  The mechanical properties of a granular sample depend frequently on the way
the packing was prepared. However, is not well understood which properties of
the packing store this information. Here we present an X-ray tomography study
of three pairs of tetrahedra packings prepared with three different tapping
protocols. The packings in each pair differs in the number of mechanical
constraints $C$ imposed on the particles by their contacts, while their bulk
volume fraction $\phi_{global}$ is approximately the same. We decompose $C$
into the contributions of the three different contact types possible between
tetrahedra -- face-to-face (F2F), edge-to-face (E2F), and point contacts --
which each fix a different amount of constraints. We then perform a local
analysis of the contact distribution by grouping the particles together
according to their individual volume fraction $\phi_{local}$ computed from a
Voronoi tessellation. We find that in samples which have been tapped
sufficiently long the number of F2F contacts becomes an universal function of
$\phi_{local}$. In contrast the number of E2F and point contacts varies with
the applied tapping protocol. Moreover, we find that the anisotropy of the
shape of the Voronoi cells depends on the tapping protocol. This behavior
differs from spheres and ellipsoids and posses a significant constraint for any
mean-field approach to tetrahedra packings.
