Suppose someone asks you to distribute 4 points on the sphere in a nice and regular way. You would probably pick vertices of a tetrahedron. A cube for 8, a dodecahedron for 12 and an icosahedron for 20. But what if someone asks you to distribute 21 points? We are out of Platonic Bodies! I will discuss an old idea of Sobolev who proposed to put them in such a way that the average of polynomials in these points coincides with the average of polynomials on the sphere for as many polynomials as possible. This recovers the Platonic Bodies and suggests many questions. I will discuss some of them, make a quick detour into Analytic Number Theory and then tell you how to do it on Graphs. This allows us, for any Graph, to define “Platonic Bodies in the Graph”. These are absolutely gorgeous, very beautiful sets – I promise you many nice pictures and many open problems!