Self-similarity in sets is characterized by the Hausdorff dimension, which is usually equal to the box counting dimension or capacity dimension, to which it  represents an upper bound when in disagreement.

 

If we cover a point set located in with a regular grid of boxes of length and call the number of boxes that contain at least one point, then for a self-similar set

 

, 

 

where is then called the box counting dimension or capacity dimension.