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Figure 2.17 The length of coastlines as a function of yardstick length. (Data from [53].)

2.4.2 'Definition'of Fractals

As the examples of the previous section illustrate, many scientists observed over several decades that natural objects or processes often have features akin to those of the von Koch curve or Menger sponge. This greatly stimulated interest in these monstrous sets; in this case, as in many others, interest in geometry was driven by its applications to nature [5].

This movement led to the publication by Mandelbrot in 1975 [1] of an essay in which he highlighted the similarities among the then-known continuous, nowhere-differentiable sets. He coined for these sets the term 'fractal', to emphasize the fact that their Hausdorff dimensions are often fractional. In the words of Dyson [54], 'fractal is a word invented by Mandelbrot to bring together under one heading a large class of objects that have [played]... an historical role... in the development of pure mathematics'.

Mandelbrot believed 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 > Dt = 0.

The above definition immediately proved unsatisfactory, in that it excluded a number of sets with properties very similar to those of sets that satisfied the definition and, therefore, which also ought to be regarded as fractals. Indeed, according to this definition, the Cantor singular function (DH = DT = 1) and the Peano plane-filling curve (Dh = DT = 2) are not fractals. Various other definitions of fractals have been proposed [e.g. 4 (p. 362)], but they all seem to suffer from the same drawback.

Perhaps by modifying slightly the list of attributes of the Cantor set (Section 2.2.1), and by relaxing or deleting some of them, one could approach, as closely as possible, to a definition of the concept of fractal. This was done by Falconer [5], in what appears to be the best approximation of this concept to date. According to this author, when one refers to a set F (of points) as a fractal, one typically has one or more of the following properties in mind:

• F has a fine structure, i.e. detail on arbitrarily small scales;

• F is too irregular to be described in traditional geometrical language, both locally and globally;

• F often has some form of self-similarity, perhaps approximate or statistical;

• usually, the 'fractal dimension' of F (defined in some way) is greater than its topological dimension;

• in most cases of interest, F is defined in a very simple way, perhaps recursively (in which case, the various stages of the iterative construction are usually referred to as prefractals).

Falconer [5] has perhaps best captured the spirit with which the above definition of fractals needs to be undertaken:

My personal feeling is that the definition of a 'fractal' should be regarded in the same way as the biologist regards the definition of 'life'. There is no hard and fast definition, but just a list of properties characteristic of a living thing, such as the ability to reproduce or to move or to exist, to some extent, independently of the environment. Most living things have most of the characteristics on the list, though there are living objects that are exceptions to each of them.

While probably agreeing in principle with Falconer's [5] perception, many authors (in the fractal geometry literature at least) tend to give preferential weight to the requirement that fractals exhibit some form of self-similarity (see above, third point in the list of attributes of fractals).3 Feder [10 (p. 11)], for example, defines a fractal as a 'shape made of parts similar to the whole in some way'. In other words, a fractal appears the same regardless of the scale of observation; its 'look' is scale invariant.4

Perhaps the most important aspect of the various definitions above is that they all consider fractals to be sets of points in Rn, i.e. geometric constructs. This feature will assume particular significance in Section 2.5, where we shall discuss nongeometric fractals.

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].

2.4.3 'Natural' versus Mathematical Fractals

Can fractals, defined as in Section 2.4.2, serve as appropriate representations of natural objects or processes? The answer to this question is (surprisingly perhaps, yet uncompromisingly) no. Strictly speaking, there are no true (mathematical) fractals in

3 It should be kept in mind, however, that self-similarity cannot be the sole defining characteristic of fractals. A straight line segment is exactly self-similar, yet it hardly qualifies as a fractal!

4 However, graphs like that of Figure 2.17 indicate that specific geometric features of fractals, such as perimeter length or surface area, depend strongly on observation scale. Instead of 'scale invariance', therefore, one might consider, with Nottale [55], that the expression 'scale-covariance' captures the essence of fractals better. However, exploration of this notion of 'scale covariance' is beyond the scope of the present chapter.

nature [e.g. 5, 57 (p. 319)]. Nevertheless, under specific conditions, physical objects, like soil aggregates or clouds, may have features that are very accurately described by fractals or prefractals. A similar situation pertains with other geometrical structures; there are no true straight lines or circles in nature, yet one would hardly claim that these concepts have absolutely no use in describing nature!

