  We present a completeness characterization of box splines on
three-directional triangulations, also called Type-I box spline spaces, based
on edge-contact smoothness properties. For any given Type-I box spline, of
specific maximum degree and order of global smoothness, our results allow to
identify the local linear subspace of polynomials spanned by the box spline
translates. We use the global super-smoothness properties of box splines as
well as the additional super-smoothness conditions at edges to characterize the
spline space spanned by the box spline translates. Subsequently, we prove the
completeness of this space space with respect to the local polynomial space
induced by the box spline translates. The completeness property allows the
construction of hierarchical spaces spanned by the translates of box splines
for any polynomial degree on multilevel Type-I grids. We provide a basis for
these hierarchical box spline spaces under explicit geometric conditions of the
domain.
