  We study the collinear splitting functions needed for next-to-next-to-leading
order calculations of jet production in the vacuum and in dense QCD matter.
These splitting functions describe the probability of a parton to evolve into
three-parton final state and are generalizations of the traditional DGLAP
splitting kernels to a higher perturbative order. Of particular interest are
the angular distributions of such splitting functions, which can elucidate the
significance of multiple parton branching for jet observables and guide the
construction of parton shower Monte Carlo generators. We find that to ${\cal
O}(\alpha_s^2)$ both the vacuum and the in-medium collinear splitting functions
are neither angular ordered nor anti-angular ordered. Specifically, in dense
QCD matter they retain the characteristic broad angular distribution already
found in the ${\cal O}(\alpha_s)$ result.
