How would you calculate the Fractal Dimension of this asymmetric Cantor Set? This construction removes the second quarter after each iteration. Picture from Wikipedia:

Wikipedia gives the Hausdorff Dimension as $\log_2 \phi$ where $\phi$ is the Golden Ratio. 
Intuitively, the dimension tells me that this set, scaled down by a factor of two, will "fit inside of itself" 1.618... times. 
My intuition is leaning on the definition of the "self-similarity" dimension though, which I realize is not the same as the Hausdorff Dimension given by Wikipedia, but I also know that for simple fractal sets like this, the Hausdorff and self-similarity dimensions usually coincide.
In my analysis class last year, we talked briefly about the definition of the Hausdorff-measure and Hausdorff-dimension, but I've found it very difficult to locate examples of people actually showing how to calculate this dimension for any but the most basic objects.

You may compute the similarity dimension as follows. The set is composed of two copies of itself - one scaled by the factor $1/2$ and the other scaled by the factor $1/4$. Thus, the similarity dimension is the unique, positive number $s$ satisfying $1/2^s + 1/4^s = 1$. Since $1/4=1/2^2$, this yields
$$\frac{1}{2^s} + \left(\frac{1}{2^s}\right)^2 = 1.$$
This is a quadratic equation in the expression $1/2^s$, which can be solved to obtain
$$\frac{1}{2^s} = \frac{-1+\sqrt{5}}{2} = \varphi.$$
Thus, $s=\log(\varphi)/\log(2).$
It's also possible to compute the dimension using a box counting technique as in this answer.
