TITLE: Geometrical objects whose volumes are fractional powers of their sizes
QUESTION [10 upvotes]: While studying properties of foams (imagine bubbly soap or microscopical grids/networks), I started wondering on the relationship between the volume occupied by the matter $V_s$ itself and the overall volume $V_a$ of a given foam sample (matter+voids). Two lengths characterize these objects: the size of each pore $l_a$ and the thickness $l_s$ of the matter that encloses such a pore.
You can picture a foam like a stack of cubes (the pores). When you suppose that each pore is fully closed by walls of matter, you obtain $V_s/V_a\propto l_s/l_a$. When you suppose that each pore is fully open, such that filaments are joined to each other and forming a sort of network, you obtain $V_s/V_a\propto (l_s/l_a)^2$. This is easy to picture, and comes naturally from the fact that basic geometrical shapes have a volume proportional to the powers of typical distances.
Now, many foams exhibit $V_s/V_a\propto (l_s/l_a)^k$, where $k$ is fractional, between 1 and 2. This means that the volumes are not proportional to integer powers of the characteristic sizes in the problem.
How is that possible? Is there a visual example of this situation? It seems related to fractals however I cannot pinpoint how.
I am looking for a non-technical answer as I am not a mathematician. I have basic knowledge on fractals.
UPDATE
Following some answers that have already been proposed, I find it difficult to imagine a fractal object that does not introduce another length (than $l_a$ and $l_s$). Iterating the fractal keeps producing more lenghts somewhat related to each other. Then, how is it possible to define only one $l_s$ in these conditions?

REPLY [2 votes]: This is going to be more of a pictorial answer, and I will leave out the math.  Hopefully it is more enlightening this way. There are two ways you can consider expanding your fractal, in 2 dimensions and in 3. If we only look at the case where our "bubbles" are actually cubes, which makes this argument simpler to see, we have these basic pictures for both types.  Let an iteration be, for 2 dimensions, dividing each area in 4 equal parts.  For 3 dimensions, let an iteration be dividing our cubes into 8 equal parts. Below are the first four iterations of both 2 and 3 dimensions.


Both of the 2 dimensional and 3 dimensional pictures are of the form you mentioned where we are "open," and therefore give a proportion of $V_s/V_a \leftrightarrow (l_s/l_a)^2$.  The other option is to use cubes with solid sides, which gives the $V_s/V_a \leftrightarrow (l_s/l_a)$ (Not pictured.)  One does not really have to stretch their imagination to see that if we take the limit the 2 dimensional squares as we iterate the fractal, we would end up with a solid square. So, to increase the power of our proportion, we want to make our 2 dimensional squares a higher iteration, BUT, we want to keep our 3 dimensional iteration lower. This will give us a higher value of $V_s$ and $l_s$, but keep our $V_a$ and $l_a$ the same.  The difference is that we know this must get us a lower power because if we let the sides, or 2 dimensional pieces tend toward the solid squares, we must get our $V_s/V_a \leftrightarrow (l_s/l_a)$ relation.  As this is, in some sense, continuous and well behaved, we see that the in between steps will give $V_s/V_a \leftrightarrow (l_s/l_a)^k$,  for $1\leq k \leq 2$. (Here is a cut away picture to show what I mean. This would be the 3 dimensional iteration done once and the 2 dimensional iteration done twice after that.)  
This tells us that for a random foam, we will get a fractional power proportion relation when we increase the number of divisions of the surfaces area of the bubbles and not divide the volume contained in them as much. One more picture, which might help.  Consider every colored section of the sphere to be a subdivision, or "2 dimensional" iteration while keeping the volume contained in them the same.