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We notice that one of the Diophantine equations, $knm=2kn+2km+2nm$, arisingin the universality originated Diophantine classification of simple Liealgebras, has interesting interpretations for two different sets of signs ofvariables. In both cases it describes "regular polyhedrons" with $k$ edges ineach vertex, $n$ edges of each face, with total number of edges $|m|$, andEuler characteristics $\chi=\pm 2$. In the case of negative $m$ this equationcorresponds to $\chi=2$ and describes true regular polyhedrons, Platonicsolids. The case with positive $m$ corresponds to Euler characteristic$\chi=-2$ and describes the so called equivelar maps (charts) on the surface ofgenus $2$. In the former case there are two routes from Platonic solids tosimple Lie algebras - abovementioned Diophantine classification and McKaycorrespondence. We compare them for all solutions of this type, and findcoincidence 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 theinterpretation of (some of) the mysterious solutions (Y-objects), appearing inthe Diophantine classification and having some similarities with simple Liealgebras.

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