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.
lupire|1 year ago
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.
throw_pm23|1 year ago
erehweb|1 year ago
kuhewa|1 year ago
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.
sitkack|1 year ago
unknown|1 year ago
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