- Fractal dimension
fractal geometry, the fractal dimension, "D", is a statistical quantity that gives an indication of how completely a fractalappears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension and none of them should be treated as the universal one. From the theoretical point of view the most important are the Hausdorff dimension, the packing dimensionand, more generally, the Rényi dimensions. On the other hand the box-counting dimensionand correlation dimensionare widely used in practice, partly due to their ease of implementation.
Although for some classical fractals all these dimensions do coincide, in general they are not equivalent. For example, what is the dimension of the
Koch snowflake? It has topological dimensionone, but it is by no means a curve: the length of the curve between any two points on it is infinite. No small piece of it is line-like, but neither is it like a piece of the plane or any other. In some sense, we could say that it is too big to be thought of as a one-dimensional object, but too thin to be a two-dimensional object, leading to the question of whether its dimension might best be described in some sense by number between one and two. This is just one simple way of motivating the idea of fractal dimension.
The many definitions
There are two main approaches to generate a fractal structure. One is growing from a unit object, and the other is to construct the subsequent divisions of an original structure, like the
Sierpinski triangle(Fig.(2))Fluctuations and Scaling in Biology. Edited by T. Vicsek, 2001] . Here we follow the second approach to define the dimension of fractal structures.
If we take an object with linear size equal to 1 residing in Euclidean dimension , and reduce its linear size by the factor in each spatial direction, it takes number of self similar objects to cover the original object(Fig.(1)). However, the dimension defined by
is still equal to its topological or Euclidean dimension. By applying the above equation to fractal structure, we can get the dimension of fractal structure (which is more or less the
Hausdorff dimension) as a non-whole number as expected.
where "(ε)" is the number of self-similar structures of linear size ε needed to cover the whole structure.
For instance, the fractal dimension of Sierpinski triangle (Fig.(2)) is given by,
Closely related to this is the
box-counting dimension, which considers, if the space were divided up into a grid of boxes of size ε, how does the number of boxes scale that would contain part of the attractor? Again,
Other dimension quantities include the information dimension, which considers how the average information needed to identify an occupied box scales, as the scale of boxes gets smaller:
correlation dimension, which is perhaps easiest to calculate,
where "M" is the number of points used to generate a representation of the fractal or attractor, and "g"ε is the number of pairs of points closer than ε to each other.
The last three can all be seen as special cases of a continuous spectrum of generalised or
Rényi dimensions of order α, defined by
where the numerator in the limit is the
Rényi entropyof order α. The Rényi dimension with α=0 treats all parts of the support of the attractor equally; but for larger values of α increasing weight in the calculation is given to the parts of the attractor which are visited most frequently.
An attractor for which the Rényi dimensions are not all equal is said to be a
multifractal, or to exhibit multifractal structure. This is a signature that different scaling behaviour is occurring in different parts of the attractor.
Estimating the fractal dimension of real-world data
The fractal dimension measures, described above, are derived from fractals which are formally-defined. However, organisms and real-world phenomena exhibit fractal properties (see Fractals in nature), so it can often be useful to characterise the fractal dimension of a set of sampled data. The fractal dimension measures cannot be derived exactly but must be estimated. This is used in a variety of research areas including physics [ B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot, and S. W. Zucker, doi-inline|10.1103/PhysRevA.39.1500|Evaluating the fractal dimension of profiles, Phys. Rev. A, 39 (1989), pp. 1500–1512.] , image analysis [P. Soille and J.-F. Rivest, [http://ams.jrc.it/soille/soille-rivest96.pdf On the validity of fractal dimension measurements in image analysis] , Journal of Visual Communication and Image Rep- resentation, 7 (1996), pp. 217–229.] , acoustics [P. Maragos and A. Potamianos, doi-inline|10.1121/1.426738|Fractal dimensions of speech sounds: Computation and application to automatic speech recognition, The Journal of the Acoustical Society of America, 105 (1999), p. 1925.] , Riemann zeta zeros cite journal|author=O. Shanker|year=2006|title=Random matrices, generalized zeta functions and self-similarity of zero distributions|journal=J. Phys. A: Math. Gen.|volume=39 |pages=13983–13997 |doi=10.1088/0305-4470/39/45/008] and even (electro)chemical processes [
Ali Eftekhari, [http://dx.doi.org/10.1149/1.1773583 Fractal Dimension of Electrochemical Reactions] "Journal of the Electrochemical Society", 2004, 151 (9), E291 – E296.] .
Practical dimension estimates are very sensitive to numerical or experimental noise, and particularly sensitive to limitations on the amount of data. Claims based on fractal dimension estimates, particularly claims of low-dimensional dynamical behaviour, should always be taken with a grain of salt — there is an inevitable ceiling, unless "very" large numbers of data points are presented.
List of fractals by Hausdorff dimension
*Mandelbrot, Benoît B., "The (Mis)Behavior of Markets, A Fractal View of Risk, Ruin and Reward" (Basic Books, 2004)
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