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Pandemic Social Distancing Can Help Explain This Geometric Challenge

By LIDA TUNESI

Researchers have come up with an answer to part of a classic geometric challenge called the anti-social jogger problem. Daily life in pandemic times can help us understand this puzzle.

Picture a cube. Now imagine you’re standing at one of the corners. You want to go for a run around your cube-shaped neighborhood, but there are neighbors on every corner. In order to properly social distance, you must create a path back to your starting spot without passing through anyone else’s corner. And you can only travel in a straight line, not even changing directions when you go over an edge.

Mathematicians have already determined that creating a path around a cube under these conditions is impossible. They’ve also shown that it can’t be done with tetrahedrons (a four-faced 3D shape, like a pyramid), octahedrons (eight faces), and icosahedrons (20 faces). But until recently, it was not known whether the challenge could be carried out on a dodecahedron, which has 12 flat faces, each one shaped like a pentagon.

The dodecahedron problem was solved by Professors David Aulicino (Brooklyn CollegeThe Graduate Center) and Jayadev Athreya (University of Washington).  Their paper in The American Mathematical Monthly showed that there is a path that allows the social-distancing jogger to make it home without breaking the rules.
 
Aulicino, Athreya and Professor Patrick Hooper (City College of New YorkThe Graduate Center) then published a second paper in Experimental Mathematics. That research showed 31 different types of routes traversing the dodecahedron, all returning to the original point without passing through other vertices. This research also showed that the other four geometric shapes–which are made up of squares and triangles–have certain kinds of symmetry that force the paths to cross more than one corner. But the dodecahedron with its pentagons doesn’t have those symmetries. 

The research made such a splash in the mathematics world that it was covered by VICE, Wired, and other media outlets. It even got tweeted by former Vermont Gov. Howard Dean.

“I’ve had so many students over the years that say,  ‘What do you mean you do math research? It’s all in the textbook,’” Aulicino told Vice. “It’s really nice to be able to say, ‘Here’s something that we didn’t know, and now we know the answer to it.’”

You can make a dodecahedron by putting two petal-shaped patterns together, and dodecahedron-shaped dice are also widely available. But if you think the problem might be easily solved with a 3D figure, a marker, and a ruler, don’t be fooled. The precision  needed for this problem required the mathematicians to model their question using “nets,” or dodecahedrons taken apart at their edges and unfolded into flat shapes, in combination with advanced computer algorithms. 

Together, cubes, tetrahedrons, octahedrons, icosahedrons, and dodecahedrons are known as Platonic solids because they were first described by Plato. More than 2,000 years later, we’re still pondering their mysteries.