It hit the news last week, that two high school kids had come up with a new proof of Pythagoras’s Theorem:
Two New Orleans high school seniors who say they have proven Pythagoras’s theorem by using trigonometry – which academics for two millennia have thought to be impossible – are being encouraged by a prominent US mathematical research organization to submit their work to a peer-reviewed journal.
I had decided to leave it alone. I had figured out enough of the story, that the two kids had done something very cool, which had then been way over-egged by the predictably clueless, Homerically lazy media. I’ve done these stories, and they can be tiring and tricky.
My friend Grant Cairns,* however, tipped it over the line, by pointing out an MAV tweet:
Good ‘ol MAV.
How a right-angled triangle is any more about “the world around us” than the number 3, God only knows. Grant summarised it as the MAV being “totally grounded in reality”. More accurately, the MAV is totally grounded by reality. To an earthbound hammer, everything is an earthbound nail.
OK, I’ve had my fun and now I must pay the price. I have to write about this sweet-absurd story.
There are, of course, and depending how you count, hundreds if not thousands of proofs of Pythagoras’s theorem. See, for example, the very long list at the brilliant cut-the-knot. Although many of these proofs can be regarded as minor variations of each other, there are still tens, probably hundreds, of fundamentally different proofs. The question is, is there a proof via trigonometry?
It is easy enough to come up with circular trig “proofs” (no pun intended). For example, we could start with the cosine rule:
Then put C = 90 and there’s your Pythagoras. The problem of course, the circularity, is that Pythagoras is first required to prove the cosine rule. More directly, we can easily get Pythagoras via the fundamental trig identity
That’s so circular, it’s basically a dot.
With the fundamental trig identity in mind, it seems plausible that any trig proof of Pythagoras must of necessity be circular. That is what Loomis bluntly declared in his famous book of Pythagoras proofs (p viii).
But, there is more, and more prior, to trigonometry than the fundamental identity. The definitions of sin and cos, for example, come down to capturing the concept of similarity in right-angled triangles, and no Pythagoras is required for that. Similarly (still no pun intended), the proofs of the double angle formulas require no Pythagoras. So, one can at least contemplate coming up with a trig proof of Pythagoras.
Calcea Johnson and Ne’Kiya Jackson, two high school students, claimed to have done exactly that. Johnson and Jackson presented their circularity-free trig proof of Pythagoras at a regional meeting of the AMS,** after which it all hit the news.
Johnson and Jackson’s proof has not been publicly presented in print, meaning plenty of people had to puzzle unnecessarily,*** but their proof appears to be correct, free of circles, and very clever. The key diagram, as reconstructed by Grant, is as follows:
In brief, the original right-angled triangle ABC is used to create a larger right-angled triangle ABD. Then, the lengths of BD and AB are calculated using geometric series, their ratio is related back to the original triangle using the sin rule and the double angle formula for sin, and Pythagoras basically pops out. Readers can enjoy working out the details, or here is a very nice presentation and discussion by MathTrain:
Johnson and Jackson’s proof is complicated but elegant. It is very nice stuff, and of course it is great for school kids to come up with this. But, there is a but.
It’ll come as no great surprise to readers here that Johnson and Jackson’s is not the first trig proof of Pythagoras. This is discussed at some length at cut-the-knot, where a 2009 trig proof by Jason Zimba is outlined. Zimba’s very careful and well-written paper is also available online, courtesy of a kind redditer.
More generally, one has to ask what it means for a proof of Pythagoras to be a “trigonometry proof”, which is also discussed at cut-the-knot. For example, Grant has suggested that any proof of Pythagoras incorporating similarity, of which there are many, might be properly called a trigonometry proof. So, it is reasonable if not very generous to suggest that Johnson and Jackson were not the first to prove something that is not a thing.
But of course what what Johnson and Jackson did is very cool, and this post is not about them. This post is about the reporters, for whom, as always with these stories, there are lessons. In brief:
1) If you’re planning on reporting on a “new proof” then first ask a mathematician.
2) If you’re too lazy to find a mathematician then first check cut-the-knot.
3) Once you realise you’re too lazy to read cut-the-knot, ask a mathematician.
Three simple rules. Which almost no reporter will ever follow.
*) The greatest Head of Mathematics in the history of LaTrobe University.
**) Kudos to AMS for giving Johnson and Jackson the platform.
***) Non-kudos to AMS for not assisting Johnson and Jackson in getting their proof clearly visible in a timely manner.