The Jackson-Johnson Proof

About six weeks ago, Ne’Kiya Jackson and Calcea Johnson gave a talk at a meeting of the American Mathematical Society in which they presented a new trigonometric proof of the Pythagorean Theorem. This was remarkable for two reasons—trigonometric proofs of this theorem were once thought to be impossible, and the two students were teenagers still in high school.

Not surprisingly, the accomplishment attracted considerable media attention. But since there was no public recording of the session and no posted paper, the details of their argument remained unclear.

Fortunately, a slide visible in one of the news reports on the event contains enough information for the proof to be reconstructed. The slide reveals an ingenious geometric construction that the students call a waffle cone, and they are able to use this, together with the law of sines and a convergent infinite sequence, to prove the theorem.

A very clear step-by-step video reconstruction of their (likely) argument has been posted online. In a nutshell, what the students seem to have done is this. Starting with an arbitrary right triangle, they reflected it along one edge to get the isosceles triangle you see on the top left of the slide. They then embedded this in a larger right triangle (the waffle cone) by drawing two lines that extend out from the original triangle and eventually meet. Within the waffle cone is an infinite sequence of similar triangles, shrinking at each step by the same factor. Summing the resulting convergent series yields the lengths of the two long sides of the waffle cone. The proof is completed by using the law of sines, applied to the angles in the constructed isosceles triangle (details in the video). Crucially, the law of sines is independent of the Pythagorean Theorem, so there is no circular reasoning involved.

The proof is original, elegant, and undoubtedly correct.

But not everyone was in a mood to celebrate. Here is Charles Murray, conjecturing that the proof will eventually be found invalid, that a publication that proudly announced it would bury this uncomfortable truth, and that Hollywood would spin yet another false narrative:

For someone like Murray, the achievement by these two young women is simply unfathomable. He has to reach for some other explanation.

A few months ago I listened to Charles Murray in conversation with Glenn Loury on the latter’s podcast. I came away convinced that Murray was a racial essentialist. I don’t believe that he can imagine a world in which we have racial parity in homicide rates, or in scientific and engineering accomplishments. It’s not just that he’s pessimistic about attaining such a state given our policy trajectory—he simply does not consider it feasible under any policy regime.

Loury is different. He is fierce critic of policies he believes to be counterproductive, and of leaders he considers misguided or worse. But his analytical starting point has always been the axiom of anti-essentialism with which he began his classic book.

That said, I do not believe that Murray should be censored or de-platformed. There is surely something to be learned from his work. Coming Apart was ahead of the curve in identifying the social disintegration that Anne Case and Angus Deaton would later document in Deaths of Despair. And what happened to Allison Stanger at Middlebury for the crime of wanting to engage Murray was both improper and self-defeating. As Killer Mike put it in his keynote at the recent FIRE Gala, free speech lets me know my enemy.

I do hope that Jackson and Johnson will consider careers in Mathematics. They clearly have the talent for it. Along the way they will encounter many people skeptical of their abilities. Most will not reveal themselves in the crass manner of Murray’s tweet. But listen to these young women speak about their great teachers and their school slogan—no excellence without hard labor—and you will see that they have the fortitude to prevail.