Hausdorff Dimension

The Hausdorff dimension, introduced in 1918 by the mathematician Felix Hausdorff, is a measure of roughness, or more specifically fractal dimension, associated with a metric space, meaning a set on which distances between members are defined; for a single point the Hausdorff dimension is zero, for a line segment it is one, for a square it is two and for a cube it is three, agreeing with ordinary topological dimension for shapes with a small number of corners. For more irregular objects, however, the same scaling and self-similarity reasoning yields non-integer Hausdorff dimensions, as with the Koch snowflake, whose component segments are each replaced by four self-similar copies at one-third the length at every iteration, giving a dimension that solves an exponential scaling equation for a non-integer value. The technique is also called the Hausdorff-Besicovitch dimension in recognition of Abram Samoilovitch Besicovitch advances in computing the dimension of highly irregular sets, and it is closely related to, and usually equivalent to, the simpler box-counting or Minkowski-Bouligand dimension.