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which, according to a formula derived by Turcotte [59], corresponds to an unspecified 'fractal' dimension D = 1.16. If an unweighted nonlinear regression is carried out on the same truncated data set, then the slope of the resulting regression line (thin solid line in Figure 2.18) equals -0.406 (i.e. D = 1.22), with an R value of 0.994. Finally, when the Pareto distribution is fitted via an unweighted nonlinear regression to the full data set (thick solid line in Figure 2.18), the slope equals -0.327 (i.e. D = 0.98). There is some waviness in the data, but the fit is quite close (R = 0.995), and is comparable to that found in the literature in situations where power-law behavior has been postulated. A consequence of this good fit is that the first five data points no longer appear artifactual or deviatory, as assumed by Cargill et al. [73] and Turcotte [59]. However, the most interesting consequence is in terms of the fractal behavior of the cumulative copper grade C. If one uses the truncated data set to estimate D and one follows Brock's [72] recommendation, then C is not a probabilistic fractal, since its linear segment extends over less than one order of magnitude in M. On the other hand, if one considers the full data set, the linear segment spans very nearly two orders of magnitude in M and one may at least consider making the case that C is a probabilistic fractal.

A similar analysis could be performed on many data sets described in the literature [e.g. 59]. This clearly highlights the fact that the evaluation of fractal dimensions, whether of geometric or probabilistic fractals, is not straightforward. It also shows that there is often some subjectivity involved in determining whether or not a given system has a Paretian behavior. These observations affect the practical use of both the geometric and the probabilistic fractals. However, the subjectivity is particularly significant for probabilistic fractals, since evidence of Paretian behavior is their sole defining feature.

A further difference between geometric and probabilistic fractals concerns the range of values that are acceptable for the Pareto exponent a in Equation (2.29). In the various illustrations of Paretian relationships associated with mathematical monsters and fractals in preceding sections of this chapter (e.g. Equations (2.2) and (2.4)), the value of the exponent was always constrained to be (inclusively) between the topological dimension of the set and the Euclidian dimension of the embedding space Rn. No such restriction applies to the parameter a in Equation (2.29) when this equation is taken simply as the mathematical formulation of a particular statistical distribution. Clearly, when Equation (2.29) is applied to data related to the size of insurance claims or to stock price fluctuations, one cannot expect a to be restricted to a certain range (e.g. 0 < a < 3) for geometrical or physical reasons. Even when Equation (2.29) is used to describe some physical attribute of a real system, it is not guaranteed that either A or a, or even x for that matter, will have a clear physical meaning and that the values of a will be geometrically constrained. Some of the most convincing illustrations of this fact are found in applications of dimensional analysis to equations describing dynamical systems. Indeed, the so-called pi-theorem, one of the cornerstones of dimensional analysis, states that if there is a relationship among N + 1 variables involving N independent dimensional units, then the relation can be expressed in terms of a single nondimensional parameter, giving a power-law dependence [74]. This single nondimensional parameter, and consequently the coefficient and exponent of the power law, may not always have a clear physical interpretation, as is illustrated by Schmidt and Housen [74] in the case of the cratering efficiency of conventional and nuclear explosives. In all these situations, and indeed for all probabilistic fractals, the range of values that the exponent a can assume is not restricted. Cases of physical systems where this exponent (after transformation whenever necessary) leads to 'fractal dimensions' >3 are routinely reported in the literature [e.g. 59, 72, 75, 76]). Such values are entirely acceptable for probabilistic fractals [e.g. 72,77], even though they would be meaningless for geometric fractals embedded in R3. This distinction appears to have been frequently overlooked in the literature, as demonstrated by the lasting, yet groundless, controversy about supposedly 'unphysical' fractal dimensions >3 for natural fractals [e.g. 78].

Because of the constraints imposed on the exponent a in the case of geometric fractals, it is clear from the above analysis that whereas a geometric fractal is automatically also a probabilistic fractal, the reverse is not true in general. In environmental science, this conclusion has particular significance with respect to the use of fragmentation models to account for the size distribution of soil particles [e.g. 76] or fragmented rocks and other geological materials [e.g. 59]. These fragmentation models are described in the following subsection.

2.4.5 Fragmentation Fractals

The earliest attempt to develop a fragmentation model that accounts for the Paretian distribution of the size of fractured or fragmented solids seems to have been Matsushita's [77] fracture cascade model (see Figure 2.19). The simplest derivation of this model, in two-dimensional space, starts with a square with a unit side length. At the first stage, the square is divided into four equal subsquares of side 1/2. One of these subsquares, chosen randomly, is shaded (hatched diagonally) to indicate that it will not be fractured or fragmented further. At the second stage, the remaining three subsquares are divided each into four equal sub-subsquares of side (1/2)2, and one of them (chosen randomly) is again shaded. The same procedure is in turn applied to the remaining 32 unshaded sub-subsquares, and so on.

