Three mathematicians have an answer to the fundamental question about straight paths on a 12-sided Platonic solid
Despite the fact that mathematicians are already more than 2000 years old [ and, perhaps, even more / approx. transl. ] analyze the structure of five regular polyhedrons (Platonic solids) - tetrahedron, hexahedron (cube), octahedron, dodecahedron and icosahedron - we still do not know much about them.
And so three mathematicians have answered one of the most basic questions about the dodecahedron.
Let's say you are standing on one of the vertices of a regular polyhedron. Is there a direct path along which one can return to the starting point without going through any of the other vertices? For four other regular polyhedra made up of squares or equilateral triangles - tetrahedron, cube, octahedron, and icosahedron - mathematicians have recently givennegative answer to this question. Any straight path starting from one of the vertices will either bump into another vertex, or will forever wind along the surface of the figure, without ever returning to the starting point. However, mathematicians did not know what to expect from a dodecahedron made up of 12 pentagons.
Now Jadev Atreya , David Olicino and Patrick Hooper have shown that there are indeed an infinite number of such paths on the dodecahedron. Their paper , published in May in the journal Experimental Mathematics, shows that these paths can naturally be divided into 31 families.
Finding a solution required the use of modern technology and the compilation of computer algorithms. “About twenty years ago, this question was out of reach; 10 years ago, it would have taken an incredible effort to write all the necessary programs; and only today all the factors came together, ”wrote Anton Zorich from the Jassi Mathematical Institute in Paris in an email.
This project started in 2016, when Atreya from the University of Washington and Olicino from Brooklyn College started playing with a set of flat shapes that folded into a regular polyhedron. During the assembly of the polyhedron, Olicino realized that the material accumulated recently on plane geometry may be useful for understanding straight paths on the dodecahedron. “We literally assembled these pieces from scattered pieces,” said Atreya. "The simple curiosity of the researchers coincided with a new opportunity."
Together with Hooper from City College in New York, the researchers figured out how to classify all straight paths that go out of one corner and come into it, bypassing other corners.
Their analysis is an "elegant solution," as Howard Mazur put it.from the University of Chicago. "This is one of those cases when I can say without hesitation: Wow, why didn't I!"
Hidden symmetries
Although mathematicians have been talking about straight paths on the dodecahedron for over a century, interest in this topic has revived in recent years thanks to new knowledge gained in the field of "transfer surfaces". Such surfaces are formed by gluing the parallel sides of a polyhedron. They have proven to be very useful for exploring a wide range of topics related to straight paths following shapes with angles - from the trajectories of billiard balls to questions about whether a single ray of light can illuminate an entire room with mirrored walls.
The basic idea in all these tasks is to unfold the shape so that it becomes easier to study the paths following it. To understand straight paths along a regular polyhedron, you can start by cutting enough edges so that they can be expanded on a plane, forming, as mathematicians say, a network. One of the networks for a cube, for example, is a "T" -shaped shape consisting of six squares.
A paper dodecahedron made in 2018 by David Olicina and Jadev Atreya to demonstrate the ability to lead a path from one vertex back to it without crossing others.
Imagine that we have made a sweep of the dodecahedron, and now we are walking along it in a certain direction. Sooner or later we will stumble upon an edge of the net, after which our path will jump to the adjacent pentagon (the one that was glued to the current one before we cut our dodecahedron). When jumping, the path simultaneously turns through an angle, the value of which is divisible by 36 in degrees.
To avoid all these jumps and turns, when we meet an edge, we could glue a new, rotated copy of the net onto it and continue walking straight. Then we will add redundancy: we will have two different pentagons, denoting the pentagon of the original dodecahedron. We have complicated our world, but we have simplified our path. We can keep adding a new network every time we need to go beyond the boundaries of our world.
By the time our path passes through 10 nets, we will rotate our original mesh by all possible angles divisible by 36, and the orientation of the next mesh we add will match the one with which we started. It turns out that the 11th network is obtained from the original by a simple shift - as mathematicians say, by transfer. Instead of gluing the 11th mesh, we can simply glue the edge of the 10th mesh to the corresponding parallel edge of the original mesh. Our figure will no longer be flat, but mathematicians believe that it "remembers" the flat geometry of its previous incarnation - so, for example, paths are considered straight if they were straight on the figure not yet glued. After we have done all possible gluing of the corresponding parallel edges, we get the so-called. transfer surface.
Atreia tattooed his favorite transfer surface, the double pentagon, on his right hand.
The resulting surface is a highly redundant representation of the dodecahedron, in which 10 copies of each pentagon are involved. And it turned out to be much more complex - it is glued together in the form of a donut with 81 holes. However, this complex form allowed the three researchers to understand the rich theory of transfer surfaces.
Faced with such a gigantic surface, mathematicians rolled up their sleeves - both figuratively and literally. After working with her for several months, they realized that the 81-hole donut surface forms an over-presentation of not only the dodecahedron, but one of the most commonly studied transfer surfaces. This is a double pentagon, which is obtained by gluing two pentagons along one of the edges, and then gluing all the parallel sides to make a donut with two holes and a large set of symmetries.
Also, this figure is tattooed on the arm of Atreya. “I already knew and loved this double pentagon,” said Atreya, who got the tattoo a year before he and Olicino started thinking about the dodecahedron.
Since the double pentagon and the dodecahedron are geometric cousins, a high degree of symmetry of the former can help to understand the structure of the latter. “This is a terrific latent symmetry,” said Alex Eskin of the University of Chicago (who advised Atreya on his doctoral dissertation 15 years ago). "That the dodecahedron has such a latent symmetry group is quite remarkable."
Jadev Atreya shares how he and his colleagues solved the long-standing problem of finding straight paths on the dodecahedron.
The relationship between these surfaces has allowed researchers to take advantage of the highly symmetric transfer surface analysis algorithm developed by Miriam Finster of the Karlsruhe Institute of Technology. By adapting its algorithm, the researchers were able to find all the direct paths on the dodecahedron that go out and return to one vertex, and classify them based on the hidden symmetries of the dodecahedron.
Atreya describes this analysis as "one of the most interesting projects in my entire career." Jadev says it's very important to constantly play with different things.
The new result suggests that even those objects that people have been studying for thousands of years, there may be secrets hiding, Eskin said. "I think that even for these three mathematicians it was a surprise that they could say something new about the dodecahedron."