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.

periodic. I believe if you mod out $\mathbb{R}^3$ by this lattice (or some sublattice of that so that you don't identify too many $n$-cells together) and consider the induced tiling of the torus, then all the quantities you consider will be reflected there. $\endgroup$ – Dan Rust Nov 23 at 12:25