  Fractal dimension constitutes the main tool to test for fractal patterns in
Euclidean contexts. For this purpose, it is always used the box dimension,
since it is easy to calculate, though the Hausdorff dimension, which is the
oldest and also the most accurate fractal dimension, presents the best
analytical properties. Additionally, fractal structures provide an appropriate
topological context where new models of fractal dimension for a fractal
structure could be developed in order to generalize the classical models of
fractal dimension. In this paper, we provide some counterexamples regarding
these new models of fractal dimension in order to show the reader how they
behave mathematically with respect to the classical models, and also to point
out which features of such models can be exploited to powerful effect in
applications.
