6 ms·

Five or ten new proofs of the Pythagorean Theorem

116 points by jhncls 2y ago

rhelz 2y ago
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.
- sitkack 2y ago
  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.
- 8bitsrule 2y ago
  >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.
- 1024core 2y ago
  Another example: the X-Prize (now named something else, I think)
- thewarpaint 2y ago
  > a job at McDonalds > far less time and effort Pick one
- khafra 2y ago
  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.
fn-mote 2y ago
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.
- deleted 2y ago
  [deleted]
- sitkack 2y ago
  Please give them some slack, they were in high school when they wrote the proofs.
  - contravariant 2y ago
    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?
    - deleted 2y ago
      [deleted]
sitkack 2y ago
I really really recommend that people watch this 60 minutes interview with the authors' of these proofs. https://www.youtube.com/watch?v=VHeWndnHuQs https://www.youtube.com/watch?v=VHeWndnHuQs What isn't stressed enough is that they both came up with their respective proofs independently.
- spidersenses 2y ago
  >What isn't stressed enough is that they both came up with their respective proofs independently. They just happened to have the same teacher...
  - sitkack 2y ago
    Think of how it must feel to be that teacher. Their school is phenomenal, https://www.cbsnews.com/news/the-inspiration-for-new-orleans-st-marys-academy-60-minutes/ 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.
tonystride 2y ago
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???
- bjordnoygbi 2y ago
  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.
  - jacobolus 2y ago
    As far as we know, a developed idea of deductively proving theorems in the style of the Elements postdates Pythagoras by about a century.
  - n4r9 2y ago
    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.
    - dr_dshiv 2y ago
      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.
user070223 2y ago
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_Trigonometry_to_Universal_Geometry https://en.wikipedia.org/wiki/Divine_Proportions:_Rational_T...
dr_dshiv 2y ago
The conclusion of this paper was so beautiful. A real feel good story.
eointierney 2y ago
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.
eointierney 2y ago
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.
- imp0cat 2y ago
  No AI content? Pretty much every other submission is about AI nowadays. Tongue firmly planted in cheek. :)
IsaacL 2y ago
I 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-demonstration-squares-proof-Pythagorean-b-square.jpg 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.

116 points by jhncls 2y ago

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.

sitkack 2y ago
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.
8bitsrule 2y ago
>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.
1024core 2y ago
Another example: the X-Prize (now named something else, I think)
thewarpaint 2y ago
> a job at McDonalds > far less time and effort Pick one
khafra 2y ago
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.

fn-mote 2y ago

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.

deleted 2y ago
[deleted]
sitkack 2y ago
Please give them some slack, they were in high school when they wrote the proofs.
- contravariant 2y ago
  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?
  - deleted 2y ago
    [deleted]

sitkack 2y ago

I really really recommend that people watch this 60 minutes interview with the authors' of these proofs. https://www.youtube.com/watch?v=VHeWndnHuQs https://www.youtube.com/watch?v=VHeWndnHuQs What isn't stressed enough is that they both came up with their respective proofs independently.

spidersenses 2y ago
>What isn't stressed enough is that they both came up with their respective proofs independently. They just happened to have the same teacher...
- sitkack 2y ago
  Think of how it must feel to be that teacher. Their school is phenomenal, https://www.cbsnews.com/news/the-inspiration-for-new-orleans-st-marys-academy-60-minutes/ 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.

tonystride 2y ago

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???

bjordnoygbi 2y ago
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.
- jacobolus 2y ago
  As far as we know, a developed idea of deductively proving theorems in the style of the Elements postdates Pythagoras by about a century.
- n4r9 2y ago
  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.
  - dr_dshiv 2y ago
    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.

user070223 2y ago

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_Trigonometry_to_Universal_Geometry https://en.wikipedia.org/wiki/Divine_Proportions:_Rational_T...

dr_dshiv 2y ago

The conclusion of this paper was so beautiful. A real feel good story.

eointierney 2y ago

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.

eointierney 2y ago

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.

imp0cat 2y ago
No AI content? Pretty much every other submission is about AI nowadays. Tongue firmly planted in cheek. :)

IsaacL 2y ago

I 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-demonstration-squares-proof-Pythagorean-b-square.jpg 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.