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New Orleans teenagers found a new proof of the Pythagorean Theorem

keith-mcnulty.medium.com

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Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#201

Post by chx »

dvt wrote: post #179
chx wrote: post #151

Earlier quoted context omitted.

You can define sine and cosine together using the functional equations S(X)C(Y)+C(X)S(Y)=S(X+Y) C(X)C(Y)−S(X)S(Y)=C(X+Y) The only solutions to this are the constant 0 functions and the sine-cosine pair.

This is super cool, I've never seen it before! Do you know what this is called so I can look up a proof/theorem on it?

I asked https://math.stackexchange.com/q/124887/6400

I wish I had a more modern summary of the papers mentioned in the linked paper

> Tannery, Fonctions d'une Variable, 1886, p. 147. Osgood, Lehrbueh der Funktionentheorie, 1912, p. 582. Van Vleck and H'Doubler, Transactions Amer. Math. Society, vol. 17 (1916), p. 30

because we spent an entire semester at the university in one class working on these two.

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#202

Post by alex_smart »

crdrost wrote: post #104
SamReidHughes wrote: post #74

Earlier quoted context omitted.

There is a very straightforward proof that involves dropping an altitude. https://www.mathopenref.com/lawofsinesproof.html You could use a similar technique to make this proof not reference trigonometric functions.

For that matter the easiest proof of Pythag that I know of involves dropping an altitude: Look at the right triangle the normal way up, clearly the area of the triangle is k c² (k = ½ sin α sin β if you like, but it just matters that it's the same nonzero k for all similar triangles). Now roll it onto its hypotenuse, drop an altitude, and observe that both subtriangles are similar to the first one, kc² = ka² + kb². T…

For this proof to be a proof, you first have to define what area is and why it should scale as kc² and why the sum of areas of the two the smaller triangles should equal the area of the larger triangle.

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#203

Post by selimthegrim »

SonOfLilit wrote: post #199
dvt wrote: post #179

Earlier quoted context omitted.

This is super cool, I've never seen it before! Do you know what this is called so I can look up a proof/theorem on it?

I'm not familiar with this result, but this comment is phrased in the language of Ordinary Differential Equations, so I'd look for a textbook on solving systems of ODEs and expect to find a technique that can prove that this is the unique solution (at least assuming differentiability of S(x) and C(x)).

You can also consider the pair of functional equations as implicitly using a Taylor expansion

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#204

Post by contravariant »

pizza wrote: post #187
contravariant wrote: post #102

Earlier quoted context omitted.

You're missing the actual theorem in your comment, but the one you are referring to is essentially the pythagorean theorem for Bregman divergences, which I think may be a bit too far removed from geometry to allow this proof to generalize.

Ah, was just a flicker in my mind. Thanks - though I will ask, is there anything like the law of sines/cosines (in high dimensional statistics) that could get you to at least the first step of the proof?

The wikipedia lists a law of cosines [1] for the Bregman divergence and someone in this thread posted a proof that basically showed the law of sines follows directly from the definition of the sine. So neither form an obstruction per se, but the generalized pythagorean theorem kind of just follows directly from the definitions you need to make to even be able to say what it means for a triangle to be rectangular.

[1]: https://en.wikipedia.org/wiki/Bregman_divergence [2]: https://www.mathopenref.com/lawofsinesproof.html

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#206

Post by scatters »

amelius wrote: post #205

How come the triangles in the picture are all congruent but they don't all look congruent.

Because the first dropped vertical is too long. The left hand angle is beta plus alpha, so it should be ninety degrees, but instead it's an obtuse angle.

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#207

Post by galaxyLogic »

FartyMcFarter wrote: post #198
qez wrote: post #185

Earlier quoted context omitted.

I wonder, are there an infinite number of proofs of the theorem, each more complex than the last? Can I rephrase what they did, make it more convoluted, and call it a new proof?

You can always insert irrelevant details into a proof, so it's easy to construct an infinite number of proofs from a single proof.

If they are irrelevant details then that should not count as a new proof, in my opinion.

