>>14383363in electromagnetics it prevents the existence of an isotropic radiator
this is a straight forward consequence of the hairy ball theorem. it is well known that as em waves get far from their source the component of the fields in the radial direction go to zero much faster than the components of the tangential/angular directions; thus really far away from the source you get a spherical surface where the only appreciable field is tangential to the surface. power density S is a vector quantity equal to the cross product of E with B. putting these facts together, you can't have S be non-zero over the entire surface of the ball.
this means that you can't build an antenna that radiates electromagnetic energy in all directions equally, which is of practical interest.
