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

TL;DR

This paper explores a Diophantine equation related to regular polyhedra and surfaces, revealing connections to Platonic solids, equivelar maps, and Lie algebra classifications, including the McKay correspondence and Y-objects.

Contribution

It provides a novel interpretation of a Diophantine equation in terms of polyhedra, surfaces, and Lie algebra structures, linking geometric and algebraic classifications.

Findings

The equation describes Platonic solids with Euler characteristic 2.

It relates equivelar maps on genus 2 surfaces to positive solutions.

Coincidence with the McKay correspondence for the icosahedron and E8.

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 $\chi=\pm 2$. In the case of negative $m$ this equation corresponds to $\chi=2$ and describes true regular polyhedrons, Platonic solids. The case with positive $m$ corresponds to Euler characteristic $\chi=-2$ 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.