  Random packings of stiff rods are self-supporting mechanical structures
stabilized by long range interactions induced by contacts. To understand the
geometrical and topological complexity of the packings, we deploy X-ray
computerized tomography to unveil the structure of the packing. This allows us
to define and directly visualize the spatial variations in the entanglement, a
mesoscopic field that characterizes the local average crossing number, a
measure of the topological complexity of the packing. We show that the
entanglement field has information that is distinct from the density,
orientational order, and contact distribution of the packing. We find that
increasing the aspect ratio of the constituent rods in a packing leads to a
proliferation of regions of strong entanglement that eventually percolate
through the system, and this is correlated with a sharp transition in the
mechanical response of the packing. We conclude with a tentative entanglement
phase diagram for the mechanical response of dense rod packings that is likely
relevant for a broad range of problems that goes beyond our specific study.
