Suppose I tile $\mathbb{R}^3$ in a ($\mathbb{Z}^3$-)translation-invariant manner.  If we insist on the tiling being regular, then we are left with only the cubic tiling.  However, suppose that we allow for any finite number of symmetry-inequivalent flags.  Then, I'm curious what tools there are to determine relations between the four quantities:

1) Average vertex degree

2) Average number of edges in a face

3) Average number of faces incident to an edge

4) Average number of faces in a volume.