Hausdorff dimension

Hausdorff dimension is a concept in mathematics introduced in 1918 by mathematician Felix Hausdorff, and it serves as a measure of the local size of a set of numbers (i.e., a "space"), taking into account the distance between each of its members (i.e., the "points" in the "space"). Applying its mathematical formalisms provides that the Hausdorff dimension of a single point is zero, of a line is 1, and of a square is 2, of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is a counting number (integer) agreeing with a dimension corresponding to its topology. However, formalisms have also been developed that allow calculation of the dimension of other less simple objects, where, based solely on its properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular sets, this dimension is also commonly referred to as the Hausdorff–Besicovitch dimension.

The Hausdorff dimension is, more specifically, a further dimensional number associated with a given set of numbers, where the distances between all members of that set are defined, and where the dimension is drawn from the real numbers, ℝ, to which two elements have been added, +∞ and −∞ (read as positive and negative infinity, respectively). The set that provides the Hausdorff dimension is called the extended real numbers, R, and a set of numbers where distances between all members are defined is termed a metric space, so that foregoing can be succinctly stated, saying the Hausdorff dimension is a non-negative extended real number (R ≥ 0) associated with any metric space.

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