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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.

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    $\begingroup$ May I ask: What is a "regular tiling"? Tiling by regular polyhedra? And could you expand upon the phrase "symmetry-inequivalent flags."? $\endgroup$
    – Joseph O'Rourke
    Commented Nov 23, 2019 at 0:19
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    $\begingroup$ Sorry that the question isn't clear. When I say tiling, I mean one in the most general sense: a regular CW-filtration of $\mathbb{R}^3$. The tiling gives us a discrete group of isometries that preserves the filtration. This group action partitions the flags into equivalence-classes, and a regular tiling is, to me, one that only has a single equivalence class. For this question, any partition of space (into non-convex bodies or otherwise) is allowed, so long as its translation invariant. $\endgroup$
    – squiggles
    Commented Nov 23, 2019 at 1:04
  • $\begingroup$ By translation invariant, presumably you mean that there exists a full-rank sublattice of the set of translations on $\mathbb{R}^3$ such that the tiling is fixed under translations in the sublattice. We would call such a tiling 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
    Commented Nov 23, 2019 at 12:25
  • $\begingroup$ That's correct. I'm equivalently interested in tools for relating these values in any regular CW-decomposition of the $3$-torus. $\endgroup$
    – squiggles
    Commented Nov 23, 2019 at 13:19
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    $\begingroup$ Well once everything is in the torus, then you can start using Euler characteristic arguments and homology to relate these quantities . $\endgroup$
    – Dan Rust
    Commented Nov 23, 2019 at 14:57

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