  We show that magnetic zero-modes of the Dirac operator on $\mathbb{R}^3$
which obey an additional non-linear equation are closely related to vortex
configurations on the 2-sphere, and that both are best understood in terms of
the geometry induced on the 3-sphere via pull-back of the round geometry with
bundle maps of the Hopf fibration. We use this viewpoint to deduce a manifestly
smooth formula for square-integrable magnetic zero-modes in terms of two
homogeneous polynomials in two complex variables.
