Fractional Dimensions

The Euclidean geometry implies integer values for the dimension of geometrical shapes. There is perhaps no physical object known to our consciousness in the world whose shape conforms to the Euclidean paradigm. Every object known to us can be broken down into smaller objects until we reach the principle of quantum mechanics where, with the present status of physics, we have to treat matter in a shapeless form. When we try to measure the length of a straight Euclidean line, its length does not depend on the length of the ruler we use to measure it with. For example, if we try to measure a 10 cm line with a 5 cm ruler we have to cover the line with 2 copies of the ruler, and if we measure the line with a 1 cm ruler we will have to cover the line 10 times with the new ruler. This relationship can be written as ${r}^{d}$, where $r$ is the ratio of the rulers' length and ${r}^{d}$ is the ratio of the number of times that we have to cover the line. In this case $r = 5$ and $d = 1$. The value of $d$ is called the Hausdorff-Besicovitch dimension (which, from now on, we simply call dimension). Therefore, we say that this line has a dimension of 1.

Imagine a square whose every side is 9 cm. If we try to cover this square with smaller squares whose side is 3 cm, we will need 9 copies of our measuring square. If we use a measuring square whose side is 1 cm, which means we are choosing an $r = 3$, we will need 81 copies of this square, which means our ratio of the number of covering squares is $81 / 9 = 9$. Therefore, ${3}^{d} = 9$, which implies that $d = 2$, or in other words the surface of the square has a dimension of 2.

When we apply this idea to a self-similar cure we get fractional values for $d$. This situation arises since a self-similar object has infinite amount of detail and no matter how small our measuring unit is, we will be ignoring some details whose lengths may actually not converge. The Koch snowflake (Figure 3-4) is a very famous self-similar shape, or in other words, fractal^3.3. The process of the construction of the curve is illustrated in Figure 3-4. Let us assume that the first level of the cure is an equilateral triangle, whose every side is 3 cm. If our measuring stick is 3 cm, we will need 3 copies of the stick to cover the whole shape, and in this case we are ignoring all the other details which result from the other levels of progression. However, if we use a measuring stick of 1 cm ($r = 3$), we can cover an extra level of detail and we will need 12 copies of our measuring stick, which implies that ${r}^{d} = 12 / 3$ or ${3}^{d} = 4$, or $d = log(4) / log(3) \approx 1.262$.

Figure 3-4: The Koch Snowflake

As described before, the phase space of Lorenz equations can also be created with such a self-similar procedure whose dimension is 2.06[31, page 126]. One way to think about a fractional dimension is to think, for example, that the Koch curve covers a space more than a straight line and less than a surface. It is also possible to have self-similar shapes whose dimension is integer like the Hilbert non-intersecting curve whose dimension is 2[43, page 10]. The progression of the Hilbert curve is illustrated in figure 3-5^3.4. In this case we are covering a two-dimensional surface with a topologically one-dimensional line.

Figure 3-5: The asymptotically self-similar Hilbert curve. Notice the effect of aliasing when the details exceed the resolution of the printer or screen.

The concept of dimensions is discussed since we believe that it is an important idea for understanding the idea of what a continuum is. For example, if we assume a very simple idea of thinking about music as a two-dimensional (time and frequency) entity, a melody can be one-dimensional, in which case it behaves like a simple line, or it can cover the whole spectrum as a white noise by having a dimension of 2. This is one way to model the continuum of tone and noise which Stockhausen sets as a criteria for electronic music[47, page 109].