K1TTT wrote:

you can have a spherically symetric static electric field as is easily
shown by gauss's law. but in order to have 'radiation' (implying em
wave propagating through space) you must have movement of some kind,
that immediately removes the spherical symetry by creating an axis
defined by the direction of movement. this is why even the
theoretical infinitesimal dipole still produces a doughnut shaped
field in free space.

That's a very clever qualitative explanation of, for, instance, why
isotropic antennas cannot exist. (without resorting to things like the
Hairy Ball Theorem, which is great for explaining polarization)