The symmetries of regular polyhedra in general relativity and contradictions with quantum mechanics

We show that every spacetime determined by a non-spherically symmetric body with the symmetry of any one of the five Platonic solids has a curvature tensor which vanishes in a neighbourhood of the centre of this symmetry. This means that every non-spherically symmetric object, be it a star, a crystal or an atom, with the exact symmetry of one of these polyhedra must have an empty Centre of symmetry. This result puts forward contradictions between General Relativity and Quantum Mechanics, when both theories are applied to the atomic scale. PACS 04.20 - Classical general relativity. PACS 04.60 - Quantum gravity.