Fractals objects

Fractal objects are quantified by their fractal dimension, dj. For linear-like stmctures, 1 < <2. FractaUy rough stmctures have a mass fractal... 

Perikinetic motion of small particles (known as colloids ) in a liquid is easily observed under the optical microscope or in a shaft of sunlight through a dusty room - the particles moving in a somewhat jerky and chaotic manner known as the random walk caused by particle bombardment by the fluid molecules reflecting their thermal energy. Einstein propounded the essential physics of perikinetic or Brownian motion (Furth, 1956). Brownian motion is stochastic in the sense that any earlier movements do not affect each successive displacement. This is thus a type of Markov process and the trajectory is an archetypal fractal object of dimension 2 (Mandlebroot, 1982). 

Fox, C. L., 22 679 Foxing, 11 409 FPAT file, 13 230, 235 Fractal gelation model, 23 63-64 Fractal objects, formation of, 23 63 Fractile distribution, 26 1021 Fractional carbonation, gallium extraction by, 12 345... 

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. 

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] ... 

Note 2 For a Euclidean object of constant density, d 3, but for a fractal object, d <3, such that its density decreases as the object gets larger. 

Note 3 For the surface area of a fractal object, s r m which s is the surface area contained within a radius, r, measured from any site or bond and d is termed the surface fractal dimension. 

In the Smale horseshoe and its variants, the repeller is composed of an infinite set of periodic and nonperiodic orbits indefinitely trapped in the region defining the transition complex. All the orbits are unstable of saddle type. The repeller occupies a vanishing volume in phase space and is typically a fractal object. Its construction is based on strict topological rules. All the periodic and nonperiodic orbits turn out to be topological combinations of a finite number of periodic orbits called the fundamental periodic orbits. Symbols are assigned to these fundamental periodic orbits that form an alphabet... 

This section attempts to examine macromolecular geometry, and in particular dendritic surface characteristics, from the perspectives of self-similarity and surface irregularity, or complexity, which are fundamental properties of basic fractal objects. It is further suggested that analyses of dendritic surface fractality can lead to a greater understanding of molecule/solvent/dendrimer interactions based on analogous examinations of other materials (e.g., porous silica and chemically reactive surfaces such as found in heterogeneous catalysts). 52 ... 

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. 

Fractal dimension, D, is another crucial property that is used to describe fractal objects and shapes. It is a measure of the amount of irregularity, or complexity, possessed by an object. For lines, 1 < D < 2 and for surfaces, 2 < D < 3. The greater the value of D, the more complex the object. As described by Avnir, 63 D is obtained from a resolu-... 

The essence of a flocculation process as described in Section 6.1 is the combining of primary particles into floccules, followed by the combining of the floccules into larger floccules, and so on. That this process can lead to a fractal object is illustrated by (he sequence of constructing clusters from a unit... 

In the real world, however, the objects we see in nature and the traditional geometric shapes do not bear much resemblance to one another. Mandelbrot [2] was the first to model this irregularity mathematically clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. Mandelbrot coined the word fractal for structures in space and processes in time that cannot be characterized by a single spatial or temporal scale. In fact, the fractal objects and processes in time have multiscale properties, i.e., they continue to exhibit detailed structure over a large range of scales. Consequently, the value of a property of a fractal object or process depends on the spatial or temporal characteristic scale measurement (ruler size) used. 

The issue of scaling was touched upon briefly in the previous section. Here, the quantitative features of scaling expressed as scaling laws for fractal objects or processes are discussed. Self-similarity has an important effect on the characteristics of fractal objects measured either on a part of the object or on the entire object. Thus, if one measures the value of a characteristic 9 (cu) on the entire object at resolution cu, the corresponding value measured on a piece of the object at finer resolution 9 (rcu) with r < 1 will be proportional to 9 (cu) ... 

This equation reveals that when measurements for fractal objects or processes are carried out at various resolutions, the log-log plot of the measured characteristic 9 (oj) against the scale oj is linear. Such simple power laws, which abound in nature, are in fact self-similar if oj is rescaled (multiplied by a constant), then 9 (oj) is still proportional to oja, albeit with a different constant of proportionality. As we will see in the rest of this book, power laws, with integer or fractional exponents, are one of the most abundant sources of self-similarity characterizing heterogeneous media or behaviors. 

When the characteristic is the average density of a fractal object, df = de + a, where de is the embedding dimension. 

To characterize the dynamic movement of particles on a fractal object, one needs two additional parameters the spectral or fracton dimension ds and the random-walk dimension dw. Both terms are quite important when diffusion phenomena are studied in disordered systems. This is so since the path of a particle or a molecule undergoing Brownian motion is a random fractal. A typical example of a random fractal is the percolation cluster shown in Figure 1.5. 

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