  Nearly all field theories suffer from singularities when particles are
introduced. This is true in both classical and quantum physics. Classical field
singularities result in the notorious self-force problem, where it is unknown
how the dynamics of a particle change when the particle interacts with its own
(self) field. Self-force is a pressing issue and an active research topic in
gravitational phenomena, as well as a source of controversies in classical
electromagnetism. In this work, we study a hidden geometrical structure
manifested by the electromagnetic field-lines that has the potential of
eliminating all singularities from classical electrodynamics. We explore
preliminary results towards a consistent way of treating both self- and
external fields.
