Example Mandelbrot

The first detailed book to describe the practice and theory of stereology was assembled by two Americans, DeHoff and Rhines (1968) both these men were famous practitioners in their day. There has been a steady stream of books since then a fine, concise and very clear overview is that by Exner (1996). In the last few years, a specialised form of microstructural analysis, entirely dependent on computerised image analysis, has emerged - fractal analysis, a form of measurement of roughness in two or three dimensions. Most of the voluminous literature of fractals, initiated by a mathematician, Benoit Mandelbrot at IBM, is irrelevant to materials science, but there is a sub-parepisteme of fractal analysis which relates the fractal dimension to fracture toughness one example of this has been analysed, together with an explanation of the meaning of fractal dimension , by Cahn (1989). 

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. 

These three cases are classical examples of fractal curves and dimensions as pioneered by Mandelbrot [96, 97] for macroscopic objects. They suggested to us possible extensions for the creation and analysis of unique spatial organizations of atoms within monomer repeat units, within branch cells, within dendrons, within a dendrimer, to give molecular-level surfaces in three dimensions of nanoscopic properties. 

Many objects in nature exhibit self-similarity, that is, they look more or less the same regardless of the degree of magnification (at least within some limits). Examples include coastlines, mountain ridges, snowflakes, leaves, branched polymers, etc. As pointed out by Mandelbrot, such self-similar objects may generally be described as... 

Unfortunately, we will not be able to delve into the scientific applications of fractals, nor the lovely mathematical theory behind them. For the clearest introduction to the theory and applications of fractals, see Falconer (1990). The books of Mandelbrot (1982), Peitgen and Richter (1986), Barnsley (1988), Feder (1988), and Schroeder (1991) are also recommended for their many fascinating pictures and examples. 

Mandelbrot s relationship (eq 18) assumes that the objects are perfectly rigid and stick immediately and irreversibly at each point of intersection (chemically equivalent to an infinite condensation rate). In fact, fractal objects are more or less compliant, and the sticking probability is always 1. These factors mitigate the criterion for mutual transparency for example, if the condensation rate is reduced, screening is less effective, and objects with D > 1.5 can interpenetrate. Because the condensation rate of silica depends strongly on pH (see Figure 3), the transparency or opacity and ultimately the film porosity can be manipulated by the addition of acid or base catalyst (24). 

Statistical fractals are generated by disordered (random) processes. An element of disorder is typical of most real physical phenomena and objects. The fact that disorder, i.e., the absence of any spatial correlation, is a sufficient condition for the formation of fractals was first noted by Mandelbrot [1]. A typical example of this type of fractal is the random-walk path. However, real physical systems are often inadequately described by purely statistical models. Among other reasons, this is due to the effect of excluded volume. The essence of this effect lies in the geometric restriction that forbids two different elements of a system to occupy the same volume in space. This restriction is to be taken into account in the corresponding modelling [10, 11]. The best-known examples of this type of models are self-avoiding random walk, lattice animals and statistical percolation. 

Over the years, many examples of continuous, nowhere differentiable functions have been published see Edgar [11 (pp. 7, 341)]. One of them, the so-called Weierstrass-Mandelbrot function, assumes a particular significance in environmental science because it constitutes the theoretical basis of the first article that used fractal geometry in connection with soils data. Burrough [19] used the Weierstrass-Mandelbrot function to describe the often erratic-looking spatial variation of soil properties along transects. 

At the same time that he coined the term fractal , Mandelbrot [4] pointed out that fractal dimensions would not suffice to provide a satisfactory description of the geometry of lacunar fractals, and that at least one other parameter, which he termed lacunarity , would be necessary. The key reason for this requirement is vividly illustrated by the fact that Sierpinski carpets (Figure 2.15) with greatly different appearances can have precisely the same fractal (similarity) dimension. Therefore, the fractal dimension alone is not a very reliable diagnostic of the geometry and properties of lacunar fractals. For physical objects, such as porous media, where the geometry of interstices and pores influences a wide range of properties, this means that any attempt to find a unique relationship between the fractal dimension of these objects and, for example, their transport or dielectric properties is most probably doomed to failure, unless one also takes lacunarity explicitly into account. 

Mandelbrot beUeved initially that one would do better without a precise definition of fractals. His original essay [1] contains none. By 1977, however, he saw the need to produce at least a tentative definition. It is the now classical statement that a fractal is a set for which the Hausdorff dimension strictly exceeds the topological dimension [4, 5,10]. For example, the Cantor set is a fractal, according to this viewpoint, since Dh = 0.631 > Z)r=0. 

As with the concept of fractals itself, a certain level of vagueness characterizes the definition of the fractal dimension. The approach advocated by Mandelbrot in 1975 [1], and reiterated in his 1982 book, is to use the expression fractal dimension as a generic term applicable to all the variants described in Section 2.3, and to use in each specific case whichever definition is most appropriate. This suggestion is adopted by a number of authors [e.g. 56]. However, it could, potentially, lead to considerable confusion if it is followed inconsistently, particularly in cases where different dimensions assume different values (see Section 2.3 for examples). Therefore, many mathematicians consider it safer to refer to specific dimensions by name, such as the correlation dimension, instead of using the generic term fractal dimension [e.g. 5]. 

Let us consider the case of deterministic fractals first, i.e. self-similar substrates which can be constructed according to deterministic rules. Prominent examples are Sierpinski triangular or square lattices, also called gasket or carpet (in d = 2) and sponge (in d = 3), respectively, Mandelbrot-Given fractals, which are models for the backbone of the incipient percolation cluster, and hierachical lattices (see for instance the overview in Ref. [21]). In this chapter, however, we restrict the discussion to the Sierpinski triangular and square lattice for brevity. 

Originally, Greek philosophers thought that the universe was continuous and that the world could be described by lines, areas, and volumes, according to the geometry (literally earth-measurement ), set down by Euclid, for example, around 300 BC. It became evident a few centuries ago that shapes are not continuous but are composed of similar but smaller shapes as they become more magnified. Thus a tree looks more complex the more it is studied on a finer scale, as shown in Fig. 5.7. This concept is the basis of fractal geometry which has been described by Mandelbrot. ... 

Gribbin, John. Deep Simplicity Brining Order to Chaos and Complexity. New York Random House, 2005. An examination of how chaos theory and related fields have changed scientific understanding of the universe. Provides many examples of complex systems found in nature and human culture. Mandelbrot, Benoit, and Richard L. Hudson. The Misbehavior of Markets A Fractal View of Financial Turbulence. New York Basic Books, 2006. Provides a detailed examination of fractal patterns and chaotic systems analysis in the theory of financial markets. Provides examples of how chaotic analysis can be used in economics and odier areas of human social behavior. Stewart, Ian. Does God Play Dice The New Mathematics of Chaos. 2d ed. 1997. Reprint. New York Hyperion, 2005. An evaluation of the role that order and chaos play in the universe through popular explanations of mathematic problems. Includes accessible descriptions of complex mathematical ideas underpinning chaos theory. 

The porosity and pore systems in cement-based composites may be one example of fractality, which is characteristic for these materials. Fractal quantity depends on the scale used to measure it for example, the fracture area of a concrete element cannot be determined in an unambiguons way if the method and scale of its determination are not given (cf. Chapter 10.5). The classification and analysis of a pore system and all quantitative results derived depend among other things on the method of observation and magnification of microscopic images (Mandelbrot 1982 Guyon 1988). 

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