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From Citizendium, the Citizens' Compendium

In mathematics, the Hausdorff dimension is a way of defining a fractional area for all dimensional figures such that the dimension describes partially the way that an area moves in the space around it. For example, a plane would have a Hausdorff dimension of 2. However, it would not make sense to give the Sierpinski triangle fractal a dimension of 2, since it does not fully occupy the 2-dimensional realm. Therefore, its Hausdorff dimension describes it mathematically, creating a relationship between the number of new self-similar sections and their scale.