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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

#101

Post by contravariant »

scythe wrote: post #52
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…

Induction isn't so bad. I'm not sure how they're getting away with the Law of Sines, though. The usual proof of LoS that I know is dependent on the existence of the circumcircle. But I don't know how to prove the existence of the circumcircle without dragging in a lot of geometry. Or you can use the area formula, which makes the proof similar to other arguments that use the area formula.

Is there not a proof using similar triangles somehow? Because the whole 'sine' part seems like a bit of a red herring, they're basically just considering a couple of ratios between different lengths, they do not use any properties of the sine function as such (in particular it does not look like they're using sin(x)^2 + cos(x)^2 = 1, which would make the proof trivial)

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

#102

Post by contravariant »

pizza wrote: post #34

It would be really interesting to see if this could be applied to other analogous scenarios that are sometimes called Pythagorean theorems, in particular I’m thinking of the Pythagorean Theorem of Information Geometry- p* = argmin p of P (a set of possible distributions) of D_KL (p||q) (where q is eg your model's distribution)

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.

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

#103

Post by girvo »

Mtinie wrote: post #95
no_butterscotch wrote: post #90

[flagged]

(Unsolicited comment and suggestion) I’ve had similar thoughts in the past but figured out I personally do better when I channel the doubt into excitement for anyone’s’ potential claim. I’m happy that a human—or humans, in this case—believe(s) they have discovered a novel way to do something and want to share it with the world. That doesn’t mean I take the claim at face value, I don’t, and want to wait for secondary…

Approaching things in the way you’ve described is a lot more joyful and enjoyable, too, I find. The world has enough to be sad about.

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

#104

Post by crdrost »

SamReidHughes wrote: post #74
scythe wrote: post #52

Earlier quoted context omitted.

Induction isn't so bad. I'm not sure how they're getting away with the Law of Sines, though. The usual proof of LoS that I know is dependent on the existence of the circumcircle. But I don't know how to prove the existence of the circumcircle without dragging in a lot of geometry. Or you can use the area formula, which makes the proof similar to other arguments that use the area formula.

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.

The diagram that's a bit involved is the angle sum diagram, you start with a right triangle (a,b,c) with some angle α, extend it to a new triangle (a,b', c') with angle α+β, then make the new triangle with angle β that you stacked on top of the original triangle into a right triangle with angle β (c', d, e) by extending the hypotenuse of the (a,b,c) triangle to a point P, basically until the angle with the hypotenuse c' is 90°. Drop a dotted line to the x-axis from P and you can work out that the dotted line is at x=cos α cos β, and its distance to a is sin α sin β. Similarly the y-coordinates give sin α cos β + cos α sin β.

As you say, you can do all of this without angles except for defining the first triangle with angle α+β, which you might not even need... We just need it here for sin(2 α) which is something like reflecting the same triangle about its hypotenuse?

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

#105

Post by erehweb »

agurjal wrote: post #89

I think you can sidestep trigonometry (and the Law of Sines) completely. You can decompose any triangle A, B, C using their construction to create smaller triangle a, b, c where A = 2abc/(b² - a²), C = c(b² + a²)/(b² - a²), and B = c. This can be shown with only similar triangles (it seems like they unnecessarily use sines in the article). It is then just algebra to show A² + B² = c²(b² + a²)²/(b² - a²)² = C². edit:…

Well sure, there are other proofs of Pythagorean theorem. The interesting part is that this is a new proof.

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

#106

Post by sbelskie »

User23 wrote: post #88
sbelskie wrote: post #82

Earlier quoted context omitted.

Perhaps the website for the school is incomplete and only listing registration fees and not tuition but the only figure I see is $750, which is not nothing for plenty of families but is hardly what I would call expensive when it comes to private education. Am I missing something? Very well could be!

Parents that care are a huge predictor of academic success. Putting your child in a private school, even an affordable one, is more effort than just going with the default option and thus serves as signal for parents caring.

Are new articles about people’s accomplishments “remiss” for not including statements about how much their parents cared about them?

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

#107

Post by Waterluvian »

hypertexthero wrote: post #94
dendrite9 wrote: post #49

I remember reading a book in high school and realizing there could be other ways to prove things that I had been taught only one way. One that particularly stood out later was using a rotating fishtank to prove the pythagorean theorum. A good friend of mine was so delighted by the example I gave him a copy of the book I found it in. https://press.princeton.edu/books/paperback/9780691154565/th... ) The relevant sectio…

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.

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

#108

Post by [deleted] »

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…

[deleted]

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

#109

Post by agurjal »

erehweb wrote: post #105
agurjal wrote: post #89

I think you can sidestep trigonometry (and the Law of Sines) completely. You can decompose any triangle A, B, C using their construction to create smaller triangle a, b, c where A = 2abc/(b² - a²), C = c(b² + a²)/(b² - a²), and B = c. This can be shown with only similar triangles (it seems like they unnecessarily use sines in the article). It is then just algebra to show A² + B² = c²(b² + a²)²/(b² - a²)² = C². edit:…

Well sure, there are other proofs of Pythagorean theorem. The interesting part is that this is a new proof.

I'm certainly no expert on proofs of the Pythagorean theorem, but if their construction is novel, then the above may be too.

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

#110

Post by robotresearcher »

yosito wrote: post #87

> doing my best to explain how Johnson and Jackson proved it using simple trigonometry. Although their proof hasn’t been published They didn't publish it, but this author is just going to take the liberty to publish their work himself? If I were one of these teenagers, this would make me angry.

The article's author is very clear whose ideas they are reporting. Neither do they aspire to be the definitive citation for the idea. They hope the girls publish, or attempt to publish, and in the process establish their priority by peer review.

This is perfectly legit in academic publishing. Credit where credit is due is the rule. You are not obliged to keep stuff secret until the originator has published, only that you attribute the idea properly.

This is good, because it means ideas can get out there and be useful without delay.

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