Plato's Fire and the Neutrino Mass Matrix

With the accumulation of many years of solar and atmospheric neutrino oscillation data, the approximate form of the 3 × 3 neutrino mixing matrix is now known. The theoretical challenge is to understand where this mixing matrix comes from. Recently, a remarkable fact was discovered that for a specific pattern of the neutrino mass matrix at a high scale, any flavor-changing radiative correction will automatically lead to the desired mixing matrix. It was also discovered that the required specific pattern at the high scale can be maintained by the non-Abelian discrete symmetry A 4 which is also the symmetry group of the regular tetrahedron, one of the five perfect geometric solids known to Plato who associated it with the element "fire". We discuss this recent development and add to it a new and very simple mechanism for the implementation of the flavor-changing radiative correction.