Two schoolgirls broke the rules of math by proving the Pythagorean theorem in an "impossible" way.

Two high school students, Kaltseya Johnson and Ne'Kiya Jackson from New Orleans, USA, presented new evidence for the famous Pythagorean theorem. The twist is that this type of proof is considered "impossible." The students relied solely on trigonometry, which is the science of measuring triangles.

THE STARRED PROBLEM

It all began back in 2022 at a school math competition when the girls encountered a certain starred problem. They were asked to prove the Pythagorean theorem using only trigonometry. Did the teacher know that this was impossible? Later, he was asked, and he replied with (as noted by the reporter) "a not-so-good smile":

- I expected the kids to just think it through.

He probably didn't know himself.

But the students unexpectedly succeeded, and the following year they presented at a meeting of the American Mathematical Society, which longtime members cannot recall ever happening before. Recently, their article was finally published in The American Mathematical Monthly, so it seems that the scientific community has acknowledged it.

The excitement surrounding the discovery (or "discovery") has gone beyond all bounds of moderation and decency. "Pythagoras would be proud," the girls "are pushing the boundaries of knowledge and revealing new horizons to humanity," write American mathematical societies on social media. The editor-in-chief of the journal states that "it is an honor for us to publish this article, which aligns with the spirit of our journal's founder." And indeed: the girls, being Black and from New Orleans, fit the mold of the infamous "narrative." Thus, criticizing them (as is customary in science) feels somewhat inappropriate. It seems like one would be offending children, women, and Black individuals all at once. Rare (and mostly anonymous) voices have emerged, claiming that the proof is not original, not accurate, complex, and, in general, does not exist.

The newly minted geniuses, while things were being sorted out, transitioned from schoolgirls to university students. Kiya is studying pharmacy (not mathematics) at the University of Louisiana, and on weekends, she indulges in her many hobbies (cats, rabbits, billiards, anime). Kaltseya has enrolled in college to become an "environmental engineer" (something between a landscape designer and an ecologist). She struggles with hobbies but spends time in theaters and is passionate about K-pop. Just like our Perelman, showing humility in everyday life. Or maybe the clever girls (there's no doubt about that) realized they had become instruments of narrative propaganda and want to distance themselves?

Let’s try to figure out what this discovery is all about, but it won’t be easy. In the scientific press, you’ll read, for instance, that even the girls' names were predicted by Providence, as there is still a popular Western musical from 1954 called "Carmen Jones," where the heroines are named just that. And there’s analysis. So we’ll have to dig into it ourselves. Well, let’s give it a shot.

FAMOUS FOR HIS PANTS

The square of the hypotenuse is equal to the sum of the squares of the legs. We remember that from childhood. Has it come in handy in life? It depends.

This is indeed a theorem, not an axiom. In other words, it needs to be proven. However, it seems that it was initially derived from practice, and little thought was given to the proof.

References to relationships in right triangles can be found in Babylonian tablets and Egyptian papyri from the 20th to 15th centuries BC. It seems that people first discovered amusing relationships between numbers, which were later called "Pythagorean triples." These are three numbers for which the rule

A (squared) + B (squared) = C (squared)

is evident. For example, 3, 4, and 5. Indeed:

Three squared (9) + Four squared (16) = Five squared (25).

It’s easy to verify: if you construct a triangle with such sides, it will turn out to be right-angled. Wow, thought the ancient Babylonians. The nuance is that they constantly had to construct various figures. And not on paper, but on the ground, with sticks and ropes. The precious arable land of Mesopotamia was always a subject of disputes; it was chopped up and divided, and land surveyors were called in. Noticing a convenient relationship, the cadastral engineers took note of it.

Pythagoras (who lived in the 6th-5th centuries BC) is said to have devised rules for finding such combinations of numbers, and where there is a rule, there is also a theorem, but it remains unknown whether Pythagoras actually had any connection to the theorem bearing his name.

No writings from Pythagoras have survived. Judging by the amount of noise he made in history, he was an extraordinary personality. Upon returning from Babylon, he devised the concept of reincarnation, often leaving his body and appearing to his students simultaneously in different parts of the city, descending alive into the underworld, claiming to hear the music of the spheres with his ears (and invented seven notes corresponding to the seven planets). The Greek thinker remains relevant even today. Conspiracy theorists write that he was a "Babylonian agent" who created a world government and infected Western civilization with destructive ideas. COVID? That was him too.

All of this is wonderful, but could a mystic, showman, and Babylonian agent substantiate a theorem? That’s the question.

No writings from Pythagoras have survived

Photo: EAST NEWS.

In the 3rd century BC, the fundamental work of Euclid, "Elements," was published, which was studied in mathematics until the 20th century (and it would still be good for students to read it, but it’s complicated), and there is a proof in it. It involves drawing squares on the triangle, a total of three squares adjacent to the three sides. It is proved that the areas of the squares are equal, thus substantiating the entire theorem. What is the square, say, of the hypotenuse, if not the area of the square adjacent to the hypotenuse? Early mathematics often thought in terms of figures rather than numbers. This is unusual for us.

The triangle drawn with squares by Euclid is called the "Pythagorean pants," which, as we know, "are equal in all directions." It really does look like pants. Schoolchildren a hundred years ago knew exactly what was being referred to, but then they stopped studying "pants," and the saying remained; eventually, it was forgotten too. I asked some modern schoolchildren (of course, my "survey" does not claim to be exhaustive), and very few had heard of "pants," and those who had thought it was multiplication tables.

EVEN EINSTEIN TRIED

Since then, the Pythagorean theorem has been proven in various ways about 400 times. No one knows the exact number. I came across a book titled "371 Ways to Prove the Pythagorean Theorem," but it clearly does not include everything. There are websites dedicated solely to proofs of the Pythagorean theorem.

Our schoolgirls have not yet made it into the statistics. Moreover, their article is titled "Five or Ten New Proofs of the Pythagorean Theorem." Whether it’s five or ten, who’s counting.

Among the greats who have worked on the right triangle are figures like Albert Einstein, and Stephen Hawking wrote a lot about it. Of course, there are proofs based on complex numbers; how could it be otherwise (it’s amusing that in St. Petersburg they say "complEksnye" numbers, while in Moscow they say "kOmplexnye," oh, this eternal debate over curbs). The theorem has even been proven by AI. Exciting!

In general, proofs can be divided into algebraic (that is, using formulas) and geometric (that is, using diagrams), although in reality, something is usually drawn first, and then various numbers are written down, meaning they are essentially mixed proofs.

However, proving the Pythagorean theorem using trigonometry is considered particularly chic. But what is "trigonometry"?

Schoolchildren are convinced that it is simply another name for geometry. The Wikipedia definition, as often in that "encyclopedia," is perfect: "Trigonometry is a branch of mathematics that studies trigonometric functions." Which cannot be disputed.