I'm trying to understand fractal dimension in the context of colloidal gels. But more on that later. I'm confused about a more fundamental thing, which I think relates to the discreteness of the objects comprising the fractal.
I read the argument about the length of a coastline here. The idea is that if you measure the length of the coastline with rulers of a certain size, halving the ruler size does not double your length measurement, in units of the number of rulers needed. If the curve that gives you the coastline had comprised of very (but not infinitesimally) small straight lines, there would be a critical ruler size below which the answer does not change. Is the fractal then defined only above a certain length scale?
So if I now have a colloidal gel, i.e. a percolating network of spherical particles with diameter $\sigma$. The spheres are considered to be bonding if the distance between two adjacent particles is less than $\lambda\sigma$, where $\lambda>1$. It is well known that some types of colloids, such as those that result from diffusion-limited aggregation, have certain fractal dimensions, depending on the dimensionality of the problem. ($d_f=1.75$ for $d=2$, etc.) Spheres are discrete objects, and yet when you zoom out, the whole thing looks like a fractal, in the sense that it has a certain roughness. Similarly to the ruler argument, if I now calculate the box-counting dimension, I get $d=3$, because I just have a collection of spheres of diameter $\sigma$. Does this mean that I have to consider "boxes" with sides larger than $\sigma$, similarly to the ruler-coastline argument? If so, how do I reconcile a maximum box size with the definition of the box-counting dimension, which involves considering the limit of vanishing box size?
Note: I am aware that there are other ways to calculate fractal dimension in colloidal systems, such as looking at how the radius of gyration scales with the number of particles in a cluster. At this point, I am just curious about this particular method.
Also note that I'm not a mathematician by training, so I may not understand proofs or definitions beyond an intro-to-proofs-class-level.
Best Answer
On the webpage to which you refer, we see a number of pictures (like the ruler and coastline example) that indicate how fractal dimension arises when we approximate an object with smaller ones. More specifically, if $E$ is a bounded set and $N_{\varepsilon}(E)$ represents the number of pieces of size $\varepsilon$ in some approximation to $E$, then the fractal dimension should be $$ \dim(E) = \lim_{\varepsilon\rightarrow0} \frac{\log(N_{\varepsilon}(E))}{\log(1/\varepsilon)}, $$ assuming this limit exist.
As you've observed, this limit can't really be carried out on physical objects. Thus, the standard interpretation in the physics literature, as I understand it, is to presume that the relationship between $N_{\varepsilon}(E)$ and $\varepsilon$ should be maintained over a broad range of values. A standard way to compute the dimension is to compute $N_{\varepsilon_k}(E)$ for some terms $\varepsilon_k$ chosen from a sequence that tends geometrically to zero. We then fit a line to the points in a log-log plot of $N_{\varepsilon}(E)$ versus $\varepsilon$. The box-counting dimension should be approximately the negative slope of that line.