Fractal dimension

from Wikipedia, the free encyclopedia

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 the case of fractals . The special thing is that the fractal dimension does not have to be an integer. There are different ways to define a fractal dimension.

Box counting dimension

With the box counting method, the amount is covered with a grid the width of the grid . If the number of boxes occupied by the crowd is the box dimension

.

In fact, you can choose other types of overlap (circles or spheres, intersecting squares, etc.) and calculate them in exactly the same way , and the result is theoretically the same, but not necessarily in numerical practice (if you cannot calculate the limit).

Yardstick method

This method is only suitable for topologically one-dimensional sets, i.e. for curves. Its length is measured by dividing it off. The intersection of a circle (or a sphere in embedding dimension 3) with the curve is in turn the new center point of the next circle. So the curve is covered with circles of the same radius. With the number and the radius of these circles, proceed as with the box counting method. In fact, the yardstick method is theoretically just a special case of the box counting method.

Minkowski dimension

If you surround a quantity with a Minkowski sausage of the thickness and measure its -dimensional volume , then a dimension equivalent to the box dimension can be defined:

,
.

Similarity dimension

Sets that consist of versions of themselves scaled down by the factor are called self-similar . For these is the similarity dimension

Are defined. Note that you don't need a Limes here.

Example: A square consists of four squares ( ) half the ( ) edge length and thus has . But even a circle does not consist of reduced circles, and the similarity dimension is not defined. The dimensions of many known fractals can be determined with it. Due to the lack of a limit, the similarity dimension is particularly simple and is therefore often the only fractal dimension that laymen can understand. This method of dimension calculation is particularly important for IFS fractals .

Hausdorff dimension

The Hausdorff dimension, or Hausdorff-Besicovitch dimension, named after Felix Hausdorff and Abram Samoilowitsch Besikowitsch , is the theoretical definition of the fractal dimension. The -dimensional Hausdorff measure takes either the value 0 or the value almost everywhere . The place at which the jump from to 0 takes place is the Hausdorff dimension.

Natural fractals

If you move away from mathematical idealization and consider quantities such as coastlines, lunar craters or simply digitized images of fractals, the limit value transition can no longer be carried out because of the finite resolution . One would always get the dimension 0 because one is considering a finite set of points. Instead, one makes use of the property of scale invariance and determines the dimension by plotting against in the so-called log-log plot . Scaled , then this plot shows the slope at least in the area of ​​smaller values . If the scaling range is sufficiently large (several decades), one speaks of natural fractals.

Theoretically equivalent definitions of the fractal dimension are no longer the same in this numerical variant. The yardstick dimension usually turns out to be larger than the box dimension.

Rényi dimensions D q

What is special about the Rényi dimensions is that they do not refer to a quantity, but to a measure (or a density). However, one can also take the point density of a set. If the box counting method is used, it not only counts whether a box is occupied or not, but also how much is in the box. The normalized content of the box is raised to the -th power and added up across all boxes:

.

For de l'Hospital's rule gives :

.

The Rényi dimension zu is the normal fractal dimension. The to is also called the information dimension and that to the correlation dimension. Dimensions that have different dimensions to 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, if defined, are surprisingly often the same size. Otherwise, inequalities are known, for example the Hausdorff dimension is always less than or equal to the box counting dimension.
  • The fractal dimension is always greater than or equal to the topological dimension.
  • The fractal dimension is always smaller than or equal to the embedding dimension.

Applications

The fractal dimension can be used in surface physics to characterize surfaces and to classify and compare surface structures.

Individual evidence

  1. Markus Bautsch: Scanning tunnel microscopic examinations of metals atomized with argon , Chapter 2.5: Fractal dimensions of surfaces , Verlag Köster, Berlin (1993), ISBN 3-929937-42-5