Appendix: Nonrandom and random fractals

     

Matthew Otwinowski

Non-linear waste rock modelling

APPENDIX 

Nonrandom and random fractals

Physical structures often look the same at different stages of enlargement. This property is often called scale symmetry or self-similarity. A good example of self-similar structures are branches of snow crystals which show the same structure at different magnifications. Objects which have this scale symmetry are often called fractals and are quantitatively characterized by a fractal dimension: a dimension that corresponds to the geometrical shape under study, and is often not an integer. Fractals fall naturally into two categories, nonrandom and random. Fractals in physics are random but we will first discuss nonrandom fractals in order to introduce the notion of a fractal dimension.

A1. Nonrandom fractals

(I) Sierpinski gasket

The Sierpinski gasket is a good toy model with features characteristic for aggregation processes and the self-similar properties of pores and cracks which play a significant role in acid rock drainage. A simple growth rule can be used to construct a fractal object. We closely follow the original presentation of the problem by Sierpinski (1919) (see also Stanley(1991))¹.

The Sierpinski gasket is defmed algorithmically as an iterative geometric aggregation process. We use black triangular tiles, each of unit mass (M_o=1) and of unit edge length (L_o=1), to construct the series of objects illustrated in Fig. A.1. At first we join vertices of three tiles to obtain a triangular object in Fig. A.1b. This object has mass M₁=3 and edge L₁=2. At each subsequent step we build a new triangular object by joining the vertices of three copies of the previously obtained object. The edge length is given by the recursive formula L_n=2L_n-1=2ⁿ and the surface area of the whole triangular object is given by L_n²=2²ⁿ. At each stage (labelled by index n; n=1,2,3, …) we analyse the density defined as

Obviously the density deceases at each iteration step. The tile in Fig. A.1a has the density ρ_o=1=3^o/2^o. The object in Fig. A.1b has the density ρ₁=3/2²=3¹/2² (we have an object of the surface area equal to four, with three tiles of unit mass). In stage two, three of the objects of mass M₁=3¹ form an object of mass M₂=3², edge L₂=4 and surface area L₂²=4²=2⁴; in this way ρ₂=3²/2⁴. Object in Fig. A.1c is obtained again by joining the three objects in Fig. A.1b to obtain an object with mass M₃=3³, surface area L₃²=8²=2⁶ and the density ρ₃=3³/2⁶. We obviously have the rule ρ_n=3ⁿ/2²ⁿ=(3/4)ⁿ and M_n=3ⁿ.

Fig. A.1. Fractal Dimension.

We may, however, try to describe the increase of mass of the subsequent objects as the function of the edge length:

If we substitute (A2) into (A1), we find

We can now drop the index n and write simply

Equation (A5) defines the fractal dimension d_f. The amplitude A is not of intrinsic interest, since it depends on the choice we made for the definitions of units for M and L. We have defined M and L so that the amplitude A is equal to one. The exponent df, on the other hand, depends on the rule that we follow when we iterate. Different rules give different exponents. The exponent df can be found by calculating the slope α (α=d_f-2) of the plot ρ(L)=L^α in Fig. A.2:

Comparing (A6) and (A5) we identify the fractal dimension of the Sierpinski gasket as

.

Fig. A.2. A log-log plot of ρ, the fraction of two-dimensional space covered by black tiles, as a function of L, the increasing linear size of the object.

The Sierpinski gasket has the dimension d_f smaller than the dimension of the embedding two-dimensional space. We see that the Sierpinski gasket occupies a fraction of two-dimensional space and has a dimension intermediate between that of a line (d=1) and a dimension of an area (d=2). Hence we use the term fractal dimension and the term fractal (coined by Mandelbrot) for an object with a fractal dimension.

We can construct the Sierpinski gasket in a slightly different way. We can take a single triangle of edge L and at every stage divide every black triangle in four and remove one triangle at the center. In this way the density decreases by a factor 3/4 at every iteration. The first three steps are illustrated in Fig. A.3. In this way, at n-th iteration we create 3ⁿ triangular holes, each with surface area L²/2²ⁿ. Without changing the external size of the object we change its density according to the rule (A4).

Fig. A.3. Construction of the Sierpinski gasket by dividing black triangles into four triangles and removing a triangle at the center of each divided triangle.

While in mathematics we can continue the procedure ad infinitum, in physics we hit the atomic limit after about 20 iterations. After six iterations we reach the resolution limit of the laser printer used to produce this page. After eight iterations we reach the resolution limit of the human eye. In practice we often detect fractal properies of physical objects by analyzing an object at different magnification scales or by increasing the resolution of our observation device. Fig. A.3a can be viewed as a picture seen by an observer for whom it is impossible to see objects on scales smaller than L. Increasing resolution twice, an observer can see the same fractal object as the object in Fig A.3b, with features on scale L/2. Increasing the resolution twice again, an observer would see features on scale L/4. Thus by improving the resolution, an observer can analyze the density for the black part of the fractal object and reconstruct the plot A.2 with the log of magnification scale instead of logL along the horizontal axis.

