Compelling evidence supports the claims of two New Orleans high school seniors who say they have found a new way to prove Pythagoras’s theorem by using trigonometry, a respected mathematics professor said, even if the students’ “really important and fantastic” achievement is not the first time trigonometry has been used to prove the theory, as their school apparently touted.
Álvaro Lozano-Robledo, of the University of Connecticut, spoke this week in a series of TikTok videos, addressing international reports about Calcea Rujean Johnson and Ne’Kiya Jackson.
Johnson and Jackson, students at St Mary’s Academy, recently gave a presentation at a regional meeting of the American Mathematical Society outlining their discovery.
The 2,000-year-old Pythagorean theorem states that the sum of the squares of a right triangle’s two shorter sides is the same as the square of the hypotenuse, the third side opposite the right angle. The notation associated with the theorem – a2+b2=c2 – is something encountered in many a geometry class.
For generations, mathematicians maintained that any alleged proof of the Pythagorean theorem based in trigonometry would constitute a logical fallacy known as circular reasoning: seeking to validate an idea with the idea itself.
In the abstract for their 18 March talk in Atlanta, at an event that drew presenters from prominent universities, Johnson and Jackson noted that the book thought to hold the largest known collection of proofs for the theorem, The Pythagorean Proposition by Elisha Loomis, “flatly states that ‘there are no trigonometric proofs because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean theorem’”.
But Johnson and Jackson said they found a way to use the trigonometry law of sines to prove Pythagoras’s theory in a way “independent of the Pythagorean trig identity sin2x+cos2x=1” – without resorting to circular reasoning.
St Mary’s issued a press release about the findings that was reported out by a local television station, WWL. Other outlets picked up the story, but skeptics scrambled to check whether Johnson and Jackson had really done something many advanced mathematicians had not managed.
As of Friday, Johnson and Jackson did not appear to have widely released their proof. The American Mathematical Society has only said it has encouraged the pair to submit their work to a peer-reviewed journal. But a YouTube account, MathTrain, reconstructed the proof using slides from Johnson and Jackson’s presentation visible in the WWL report.
Lozano-Robledo reviewed MathTrain’s reconstruction, broke it down in his own video and concluded that the students had done what they said.
In a follow-up video, he summarized how the proof involved “a fractal of similar triangles” as well as “infinite series” to compute the shapes’ sides.
“It’s so ingenious,” Lozano-Robledo said. “The proof itself is just so beautiful and so elegant.”
But Lozano-Robledo also said people who pointed to at least one other trigonometric, noncircular proof of Pythagoras’s theory were correct to do so.
Jason Zimba, then at Bennington College in Vermont, established in 2009 that sin2x+cos2x=1 could be derived independently of the Pythagorean theorem, though he took a different route.
In text under his video, Lozano-Robledo said it was not Johnson and Jackson’s fault that people had the impression they were claiming to have done something not done in more than 2,000 years. He said the students did not say that in their abstract.
Johnson and Jackson’s abstract said: “In the 2,000 years since trigonometry was discovered, it’s always been assumed that any alleged proof of Pythagoras’s Theorem based on trigonometry must be circular.”
St Mary’s Academy then issued a press release which led with that sentence, under a title containing the words “unprecedented research”.
Nonetheless, Lozano-Robledo made clear that by all indications Johnson and Jackson had arrived at a valid new Pythagorean theorem proof, an accomplishment worthy of celebration. The eastern section of New Orleans often makes the news for reasons including deadly violence or political corruption, but now two children there have apparently made a discovery of which many researchers dream.
Lozano-Robledo said new proofs occasionally give mathematicians insights that can be applied in other settings or equip researchers with a tool to prove other theorems previously thought unprovable. One of the main points of teaching established knowledge, he said, is to inspire new solutions.
“It’s just so invigorating – it makes me so happy to see teenagers thinking about new math and coming up with something really important and something fantastic,” Lozano-Robledo said. “It’s just great.”
In an interview published on Thursday by the Times-Picayune, Jackson said she and Johnson began plotting out their proof thanks to a math contest set for the Christmas break.
“We were the only students that answered the bonus question,” Jackson said. “It asked you to create a proof for the theorem and asked a bunch of questions about how you would move from one step to the next.”
Jackson said she planned to study pharmacy or anesthesiology at college. Johnson, who has more than 25 acceptance letters and about $1.2m in scholarship offers, said she wanted to be an environmental engineer.
“I’ve always had a passion for problem solving,” she said.
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