  We investigate the variation of the fine-structure constant, {\alpha}, in
symmetron models using N-body simulations in which the full spatial
distribution of {\alpha} at different redshifts has been calculated. In
particular, we obtain simulated sky maps for this variation, and determine its
power spectrum. We find that in high-density regions of space (such as deep
inside dark matter halos) the value of {\alpha} approaches the value measured
on Earth. In the low-density outskirts of halos the scalar field value can
approach the symmetry breaking value and leads to significantly different
values of {\alpha}. If the scalar-photon coupling strength {\beta}{\gamma} is
of order unity we find that the variation of {\alpha} inside dark matter halos
can be of the same magnitude as the recent claims by Webb et al. of a dipole
variation. Importantly, our results also show that with low-redshift symmetry
breaking these models exhibit some dependence of {\alpha} on lookback time (as
opposed to a pure spatial dipole) which could in principle be detected by
sufficiently accurate spectroscopic measurements, such as those of ALMA and the
ELT-HIRES.
