Are you really talking about a "fractal" dimension, or do you mean a "fractional" dimension? I could understand the concept of the latter, but don't know what the first would be.
When studying fractals (strange attractors, etc.) we find structures that do not fit well into our preconceived classes of 'curves', 'surfaces', etc.
The concept of dimension of a space has several meanings. We can say that a vector space has dimension
n, if there are
n linearly independent vectors such that they span the whole space. On a manifold (think: hypersurface) of dimension
n, there are
n local bijections between the manifold and a vector space of the same dimension. A connected topological space has dimension 1, if it is separated in two disconnected parts by removing one of its elements... All these examples have integer dimension and are quite reasonable and even intuitive.
Enter fractals. Consider
Cantor's set, formed by all numbers with no 1's in their expansion on base 3 (for a nicer, geometrical definition and diagram, see the link). This is certainly not a line with dimension 1, neither a discrete set of points with dimension 0
[1]. It seems we need a more general concept of dimension, accomodating rational (and, perhaps, even real) values for the dimension of a set. Hausdorff thought about this around 1920, so now we can manipulate noninteger dimensions. I was deliberately vague in my post, because there are several definitions (I will give a brief note below) so I used the general term 'fractal dimension'. This is usually (even in some texts) used as a synonym of one (or, worse, more) of the following definitions:
The first idea (and the simpler one) is the concept of capacity dimension. Assume you have a set
A embedded in 3D space. We ask ourselves for the number of cubes of side
x needed to cover
A, that is, so that each point in A is at least in one of the cubes. Let N(x) be that number. If the set A is a curve of lenght L, a line segment for instance, it is clear that, if x -> 0, we wil get N(x) ~ L/x. If A is a surface, N(x) ~ S/x^2. If it is a volume V, N~V/x^3, etc. In the more general situation of a manifold of dimension
d, embedded in a dimension
n>d, the number of hypercubes, when x -> 0, will be proportional to x
-d. Precisely, we have
This is not an integer sometimes. For the Cantor Set, if we take x=3
-k, it is clear that N(x)=2
k (clear, that is, if you follow the previous link and see the geometrical interpretation). So the Cantor set has
capacity dimension d = lim_{k->oo} log (2^k) / log (3^k)=log(2)/log(3) ~ 0.6309. Cantor's Set is a bit more than halfway between a point and a line.
Hausdorff's original concept is a bit more involved. I'm not going to define it here (if you are interested, search for 'Hausdorff dimension'). Suffice it to say that d_H >= d, and that both are the same in many physical cases. It is difficult to calculate, a lot of computer time, so another definition is sometimes invoked:
pointwise dimension.
When a set has noninteger dimension, we call it 'fractal'. Hence the name fractal dimension. Examples of complicated fractals arise when studying stability of differential equations (this is what is meant by 'chaos'). A differential equation is said to be stable if (roughly, the real definitions are more subtle) two solutions, initially close together (similar initial values), will keep together as time passes. (Linear equations are an example of this).
If, however, we do not have linear equations, then all bets are off. Anything can happen. Fluids are an example of such behaviour. They are governed by Navier-Stokes's equation, which is quite complex and definitely nonlinear. This is why the wheather guys on TV miserably fail once and again. Fluids are intrinsically unstable, even if we know the initial conditions with a very high accuracy, our solution may be far off from the real one, even if the 'real' solution has very similar starting conditions. (Butterfly effect: the disturbance on the initial conditions by the butterfly may be enough). Studying (generally with numerical methods) these equations, we arrive at 'phase maps' with fractal behaviour. Example:
Lorenz's Attractor, which arose precisely in atmospheric problems.
As I said, I was deliberately vague in my previous post. I didn't specify which of these definitions would correspond to a noninteger
n in the formula for the volume of the
n-sphere. Frankly, the reason for this is that I hadn't even thought of plugging noninteger values in it (until AV pointed it out). It probably makes sense in some way, it is continuous. But you would have to define very carefully 'sphere' to allow for those values.
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[1] Note for the initiated: its Lebesgue measure is, however, 0, even though it has uncountably infinite elements.