At the nth stage, there are 3n-1 newly shaded squares of side rn = 2-n. For finite n >> 1, the cumulative number N(rn) of shaded squares of side length greater than rn = 2-n may be expressed as follows [77]:

N(r„) = 1 + 3 + 32 + ... + 3n-1 a 3n = r„-Dfragm (2.30)

Equation (2.30) amounts to a power-law relationship between N(rn) and rn, so, as n ^to, it defines a probabilistic fractal. Following Kaye [43], it is occasionally called a fragmentation fractal, and Dfragm is termed the fragmentation fractal dimension.

UJJJJ-LLLLJULU iJJJLLUJUU-L-1 U-L-LLU

Figure 2.19 Fifth iteration in Matsushita's fracture cascade in two dimensions. The hatched squares are arranged regularly for convenience. (Modified from [77].)

UJJJJ-LLLLJULU iJJJLLUJUU-L-1 U-L-LLU

Figure 2.19 Fifth iteration in Matsushita's fracture cascade in two dimensions. The hatched squares are arranged regularly for convenience. (Modified from [77].)

There is an interesting connection between Matsushita's [77] fracture cascade model and some of the mathematical monsters (or geometric fractals) of Section 2.2. For example, if instead of dividing the initial square in Figure 2.19 into four equal subsquares and shading one, we divide it into nine equal subsquares and shade one, it is easy to see that the collection of shaded squares that we are producing in this manner corresponds to the square holes that are punched during the attrition process leading to the Sierpinski carpet (see Section 2.2.3). The exponent Dfragm of Equation (2.30), under these conditions, is equal to ln 8/ln 3 = 1.89, which is the value of the Hausdorff and similarity dimensions of the Sierpinski carpet itself (see Section 2.3.2).

The fracture cascade has been considered so far in two-dimensional space. To be applicable to the fragmentation of solid bodies, the model has to be extended to R3. In this case (see Figure 2.20), we could start with a cube with unit side length, divide it into eight subcubes, hatch three, apply the same procedure to each of the remaining five subcubes, and so on. The resulting value of Dfragm would be ln 5/ ln 2 = 2.322. More generally [e.g. 38], one could consider that the initial cube is divided into b3 equal-sized subcubes of side length 1/b, and that i randomly chosen subcubes are hatched, where b and i are arbitrary integers satisfying 1 < b and 1 < i < b3. At the

Figure 2.20 Schematic representation of a fracture cascade in three dimensions. Three cubes are hatched (i.e. not fragmented further) at each fragmentation step.

Figure 2.20 Schematic representation of a fracture cascade in three dimensions. Three cubes are hatched (i.e. not fragmented further) at each fragmentation step.

nth iteration of this procedure, the cumulative number of hatched cubes larger than r„ = 1/bn is

N(r,) = i[1 + (b3 - i) + (b3 - i)2 +... + (b3 - i)"-1] a (b3 - i)" = r,

Dfragm

The power-law relationship of Equation (2.31) may be expressed differently if one introduces a 'probability of fragmentation' pc = (b3 - i)/b3, representing the fraction of the nth-order cubes of size b-" that are further fragmented. Under these conditions, Dfragm in Equation (2.31) becomes

Dfragm ln(b3pc ) ln b

The inequality 1 < i < b3 (see above) implies that 1/b3 < pc < 1, i.e. that 0 < Dfragm < 3. This restriction on the range of values taken by Dfragm is a definite sign that the model described above is not satisfactory as a 'physical' explanation of the general Pareto distribution, since values of a in Equation (2.29) as high as 4 or 5 are routinely measured in practice [e.g. 59, 76]. On the other hand, the inequality Dfragm < 3 has been considered by some to be an indication that geometric fractal concepts are appropriate to describe any fragmented material [59]. This viewpoint deserves further analysis, since it assumes that every fragmented material was created by a process similar to that shown in Figures 2.19 and 2.20. If fragments or particulate matter are characterized using Equation (2.29), then the exponent a characterizes only the size of the individual fragments. It provides no quantitative insight concerning the geometry of the arrangement of these same fragments. By piling up or aggregating the fragments in specific ways, it may be possible to create with them various pore, mass or surface fractals. There are probably infinitely many ways to do this. However, there are also many ways to assemble the fragments to create nonfractal structures. One such structure is in fact illustrated in Figure 2.20. Indeed, it is always possible to assemble the fragments resulting from a given fracture cascade in such a way as to reconstitute the original cube, which is definitely not fractal! Arguments that the dimension found when the size distribution of a fragmented material is characterized using Equation (2.29) should obey the inequality a < 3 (because Dfragm < 3) are, thus, without basis.

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