  Clusters of galaxies, filled with hot magnetized plasma, are the largest
bound objects in existence and an important touchstone in understanding the
formation of structures in our Universe. In such clusters, thermal conduction
follows field lines, so magnetic fields strongly shape the cluster's thermal
history; that some have not since cooled and collapsed is a mystery. In a
seemingly unrelated puzzle, recent observations of Virgo cluster spiral
galaxies imply ridges of strong, coherent magnetic fields offset from their
centre. Here we demonstrate, using three-dimensional magnetohydrodynamical
simulations, that such ridges are easily explained by galaxies sweeping up
field lines as they orbit inside the cluster. This magnetic drape is then lit
up with cosmic rays from the galaxies' stars, generating coherent polarized
emission at the galaxies' leading edges. This immediately presents a technique
for probing local orientations and characteristic length scales of cluster
magnetic fields. The first application of this technique, mapping the field of
the Virgo cluster, gives a startling result: outside a central region, the
magnetic field is preferentially oriented radially as predicted by the
magnetothermal instability. Our results strongly suggest a mechanism for
maintaining some clusters in a 'non-cooling-core' state.
