What is "dimension" in Wolfram's Physics?

Question

In The Notion of Dimension, which is section 4.5 in Wolfram's Physics Project, they describe a kind process for estimating "dimension".

Suppose I have a metric space $(X, \epsilon)$ for which I am free to induce opens balls up to radius $r$. Suppose $(X, \mathcal{F}, \mu)$ is a measure space on the same set $X$. I can parametrize our measure by the radius, which in the linked article considers asymptotically $\mu_r \sim \mathcal{O}(r^d)$ where $d \in \mathbb{R}_{\geq 0}$ is a kind of dimension. In my personal notes I have been calling this "scale dimension", but I am now looking to connect my thinking with the literature.

The informal notion of "Haussdorf dimension" given on Wolfram MathWorld looks highly similar. It is given in the form $d = \frac{\ln N}{\ln s}$ where $N$ and $s$ are some parameters. But more generally it seems to be defined in terms of Hausdorff measure (according to wiki). I may be confused about the definition of Hausdorff measure here, but my thinking is that (hyper)graphs are countable so they would always have Hausdorff measure of zero for any $r$. Thus the Hausdorff dimension of a graph would be zero, which is not what we are describing above with "scale dimension".

Minkowski-Bouligand dimension seems to consider arbitrarily small radii in defining dimension. With these graphs our radius can be chosen as a count representing path lengths, in which case an "arbitrarily small $r$" doesn't make a lot sense.

So this notion of dimension being used here on graph/hypergraphs seems similar to Hausdorff dimension, but in a countable setting.

What is the notion of dimension being used in Wolfram's Physics project?

Answer 1

There's not much more to this than what the writer is telling you plainly: in some particular unstated example the quantity $\mu_r$ is asymptotic to $r^d$, and in the writer's personal notes the number $d$ is called "scale dimension". You can happily convince yourself that the Euclidean plane has scale dimension equal to $2$, and in general Euclidean $n$-space has scale dimension equal to $n$. And if you know any hyperbolic geometry you can convince yourself that $\mu_r$ is asymptotic to $e^r$ and hence scale dimension is undefined (or is infinite, depending on your taste).

Your account of Hausdorff dimension leaves out all of the subtleties of its full definition, regarding limits of various parameters as other parameters approach $0$ and/or $\infty$. So you should be rather skeptical of any surface similarities with "scale dimension" that you might perceive, until you have a very solid understanding of the full definition of Hausdorff dimension.

Answer 2

You asked

So this notion of dimension being used here on graph/hypergraphs seems similar to Hausdorff dimension, but in a countable setting.

What is the notion of dimension being used in Wolfram's Physics project?

Probably Wolfram could give you a better answer, but if you read the section carefully, it is a heuristic attempt at a non-rigorous notion of dimension that is based roughly on $\sim \mathcal{O}(r^d)$ behavior. It agrees with other definitions of dimension for graphs which are modeled on Euclidean spaces. It is certainly not meant to be a formal definition of dimension such as those defined in mathematics. This is physics after all.

Also, the Hausdorff dimension you refer to is only defined in the context of a metric space. There is no metric space in the Wolfram Physics Project models.