How can we find the "dimension" of a non-continuous "tiled" space? When talking about dimension we often do this with regards to some topological space, in these cases we can use things like Lebesgue covering to extract a measure of dimension.
However spaces may well be composed of many "cells" connected to each other (even the universe we're in may well be composed of discrete-albeit-it-extremely-tiny cells). In these spaces topological arguments for dimensionality don't work as near any points in these cell spaces there's only finitely many points, so notions of "open" and "closed" don't transfer well.
One might be tempted to define the dimension of such spaces as the "dimension" of the cells, but the cells are more of an abstract thing. For say a cube tiling, the existence of "cubes" isn't so much a thing as the idea that from any cube you can reach its neighbours by going to 6 possible neighbours, we get the usual 3 dimensions by recognizing pairs of neighbours as being on the "same line".
However this definition only works for spaces "tiled" using cubes. For example we could easily tile 3-dimensional space with (certain irregular) tetrahedra, but tetrahedra only have 4 neighbours so clearly "number of neighbours" cannot be used to be measure dimension. There may of course be many irregular "tilings" of space that have arbitrary neighbouring connections.
My question is, is there a way given a tiling, which is essentially just a (potentially infinite) connected graph (where vertices are cells, and edges indicate neighbours), do we have ways to define and derive dimension? (i.e. In topological spaces we can use covering dimension to determine the dimension of a space)
 A: One option for infinite graphs is to measure the number of balls within a fixed distance $r$ of a vertex. This is analogous to the idea that if we have a point in $d$-dimensional space, the volume of the ball of radius $r$ is proportional to $r^d$.
So if we let $N_{\le r}(v)$ be the set of all vertices at distance at most $r$ from $v$ in a graph, we can define the dimension of the graph as
$$
   \lim_{r \to \infty} \log_r | N_{\le r}(v)|.
$$
We hope that this is independent of the choice of $v$.
This gives reasonable results for many infinite graphs that look like they ought to have integer dimension. For infinite graphs that "look like" a well-known fractal, such as Sierpinski graphs, it often gives the fractal dimension. All infinite connected graphs have dimension at least $1$.
Many graphs will have a dimension of $\infty$ by this definition. For example, this will happen if any vertex has infinitely many neighbors. Even in a locally finite graph, this can happen: take an infinite binary tree, or the graph corresponding to a tiling of hyperbolic space.
For finite graphs, we run into trouble because eventually, $N_{\le r}(v)$ will stop growing, and the limit will be $0$. (Intuitively: from far away, any finite graph looks like a point, and points are $0$-dimensional.) We can ask about the growth rate of $|N_{\le r}(v)|$ for values of $r$ that are not too small but not too large, either - but this is more of a heuristic than a definition.
