Untangling Complex Systems, page 76
(Strogatz 1994).
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K 0
K 1
K 2
K 3
K 4
FIGURE 11.4 The first four steps in the construction of the Koch curve. It has been obtained by using the
“Examples of Fractals” model, Complexity Explorer project, http://complexityexplorer.org.
K is achieved by replacing the middle third of each line segment of K by the other two sides of
n
n−1
an equilateral triangle. In the end, after an infinite number of iterations, we obtain K∞ that is the
Koch curve. It consists of an infinite number of segments, and it has an infinite length, although it is confined to a finite region of the plane.
TRY EXERCISE 11.3
It is evident that the Cantor set and the Koch curve have extraordinary properties. The Cantor set
consists of an infinite number of segments, but its length approaches zero. Moreover, it is not con-
tinuous. The Koch curve has an infinite length, and it is not differentiable. They are “monsters”! Try
to imagine what happens if you build the Koch curve on the three sides of an equilateral triangle
(see Figure 11.5). We obtain the so-called Koch snowflake. It is a “geometrical monster” because it has an infinite perimeter that encompasses a finite area.
The tools of Analysis cannot deal with fractals. In fact, they are not smooth curves; sometimes,
they are neither continuous curves. Hence, they are not differentiable everywhere. Isaac Newton
and Gottfried Leibniz, when inventing Calculus, in the middle of the seventeenth century, proposed
the fundamental idea of approximating any smooth function by tiny little segments of straight lines
(see Figure 11.6). This approach is reasonable. It suffices to zoom into any of such smooth curves, to understand why. However, the intuition of Newton and Leibniz becomes useless in the case of
fractals. In fact, a fractal is a geometric figure that does not become simpler when analyzed into
smaller and smaller parts. Straight lines cannot describe a fractal on each of its points.
Ks0
Ks1
Ks2
Ks3
Ks4
FIGURE 11.5 The first four steps for constructing the Koch snowflake.
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FIGURE 11.6 The smooth gray function is approximated by black segments. The shorter the segment, the
better the approximation.
11.3 FRACTAL DIMENSION
What is the dimension of the Cantor set that is a set of points? And what is the dimension of
the Koch curve that has an infinite length? If we think about Euclidean geometrical objects, it
is easy the determination of their dimensions. For instance, segments are one-dimensional. But
even any smooth curve is one-dimensional because every point on it is determined by one num-
ber, which is the arc length from some fixed reference point on the curve. The dimension can
be defined as the minimum number of coordinates needed to describe every point in the object.
Therefore, planes and smooth surfaces are two-dimensional; solids are three-dimensional. In
fact, what happens if we bisect a one-dimensional object, such as a segment? We obtain two
equal half-sized segments. What happens if we trisect the original segment? We obtain three
equivalent copies (see the top left part of Figure 11.7). Now, let us consider a square. What happens if we bisect both sides of the square? We obtain four copies of the original square. What
happens if we trisect each side? We obtain nine copies (see the bottom left part of Figure 11.7).
Now let us consider a cube. What happens if we bisect each side of the cube? We achieve eight
copies of the original cube. What if we trisect each side? We obtain 27 copies (see the right
part of Figure 11.7). From these observations, we infer that the number of copies N depends on the scale factor r and the dimension d (known as the Hausdorff dimension), according to the
following equation:
N rd
= [11.3]
In the case of a line, d = 1; in the case of a square, d = 2, and in the case of a cube, d = 3. In fact, 21 = 2, 31 = 3, 22 = 4, 32 = 9, 23 = 8, 33 = 27.
We can, now, apply equation [11.3] for determining the dimension of the Cantor set represented
in Figure 11.3. The scale factor is r = 3 because we trisected the initial segment into three equivalent parts. Then, we obtained two copies, N = 2, because we removed the middle third. Therefore, the
dimension of the Cantor set is:
N
log( )
log(2)
d =
=
≈ 0.63 [11.4]
r
log( )
log(3)
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r = 2
r = 3
r = 3
r = 2
r = 2
r = 3
FIGURE 11.7 What happens to a segment (top left), a square (bottom left) and a cube (on the right) when
they are bisected ( r = 2) and trisected ( r = 3).
Puzzling! The dimension is not an integer! And it is between 0 and 1. In fact, if we take a moment
to reflect, we realize that d ≈
0.63 is reasonable because the Cantor set is not a simple point (with
d = 0), but neither a line (with d = 1). It is in between a point and a line.
And what about the Koch curve? Let us apply equation [11.3], again. For the Koch curve, the
scale factor is 3, because we trisect the initial segment. The number of copies is 4 because the
middle part is replaced by two equivalent segments. Therefore:
N
log( )
log(4)
d =
=
≈ 1.26 [11.5]
r
log( )
log(3)
Amazing! The dimension is not an integer, again. It is larger than 1 because the Koch curve has an
infinite length. But it is smaller than 2 because it does not have an area.
