Source: Quanta, Aug 2020
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.
The solution required modern techniques and computer algorithms. “Twenty years ago, [this question] was absolutely out of reach; 10 years ago it would require an enormous effort of writing all necessary software, so only now all the factors came together,” wrote Anton Zorich, of the Institute of Mathematics of Jussieu in Paris, in an email.
Although mathematicians have speculated about straight paths on the dodecahedron for more than a century, there’s been a resurgence of interest in the subject in recent years following gains in understanding “translation surfaces.” These are surfaces formed by gluing together parallel sides of a polygon, and they’ve proved useful for studying a wide range of topics involving straight paths on shapes with corners, from billiard table trajectories to the question of when a single light can illuminate an entire mirrored room.
In all these problems, the basic idea is to unroll your shape in a way that makes the paths you are studying simpler. So to understand straight paths on a Platonic solid, you could start by cutting open enough edges to make the solid lie flat, forming what mathematicians call a net. One net for the cube, for example, is a T shape made of six squares.
a highly redundant representation of the dodecahedron, with 10 copies of each pentagon. And it’s massively more complicated: It glues up into a shape like a doughnut with 81 holes.
To tackle this giant surface, the mathematicians rolled up their sleeves — figuratively and literally. After working on the problem for a few months, they realized that the 81-holed doughnut surface forms a redundant representation not just of the dodecahedron but also of one of the most studied translation surfaces.
Called the double pentagon, it is made by attaching two pentagons along a single edge and then gluing together parallel sides to create a two-holed doughnut with a rich collection of symmetries.
Because the double pentagon and the dodecahedron are geometric cousins, the former’s high degree of symmetry can elucidate the structure of the latter.
The relationship between these surfaces meant that the researchers could tap into an algorithm for analyzing highly symmetric translation surfaces developed by Myriam Finster of the Karlsruhe Institute of Technology in Germany.
By adapting Finster’s algorithm, the researchers were able to identify all the straight paths on the dodecahedron from a corner to itself, and to classify these paths via the dodecahedron’s hidden symmetries.