My question has to do with scaling laws, i.e., how surface areas scale as the square of the linear dimension, and volumes scale as the cube of the linear dimension. And, therefore, volumes should scale as the 3/2 power of areas.
Examples: (1) Suppose you have a cube in which each edge measures 5 meters. By what factor will the volume increase if you double the length of each edge? Answer: The volume increases by 2^3 or a factor of 8. (The 2 came from the factor by which we're increasing the length of each edge. And the 5 meters was irrelevant; the volume would increase by a factor of 8 regardless of the starting size.) (2) What is the volume of a cube whose sides each have an area of 9 square centimeters. Answer: 9^(3/2) = 27 cubic centimeters. (One could also answer this by taking the square root of 9 to determine that each edge is 3 cm long, and then cube that to get 27 c.c., but that square root and cubing is equivalent to raising to the 3/2 power.)
My problem: Suppose you have a square divided into 3x3 or 9 sub-squares. Remove the center sub-square; we can think of that as a pore or a void, and it occupies 1/9 (11.1%) of the area. We can say that the porosity is 11.1%.
Now lets move up to three dimensions. Instead of a square, we have a cube divided into 3x3x3 or 27 sub-volumes. Create a void by removing the center sub-cube. Your porosity is then 1/27 (3.7%). Consistent with our scaling laws, we could have also gotten that result by raising 1/9 to the 3/2 power, which also gives you 3.7%. So the scaling law seems to work for this kind of problem.
Now the question is, Why can't you do the same thing with the non-void (solid) portion? The solid part occupies 88.9% of the square and 96.3% of the cube. But one cannot go from 89.9% to 96.3% with the same 3/2-power scaling that worked so well for the empty area. Why not? Why the difference in doing this with the solid part vs. the empty part?
You defined the "solid portion" as the difference of two volumes, one of scale $1$, the other of scale $\frac{1}{3}$. So the total volume in each case is $1^{d} - (\frac{1}{3})^{d}$, where $d$ is the dimension. You're asking why $$(1^{d_1} - (\frac{1}{3})^{d_1})^{\frac{d_2}{d_1}} \neq 1^{d_2} - (\frac{1}{3})^{d_2} $$ where $d_2$ = 3 and $d_1 = 2$. Generaly speaking, exponentiation does not distribute over sums or differences. Ie. $(x-y)^d \neq x^d - y^d$.
Informally speaking, the "scaling law" works only for simple objects such as cubes, or cube-shaped voids (which can be thought of as a "negative" cube). Or things that can be assembled to form a simple object.
The "solid" portion is not a simple object (a cube), but is the difference between two simple objects (larger cube minus smaller cube). So the scaling law doesn't work. It works for the cubes, but not their difference.