Geometric fractals, self-similarity

Fractal theory is a relatively new field of geometry, formulated by Mandelbrot [196] for irregular rough-surfaced objects. The major properties of such objects are the dependence of the measured length (perimeter), surface, or volume on the scale of measurement and geometrical self-similarity... 

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. 

The word fractal was coined by Mandelbrot in his fundamental book.1 It is from the Latin adjective fractus which means broken and it is used to describe objects that are too irregular to fit into a traditional geometrical setting. The most representative property of fractal is its invariant shape under self-similar or self-affine scaling. In other words, fractal is a shape made of parts similar to the whole in some way.61 If the objects are invariant under isotropic scale transformations, they are self-similar fractals. In contrast, the real objects in nature are generally invariant under anisotropic transformations. In this case, they are self-affine fractals. Self-affine fractals have a broader sense than self-similar fractals. The distinction between the self-similarity and the selfaffinity is important to characterize the real surface in terms of the surface fractal dimension. 

For a limited discussion of fractal geometry, some simple descriptive definitions should suffice. Self-similarity is a characteristic of basic fractal objects. As described by Mandelbrot 58 When each piece of a shape is geometrically similar to the whole, both the shape and the cascade that generate it are called self-similar. Another term that is synonymous with self-similarity is scale-invariance, which also describes shapes that remain constant regardless of the scale of observation. Thus, the self-similar or scale-invariant macromolecular assembly possesses the same topology, or pattern of atomic connectivity, 62 in small as well as large segments. Self-similar objects are thus said to be invariant under dilation. 

The replacement rule we have used so far to generate geometric fractals creates isotropic fractals. In other words, the property of geometric self-similarity is... 

Due to the characteristic self-similar structure of the CCA-clusters with fractal dimension df 1.8 [3-8, 12], the cluster growth in a space-filling configuration above the gel point O is limited by the solid fraction Oa of the clusters. The cluster size is determined by a space-filling condition, stating that, up to a geometrical factor, the local solid fraction Oa equals the overall solid concentration O ... 

Disordered porous media have been adequately described by the fractal concept [154,216]. It was shown that if the pore space is determined by its fractal structure, the regular fractal model could be applied [154]. This implies that for the volume element of linear size A, the volume of the pore space is given in units of the characteristic pore size X by Vp = Gg(A/X)°r, where I), is the regular fractal dimension of the porous space, A coincides with the upper limit, and X coincides with the lower limit of the self-similarity. The constant G, is a geometric factor. Similarly, the volume of the whole sample is scaled as V Gg(A/X)d, where d is the Euclidean dimension (d = 3). Hence, the formula for the macroscopic porosity in terms of the regular fractal model can be derived from (65) and is given by... 

Fractals are geometric structures of fractional dimension their theoretical concepts and physical applications were early studied by Mandelbrot [Mandelbrot, 1982]. By definition, any structure possessing a self-similarity or a repeating motif invariant under a transformation of scale is caWcd fractal and may be represented by a fractal dimension. Mathematically, the fractal dimension Df of a set is defined through the relation ... 

Different relaxation functions are derived assuming that the actual (physical) ensemble of relaxation times is confined between the upper and lower limits of self-similarity. In this respect, the temporal fractal differs from a geometrical fractal (e.g., Cantor dust) for which only an upper limit (i.e., the initial segment before its subdivision) is assumed to exist. It is predicted that at times, shorter than the relaxation time at the lowest (primitive) self-similarity level the... 

In modern language assumption (L3) states that the surface reconstructs in a self-similar way. Though the exact mathematical specification of this self-similarity is an open problem, the appearance of the Elovich equation seems essentially related to a fractal nature of the reconstructed surface this property is immediately understood by observing that the number of molecules accomodated on a given finite geometric surface area diverges as t —> -poo. 

Roughly speaking, fractals are complex geometric shapes with fine structure at arbitrarily small scales. Usually they have some degree of self-similarity. In other words, if we magnify a tiny part of a fractal, we will see features reminiscent of the whole. Sometimes the similarity is exact more often it is only approximate or statistical. 

Fractal A geometric figure that is self-similar that is, any smaller piece of the figure will have roughly the same shape as the whole. 

Fractals - Geometrical objects that are self-similar under a change of scale i.e., they appear similar at all levels of magnification. They can be considered to have fractional dimensionality. Examples occur in diverse fields such as geography (rivers and shorelines), biology (trees), and solid state physics (amorphous materials). 

The exponent v has often been coimected with the fractal dimension of the eleetrode surface, but this connection is not necessary. Pajkossy and coworkers (76, 77) have shown, however, that speeifie adsorption effects in the double layer neeessarily do appear for such a dispersion. We can cormect the exponent to the fractality of the dynamical polarization and show that the polarization is self-similar in time, in contrast to the self-similar geometrical sfructure. 

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