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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

#261

Post by alex_smart »

eneim wrote: post #220

I think this approach briliant. However, I believe it can be simplified to avoid depending to trigonometry and geometry series, while keeping the original idea. I have no idea if it is similar to any existing proof of the theorom, but I put my thought into a post [1]. [1] https://eneim.notion.site/On-a-new-proof-of-the-Pythagorean-... .

There really isn't any trigonometry here apart from sine rule, and because sine rule is derived from similar triangles, anything you can say using sine rule, you can also say using similar triangles.

The trigonometry thing is simply a marketing gimmick for this proof. There is no more or less trigonometry in this proof than there is in Einstein's proof. In fact, you can just taken Einstein's construction and reformulated that proof in their language by using sine rule instead of similar triangles. But then the gimmick would be too obvious.

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

#262

Post by mturmon »

chx wrote: post #151
pramodbiligiri wrote: post #144

Earlier quoted context omitted.

The sine function can be defined in terms of its own behavior, using its first-order differentiation and no reference to triangles. See this detailed article on sine. https://betterexplained.com/articles/intuitive-understanding... There’s section there titled Part 2: Understanding the definitions of sine.

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.

I did not know that one! It's a more complex version of the well-known functional definition of the exponential function, i.e. the unique continuous function satisfying

  E(x) E(y) = E(x + y)
and a normalization, E(1) = (whatever).

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

#263

Post by alex_smart »

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.…

The trigonometry thing is simply a marketing gimmick for this proof. There is no more or less trigonometry in this proof than there is in Einstein's proof. In fact, you can just taken Einstein's construction and reformulated that proof in their language by using sine rule instead of similar triangles. But then the gimmick would be too obvious.

Somehow the second gimmick (the infinite series construction instead of Einstein's elegant and simple construction) makes our monkey brains not notice the first gimmick.

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

#264

Post by alex_smart »

nolamark wrote: post #54

for completeness, here is a reference to the AMS presentation. https://meetings.ams.org/math/spring2023se/meetingapp.cgi/Pa... [my post updated to include abstract] Abstract In the 2000 years since trigonometry was discovered it's always been assumed that any alleged proof of Pythagoras’s Theorem based on trigonometry must be circular. In fact, in the book containing the largest known collection of proofs (The Pythag…

The trigonometry thing is simply a marketing gimmick for this proof. There is no more or less trigonometry in this proof than there is in Einstein's proof. In fact, they could just as well have taken Einstein's construction and reformulated that proof in their language by using sine rule instead of similar triangles. But then the gimmick would be too obvious.

Somehow the second gimmick (the infinite series construction instead of Einstein's elegant and simple construction) makes our monkey brains not notice the first gimmick.

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

#265

Post by doublerebel »

deathlight wrote: post #233
fnordpiglet wrote: post #130

Earlier quoted context omitted.

Is there more to trigonometry? I’m not a abstract math person, so forgive the ignorance, but my understanding was all trigonometric functions derive from ratios of angles and lengths of triangles so in the end each occurrence of a trigonometric function can be replaced by the corresponding ratios in some triangle. There are other ways to construct things, such as power series representations, etc, but even these must…

Sine and cosine can take as their input any real number including negatives and including very large positive numbers. Their outputs can also be negative numbers between negative 1 and 1 if they have real inputs. None of this necessarily makes any sense if you're considering a purely geometric naive interpretation in terms only of ratios of lengths. You have to introduce concepts like modulo the angle in a circle and…

4 dimensional triangles are the same as 3 dimensional lines. They don’t exist in the 4th dimension any more than they exist in any dimension >= 3. You would need a fourth side/point in the polygon in order for it to have any position in that dimension.

(It could be a triangle in dimensions 2-4 from our perspective but to the triangle it only has 3 dimensions any way you arrange it.)

Or you can bend a triangle in another dimension(s), but then it’s not a triangle by the commonly accepted definition. (E.g a 270° “triangle” on a sphere)

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

#266

Post by SamBam »

kzz102 wrote: post #247

It's a really nice idea. I wouldn't call it a proof in the rigorous sense, because you need to define trigonometric functions first and be careful you don't use Pythagorean theorem to avoid circular logic. It's however perfectly fine to call it a proof for students, which is about deducing logical relation between statements. I am however, against media hype of this type of student achievements. This is very nice for…

The only definition of the trigonometric function required is that sin (x) = opp/adj. It's just a ratio, nothing else. The students could have called this ratio "snoo" and the proof would have still worked.

The proof is rigorous. The original article explains this, and why it's not circular.

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

#267

Post by option »

teo_zero wrote: post #244
option wrote: post #124

how does one define sin without Pythagorean theorem? E. g. need to prove that it only depends on angle and not triangle size.

You don't need to prove anything to define sin.

the definition of function in mathematics requires that same input always gives same output for all valid inputs.

So yeah, to define sin as a function one absolute must prove that it only depends on angle and not triangle size.

This proof is done via similar triangle properties (same angles => same proptions of sides) btw.

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

#269

Post by 908B64B197 »

threatofrain wrote: post #64
uberman wrote: post #59

Earlier quoted context omitted.

How would it be remiss to omit these factors, several of which seen irrelevant. Are you suggesting that their proof be discounted because it might be divinely inspired?

The article is highlighting the inspirational quality of two black women from an underachieving area; it's not just about the proof. The inspirational story is an important element. Adding that these two youths came from a private school certainly relates to the "underachieving area" part of the story.

> Adding that these two youths came from a private school certainly relates to the "underachieving area" part of the story.

I don't want to take anything away from these two; the proof is novel and interesting and even if it wasn't novel after all it's still incredible to see high school students interested in math and capable of that level of reasoning.

But yeah, that the two researchers (they've earned the tittle by giving a conference talk!) are from a private, selective school undermines a lot the "bad area and underachieving" narrative. It reminds me of a tech company who had a panel about their black engineers, pointing to the gap between the proportion of black SWE in the bay area relative the the percentage of people who identify as black. Three out of the four panelists were Nigerian born. When discussing their path to tech, one of them explained it was hard for him to convince his parents that he was not going to study surgery like his father and uncle. I assumed it would have resonated with anyone from an "underachieving area" in the audience...

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

#270

Post by starchild_3001 »

ummonk wrote: post #152
starchild_3001 wrote: post #79

Ok, I read through the proof, and I think I understood it. Thanks for posting this! Until the authors' work is submitted to a journal and reviewed, it's hard to say everything claimed here is definitely correct & new. Update: Nice video on the proof: https://www.youtube.com/watch?v=nQD6lDwFmCc I like what they did :) Seems legit to me. Btw, law of sines can be proven independently of pythagoras. So using that as a st…

It's definitely correct, and quite trivial to verify. It's possible it has been discovered before, but none of the proof compilations I've seen (e.g. cut-the-knot) has it, and the trigonometric proofs I can find involve using angle-sums ( https://forumgeom.fau.edu/FG2009volume9/FG200925.pdf ). This is definitely a much more elegant proof than the angle-sum proofs.

Correctness might be there. But there are also some grandiose claims in the abstract ("“There are no trigonometric proofs, because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean Theorem.”) and the argument being novel. I don't have a way to verify these two. That's what the peer review is for.

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update: This video refers to another trigonometric proof that doesn't rely on sin^2 + cos^1 = 1. https://www.youtube.com/watch?v=p6j2nZKwf20

I would write a more modest abstract for this work. Just my $0.02.

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