Two years ago, a pair of high school classmates each crafted a mathematical marvel, a trigonometric proof of the Pythagorean theorem. Now, they are unveiling 10 more. For over 2,000 years, such proofs were deemed impossible. However, undeterred, Ne’Kiya Jackson and Calcea Johnson published their new proofs on October 28 in the American Mathematical Monthly.

“Some people believe that you must spend years in academia before you can produce new mathematics,” says mathematician Álvaro Lozano-Robledo of the University of Connecticut in Storrs. But, he notes, Jackson and Johnson show that “you can make a significant impact even as a high school student.” Jackson is now a pharmacy student at Xavier University of Louisiana in New Orleans, while Johnson is studying environmental engineering at Louisiana State University in Baton Rouge.

Mathematical proofs are sequences of statements that demonstrate whether an assertion is true or false. Pythagoras’ theorem — a² + b² = c², relating the length of a right triangle’s hypotenuse to the lengths of its other two sides — has been proven many times using algebra and geometry. But in 1927, mathematician Elisha Loomis claimed that the feat could not be achieved using trigonometry, a subset of geometry dealing with the relationships between angles and side lengths of triangles. He thought that Pythagoras’ theorem is so fundamental to trigonometry that any trigonometry-based attempt to prove it would have to assume its truth, leading to circular logic.

Jackson and Johnson conceived their first trigonometry-based proof in 2022, while seniors at St. Mary’s Academy in New Orleans, a Catholic school primarily attended by young Black women. At that time, only two other trigonometric proofs of Pythagoras’ theorem existed, presented by mathematicians Jason Zimba and Nuno Luzia in 2009 and 2015, respectively. Working on these early proofs “ignited the creative process,” Jackson says, “and from there we developed additional proofs.”

After formally presenting their work at an American Mathematical Society meeting in March 2023, the duo aimed to publish their findings in a peer-reviewed journal. “This proved to be the most challenging task of all,” they stated in the paper. In addition to writing, the duo had to acquire new skills, all while entering college. “Learning how to code in LaTeX [a typesetting software] is not straightforward when you’re also trying to write a 5-page essay with a group, and submit a data analysis for a lab,” they wrote.

Nevertheless, they were determined to complete what they had started. “It was crucial to me to have our proofs published to confirm that our work is correct and respectable,” Johnson says.

According to Jackson and Johnson, trigonometric terms can be defined in two different ways, which can complicate efforts to prove Pythagoras’ theorem. By focusing on just one of these methods, they developed four proofs for right triangles with sides of varying lengths and one for right triangles with two equal sides. Among these, one proof particularly stands out to Lozano-Robledo. In it, the students fill one larger triangle with an infinite sequence of smaller triangles and use calculus to determine the lengths of the larger triangle’s sides. “It looks unlike anything I’ve ever seen,” Lozano-Robledo says.

Jackson and Johnson also leave five more proofs “for the interested reader to explore,” they wrote. The paper includes a lemma — a sort of stepping-stone to proving a theorem — that “provides a clear path towards the additional proofs,” Johnson says. Now that the proofs are published, “others might take the paper and generalize those proofs, or extend their ideas, or use their ideas in other ways,” Lozano-Robledo says. “It just opens up many mathematical discussions.”

Jackson hopes that the paper’s publication will inspire other students to “realize that obstacles are part of the journey. Persevere, and you might achieve more than you ever imagined.”

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