Let us consider another famous fractal: the Sierpinski gasket. To construct it, we start with an equi-
lateral triangle ( S in Figure 11.8). We bisect each side, and we link each midpoint through a segment.
0
We obtain another equilateral triangle that is inverted with respect to the first. We remove it, and we
obtain S . We repeat this step for the remaining three triangles, and we have . And so on. After repeat-
1
S 2
ing an infinite number of times that procedure, we achieve the Sierpinski gasket. The scale factor is 2
because we bisect the sides of the triangles. The number of copies is 3. Hence, the dimension is
N
log( )
log(3)
d =
=
≈ 1.58 [11.6]
r
log( )
log(2)
S 0
S 1
S 2
S 3
Sn
…
FIGURE 11.8 The Sierpinski gasket, and the way it is constructed.
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Again, the dimension is between 1 and 2, as in the case of the Koch curve. However, the Sierpinski
gasket has a dimension larger than that of the Koch curve because it is closer to a two-dimensional
object, as a triangle is.
TRY EXERCISES 11.4 THROUGH 11.7
11.4 FRACTALS THAT ARE NOT PERFECTLY SELF-SIMILAR
It is possible to create fractals also by calculating a simple non-linear equation iteratively. The foun-
dations of this methodology were laid by Gaston Maurice Julia, 4 at the beginning of the twentieth century. He worked on the iteration of polynomials and rational functions. For example, he studied
the following quadratic equation:
f z
( ) z 2
=
+ c [11.7]
In [11.7], c is a constant and z = a + ib is a complex number. Starting from an initial point z , Julia calculated if
0
f( z) diverged to infinity or approached zero after many iterations:
f z
( n) = f z
( n 1
− ) = … = f z
( 1) = f z
( 0). He repeated these iterations for a heap of points in the complex
plane, and he found that some of them tended to a limiting position, whereas others never settled
down, but diverged. However, Julia could not visualize the shapes of the two kinds of sets because,
at his time, there were no computers available. In the 1970s, the Polish-born, French-educated math-
ematician Benoit Mandelbrot (1924–2010), working at IBM’s research center in New York (USA)
and having the most powerful computers at his disposal, revived Julia’s work. He developed the first
not perfectly self-similar fractal, the famous “Mandelbrot set.” He solved equation [11.7] iteratively,
always starting from the seed z
0, and for different complex
, the
0 =
c values. If, after n iterations, zn
so-called orbit of z , did not tend to infinity, the seed (0,
0
c) belonged to the Mandelbrot set. Otherwise,
it did not. Thus, the Mandelbrot set is a record of the fate of the orbit of z
0 under iteration of z 2 + :
0 =
c
the c numbers are represented graphically and colored in different ways depending on the fate of the
orbit of 0. An example is shown in Figure 11.9. All the c values in the black region cause z to stay finite.
All the c values outside the black region cause z to go to infinity. The shades of gray are proportional to the speed with which the value of z goes to infinity. The speed increases by going from light gray to
gray up to dark gray. In Figure 11.9, there are different magnifications of the initial set. It is evident that the Mandelbrot set is self-similar. However, the self-similarity is not perfect, like in the case of the
fractals we studied in the previous paragraphs.
x2
1
x4
x8
x16
x32
−1
0
−1
x64
x128
x256
x512
x1024
x2048
FIGURE 11.9 Increasing magnification of the Mandelbrot set built by using “Easy Fractal Generator” avail-
able at the website http://www.easyfractalgenerator.com/Home.aspx.
4 Gaston Maurice Julia (born in 1893, Sidi Bel Abbès, Algeria, and died in 1978, Paris, France) was one of the main inventors of iteration theory.
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(a)
(b)
(c)
FIGURE 11.10 Julia set when c = −0 1
. 2 + 0. i
75 in (a) and a magnification of it in (b); Julia set when
c = 0 4
. − 0. i
3 in (c).
TRY EXERCISE 11.8
There is another famous fractal that is based on the iteration of the quadratic equation [11.7]: it is the
Julia set. For the construction of the Julia set, it is necessary to fix the value of the constant c and
calculate the fate of all possible seeds in the plane of the complex numbers. Those seeds that do not
escape to infinity, but get closer to the origin of the axes, form the black filled Julia set. Of course,
there are as many Julia set as are the possible values of c.
