Teen Mathematicians Tie Knots Through a Mind-Blowing Fractal
upstart writes:
Teen Mathematicians Tie Knots Through a Mind-Blowing Fractal:
In the fall of 2021, Malors Espinosa set out to devise a special type of math problem. As with any good research question, it would have to be thought-provoking, its solution nontrivial - something others would want to study. But an additional constraint stumped him. Malors, then a graduate student in mathematics at the University of Toronto, wanted high school students to be able to prove it.
For years, Malors had been running summer workshops for local high schoolers, teaching them about basic ideas in mathematical research and showing them how to write proofs. But a few of his students seemed ready to do more - to find out what it means to domath when there is no answer key. They just needed the right question to guide them.
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Menger's statement didn't distinguish between homeomorphic curves. His proof only guaranteed, for instance, that the circle could be found in his sponge - not that all homeomorphic knots could be, their loops and tangles still intact. Malors wanted to prove that you could find every knot within the sponge.
It seemed like the right mashup to excite young mathematicians. They'd recently had fun learning about knots in his seminar. And who doesn't love a fractal? The question was whether the problem would be approachable. "I really hoped there was an answer," Malors said.
There was. After just a few months of weekly Zoom meetings with Malors, three of his high school students - Joshua Broden, Noah Nazareth and Niko Voth - were able to show that all knots can indeed be found inside the Menger sponge. Moreover, they found that the same can likely be said of another related fractal, too.
"It's a clever way of putting things together," said Radmila Sazdanovic, a topologist at North Carolina State University who was not involved in the work. In revisiting Menger's century-old theorem, she added, Malors - who usually does research in the disparate field of number theory - had apparently asked a question that no one thought to ask before. "This is a very, very original idea," she said.
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