Two of the ways in which true fractals can fail to represent physical objects are illustrated by the attempt to use the Menger sponge as a model of soil aggregates. First, soil aggregates have a porosity strictly smaller than unity and a mass density different from zero (otherwise there would be no aggregates to speak of!), but the Menger sponge has both porosity equal to unity and zero density (Section 2.2.3). Second, even if soil aggregates exhibit self-similarity over a range of observation scales (e.g. in thin sections observed under the microscope at different magnifications), this self-similarity eventually disappears if the aggregates are viewed at sufficiently small observation scales (e.g. in the extreme, at subatomic scales) or, at the other end of the spectrum, at observation scales commensurate with the size of the aggregate itself. By contrast, the Menger sponge has a fine structure at arbitrarily small scales.

For these two reasons, the Menger sponge cannot serve as a model of soil aggregates. In discussing Chepil's [52] results, however, we have seen that the prefractals associated with the Menger sponge (i.e. the sets of points obtained at intermediate steps in the iterative procedure which leads ad infinitum to the Menger sponge) have properties that are closely related to those of real aggregates (see Figure 2.16). These prefractals have a porosity < 1 and a mass density >0. Analysis of other examples, such as coastlines, clouds or landscapes, can lead to the same conclusion: fractal geometry is never an exact description of nature and often the geometry of a prefractal is a closer approximation to a physical object than is its associated fractal. Although physical systems are commonly referred to as 'natural fractals' (and we shall uphold that usage in the following to conform to standard usage), the expression 'natural prefractals' would probably be far more appropriate.

When describing the geometry of a given soil aggregate with a prefractal of the Menger sponge, one has to decide on the iteration step with which this prefractal is associated. To this iteration step corresponds a dimension rn (the individual subcube size), which represents the lower value of the range of scales at which the prefractal exhibits self-similarity. This length, when related to natural fractals, is usually referred to as the inner cutoff length [e.g. 57]. Physically, in the case of soil aggregates, this length is associated with the size of individual particles. On the other hand, since a soil aggregate is necessarily of finite size, there must be an upper limit to the range of scales at which it may be observed. For a given system, this upper limit, referred to as the upper or outer cutoff length, may be the actual size of the system itself. However, it may be more accurate to consider that it corresponds to the largest scale at which the system displays self-similarity.

The existence of an inner cutoff length has important consequences with respect to the evaluation of the dimensions of natural fractals. All the dimensions described in Section 2.3, except the similarity dimension, require a passage to a limit (e.g. limit to vanishingly small ball radius, box side or divider length). For physical reasons, the use of a limit is precluded for natural fractals. In consequence, the definitions of these dimensions have to be slightly modified when one applies them to natural fractals. For example, the box-counting dimension DBC can no longer be defined as in Equation (2.15). At best, one would be able to determine the number of boxes NS(F) only in the range of S values between the inner and outer cutoff lengths. In practice, the range of S values accessible via measurement may be much narrower. Under these conditions, Equation (2.15) may best be replaced by defining DBC(F) as the slope of the graph of ln NS(F) versus - ln S over a sufficiently large range of S or, equivalently, by finding a best-fit value of DBC in the power-law relationship NS(F) a S-dbc.

A direct consequence of this necessary change in the definition of the various dimensions is the fact that equalities or inequalities between dimensions that have been proven mathematically in the limit of vanishingly small ball radius, grid size or divider length may no longer be valid when such limits are not taken. Furthermore, the finite length of the 'yardsticks' may create serious practical difficulties. One of the key ones associated with the divider method (see Section 2.3.4), for example, relates to the existence of a remainder. This remainder stems from the fact that, most often, a noninteger number of steps is required to cover a given curve. At present there is no general consensus on how to deal with this remainder [e.g. 39].

In addition to the above problems, the existence of an inner cutoff length raises the question of what is 'fractal' in natural fractals. To understand this point, it is useful to take once again the example of the Menger sponge. As defined in Section 2.2.3, the Menger sponge is a set of points in R3 with (Lebesgue) measure equal to zero. One may also consider that there is another set of points that is closely associated with this first one: the points that were removed during the iterative attrition process leading to the Menger sponge. In other words, to the 'solid' structure of the Menger sponge is associated a volume of 'voids'. If one decides to interrupt the iterative process leading to the Menger sponge, then the 'solid' structure (no longer of measure equal to zero!) and the voids now have an interface. This feature is common to many natural fractals. In general, if the interface is fractal and scales like the mass of solids (i.e. follows a power law, or Paretian relationship, and has the same power-law, or Pareto-, exponent), the system is termed a mass fractal [e.g. 58]. If void (or pore) space and surface happen to scale alike, then the system is called a pore fractal, whereas if only the surface is fractal, then the system is called a surface fractal, or boundary fractal. In each case, one may use these fractals to characterize the dimensions introduced in Section 2.3, leading, for example, to 'box-counting pore fractal dimensions' or 'correlation surface fractal dimensions'.

