  We study the rank 4 linear matroid $M(H_4)$ associated with the 4-dimensional
root system $H_4$. This root system coincides with the vertices of the
600-cell, a 4-dimensional regular solid. We determine the automorphism group of
this matroid, showing half of the 14,400 automorphisms are geometric and half
are not. We prove this group is transitive on the flats of the matroid, and
also prove this group action is primitive. We use the incidence properties of
the flats and the {\it orthoframes} of the matroid as a tool to understand
these automorphisms, and interpret the flats geometrically.
