Mathematicians Report New Discovery About the Dodecahedron | Mathematics - Quanta Magazine

Erica Klarreich, Freelance Mathematics and Science Journalist - Berkeley California argues, Three mathematicians have resolved a fundamental question about straight paths on the 12-sided Platonic solid.

An illustration of an ant walking in a straight line around a dodecahedron.
Photo: Samuel Velasco/Quanta Magazine

Even though mathematicians have spent over 2,000 years dissecting the structure of the five Platonic solids — the tetrahedron, cube, octahedron, icosahedron and dodecahedron — there’s still a lot we don’t know about them.

Now, a trio of mathematicians has resolved one of the most basic questions about the dodecahedron. 

Suppose you stand at one of the corners of a Platonic solid. Is there some straight path you could take that would eventually return you to your starting point without passing through any of the other corners? For the four Platonic solids built out of squares or equilateral triangles — the cube, tetrahedron, octahedron and icosahedron — mathematicians recently figured out that the answer is no. Any straight path starting from a corner will either hit another corner or wind around forever without returning home. But with the dodecahedron, which is formed from 12 pentagons, mathematicians didn’t know what to expect. 

Now Jayadev Athreya, David Aulicino and Patrick Hooper have shown that an infinite number of such paths do in fact exist on the dodecahedron. Their paper, published in May in Experimental Mathematics, shows that these paths can be divided into 31 natural families...

By the time our path has traveled through 10 nets, we’ve rotated our original net through every possible multiple of 36 degrees, and the next net we add will have the same orientation as the one we started with. That means this 11th net is related to the original one by a simple shift — what mathematicians call a translation. Instead of gluing on an 11th net, we could simply glue the edge of the 10th net to the corresponding parallel edge in the original net. Our shape will no longer lie flat on the table, but mathematicians think of it as still “remembering” the flat geometry from its previous incarnation — so, for instance, paths are considered straight if they were straight in the unglued shape. After we do all such possible gluings of corresponding parallel edges, we end up with what is called a translation surface. 
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Additional resources
doi.org/10.1080/10586458.2020.1712564

Source: Quanta Magazine