Stage 1

Stage 2

Stage 1

Stage 3

Figure 2.3 First stages of the construction of Peano's space-filling curve. At each step, each line segment is replaced by nine line segments scaled down by a factor of 1/3.

one obtains stage 2. Independently of the direction (horizontal or vertical) of the line segments in stage 1, the scaling factor is equal to 1/3. At the next stage, the scaling factor is equal to 1/9, and so on. If one assumes that the length of the original line segment constituting the initiator is one, then it is easy to calculate the length of the curves at each stage. In stage 1, there are nine line segments of length 1/3 and the total length of the curve is three. In stage 2 there are 92 line segments of length 1 /32, amounting to a total length equal to nine. Expressed as a general rule, in each step of the construction, the resulting curve increases in length by a factor of three. In stage n, the length is thus 3".

If one pursues the above construction procedure to the limit as n ^ro, the number of straight-line segments and their total length tend to infinity. The result is generally termed Peano's curve and is of great interest mathematically because it is nowhere differentiate, i.e. it does not allow a tangent at any of its points (which are all 'corners'). As with the Cantor set (but for different reasons), it is impossible to visualize the structure of this curve. All one can see, at any scale, is a completely 'filled out' square, which does not look in the least similar to the early steps of the construction (see Figure 2.3). Nevertheless, the Peano curve, like P, contains exact copies of itself [e.g. 14, 15]. The same general characteristics are exhibited by Hilbert's [13] plane-filling curve, except for the fact that it is self-avoiding, i.e. it never intersects or touches itself.

2.2.2 The Triadic Von Koch Curve

The Swedish mathematician von Koch introduced in 1904 what is now called the (triadic) von Koch curve [e.g. 5, 14 (pp. 89-93)]. The construction of this curve, illustrated in Figure 2.4, starts with a line segment, /0, of unit length. This initiator is replaced by the generator shown as the curve /i, consisting of four segments of length 1/3. At the next stage, /2 is constructed by replacing each line segment in /1 by a properly scaled-down version of the generator. The scaling factor is equal to 1/3. The curve /2 consists of 42 = 16 segments each having a length of 1/32 = 1 /9, so that the total length of I2 is equal to (4/3)2 = 16/9.

At the nth stage, there are 4" segments of length <5n = 3-n, with a total length of L(/n) = (4/3)". This total length L(/n) may be expressed differently, in a way that makes explicit the dependence of L(/n) on <5n. Taking the exponential of the natural logarithm of the total length L(/n) = (4/3)" and substituting the value of n (= -ln <5n/ ln 3) obtained by solving the equality ln <5n = ln 3-n [e.g. 5 (p. 17)], one obtains

Equation (2.2) corresponds to a power-law (or 'Paretian') relationship between L and <5n. Similarly, the number Nn of segments of length <5n is also a power-law function of <5n. Indeed, the equality L(/n) = Nn<5n implies that Nn = <5n-ln4/ln3. We shall return to these important power-law dependencies in later sections of this chapter.

Even though we cannot represent it graphically, it is clear that, if we carry on the iterative procedure illustrated in Figure 2.4 to the limit where n ^ro, the resulting



Figure 2.4 Construction of the triadic von Koch curve. At each step, the number of line segments increases by a factor of 4.



Figure 2.4 Construction of the triadic von Koch curve. At each step, the number of line segments increases by a factor of 4.

curve (the von Koch curve) has an infinite number of vanishingly small segments and has a total length tending to infinity.

If, in Figure 2.4, one takes the part of I4 that corresponds to the interval [0,1/3] of the initiator, and scales this part up horizontally and vertically by three, one obtains I3. In the limit n ^-<x>, however, this same scaling up of any segment of In would reproduce the von Koch curve itself. Furthermore, like the Peano-Hilbert space-filling curves, the von Koch curve is nowhere differentiable. All of these curves have a topological dimension equal to unity; a simple stretching operation (or 'rectification') transforms them into an infinite straight line.

At first sight, the von Koch curve appears less monstrous than the Cantor set or Peano's space-filling curve. Nevertheless, it has astonishing properties, which challenge the traditional concepts of dimension. Indeed, since the von Koch curve is infinite in length and contains exact replicates of itself, any scaled-down sub-image is also of infinite length [3]. This leads to the conclusion that for any two points on the curve, no matter how close they are, the curve between them is of infinite length! In addition, if one joins together three initiators like /0 (in Figure 2.4) to form a triangle and one performs on each initiator the iteration procedure of Figure 2.4, the resulting geometrical construct, commonly known as the von Koch 'island' or 'snowflake', has infinite length, even though it fits within a finite area!

2.2.3 The Sierpinski Carpet and the Menger sponge

Cantor [8] remarked that examples similar to the (Cantor) set P 'can be easily constructed for higher dimensions'. One such example is the Sierspinski carpet. The iterative procedure leading to this carpet starts with a square initiator of side length 10 (Figure 2.5). In the next step, one divides the initiator in nine smaller squares of side length 10/3 and removes the central square. When one applies the same procedure ad infinitum to the remaining squares, a structure known as the Sierpinski carpet is obtained [e.g. 10 (p. 25)], which is Cantor-like in many respects, even though its initiator is two-dimensional. Like the Cantor set, the Sierpinski carpet is a self-similar, uncountable set with a topological dimension equal to zero. The two-dimensional Lebesgue measure of the Sierpinski carpet, i.e. practically, its area in the plane, is equal to zero, so the traditional concept of area does not provide a very useful description of the spatial coverage of the Sierpinski carpet. Some other type of dimension is needed.

Figure 2.5 First stages of the construction of the Sierpinski carpet. /0 is the initiator. /1 is the generator.

When the iterative process used for the construction of the Sierpinski carpet is generalized to three dimensions and when the initiator is a cube, one obtains a geometrical structure of particular interest in geophysics and soil science, the Menger sponge (Figure 2.6). The construction of the Menger sponge is very similar to that of the Sierpinski carpet. Each square face of the cube is treated in exactly the same way as the square initiator of the Sierpinski carpet. This time, extracting a square shape involves punching a hole directly through the cube at right angles to the face concerned. Thus, at the first stage /1, three holes are punched through. This leaves 20 subcubes at one-third scale, each of which is repeatedly subdivided to create, ad infinitum, the hollowed structure of the Menger sponge. Another way to describe the recursive construction process is to consider that, at the nth stage, each cube of size

Figure 2.6 First three stages in the construction of the Menger sponge.

ln-1 is divided into 27 equal cubes of size ln = ln-1 /3, and that the central small cube is removed along with the six cubes with which it shares faces.

This iterative construction may be envisaged from a physical standpoint. This produces mathematical relationships that will be useful later on. If the initiator is a cube of side length l0, made of some material of uniform mass density p0, the firstorder structure (obtained after one iteration) would have a porosity (void volume divided by total volume) equal to 7/27 and a mass density p1 equal to 20p0/27. After the second iteration, theporosity 02 would increase to 329/729 («0.45) and the density p2 would decrease to 400p0/729 («0.55p0). At the nth iteration, porosity and density would be given by

0 0

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