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Comment by joe_the_user | Hacker News Reader

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 the sum of geometric series' to derive the length of the original triangle's hypotenuse. It certainly seems clever.

I'm still confused what axioms they're effectively using relative to the usual Pythagorean theorem proofs - most of these use the formula for area of a right triangle and this seemingly doesn't. On the other hand, it seems an infinite construct would require things like the axiom of induction, which may or may not be included in axiom of axiomatic geometry.

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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 law of sines is required. But the proof of this law [1] also boils down to nothing more than the ratio definition and some algebra.

So afaict no circular logic is being used, but at the same time it doesn't seem to be doing anything previously thought to be impossible, unless there was a previous belief that the law of sines could not be used in a proof, which would be a strange belief to hold. I see it simply as a creative, unexpected proof.

[1] https://www.youtube.com/watch?v=4xO8xqLyEbA

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.

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 necessarily be replaceable by the ratio of angles and lengths of some triangle. What am I missing?

> What am I missing?

Nearly all the nontrivial results of trigonometry do in fact rest on the pythagorean theorem. The trig identities you learned in high school, as well as more advanced results like power series, etc. These results would be inadmissible.

So the “uses trigonometry” part of this story feels like an attempt to manufacture mystery and hype. Which is a shame, because the geometric series construction is imo the interesting part, and can stand on its own merits.

Apparently (found this while reading an article on the girls' accomplishments) somebody proved sin^2x + cos^2x = 1 without using the Pythagorean theorem in 2009: https://forumgeom.fau.edu/FG2009volume9/FG200925index.html.

I don't think the "uses trigonometry" part is hype. They do use the definitions and law of sines, they just cleverly avoid the parts of trigonometry that depend on the Pythagorean theorem.

I address this in my OP. If you watch the video of the proof, you will see that the "law of sines" is 1 step away from the ratio definition of sin. You just drop one altitude, apply the definition again to the similar triangles, and re-arrange. It is almost content free as a result -- I see no reason using this in a proof would have special significance. For example, the standard proof using similar triangles (https://sumantmath.wordpress.com/2020/08/16/proof-of-pythago...) is implicitly using the law of sines.

The hype part is the implication that impossible trig barrier was shattered by their proof.

Not at all - the hype part is that 2 teenagers derived a uniquely elegant proof that other highly-trained mathematicians had thus far failed to do so.

Any other claims seem to have been added by the media, not the teenagers themselves.

That is what I was referring to. I wasn’t accusing the teenagers of anything.

This theorem is essentially the Pythagorean theorem, so … a proof of one is a proof of the other.

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 analytic coordinate system for it all to square with normal naive intuition.

In fact the sign and cosine can take as their inputs any and produce as their output any complex number. You have to come up with some very interesting triangles to make this makes sense. I'm sure it might be doable but they would potentially be four dimensional triangles and I haven't explored that concept very deeply.

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)

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.

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.

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

Yeah, the Math Overflow answers are a bit sparse, and I think the Euler formulation (while clever) is a bit of a red herring and might be circular. Trying to slowly go through that 1917 paper, thanks for linking!

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

Also related: doi:10.1007/s00010-003-2700-z

f(x)f(y)f(z) = f(x) + f(y) + f(z) sort of implies f(x) is tan(x) -- it's actually tan(kx + (1-k) pi/3).

Sorry I left out the condition of x + y + z = π

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).

> 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.

The Pythagorean theorem is the most frequently re-proved thing in the history of math, with hundreds of published proofs and entire books dedicated to collecting them. The reason to come up new proofs these days is solely for the novelty, not because we have a need for a simpler proof. The fact that it's such a popular subject for proofs is why novel proofs are inherently interesting, regardless of their complexity.

> The reason to come up new proofs these days is solely for the novelty, not because we have a need for a simpler proof.

On the contrary, every new way to prove a known theorem has the potential to be applicable in other areas of the same or related fields, extending the mathematicians' toolset with new instruments. These new methods often serve as a seed for new discoveries.

The mistake is seeing this as a constructed thing. Math is already there, we only uncover it. If they revealed a previously unseen chamber in the Great Pyramid or something, you wouldn't say "aw nuts, that overcomplicates our existing knowledge of the structure".

Hey, come to think of it, I'd be perfectly happy with it being the corner of a solid cube[*] of rock ploinked diagonally down into the sand; the whole idea of interior chambers and tunnels does kinda make it feel too fussy, not as clean and sleek as it could be...

___

[*]: Dangit, no, wrong shape.

For applied maths I agree with you. But when exploring novel and innovative approaches I disagree. There is so much we can learn from finding new ways to look at the world. The initial take may be complex or clever, but it can lead to a deeper understanding that allows for later simplification of entire subject areas.

I disagree.

'Aesthetic' appearance in math is important in helping drive mathematical innovation and help new human beings derive pleasure from that wonderful field.

Programming is kind of an applied mathematics where efficiency does matter because it's a tool, a means to an end.

Not to say that people can't find aesthetics in programming, nor that they shouldn't, rather in math at least the pleasure of discovering a new way of doing/proving something is the end in of itself.

It's pleasure for me to see another way to do or prove something; I can only imagine the feelings this teenager got from actually making a discovery.

I find simplicity aesthetically appealing. Obviously some other people differ.

Well you're in luck since these girls' proof is pretty simple so should be pretty pleasing for you!

wouldnt downvote this because it makes sense in some cases.

while programming and math overlaps in so many places, i think mathemathicians are quite a separate species and KISS is often not on the menu.

a fun, non math heavy book with a window into this i enjoyed is https://en.wikipedia.org/wiki/Uncle_Petros_and_Goldbach%27s_...

really just an arts/hobby vs job/career disagreement. and gosh, they’re teenager students: we can’t suck all the joy out of school!

but related, i wonder what a mathematics demo party would look like.

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?

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.

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)

Yes, if they droped height h to the side of isoscales triangle, then from similar triangles they would have h/A=c/C and h/b=2a/c and would get the result without mention of sine rule or areas.

,,In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M.''

It would be interesting to see if the original proofs work with non-complete metric spaces or not, as probably this proof doesn't.

https://en.wikipedia.org/wiki/Complete_metric_space

https://sharegpt.com/c/qRsum0c

Here is a proof of Pythagoras using a geometric series:

https://www.cut-the-knot.org/pythagoras/Proof100.shtml

The proof is by John Arioni and also features an infinite number of similar triangles.

I don't see any Cosine being mentioned in the proof. I don't see any circular logic here.

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