Fractal dimensions Hausdorff

Herein, ds is the space dimension, i.e. 2, and df is the fractal dimension (Hausdorff dimension) of the 2D aggregated particle cluster. Typical values for Hausdorff dimensions are e.g. 1.44 for diffusion limited cluster aggregation (DLCA) and 1.55 for reaction limited cluster aggregation (RLCA) [22, 31], Assuming isolated 2D aggregates which were formed by interfacial particle-particle aggregation, an exponent of about —6 is estimated. Close to the two- dimensional sol-gel transition, the system should behave like a percolated network and the corresponding exponent is determined to —9.5 [31],... 

The properties characteristic to fractal objects were mentioned first by Leonardo da Vinci, but the term fractal dimension appeared in 1919 in a publication by Felix Hausdorff [197], a more poetic description of fractals was given by Lewis Richardson in 1922 [198] (cited by [199]), but the systematic study was performed by Benoit B. Mandelbrot [196], Mandelbrot transformed pathological monsters by Hausdorff into the scientific instrument, which is widely used in materials science and engineering [200-202]. Geometrical self-similarity means, for example, that it is not possible to discriminate between two photographs of the same object taken with two very different scales. 

For fractal systems, the Hausdorff-Besicovitch dimension is equal to the similarity dimension, that is, df - d . We consider the triangular Sierpinski carpet as an example (Fig. 4). The iteration process means that the triangle is replaced by N = 3 triangles diminished with similarity coefficient K = 1 /2. Thus, the fractal dimension and the triangular Sierpinski carpet similarity dimension are given by... 

Falconer (1990) discusses other fractal dimensions, the most important of which is the Hausdorff dimension. It is more subtle than the box dimension. The main conceptual difference is that the Hausdorff dimension uses coverings by small sets of varying sizes, not just boxes of fixed size e. It has nicer mathematical properties than the box dimension, but unfortunately it is even harder to compute numerically. 

Figure 4.9. The A = 123 Sierpinski gasket, a two-dimensional uncountable set with zero measure and Hausdorff (fractal) dimension 3/ 2 = 1.584962... the companion Euclidean lattice referred to in the text is a space filling triangular lattice of (interior valency v = 6.

Figure 4.15. The N = 12 first generation Menger sponge, a symmetric fractal set in three dimensions of Hausdorff (fractal) dimension tn 20/fn 3 = 2.7268. is configuration (1). Calculations based on (1) and the additional configurations (2-7) are discussed in the text.

To begin, recall that, in general, spaces can be characterized by three quantities dg, the dimension of the embedding Euclidean space, df the Hausdorff or fractal dimension, and ds the spectral or fracton dimension. A key to what follows is that for Euclidean spaces, these three dimensions are equal [57,58]. 

A fractal object such as a C curve may have some unusual properties. The properties are that it has a fractal dimension but this fractal dimension is not a fraction. In the case of the C curve it is equal to two. The reason this object is still a fractal relates to a definition of fractal dimension. First, one defines two concepts of dimensions the topological dimension, which corresponds to our usual concept of a dimension, and a so-called Hausdorff-Besicovic dimension. If for a given object the two dimensions defined are different, the object is said to have a fractal dimension. In the case of the... 

Certain real systems seem to be described by OLA, notably electrodeposition on a sharp point (39) and dielectric breakdown (33,40). The second class involves cluster formation by the homogeneous aggregation of a collection of two clusters of comparable size (37, ) (cluster-cluster aggregation, CA) and the resultant aggregate has a more open structure and lower fractal dimension, D = 1.4 ( d = 2) and 1.8 (d = 3). Real smoke ( ) and colloids (41) seem to have D = 1.8 this is a satisfying verification of the model. A process that has not, however, been included in the simulations is rearrangement within the clusters. This would lead to denser structures with higher Hausdorff dimensions ( ). 

Nevertheless, the determination of the fractal dimension from a data set thought to be chaotic is often of interest. A number of different dimensions exist in the literature, including the Hausdorff dimension, the information dimension, the correlation dimension, and the Lyapunov dimension. Which of these is the true fractal dimension Of the ones in this list, the information dimension, Di, has the most basic and fundamental definition, so we often think of it as the true fractal dimension. Because the information dimension is impractical to calculate directly, however, most investigators have taken to finding the correlation dimension, Dq, as an estimate of the fractal dimension. Grassberger and Procaccia published a straightforward and widely used algorithm for the calculation of the correlation dimension. On the other hand, the Lyapunov... 

Fractals are self-similar objects invariant with respect to local dilatations, i.e., objects that reproduce the same shape during observation at various magnifications. The concept of fractals as self-similar ensembles was introduced by Mandelbrot [1], who defined a fractal as a set for which the Hausdorff-Besicovitch dimension always exceeds the topological dimension. The fractal dimension D of an object inserted in a d-dimensional Euclidean space varies from 1 to d. Fractal objects are the natural filling of the gap between known Euclidean objects with integer dimensions 0,1, 2, 3. .. The majority of naturally existing objects proved to be fractal, which is the main reason for the vigorous development of the methods of fractal analysis. 

