Diophantine equations, Platonic solids, McKay correspondence, equivelar maps and Vogel's universality

Abstract

We notice that one of the Diophantine equations, $knm=2kn+2km+2nm$, arising in the universality originated Diophantine classification of simple Lie algebras, has interesting interpretations for two different sets of signs of variables. In both cases it describes "regular polyhedrons" with $k$ edges in each vertex, $n$ edges of each face, with total number of edges $|m|$, and Euler characteristics $χ=\pm 2$. In the case of negative $m$ this equation corresponds to $χ=2$ and describes true regular polyhedrons, Platonic solids. The case with positive $m$ corresponds to Euler characteristic $χ=-2$ and describes the so called equivelar maps (charts) on the surface of genus $2$. In the former case there are two routes from Platonic solids to simple Lie algebras - abovementioned Diophantine classification and McKay correspondence. We compare them for all solutions of this type, and find coincidence in the case of icosahedron (dodecahedron), corresponding to $E_8$ algebra. In the case of positive $k$, $n$ and $m$ we obtain in this way the interpretation of (some of) the mysterious solutions (Y-objects), appearing in the Diophantine classification and having some similarities with simple Lie algebras.