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 , we will need 81 copies of this square, which means our ratio of the number of covering squares is . Therefore, , which implies that , 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 . 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 (), we can cover an extra level of
detail and we will need 12 copies of our measuring stick, which implies that
or , or
.

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.

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].