Mathematicians have spent over 2,000 years dissecting the structure of the five Platonic solids – the tetrahedron, cube, octahedron, icosahedron, and dodecahedron – but there is still a lot we don’t know about them.
Now a trio of mathematicians have solved one of the most fundamental questions about the dodecahedron.
Suppose you are standing at one of the corners of a Platonic solid. Is there a straight path you could take that would eventually bring you back to your starting point without going through any of the other corners? For the four Platonic solids constructed from equilateral squares or triangles – the cube, tetrahedron, octahedron, and icosahedron – mathematicians recently understood that the answer is no. Any straight path starting from a bend will collide with another bend or curl up forever without returning home. But with the dodecahedron, which is made up of 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 actually exist on the dodecahedron. Their paper, published in May in Experimental mathematics, shows that these paths can be divided into 31 natural families.
The solution required modern techniques and computer algorithms. “Twenty years ago, [this question] was absolutely out of reach; 10 years ago, it took a huge effort to write all the necessary software, so it’s only now that all the factors are in place, ”wrote Anton Zorich, of the Jussieu Institute of Mathematics in Paris, in an email.
The project began in 2016 when Athreya, from the University of Washington, and Aulicino, from Brooklyn College, began playing with a collection of card stock cutouts that fold into Platonic solids. As they constructed the various solids, Aulicino realized that a body of recent research on plane geometry might be just what they would need to understand the straight paths on the dodecahedron. “We literally put these things together,” Athreya said. “So it was a sort of idle exploration that meets an opportunity.”
In collaboration with Hooper, of the City College of New York, the researchers discovered how to classify all the straight paths from one corner to itself while avoiding the other corners.
Their analysis is “an elegant solution,” said Howard Masur of the University of Chicago. “It’s one of those things where I can say, without any hesitation, ‘Damn, oh, I wish I had done that!’ ‘
Although mathematicians have speculated on the straight paths on the dodecahedron for over a century, there has been a resurgence of interest in the subject in recent years following advances in the understanding of “translation surfaces.” These are surfaces formed by gluing the parallel sides of a polygon together, and have been found useful for studying a wide range of topics involving straight paths on shapes with corners, from pool table trajectories to the question of when one light can illuminate an entire room in mirror.
In all of these problems, the basic idea is to unfold your shape in a way that simplifies the paths you are studying. So, to understand straight paths on a Platonic solid, you can start by cutting edges open enough that the solid is flat, forming what mathematicians call a fillet. A net for the cube, for example, is a T shape made up of six squares.
Imagine that we have flattened the dodecahedron, and now we are walking along this flat shape in a chosen direction. Eventually we will reach the edge of the net, at which point our path will jump to a different pentagon (the one that was glued to our current pentagon before cutting the dodecahedron). Each time the path jumps, it also rotates by a multiple of 36 degrees.
To avoid all this jumping and spinning, when we hit an edge of the rule, we could instead paste on a new rotated copy of the rule and continue straight into it. We added some redundancy: we now have two different pentagons representing each pentagon on the original dodecahedron. So we made our world more complicated, but our path became easier. We can keep adding a new network whenever we need to expand beyond the limits of our world.