Fractals

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Encyclopedia of Nonlinear Science

Many spatial patterns and objects in nature are either irregular or fragmented to such an extreme degree that it is very hard to describe their form. For example, the coastline of a typical oceanic island or of a continent is neither straight, nor circular, nor elliptic, and no other classical curve can serve to describe it. Similarly, there is no Euclidean surface to capture adequately the boundaries of clouds or of rough turbulent wakes. To fill this gap, Benoit Mandelbrot proposed to use a family of shapes called fractals for geometric representation of the shapes mentioned above and many other irregular patterns. The word fractal derives from the Latin fractus (a n adjective from the verb frangere, meaning “to break”), capturing the irregular characteristics of geometrical objects it describes. One can think of a fractal as an irregular set consisting of parts similar to the whole. It means that a fractal is a rough or fragmented geometrical shape that can be subdivided into parts, each of which is (at least approximately) similar at reduced size to the whole. Roughness present at any resolution of a fractal object distinguishes it from Euclidean shapes. The transition from a Euclidean geometry to a fractal one reflects the conceptual jump from translational invariance of traditional Euclidean shapes to continuous scale invariance of fractal objects. The concept of fractals is useful for describing various natural objects, such as clouds, coasts, and river or road networks. However, the fractal structure of natural objects is only observed over a limited range of scales, beyond which the fractal description breaks down.

During the study of geometry of irregular patterns, it became clear that a proper understanding of irregularity or fragmentation must, among other things, involve an analysis of the intuitive notion of dimension. Topological dimension cannot properly describe strongly irregular fractal structures. Let us consider the simplest example of a fractal, namely the Cantor set, see Figure 1. For the construction of the Cantor set an iteration procedure is used. At the zeroth level, the construction of the triadic Cantor set begins with the unit interval, that is, all points on the line between 0 and 1. The first level is obtained from the zeroth one by deleting all points between and that is the middle third of the initial interval. The second level is obtained from the first level by deleting the middle third of each remaining interval at the first level. In general, the next level is obtained from the previous one by deleting the middle third of all intervals obtained from the previous level. This process continues ad infinitum. The result is a collection of points with topological dimension d_t=0 since its total measure (length) is zero.

It is seen that the notion of dimension from Euclidean geometry is not very useful for the Cantor set and similar objects since it does not distinguish between the rather complex set of elements and a single point that also has a vanishing topological dimension. To cope

with this degeneracy, concepts of fractal dimensions were introduced for quantifying such fractal sets. Many paradoxical aspects of the Cantor set can be traced to the fact that it is dimensionally discordant. Furthermore, the discordant character of basic fractals is not at all a minor nuisance. Rather, it is such a basic feature that we shall use it to give a tentative definition of the concept of fractal. The simplest nontrivial dimension that generalizes the topological dimension is the so-called fractal dimension, which in this context can be defined as follows:

Here, N_n is the number of observable elements at the nth level, l_n is the measure (length) of the smallest element at the nth level, and 1/l_n is the resolution. For the triangle Cantor set at the nth level, the set consists of N_n=2ⁿ segments, each of which has length Therefore, the capacity dimension of the triangle Cantor set equals

We see that fractal dimension is noninteger. Mathematically, we can say that a fractal is a set of points whose fractal dimension exceeds its topological dimension.

Felix Hausdorff developed another way to quantify a fractal set. He suggested examining the number N(ε) of small intervals needed to cover the set at a scale ε. The Hausdorff dimension is defined by considering the quantity

which is constructed by summing d-dimensional volumes of balls of radii ε_m not exceeding ε that cover the fractal set. The “inf” means that all partitions of balls of radius less than ε that cover the set are considered and the one that gives the smallest possible value for the sum is kept. Hausdorff demonstrated that there is a special value D_H for the exponent d, called the Hausdorff dimension, such that for d<D_H measure M=∞ as ε→0 and for d>D_H measure M=0 in the limit ε→0.

