How to imagine a fractal dimension? I am interested in fractal dimensions and more or less familiar with the self similarity, boxcounting and Hausdorff dimension.
If someone asked me “What is a fractal dimension?”, my answer would be something like “Well, a subset $M \subset \mathrm{R}^n$ should have dimension $d$, if it – informally sayed – “locally looks” like the $\mathrm{R}^d$. This means a plane has dimension $2$ and as a sphere “looks locally similar” to a plane, a sphere has dimension $2$ as well. Furthermore a point should have dimension $0$ and therefore countable union of points as well. Lines have dimension $1$ and as the graph of a differentiable function looks similar (locally at least!), it should have dimension $1$ as well; even the graph of only piecewise differentiable functions should have dimension $1$.
But then, there are sets which do not look similar to $\mathrm{R}^d$ for any integer $d$. The Sierpinski triangle for example is somehow (optically, at least – and at any zoom factor!) “more than a line“ but “less than a plane“, and one can start thinking of a definition of a fractal function. A good definition should meet the criteria above (for example the boxcounting dimension of some countable sets is not zero, while the Hausdorff dimension always is).”
First of all: Is this interpretation a good idea? Do I get the motivation for fractal dimensions correct? And more importantly:
I stated above that the graph of piecewise differentiable functions, which haves only countable „sharp bends“, should have dimension $1$. However the Sierpinski triangle also has only countable „sharp bends“, but a fractal dimension of $\frac{\ln 3}{\ln 2} \neq 1$. Where is the difference here?
I am not interested in any formal explanation why a certain fractal dimension of a graph of a piecewise differentiable function is lower than the fractal dimension of the Sierpinski triangle. The important thing is that I have build up an informal description of a fractal dimension which (at least this is what I am hoping for) is good enough to somehow find an explanation for my example above. If there is no such explanation, of course feel free to explain why there is not and provide me with a better imagination, if possible. (Although that would imply my descriptions sucks, which would be somewhat sad.)
 A: First, it seems that the first "intuitive" definition of dimension you have stated, is close to the definition of manifold dimension. Ultimately, manifold dimension counts the number of parameters necessary to identify a point. It does this by locally using Euclidean space (where Cartesian coordinates provide such a framework). Hausdorff dimension, on the other hand, uses a completely different method to describe the location of a point. A great example is the topologists space filling curve $\gamma: I \to I\times I$. Using the space filling curve, one can identify point using only one parameter. This function, however, is not a homeomorphism and does not provide a 1D manifold structure on $I\times I$. There are many curves which look, sort of in between this type of space filling curve and those whose images are manifolds, such as the Serpinski triangle. In these cases, one expects to get a dimension in between integer dimensions. The Hausdorff dimension measures this by detecting how much room the set takes up in the ambient space. It does this using open covers and measures how much of the set fills the open cover as the total volume of the cover approaches zero.
A: Yes, this is a good motivation.
One way to see that fractals are 'between' dimensions is the following:
If you scale a line segment of length 1 by 3, its length is 3.
If you scale a polygon of area 1 by 3, its area is 3^2.
If you scale a polyhedron of volume 1 by 3, its volume is 3^3.
If you scale the Cantor set on the unit interval by 3, its 'size' only doubles.
If you scale the Sierpinski carpet by 3, its 'size' is 8 times larger.
Note that the dimension is found in each case by taking logs base 3.
