In May, a trio of mathematicians solved a seemingly simple question about the dodecahedron, but it remained unanswered: imagine one of the vertices of the dodecahedron, a solid with twelve faces and twenty vertices. Is there a direct line that allows us to return to our original position, without going through any of the other nineteen vertices?
The three mathematicians, Jayaved Athreya, David Aulicino and Patrick Hooper, showed in their study, published in the newspaper Experimental Mathematics, that there are an infinite number of these lines in the dodecahedron, contrary to the rule of the other four Platonic solids, since these lines do not exist in the tetrahedron, cube, octahedron and icosahedron.
However, the discovery was not easy: it was necessary to adopt modern techniques and computer algorithms. According to Anton Zorich, a mathematician at the Jussieu Institute of Mathematics in Paris, “this process would have been unthinkable twenty years ago”. Even ten years ago, “it would require a huge effort to write all the necessary software.”
The research project started in 2016, when Athreya and Aulicino were assembling geometric figures with cutouts of letters. While assembling and disassembling the solids, they had the idea of answering the question that remained unanswered about the dodecahedron. They joined Hooper and started a research that would go on for four years.
Mathematicians have been speculating about these direct lines in the dodecahedron for over a hundred years, but interest in this topic has reappeared in recent years, after new discoveries were made about “surface translation”, a geometry technique that is characterized by the formation of surfaces by gluing the parallel sides of a polygon.
The end result of the experiment was a representation of the dodecahedron with ten copies of each pentagon, which resembles “a donut with eighty-one holes”. When analyzing this huge surface, mathematicians came to the conclusion that it also represented one of the most studied surface translations by mathematicians – the “double pentagon”, created by bonding two pentagons.
Given that the double pentagon and the dodecahedron are “geometric cousins”, the high level of symmetry of the first helped to elucidate the structure of the second. The relationship between these surfaces allowed researchers to use an existing surface analysis algorithm. Through this algorithm, the specialists identified and classified all the direct lines that allow to go to and from the same vertex without going through another vertex.
The exception to the rule
This study proved that the dodecahedron is the only Platonic solid in which it is possible to draw direct lines that leave a vertex and return to it, without going through other vertices.
According to Aulicino, this exception is justified because “there are certain symmetries that the dodecahedron does not have, but that the other solids have.” This “lack of symmetries” is what allows these direct lines to be drawn in the dodecahedron and prevents others from having them.
“Platonic solids” have been studied for over two thousand years. Plato associated each of these with a natural element – the cube with the earth, the octahedron with the air, the icosahedron with water and the tetrahedron with fire. The fifth solid – the dodecahedron – was “used by the gods to form the constellations of the sky”, according to the Greek philosopher. Now, after 2380 years, a trio of mathematicians has returned to grant a special status to the geometric solid.
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