Haussdorff 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 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.

In mathematical terms, the Hausdorff dimension generalizes the notion of the dimension of a real vector space. That is, the Hausdorff dimension of an n-dimensional inner product space equals n. This underlies the earlier statement that the Hausdorff dimension of a point is zero, of a line is one, etc., and that irregular sets can have noninteger Hausdorff dimensions.

Examples of non-integer Haussdoff dimensions:

• Countable sets have Hausdorff dimension 0
• The Euclidean n-dim space have Hausdorff dimension n
• Space-filling curves like the Peano and the Sierpiński curve have the same Hausdorff dimension as the space they fill.
• The Koch curve have Hausdorff dimension ln(2)/ln(3)
• Sierpinski triangle Hausdorff dimension ln(3)/ln(2)

The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or Minkowski-Bouligand dimension.

See also

Dynamical systems

Papers

• F. Hausdorff (March 1919). Dimension und äußeres Maß. Mathematische Annalen 79 (1-2): 157-179.
• Hutchinson, John E. (1981). Fractals and self similarity. Indiana Univ. Math. J. 30 (5): 713-747.
• Dodson, M. M., & Kristensen, S. (2004). Hausdorff dimension and Diophantine approximation. Fractal geometry and applications: a jubilee of Benoıt Mandelbrot, (Part 1), 305-347.