What is the fractal dimension of a cauliflower?

Question

I tried to calculate it's dimension using mass. Here's how I did, I took a small cauliflower and measured it's mass = 105g, then I divided it into 12 similar looking branches and calculated the mass of one of the branch (scaled down version) and got 15g, Then I found 15/105 = 1/7. Then since I scaled it down by 1/12 so it put the number in the log formula which is log base 12 of 7 and got 0.78. So the fractal dimension of cauliflower = 0.78 however I think this is wrong cause on wiki, fractal dimension of cauliflower is given as 2.88..And also for a cauliflower to be a fractal it's dimension should exceed it's topological dimension of 3 which is clearly not the case here. Where am I wrong? Please correct me!

Answer 1

The version of dimension that you seem to be looking to compute is the similarity dimension. Roughly speaking, if a set $F$ is the (disjoint) union of $n$ sets, each of which is congruent to the original set scaled down by some factor $r$, then the similarity dimension is given by $$ \dim_s(F) = -\frac{\log(n)}{\log(r)} \tag{1}$$ (note that this will be positive, since we are assuming that $r<1$ so that we are scaling down). This comes from the idea that the "volume" (or measure) of $F$ is given by $$ \operatorname{vol}(F) = n \operatorname{vol}(rF) = nr^d \operatorname{vol}(F) \implies \log(\operatorname{vol}(F)) = \log(n) + d \log(r) + \log(\operatorname{vol}(F)),$$ which we can solve for $d$ in order to get the formula above. The point is that $d$ is the magic exponent that tells us how to scale the radius of a set, and $n$ is like some kind of "normalizing constant". If this doesn't make sense, don't worry about it---the key result is the formula at (1).

More generally, if $F$ is made up of self-similar pieces, not all of which are the same size, then the similarity dimension is the unique real solution $d$ to the Moran equation: $$ 1 = \sum_{j=1}^{J} r_j^d, $$ where $F$ is the disjoint union of $J$ copies of itself, scaled by the factors $r_j$.

Now, this is an abstract mathematical definition of dimension, built to deal with ideal mathematical objects, which don't really exist anywhere in the world. When you try to compute the dimension of a cauliflower, there are a lot of variables that are going to make the computation more difficult to deal with. When you break the plant into 12 similar pieces, how accurate is your measurement of the scaling factor between the original plant and the smaller pieces? Is the mass really a good proxy for volume? Or do you want to measure the dimension with respect to mass, which is a different kind of measure (though, perhaps, related---especially if the density is constant; is it?)? Do you really need to compute the mass/volume at all? Does it matter that the cauliflower does not take a consistent dimension (for example, I would imagine that the stalk is genuinely three dimensional, while the florets are lower dimensional)? These kinds of experimental errors and physical phenomena can make it hard verify mathematical results empirically.

That being said, there does seem to be an error in your computation. You said that you divided the cauliflower into 12 pieces (are they all about the same size?), then weighed them. But the formula deals with the scaling ratio, not the original or resulting volume. If the original cauliflower was 6 inches tall, and each of your 12 pieces is 2 inches tall, then the scaling ratio is $\frac{1}{3}$ (not $\frac{1}{7}$, which is what you got by measuring the weight). That is, you need to compare the lengths, not the volumes.

Answer 2

It depends on what you are looking for in the fractal dimension...

Let me explain. Suppose you are looking for the fractal dimension of a cube (since cubes are nice and simple). Then the length of the cube gets halved 2 and the mass gets divided by 8 (scaled down by 8). Hence the fractal dimension of the volume of the cube is log$_2$8=3. However, the fractal dimension of the surface area is different. Instead, we scale down to half the size but the surface area (of a single cube) reduces to a quarter of the large cube (scaled down by 4). Hence the fractal dimension if log$_2$4=2, as expected.

Now onto your question. The number of shapes you have divided into is 12 (so it has been scaled down by 12). Assuming that the shape is indeed similar, we have (length$^3$ $\propto$ mass) so the length is scaled down by roughly the cube root of 7 $\approx$ 1.912. log$_{1.912}$12 is roughly 3.833.

EDIT: In retrospect, 12 similar-looking branches should mean that each one has a mass of 105/12 grams. This is bad. I think, instead of mass, you measure the length of each one compared to the length of the original. Also, looking it up online yields a fractal dimension of 2. The thing is, the topological dimension of a 3d object is not 3, but rather 2.