Now the reader may ask, why do we need more than one fractal dimension? Actually, as is seen below, there is an infinite set of fractal dimensions. The reason for this number of quantities describing a single fractal object is the fractal form. The form of Euclidean shapes is well described by topology. However, as was noted by Mandelbrot the concept of form possesses mathematical aspects other than topological ones. Topology is the branch of mathematics that teaches us, for example, that all pots with two handles are of the same form because, if they were made of an infinitely flexible clay, each could be molded into any new opening, closing up any old one. Obviously, this particular aspect of the notion of form is not useful in the study of individual coastlines, since it simply indicates that they are all topologically identical to each other (and to a circle). This identity is underlined by the fact that the topological dimension in each case equals 1. By way of contrast, it will be seen that coastlines of different “degrees of irregularity” tend to have different fractal dimensions. Differences in fractal dimensions express differences in nontopological aspects of form, which can be called the fractal form.

Now let us consider the cumulative distribution function of mass along the Cantor set. Let the initial line’s length and mass both be equal to 1, and define the cumulative distribution function for the abscissa x as being the mass contained between 0 and x. It is assumed that the total mass of the pattern is not changed while building the Cantor set. It means that we do not take away one-third of each line, but divide it in to two parts and compress them until the length of each part equals one-third of the initial line length at the previous step. Iterating this procedure up to infinity, we obtain the massive Cantor set. Because there is no mass in the intermissions, the distribution function remains constant along almost the whole length of the unit interval. However, since hammering does not affect the total mass, the distribution must manage to increase somewhere from the point of coordinates (0, 0) to the point of coordinates (1, 1). It increases over infinitely many, infinitely small, highly clustered jumps corresponding to the points of the Cantor set. The plot of the cumulative distribution function shown in Figure 2 is called the Devil’s staircase. It is the plot of a continuous function on the unit interval, whose derivative is 0 almost everywhere, but it rises from 0 to 1 (see Figure 2). The cumulative sums of the widths and of the heights of the steps are both equal to 1, and one finds in addition that this curve has a well-defined length equal to 2. A curve of finite length is called rectifiable and is of dimension D=1. This example has the virtue of demonstrating that sharp irregularities do not necessarily prevent a curve from being of dimension D=1, as long as they remain sufficiently few and scattered.

You can see from Figure 2 that the Devil’s staircase is a function which is not differentiable at the Cantor set points, but has zero derivative everywhere else. Because a Cantor set has measure zero, this function has

zero derivative practically everywhere and rises only on Cantor set points.

Now we consider the fractal whose dimension, in contrast to the Cantor set, exceeds unity. As an example we consider the Koch snowflake; see Figure 3. This is a fractal, also known as the Koch island, which was first described by Helge von Koch in 1904. It is built by starting with an equilateral triangle, removing the inner third of each side, building another equilateral triangle at the location where the side was removed, and then repeating the process indefinitely (see Figure 4). The border of the snowflake is a curve. Indeed, its area vanishes, but on the other hand each stage of its construction increases its total length by the ratio 4/3. Thus, the limit border is of infinite length. It is also continuous, but it has no definite tangent in almost all of its points because it has, so to speak, a corner almost everywhere. Its nature borders on that of a continuous function without a derivative. This object has a fractal nature. The capacity dimension of the Koch snowflake equals Though the total length of the border is infinite, the area

of the Koch island originated from the initial triangle with unit side is finite and equals

Fractals arising in nature usually have a more complex scaling relation than simple fractals (like the Cantor set) described above. The single fractal dimension is not enough to describe their structure, because these fractal sets usually involve a range of scales that depend on their location within the set (the simplest example is the Cantor set with some measure distributed on it). Such fractals are called multifractals.

The multifractal formalism is a statistical description that provides global information on the self-similarity properties of fractal objects. For practical realization of the method, one primarily covers the fractal set under study by a regular array of cubic boxes of some given mesh size l. The measure of weight p_n of a given box n is then defined as the sum of the measure of interest within the box. A simple fractal dimension α is defined by the relation:

Multifractal analysis is a generalization in which α may change from point to point and is a local quantity. The standard method to test for multifractal properties consists in calculating the moments of order q of the measure p_n defined by

Here n(l) is the total number of non-empty boxes. Varying q, one can characterize the inhomogeneity of the pattern, for instance, the values of moments with large q are controlled by the densest boxes. If scaling holds, then the generalized dimension D_q is defined by the relation:

For instance, D₀, D₁, and D₂ are called capacity (defined above), information, and correlation dimensions, respectively.