Looking at a mathematical fractal we could double the magnification infinitely many times and at every stage we would see the same object. For this reason fractal structures are said to be scale-invariant or self-similar. We use the self-similarity property to analyze the fractal dimension of fragmented rock. For physical fractal objects the self-similarity ceases to exist at sufficiently large magnifications. The fractal nature of physical objects can be often observed in simple experiments over 2-4 orders of magnitude. When the slope of the logρ vs. logL plot changes at a certain point we say that a physical object is multifractal.

(II) Menger sponge

The Menger sponge is a fractal object embedded in three dimensions. The Menger sponge has cubic voids on all scales smaller than the size of the initial cube used for the iterative construction. It has been used as a simple fractal model for porous media. The iterative algorithm is illustrated in Fig. A.4. A solid cube is of unit dimensions and has square passagees with dimensions r₁=1/3 cut through the centers of the six sides. The six cubes in the cubes with dimensions r₁=1/3 are removed as well as the central cube. Twenty out of 27 cubes with dimensions r₁=1/3 are retained so that N₁=20. At second order r₂=1/9 and 400 out of 729 cubes are retained so that N₂=400. The value of fractal dimension is d_f=2.7268. The Menger sponge has been used as a model for flow in porous media with a fractal distribution of porosity. We do not consider however the Menger sponge, and other similar geometric models with large porosity values, to be a good representation of physical situation in porous media. Many rocks show a nonzero permeability at porosity values below 10%, much smaller than mathematical models using geometric fractals. (See also discussion in Chapter II).

Fig. A.4. Menger sponge.

A2. Random fractals

Physical objects are random fractals. This means that parts of a fractal are randomly arranged. One could easily construct a random fractal by randomly rotating black tiles in the Sierpinski gasket and slightly deforming individual tiles. There is experimental evidence indicating that porous rock has a fractal porous structure. Fig. A.6 shows the structure of sandstone observed by using an electron scanning microscope. The difficulty of ordering different magnifications is an indication of self-similarity of pore structure. For a given magnification scale the surface area of the observed features can be measured by using grids of increasing resolution. At each resolution scale, the area S_n of the analyzed features can be measured in units of the elementary squares of a grid. We count all squares which contain the elements of the analyzed structure. Next we choose many different local origins for our window boxes to obtain good statistics. The results of the measurments at different scales can then be expressed in units L_min² of the area of the squares in the finest grid. The results can be plotted as logM vs. log (L_n/L_min) and the fractal dimension is obtained as the slope d_f (see formula (A3)).

Fig. A.6. Fractal structure of sandstone.

It is worth noting that the fractal dimension of an ensemble of fractal objects is not necessarily a fractal object on all scales. Physical objects may show a crossover from the fractal character at small length scales to a homogeneous behaviour at large length scales. This property is illustrated in Fig. A.7 by a hexagonal lattice of Sierpinski gasket cells of size ξ. On a length scale smaller than ξ, we see a fractal structure. On length scales larger than ξ, we see a homogeneous system which is translationally invariant for translations by integer multiplicities of ξ.

Fig. A.7. Lattice of Sierpinski gaskets.

Now imagine a situation in which we spoil the perfect order of the lattice in Fig. A.7. Lets us randomly rotate and translate individual cells, thus arranging them in a random manner. Such a system becomes invariant (in a statistical sense) with respect to translations by an arbitrary distance, i.e. the system looks the same from any reference point. The contact with the properties of a waste rock pile can be made if we interpret each of the hexagonal cells as an object analogous to a porous rock particle of size ξ - we can call each cell a "Sierpinski particle". To make our construction more realistic we break the individual cells into fragments which have a size distribution with a fractal dimension d_r different than the fractal dimension d_f of pores in individual "Sierpinski particles". Such an ensemble has multifractal features. On length scales greater than the maximum size of particles, the system is homogeneous. On scales smaller than the size of the largest particles but greater then the pore size, the system has the fractal dimension d_r. Individual particles have the fractal dimension d_f. The process of rock fragmentation is often described in terms of formation and growth of cracks. Because there is experimental evidence that pores and preexisting cracks (cracks which exist before blasting) can be characterized by the fractal dimension, one might expect a relation between the fractal dimensions of pores and cracks and a fractal dimension describing the particle size distribution. To our knowledge no analysis of this problem has been performed.

For a more rigorous presentation based on the topological concepts and Haussdorf dimension we refer the reader to Mandelbrot:

Mandelbrot (1983). The fractal geometry of nature, Freeman, San Francisco (1983); 

Mandelbrot, & B.B. (1989). An Overview of the Language of Fractals, in Avnir, & D. The Fractal Approach to Heterogeneous Chemistry, Wiley.

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¹ Stanley, H. E. (1991). A. Bunde, & Havlin S. (ed.), Fractals and disordered systems. New York: Springer.