Five or ten new proofs of the Pythagorean Theorem

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.

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

> 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.

>What isn't stressed enough is that they both came up with their respective proofs independently.

They just happened to have the same teacher...

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.

>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.

Another example: the X-Prize (now named something else, I think)

> a job at McDonalds

> far less time and effort

Pick one

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.

No AI content? Pretty much every other submission is about AI nowadays.

Tongue firmly planted in cheek. :)

Please give them some slack, they were in high school when they wrote the proofs.

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.

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.

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.