The Hausdorff Dimension calculator computes the Hausdorff dimension for geometric shapes typically represented by fractal graphics.

Description

The Hausdorff dimension is a mathematical concept that is a measure of the local size of a set of numbers (i.e., a "space").  The Housdorff dimension takes incorporates the distance between each of its members (i.e., the "points" in the "space").

The mathematical definition of the Hausdorff dimension defines a single point as having Housdorff dimension of 0, a line having Housdorff dimension of 1, a square having Housdorff dimension of 2, and a cube having Housdorff dimension of 3. The sets of points thus defined must be sets of points that define shapes that are smooth, or a shape that has a small number of corners—the shapes of traditional geometry and science, The  Hausdorff dimension was defined to be an integer agreeing with a dimension corresponding to its topology. Formalisms were developed that calculate the dimension of less simple objects, based solely on the object's properties of scaling and self-similarity. Some 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, plus +∞ and −∞. The set that provides the Hausdorff dimension is called the extended real numbers, R.  R is a set of numbers where distances between all members are defined and is termed a metric space. Re-stated, the Hausdorff dimension is a non-negative extended real number (R ≥ 0) associated with any metric space.

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.

A good visual example is the Koch curve which is constructed from an equilateral triangle. In each iteration, the triangular component line segments are divided into 3 segments of unit length and the newly created middle segment becomes the base of a new equilateral triangle that points outward. The original base segment of this newly created triangle is then deleted to leave a final object from the iteration of unit length of 4.That is, after the first iteration, each original line segment has been replaced with N=4 segments, where each self-similar copy is 1/S = 1/3 as long as the original. Stated another way, we have taken an object with Euclidean dimension, D, and reduced its linear scale by 1/3 in each direction, so that its length increases to N=S^D. This equation is easily solved for D, yielding the ratio of logarithms (or natural logarithms) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects.

The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or Minkowski–Bouligand dimension.¹

1. ^ http://en.wikipedia.org/wiki/Hausdorff_dimension