Yet another important property of fractals which distinguishes them from traditional Euclidean objects is that at least three dimensions have to be determined, namely, d, the dimension of the enveloping Euclidean space, df, the fractal (Hausdorff) dimension, and d the spectral (fraction) dimension, which characterises the object connectivity. [For Euclidean spaces, d = d = d this allows Euclidean objects to be regarded as a specific ( degenerate ) case of fractal objects. Below we shall repeatedly encounter this statement] [27]. This means that two fractal dimensions, d( and d are needed to describe the structure of a fractal object (for example, a polymer) even when the d value is fixed. This situation corresponds to the statement of non-equilibrium thermodynamics according to which at least two parameters of order are required to describe thermodynamically nonequilibrium solids (polymers), for which the Prigogine-Defay criterion is not met [28, 29]. 

Linear polymer macromolecules are known to occur in various conformational and/or phase states, depending on their molecular weight, the quality of the solvent, temperature, concentration, and other factors [1]. The most trivial of these states are a random coil in an ideal (0) solvent, an impermeable coil in a good solvent, and a permeable coil. In each of these states, a macromolecular coil in solution is a fractal, i.e., a self-similar object described by the so-called fractal (Hausdorff) dimension D, which is generally unequal to its topological dimension df. The fractal dimension D of a macromolecular coil characterises the spatial distribution of its constituent elements [2]. 

Fractals can be defined in two different ways. The first definition considers a fractal to be a structure that is self-similar at any scale. The second definition considers a fractal to be a stmcture with a noninteger Hausdorff dimension. Let M be the fractal mass, i.e., the number of points of the fractal. If the mass density is constant, the mass is proportional to the fractal volume. The latter is proportional to, where L is the fractal length and the Hausdorff, or fractal, dimension. Both definitions... 

At present there are several methods of filler structure (distribution) determination in polymer matrix, both experimental [10, 35] and theoretical [4]. All the indicated methods describe this distribution by fractal dimension of filler particles network. However, correct determination of any object fractal (Hausdorff) dimension includes three obligatory conditions. The first from them is the indicated above determination of fiiactal dimension numerical magnitude, which should not be equal to object topological dimension. As it is known [36], any real (physical) fractal possesses fiiactal properties within a certain scales range. Therefore, the second condition is the evidence of object self-similarity in this scales range [37]. And at last, the third condition is the correct choice of measurement scales range itself As it has been shown in Refs. [38, 39], the minimum range should exceed at any rate one self-similarity iteration. 

Fractal or Hausdorff-Besicovitch dimension of a pattern Diameter of an object... 

Fractal structures are self-similar in that the two-point density-density correlation function and their essential geometric properties are independent of the length scale [59,61-63]. In d-dimensional space, they can be characterized by fractal or Hausdorff-Besicovitch dimension Df [61,63,64]. The... 

Fractal Dimension Measure of a geometric object that can have fractional values. It refers to the measure of how fast the length, area, or volume of an object increases with a decrease in scale. Fractal dimension can be calculated by box counting or by evaluating the information dimension of an object. Generator Collection of scaled copies of an initiator. Hausdorff-Besicovitch Dimension Mathematical statement used to obtain a dimension that is not a whole number, commonly written as d = log (N)/ log (r). 

At present there are several methods of filler structure (distribution) determination in polymer matrix, both experimental [10, 35] and theoretical [4]. All the indicated methods describe this distribution by fractal dimension of filler particles network. However, correct determination of any object fractal (Hausdorff) dimension includes three obligatory conditions. The first from them is the indicated above determination of fractal dimension numerical magnitude, which should not be equal to object topological... 

It is known [3], that macromolecular coil in various polymer s states (solution, melt, solid phase) represents fractal object characterized by fractal (Hausdorff) dimension Df. Specific feature of fractal objects is distribution of their mass in the space the density p of such object changes at its radius R variation as follows [4] ... 

This function is normahzed to take the unit value for 0 = 2n. For vanishing wavenumber, the cumulative function is equal to Fk Q) = 0/(2ti), which is the cumulative function of the microcanonical uniform distribution in phase space. For nonvanishing wavenumbers, the cumulative function becomes complex. These cumulative functions typically form fractal curves in the complex plane (ReF, ImF ). Their Hausdorff dimension Du can be calculated as follows. We can decompose the phase space into cells labeled by co and represent the trajectories by the sequence m = ( o i 2 n-i of cells visited at regular time interval 0, x, 2x,..., (n — l)x. The integral over the phase-space curve in Eq. (60) can be discretized into a sum over the paths a>. The weight of each path to is... 

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