# Humans Have Pondered These Geometric Shapes for 2,000 Years. These Two Professors Shed Light on Their Mysteries

Dodecahedrons. (Credit: Getty Images.)

By BETH HARPAZ
Editor of SUM

Geometry research doesn’t often get covered by mainstream news. But CUNY mathematicians who solved a classic problem saw their work featured by VICE and Wired, and they even earned a tweet from former Vermont Governor Howard Dean

The challenge involved charting a straight-line trajectory around 3D shapes known as Platonic solids, starting and ending from the same vertex while avoiding all other vertices. Professors David Aulicino (GC/Brooklyn College, Mathematics) and Jayadev Athreya (University of Washington) showed that the challenge can be solved for dodecahedrons (which look like 12-sided dice), but not for the other Platonic solids (cubes, tetrahedrons, octahedrons, and icosahedrons). Working with Professor Patrick Hooper (GC/City College, Mathematics), they achieved a second breakthrough, coming up with 31 different types of trajectories around the dodecahedron.

Aulicino and Hooper spoke to The Graduate Center about their collaboration and their breakthrough.

Aulicino: Jayadev (Athreya) and I had found the closed trajectory on the dodecahedron and we were trying to classify all of the trajectories. We had hit a wall and couldn't get past the technical and computational challenges that we faced. Pat (Hooper) and I have known each other for years and we'd catch up with each other during weekly seminars at the GC. I was explaining to him all of these problems, and for him, he was able to work right through them. He joined our collaboration and we completed the classification.

Professors David Aulicino and Patrick Hooper

Hooper: The discovery David and Jayadev made was really beautiful, and I became pretty excited once I heard there was more to be done. Prior to the collaboration, I had spent a lot of time contributing to a computer program to draw and analyze geometric surfaces like these. It was great that this tool I had helped develop could be used here.

GC: Was there a eureka moment for you and if so, what was that like?

Aulicino: There were a lot of small steps that added up to the final result. Especially with the classification of 31 trajectories, we had so many missteps that gave us the wrong number that when we finally got the right answer, we didn't know until after we checked everything 10 more times.

GC: What does it feel like to solve a big puzzle like this?

Aulicino: It was really gratifying to get a complete answer to this problem. In this case, we can theoretically find every single closed trajectory on the dodecahedron. Of course, it would take forever, but we know exactly how to get every one as a result of our work. Personally, I like the idea of finishing off a problem, and this fit into that desire.

Hooper: I don’t think it is reasonable to say we solved a big puzzle. We did realize that there were things not understood about the dodecahedron (and by extension, the Platonic solids) and we figured out how to understand these things. These are beautiful objects, and it was a lot of fun to delve more deeply into them.

GC: Why are Platonic solids so fascinating to us 2,000 years after Plato, and what is it about them that feels so cosmic?

Hooper: The Platonic solids are accessible because anyone can recognize their beauty. For mathematicians, the amazing thing about the Platonic solids is that you have a natural concept of a maximally symmetric polyhedron, and surprise: There are only five solids that realize this maximal symmetry. Symmetry is a kind of beauty, and I think especially given historical connections between art and science, the Platonic solids have featured prominently in the minds of famous scientists including Leonardo da Vinci and Kepler.

GC: What advice do you have for fellow researchers looking to break through the noise and get more attention for their discoveries?

Aulicino: I would say don't do it for the press. I wanted to solve a fun problem and I'm humbled that the result has gotten the attention that it did. It can be very random whether or not a result receives attention. The focus should always be to do work that you love and care about.

Hooper: I think this experience will change how I prioritize research topics. There is a lot of super interesting mathematics done all the time, but most requires a lot of mathematical background to appreciate. Accessibility needs to be a priority for mathematicians seeking attention.

GC: Professor Aulicino, tell us about your work with high school students.

Aulicino: When I was in high school in Armonk, NY, I  got to work on a research project on graph theory with a SUNY Purchase professor, Marty Lewinter (in fact, he is an alumnus of both Brooklyn College and The Graduate Center). He inspired me to pursue math. I grew tremendously from this experience. I very much believe in “paying it forward.” It’s a really special opportunity for me to work with the high school kids that I’m working with in particular. They are from the STAR Early College school in Flatbush whose students are predominantly from backgrounds that are underrepresented in STEM. The students are at this school exactly because of their interest in science and math, so they are very smart. It’s been a pleasure to teach them about math and math research because I’m the first one that’s showing them that math is a vibrant and active pursuit.

Beth Harpaz is the editor of SUM. Follow her on Twitter at @literarydj.

Submitted on: OCT 13, 2020

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