Platonic solids symmetry in particle physics

Non-linearity of the Yang-Mills fields caused by the commutator in the Lagrangian calls attention to symmetries helping to reveal important features of the system, while obviating the need of actual solutions. To this end, a charge fixing procedure is carried out by means of counting photon oscillations between test particles moving along trajectories that form Platonic solids shaped patterns. With the infinite-range electromagnetic interaction as its origin, the procedure defines a class of interactions allowed for particle physics. Besides the familiar electromagnetic, weak and strong interactions, the scheme suggests an unusual force that cannot be detected with conventional means though possibly contributes to gravitation.