In multifractal analysis, one also determines the number N_α(l) of boxes having similar local scaling characterized by the same exponent α. Using it, one can introduce the multifractal singularity spectrum f(α) as the fractal dimension of the set of singularities of strength α according to the following relation:

There is a general relationship between a moment of order q and a singularity strength α, expressed

mathematically as a Legendre transformation:

Fractal sets can be impressive. For example, see Figure 5 where a number of so-called Julia sets is shown. (See also color plate section.)

Mathematically, the Julia set can be introduced as follows. Take some function f(z) and consider the sequence obtained when f(z) is iterated starting from the point z=a:

Depending on the initial condition a and form of the function f, it may happen that these values stay small or they do not, that is, repeatedly applying f to yield arbitrary large values. So the set of all

numbers (initial conditions) is partitioned; into two parts, and the Julia set associated with the function f is the boundary between these sets. The “filled” Julia set includes those numbers z=a for which the iterates of f applied to remain bounded. If one considers complex numbers rather than real ones, it is the complex plane that is partitioned into two sets, and the resulting picture can be quite striking; see Figure 5. That is an example of iterating a quadratic function defined in the complex plane. Linear functions do not yield interesting partitions of the complex plane, but quadratic and higher-order polynomials do.

Consider the most studied family of quadratic polynomials f(z)=z²+μ parametrized by a complex variable µ. As μ varies, the Julia set will vary on the complex plane. Some of these Julia sets will be connected, and some will be disconnected. Those values of μ for which the Julia set is connected are called the Mandelbrot set in the parameter plane. The boundary between the Mandelbrot set and its complement is often called Mandelbrot separator curve. The Mandelbrot set is the black shape in Figure 6 (see also color plate section). The Mandelbrot set is the set of points in the complex µ-plane that do not go to infinity when iterating the map starting with z=0. One can avoid the use of the complex numbers by substitution z=x+iy and μ=a+ib, and computing the orbits in the ab-plane for two-dimensional mapping:

with initial conditions x=y=0 or equivalently x=a, y=b.

Now let us consider a natural phenomenon exhibiting fractal properties, namely, Brownian motion. Consider a fluid mass in equilibrium, for example, water in a glass; all the parts appear completely motionless. However, if one puts a small enough particle into the water, it is observed that instead of rising or descending regularly (depending on the density of the particle),

it moves with a perfectly irregular movement. The segment of motion of a colloidal particle is presented in Figure 7 as it is seen under the microscope. The successive positions of a particle in equal time intervals are marked by points, then joined by straight lines having no physical reality whatsoever. If this particle’s position were marked down 100 times more frequently, each segment would be replaced by a polygon relatively just as complicated as the whole drawing, and so on. Here, we have the natural example of the scaling property of the colloidal particle’s motion. It is easy to see that in practice the notion of tangent is meaningless for such curves. This property of the trajectory reminds one of the Koch’s snowflake described above, because when a Brownian trajectory is examined increasingly closely, its length increases without bound. Thus, we see that the trajectory of a colloidal particle has features peculiar to fractal objects, and we can qualify Brownian motion as being fractal and giving us a natural example of a fractal pattern.

VITCHESLAV KUVSHINOV AND ANDREI KUZMIN

See also Brownian motion; Dimensions; Multifractal analysis

Barnsley, M. 1988. Fractals Everywhere, Boston: Academic Press

Devaney, R. & Keen, L. (editors). 1989. Chaos and Fractals: The Mathematics Behind the Computer Graphics, Providence, RI: American Mathematical Society

Falcone, K.J. 1985. The Geometry of Fractal Sets, Cambridge and New York: Cambridge University Press

Feder, J. 1988. Fractals, New York and London: Plenum Press

Mandelbrot, B.B. 1988. Fractal Geometry of Nature, New York: Freeman

Sornette, D. 2000. Critical Phenomena in Natural Sciences. Chaos, Fractals, Self-organization, and Disorder: Concepts and Tools, Berlin and New York: Springer

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