I guess it might be possible to have a "canonic form" of a proof so that if two proofs can be reduced to the same canonic proof then they are in fact the same proof.

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#208

Post by mananaysiempre »

Waterluvian wrote: post #107
hypertexthero wrote: post #94

Earlier quoted context omitted.

This brings to mind the visual solution to calculating triangle area in James Somers post “I should have loved biology”: > In his “Mathematician’s Lament,” Paul Lockhart describes how school cheapens mathematics by robbing us of the questions. We’re not just asked, hey, how much of the triangle takes up the box? > That’s a puzzle we might delight in. (If you drop a vertical from the top of the triangle, you end up wi…

I vividly remember math class one year boring me to death so bad that I distracted myself with my own puzzles like this. It was when I discovered the Fibonacci sequence inside Pascal’s triangle. I didn’t think this was a new discovery but it was new to me and it felt like lightning. I think that might have been an early glimpse of my later discovery that all my best learning would be done outside school.

> [A]ll my best learning would be done outside school.

Choose one:

- Experience of discovery and survival of curiosity to adulthood;

- Set of job-relevant skills well defined by names of subjects;

- Standardized testing and easily comparable grades.

(In my admittedly limited teaching experience.)

I would guess that the last point will always get chosen, because it’s bureaucracy-friendly, and a bureaucracy makes the choice. But one of my most bizarre experiences is (some) HN readers being quite vocal about their support for it as well, where I haven’t seen it be anything but harmful. The bullshit admission process at US colleges might be to blame—I’m really not sure.

References: Lockhart’s “Lament”[1], of course, for describing the feelings that (good) teachers have on this subject; Quinn’s “Revolution in mathematics”[2], as a more clinical analysis of how the bureaucracy won and got to basically redefine what “mathematics” even means for the majority of the population (in a way that’s as hopelessly obsolete as it is intensely harmful to the subject proper). The point shouldn’t be specific to mathematics, but it’s what I have the references for.

[1] https://www.maa.org/external_archive/devlin/devlin_03_08.htm...

[2] http://www.ams.org/notices/201201/rtx120100031p.pdf

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#209

Post by cabalamat »

jonahx wrote: post #99
joe_the_user wrote: post #20

I have read about this proof for a bit and this is the first write-up that gives the slightest details. The phrase "using trigonometry" is confusing. What they do is assume functions sine and cosine exist, as normally defined, as ratios of triangle values, without assuming these have the various Pythagorean-theorem derived properties. They then construct an infinite series of nested triangles and use the formula for…

It's a very clever proof. Agreed "using trigonometry" is potentially misleading. After reading the proof, the only 2 senses in which "trignometry" is being used are: 1. The term "sin a" is used to denote the ratio opposite / hypotenuse. But this can be considered a purely notational convenience. They could have called it "foo a" and nothing would change, or they could have inlined the referred-to ratio everywhere. 2.…

> It's a very clever proof.

Which makes it a bad one, IMO. In maths, as in programming, one should go with the simplest way that works.

Re: New Orleans teenagers found a new proof of the Pythagorean Theorem

#210

Post by capableweb »

majormajor wrote: post #18
capableweb wrote: post #11

Earlier quoted context omitted.

"Here's How" is usually used for clickbait stuff and adds nothing of substance to the title itself. New Orleans Teenagers Found a New Proof of the Pythagorean Theorem is both shorter and less editorial than Here’s How Two New Orleans Teenagers Found a New Proof of the Pythagorean Theorem

I get why that's the default behavior but "adds nothing of substance" is a huge generalization that fails in this case. An article with zero details about the proof could easily be titled "New Orleans Teenagers Found a New Proof of the Pythagorean Theorem" but couldn't accurately be titled "Here’s How Two New Orleans Teenagers Found a New Proof of the Pythagorean Theorem." Whereas here it says it'll have more details…

> (although TBH this is less a "here's how they found" and more a "here's what they found", if I'm being extra pendantic)

Yeah, I guess I'm a bit extra pedantic when it comes to the titles, because I agree the most with this, the article doesn't seem to actually go into how they found it out, meaning the original title was misleading after all.

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