2.4.4 Is 'Power Law' Equivalent to 'Fractal'?

Repeatedly in the preceding sections, the analysis of the geometrical properties of mathematical and 'natural' fractals has resulted in power-law or Paretian relationships between selected parameters. At no point in the text has this existence of a Paretian relationship been presented as a defining characteristic of fractals; it was instead obtained as a consequence of fractal geometry. A number of authors, however, consider that the essence of fractals is not the notion of an underlying geometry, as predicated above, but the evidence of Paretian behavior. From this viewpoint, a fractal is a set for which some statistical distribution function is a power law [e.g. 59]. Some authors [e.g. 60] label the latter fractals as probabilistic, to distinguish them from the geometric fractals, defined in Section 2.4.2.

To understand clearly the connections between these two types of fractals, it is worth reviewing briefly the development of the Pareto distribution and its relation to other statistical distributions in common use.

The Pareto distribution is named after the Italian-born Swiss professor of economics, Vilfredo Pareto (1848-1923). Originally, it dealt with the distribution of income over a population and may be stated as follows [61-66]:

where N is the number of persons having income >x. A and a are positive, real parameters. The relation of Equation (2.29) is now usually referred to as the 'Pareto distribution of the first kind', to distinguish it from alternative forms [e.g. 67 (p. 234)]. When applied to discrete data (e.g. the length of words), it is also often referred to as the Zipf distribution [60, 64]. In the following, we shall simply call it the Pareto distribution.

When N is plotted as a function of x, for given values of A and a, the distribution of Equation (2.29) is characterized by a very long right tail. Over the years, this Paretian behavior has been observed in relation with many socio-economic and other naturally occurring quantities. Examples [e.g. 67, 68] are the distributions of city population sizes, insurance claims, occurrence of natural resources, stock price fluctuations, size of firms, and of error clusters in communication circuits, to list only a few.

Usage of the Pareto distribution to describe data exhibiting very long right tails has been criticized by many researchers on the grounds that the Pareto distribution is not the only distribution with a very long right tail and that, often, it does not convincingly outperform its competitors. Macaulay [69], in particular, argues that 'the approximate linearity of the tail of a frequency distribution charted on a double logarithmic scale signifies relatively little, because it is such a common characteristic of frequency distributions of many and various types'. Indeed, in many cases, the exponential [38, 70], Weibull [38, 59] and lognormal [60] distributions mimic the Pareto distribution over certain ranges, or even provide a statistically better fit to data than the Pareto distribution [e.g. 65, 71].

In that context, whether or not one uses the Pareto distribution to fit experimental data often appears linked to one's belief in the universality of the Paretian behavior, and to the deviation from Paretian behavior that one is willing to tolerate. In some well-publicized cases of the use of the Pareto distribution in the literature [e.g. 60 (figure 4.7) and discussion in 38 (p. 205)], there is enough deviation from linearity in log-log plots, and the deviation is systematic enough over the whole range of experimental data, that skepticism in the applicability of the Pareto distribution is warranted. Yet the authors of the original articles consider a unique Pareto exponent appropriate, implying that the systems under study exhibit power-law behavior. A related issue concerns the method one uses to fit the Pareto distribution to experimental data. There are an array of methods available, including weighted or unweighted linear regression of log-transformed data, and nonlinear regression. Each of these methods may be applied to the full data set, or to a portion thereof, corresponding to points that are selected because they appear to fall on a straight line on a log-log plot. When the latter approach is adopted, one is faced with the problem of determining how big a range this linear segment should span to provide any confidence that the data indeed exhibit a power-law behavior. Brock [72] argues that data points showing linearity should span at least two or three orders of magnitude in the abscissa before one tries to fit to them with the Pareto distribution. A similar recommendation is made by other authors [e.g. 38].

Figure 2.18 illustrates some of the practical difficulties frequently encountered when trying to determine if the Pareto distribution adequately describes experimental data, and when evaluating its parameters. Originally, Cargill et al. [73] fitted a power law to their data via weighted linear regression, after elimination of the leftmost five data points (considered to be artifacts resulting from a less efficient extraction of copper from the ore). The weighting is proportional to the amount of ore. The Cargill et al. [73] regression line (dashed line in Figure 2.18) has a slope of -0.386,

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