Fractal dimension

In mathematics, the fractal dimension of a set is a generalization of the concept of dimension of geometric objects such as curves ( one-dimensional) and surfaces ( two-dimensional), especially in fractals. The special is that the fractal dimension must be an integer. There are different ways to define a fractal dimension.

Boxcounting dimension

When Boxcounting method is to cover the amount with a grid of mesh width. If the number of occupied boxes from the crowd, so is the box dimension

In fact, you can other types of coverings (circles or spheres, overlapping squares, etc.) will select and calculate exactly, and the result is theoretically the same, in numerical practice ( if the limit can not calculate) but not necessarily.

Yardstick method

This method is suitable only for topologically one-dimensional quantities, ie for curves. Measure the length by Abzirkeln. The point of intersection of a circle (or sphere of embedding dimension in 3) with the new curve is again the center of the next circuit. The curve is overlapped with the circles of the same radius. With the number and the radius of these circles, the procedure continues as in the Boxcounting method. In fact, the yardstick method is theoretically only a special case of Boxcounting method.

Minkowski dimension

Surrounds you a lot with a Minkowski sausage of thickness and measures their - dimensional volume, so can use it to define an equivalent to box dimension dimension:

Similarity dimension

Quantities that consist of scaled down by a factor versions of themselves, called self-similar. For this is the similarity dimension

Defined. Note that you need not limit here.

Example: A square consists of four squares () to half () on a side and has it. But a circle is not scaled circles, and the similarity dimension is not defined. The dimension of many famous fractals can, however, determine that. Due to lack of education Limes the similarity dimension is particularly simple and is therefore often the only understandable for laymen fractal dimension. This method of calculation dimension suggests itself especially at IFS fractals.

Hausdorff dimension

The Hausdorff dimension or Hausdorff Besicovitch dimension, named after Felix Hausdorff and Abram Samoilowitsch Besikowitsch, the measure-theoretical definition of the fractal dimension. The -dimensional Hausdorff measure takes almost anywhere on either the value 0 or the value. The point at which the step of taking place by 0, the Hausdorff dimension.

Natural fractals

When moving away from the mathematical idealization, and consider quantities such as coastlines, moon crater or just digitized images of fractals, it can not be carried out due to the finite resolution of the limit transition. One would always get the dimension 0 because we consider a finite set of points. Instead, one makes use of the property of scale invariance advantage and determines the dimension by plotting against the so-called log-log plot. Scales, then assigns this plot, at least in the region of small values ​​in the slope. Is the range of scales sufficiently large (several decades), one speaks of natural fractals.

Theoretically equivalent definitions of the fractal dimension are not equal in this numerical variation. Thus, the Yardstick dimension proves mostly as larger than the box dimension.

Rényi dimensions Dq

The special feature of the Rényi dimensions is that they are not on a lot, but to an extent ( or density ) refer. However, one can also take the point density of a set. Judging from the Boxcounting method, so is not only whether a box is occupied or not, but also how much is in the box. The normalized contents of the box is raised to the nth power, and summed over all boxes:

For supplies the rule of l' Hospital:

The Rényi dimension to the normal fractal dimension. The to is also called information dimension and the correlation dimension to. Dimensions, which have different dimensions up, are also called multifractals.

Properties and relationship between the dimensions

• The fractal dimension of a set is greater than or equal to the dimension of a subset.
• All fractal dimensions of an object are, if defined, surprisingly often the same size. Otherwise inequalities are known, then the Hausdorff dimension, for example, is always less than or equal to the Boxcounting dimension.
• The fractal dimension is always greater than or equal to the topological dimension.
• The fractal dimension is always less than or equal to the embedding dimension.

Applications

The fractal dimension can be used in surface physics to the characterization of surfaces and for the classification and the comparison of surface structures.