[HN Gopher] Five or ten new proofs of the Pythagorean Theorem
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Five or ten new proofs of the Pythagorean Theorem
Author : jhncls
Score : 56 points
Date : 2024-10-29 20:40 UTC (2 days ago)
(HTM) web link (www.tandfonline.com)
(TXT) w3m dump (www.tandfonline.com)
| rhelz wrote:
| 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 wrote:
| 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 wrote:
| >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.
| fn-mote wrote:
| 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.
| sitkack wrote:
| Please give them some slack, they were in high school when they
| wrote the proofs.
| contravariant wrote:
| 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_?
| throw_pm23 wrote:
| I think this is beautiful work by any standards. It is
| definitely _not_ a central research topic within mainstream
| modern mathematics, but it is accessible and well-known,
| which is exactly what makes good high-school level
| mathematics.
|
| Math as a career is brutal, I don't think an ambitious kid
| would get demotivated so easily, but if you protect them
| from early frustration, you may set them up for bigger
| frustrations later.
| erehweb wrote:
| 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.
| kuhewa wrote:
| > "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.
| sitkack wrote:
| 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.
| spidersenses wrote:
| >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 wrote:
| 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.
| tonystride wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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.
| sitkack wrote:
| 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.
| dr_dshiv wrote:
| 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_
| pyuser583 wrote:
| There was some weirdness to him too. He basically founded a
| religion. There were some strange ideas about the hierarchies
| of food, etc.
| user070223 wrote:
| 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...
| dr_dshiv wrote:
| The conclusion of this paper was so beautiful. A real feel good
| story.
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