What is meant by the fractal dimension of a Mandelbrot set?

Question

I have recently undertaken a project that looks at the Mandelbrot set. I have been able to write software that estimates the area of the set, and have extended this to apply confidence limits to this. However, the final part of the project is to estimate the fractal dimensions of the set. Can anyone please advise as to what exactly fractal dimensions are, and especially in relation to the Mandelbrot set?

Answer 1

The central idea of the fractal dimension $d$ of a set is that if you magnify the set by a factor of $\alpha$, the natural size measurement for the set increases by a factor of $\alpha^d$.

• If you magnify a finite set of points by $\alpha$, the points move further away from each other, but the count remains unchanged (it changed by a factor of $\alpha^0$).
• If you magnify a curve by $\alpha$, the length of the curve increases by a factor of $\alpha^1$.
• If you magnify a surface by $\alpha$, the area of the surface increases by a factor of $\alpha^2$.
• If you magnify a solid by $\alpha$, the volume of the surface increases by a factor of $\alpha^3$

Hausdorff measure extends this concept to all non-negative real exponents. For any $d \ge 0$, the Hausdorff measure $H^d$ of a set increases by a factor of $\alpha^d$ when the set is magnified by a factor of $\alpha$. $H^d$ has the following properties for a well-behaved set $S$:

• $0 \le H^d(S) \le \infty\qquad$ ($\infty$ is a valid measure value.)
• if $d < e$, and $H^e(S) > 0$, then $H^d(S) = \infty$.
• if $d < e$, and $H^d(S) < \infty$, then $H^e(S) = 0$.

(There are sets for which the Hausdorff measure is undefined, but these are highly esoteric, so bad that they cannot be constructively defined. We'll ignore them for this.)

The latter two properties show that for each set $S$, there is some value $e$ such that for all $d < e, H^d(S) = \infty$ and for all $d > e, H^d(S) = 0$. Only $H^e(S)$ can take on any other value (though it may also be $0$ or $\infty$, depending on $S$). This $e$ is defined to be the Hausdorff dimension of $S$.

While Hausdorff dimension is a theoretical favorite, it can be hard to calculate. For this reason some other variants of fractal dimension have been developed that can be more easily calculated. These mostly agree with Hausdorff dimension but may differ on certain sets.

Concerning the Mandelbrot set. You've already stated that you can calculate its area, which is not $0$, and is not $\infty$, so ...