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.

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.