TRY EXERCISE 11.9
Two types of Julia sets are depicted in Figure 11.10. The one in picture (a) is an example of a connected Julia set because the black parts are all connected. It is called the “Douady’s rabbit,” after
the French mathematician Adrien Douady. It looks like a fractal rabbit because it has a main body
with two ears attached. By zooming in, we see pairs of ears along the borderlines, at every scale (a
magnification is reported in [b]). In (c), there is an example of a “cloud” Julia set because it consists
of infinitely many pieces, each of which is a single point. These points pile up and form a structure
that looks like a Cantor set.5
TRY EXERCISE 11.10
11.5 THE FRACTAL-LIKE STRUCTURES IN NATURE
Many natural forms, in both the animate and the inanimate worlds, look like fractals. Examples are
ferns, Romanesco broccoli, flames, clouds (see Figure 11.11), but also coastlines, leaf veins, blood vessels in lungs, rivers, mountain ranges, plant roots, neural networks, and many more.
Even the crowns of trees are beautiful examples of fractal-like structures. In fact, if we zoom in one of
them, we obtain images that maintain the structural complexity of the overall crown (see Figure 11.12).
The first who had the idea that many natural forms have fractal structures was Benoit Mandelbrot
(Mandelbrot 1982). Mandelbrot coined the name “fractal” from the Latin word fractus that means
“broken” or “fractured.” This term was chosen in explicit cognizance of the fact that the irreg-
ularities found in fractal sets are often strikingly reminiscent of the fracture surfaces in metals
(Mandelbrot et al. 1984). Mandelbrot was attracted by the beauty and irregularity of fractal-like
natural forms and was frustrated by the inability of usual differential geometries6 to describe them properly. Mandelbrot was wondering himself: “Why is geometry often described as ‘cold’ and
‘dry’?” Mandelbrot (1982) replied, “One reason lies in its inability to describe the shape of a cloud,
5 The Julia set is of the “connected-type” (said also “filled-type”) when c belongs to the Mandelbrot set. If c escape from the Mandelbrot set, the corresponding Julia set is of the “Cantor-type” or “cloud-type.”
6 The non-fractal geometries are the Euclidean (built on the plane), the spherical (relative to surfaces with positive curvatures) and the hyperbolic (relative to surfaces with negative curvatures) geometries.
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FIGURE 11.11 Pictures of natural fractal-like structures: ferns, broccoli, flames, and clouds.
(0)
(Z1)
(Z2)
(Z1)
(Z2)
(Z3)
(Z3)
(Z4)
(Z4)
FIGURE 11.12 Four successive magnifications (Z1–Z4) of the original image (0) of a tree.
a mountain, a coastline or a tree. Clouds are not spheres, mountains are not cones, coastlines are not
circles and bark is not smooth, nor does lightning travel in a straight line.”
To be honest, the natural forms are fractal-like and not perfect fractals (Shenker 1994). An infinite
number of iterations generates the true fractals. Genuine fractals are analogous to irrational num-
bers, which are infinite sequences of numbers. There are irrational numbers that can be expressed
just by one symbol. For example, 2, or π, or e. But, there are much more irrational numbers that
cannot be compressed. Similarly, some fractals can be described by a short algorithm; for instance,
the Cantor set or the Koch curve. But, there are also fractals that are not exactly self-similar. In other
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words, their self-similarity is approximate or just statistical. Natural forms that look like fractals
have always a self-similarity that is not exact. Moreover, their self-similarity holds on only for a
finite number of spatial scales, not ad infinitum. In fact, whatever is the object, either a cloud, or a
tree, or else, after several magnifications, the object loses its original identity, and what we see are
molecules or subatomic particles. On the other hand, if we look at the same object but, in a scale,
quite larger than its dimensions, we lose its details and roughness, and we can describe it by using
the tools of the traditional geometries and analysis.
TRY EXERCISE 11.11
11.6 THE DIMENSIONS OF FRACTALS THAT ARE
NOT PERFECTLY SELF-SIMILAR
There are a lot of different methods for determining the dimension of fractals that are not perfectly
self-similar (Falconer 1990). One method used commonly is the “Box Counting.” Let us imagine
having the picture of a fractal or fractal-like structure on a plane. We cover it with a grid of squares
or boxes having size l smaller than the dimensions of the fractal. The number of boxes required to
cover the object is proportional to its dimension. For example, the number of squares of side l, N( l), needed to cover the black tick curve of Figure 11.13, being of total length L (> l), is L
N l
( ) ∝ [11.8]
l
On the other hand, the squares necessary to cover the grey spot of Figure 11.13, having area Ar, is Ar
N l
( ) ∝
[11.9]
l 2
Similarly, for a fractal or fractal-like object having dimension D, the number of boxes, required to
overlay it, is N l ∝ 1 lD
( )
/ . The Box Counting method requires that we repeat the calculation of N( l),
many times, by changing the box size, l. Then, we plot the log( N( l)) versus log(1/ l). We fit the data by a straight line, determined by the least-squares method. The slope of the straight line gives us D, because
N
log( ( l))
