Around the turn of the century, the intuitive notion of dimension was made mathematically rigorous for the first time. The new definitions did not simplify matters however, because there are now more then ten different definitions of dimension: topological dimension, Hausdorff dimension, capacity dimension, self-similarity dimension, information dimension, box-counting dimension, and more. Many of these concepts were inspired by Felix Hausdorff's pioneering work. In 1919, he introduced the Hausdorff dimension. Although important for theoretical purposes, the Hausdorff dimension of most objects is too difficult to compute to be practical.

To define the Hausdorff dimension, we need a few concepts from Euclidean geometry. The distance between points (x₁,...,x_n) and (y₁,...,y_n) (in Euclidean n-dimensional space Eⁿ) is the square root of (x₁-y₁) ² + ... + (x_n - y_n) ². The open ball centered at x with radius r is the set of all points in...