Hausdorff Dimension | Research & Encyclopedia Articles

Hausdorff Dimension

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 Eⁿ that are a distance less than r from x. An open subset is a union of open balls.

The diameter of an open subset is a number k such that all pairs of points in the set are less than a distance k apart and k is the smallest real number with this property. An open cover of a subset A in Eⁿ is a collection of open sets whose union contains A. Such a cover has diameter k if every open set in the cover has diameter less than or equal to k. The d-dimensional Hausdorff measure of a subset A is a real number, denoted by H^d(A), with two properties. First, if y is a number such that for every k, there is a k-diameter open cover of A such that the sum of the dth powers of the diameters of the open cover is less than y, then y is greater than or equal to H^d(A). Second, H^d(A) is greater than or equal to any number that satisfies the first property. Hausdorff proved that if A is a smooth curve, then H¹(A) is the length of A. If A is a smooth surface, then H²(A) is the area of A. Also, there exists a number called D(A), such that if x is less than D(A) then H^x(A) is infinity and if x is greater than D(A) then H^x(A) is zero. This number, D(A), is called the Hausdorff dimension of A.

For many fractals, such as the snowflake curve, the Menger curve, Sierpinski's triangle and carpet, the Hausdorff dimension is equal to the self-similarity dimension (which is also equal to the capacity dimension).