27/10/2020

Mathematicians unravel the secret of the dodecahedron.

Ilustração: Quanta Magazine

There are more questions about the five Platonic solids—tetrahedron, cube, octahedron, icosahedron, and dodecahedron—than our vain philosophy dreams of. Just ask the mathematicians, who have spent over 2,000 years trying to decipher the secrets these structures hold. Recently, researchers Jayadev Athreya of the University of Washington, David Aulicino of Brooklyn College, and Patrick Hooper of the City College of New York solved one of the most basic questions about the dodecahedron. Suppose you are at one of the corners of a Platonic solid. Is there a straight path you could take back to your starting point without passing through any of the other corners?

In the four Platonic solids constructed from squares or equilateral triangles, mathematicians discovered that the answer is no. Regardless of the possible alternatives, any straight path starting from a curve will either reach another curve or rotate indefinitely without returning to the starting point. But with the dodecahedron, formed by 12 pentagons, mathematicians showed that there is an infinite number of paths.

The article that brought the mathematical breakthrough was published in May in Experimental Mathematics, proving 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. Ten years ago, it would have required an enormous effort to write all the necessary software, so only now have all the factors come together,” Anton Sorich, from the Institute of Mathematics of Jussieu-Paris, told Quanta Magazine .

The project began in 2016, starting with a playful experiment using a collection of cardboard cutouts that fold into Platonic solids. "It was a kind of idle exploration that found an opportunity," Athreya told the magazine.

Speculation about possible straight paths in the dodecahedron has gained momentum in recent years, following gains in the understanding of "translation surfaces." This type of surface is formed by gluing together parallel sides of a polygon and has proven useful for studying a wide range of topics involving straight paths in shapes with corners, from billiard table trajectories to the question of when a single light can fully illuminate a mirrored room.

These problems share the basic idea of unfolding their shape so that the paths you are studying are simplified. To understand straight paths in a Platonic solid, you can start by cutting edges open enough to bring the solid flat, forming what mathematicians call a lattice. A lattice for the cube, for example, is a "T" shape made of six squares.

Want to know more about the discovery? Check out the video!