In the fall of 2021, Malors Espinosa, an alumnus and current post-doctoral fellow in the Department of Mathematics, 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 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 do math when there is no answer key. They just needed the right question to guide them.
Malors finally found one while reading a textbook about chaos. In its pages, he came across a familiar object: a fractal, or self-similar shape, called the Menger sponge, which has a simple but elegant construction. First divide a cube into what looks like a Rubik’s cube. Remove the cube in the very center, along with the center cube of each of the six faces. Then repeat this process for each of the 20 remaining cubes. And repeat. And repeat. You’ll quickly see why the resulting fractal is called a sponge: With each iteration, its pores multiply exponentially.
Ever since Karl Menger introduced his fractal sponge nearly a century ago, it has captured the imaginations of professional and amateur mathematicians alike. One reason: It looks cool. In 2014, hundreds of math enthusiasts participated in a global effort, called MegaMenger, to build finite, 200-pound versions of the sponge out of business cards. Because of its porous, foam-like structure, the sponge has also been used to model shock absorbers and exotic forms of space-time.
But most important, the fractal possesses various counterintuitive mathematical properties. Continue to pluck out ever smaller pieces, and what started off as a cube becomes something else entirely. After infinitely many iterations, the shape’s volume dwindles to zero, while its surface area grows infinitely large. Such is the weirdness of fractals: hovering somewhere between dimensions, occupying space without truly filling it.
When he first defined his sponge in 1926, Menger also proved that any conceivable curve — simple lines and circles, structures that look like trees or snowflakes, fractal dusts — can be deformed and then embedded somewhere on the sponge. They can be made to wind their way along the sponge’s convoluted contours without ever leaving its surface, hitting a hole, or intersecting themselves. The sponge, Menger wrote, was therefore a “universal curve.”
But this, Malors later realized, raised a new question. Menger had proved that you could find a circle in his sponge. But what about objects that were equivalent, in a certain sense, to the circle? Consider a mathematical knot: a string that’s been twisted and tied up, its ends then closed to form a loop. From the outside, it might look like a tangled mess. But an ant walking along it would eventually find itself back where it started, just as it would on a circle. In this way, every knot is equivalent, or “homeomorphic,” to a circle.
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.
Read the complete article in Quanta Magazine.