A fractal is self-similar. But it is not true, that if an object is
self-similar, then it is fractal. Consider the line, square and cube in
fig. 6. Each of them is broken into small copies
of themselves reduced by a scaling factor [8].
Figure 6: Euclidean objects
The relation between the number of rescaled objects and the
reduction factor
is a power law:
is the self-similarity dimension.
for the line,
for the square and
for the cube. They
are identical with their Hausdorff-Besicovitch dimension
.
Furthermore they are equal to the topological dimension
.
Thus they are not fractal.
The Cantor set (fig. 7 top) is generated by removing the
middle third of a line. From the remaining lines the middle thirds are
removed again. Continuing this removal infinitly often one gets the
Cantor set. It consists of an uncountable infinite number of points
which are all disconnected. Thus the topological dimension is .
Figure 7: Fractals
A similar procedure leads to the Sierpinski gasket
(fig. 7 center), where a rescaled triangle is
removed from the middle of the original triangle. Repeating this
procedure one gets the Sierpinsky gasket in the limit, which is a line
and has topological dimension of .
The construction of the Peano curve (fig. 7 bottom)
starts with a line. In each step one line segment is replaced by
line segments scaled down by a factor of
. In the limit one gets
a line with topological dimension
.
Equ. (9) also holds for the objects in
fig. 7. Resolving for the self-similarity
dimension becomes:
The self-similarity dimensions of the objects in fig. 7 are equal to their Hausdorff-Besicovitch dimensions and exceed their topological dimensions. Hence they are fractals.
The fractal dimension of the Cantor set is and lies between
and
. So its more than a normal set of points but less than a line.
The Sierpinski gasket with fractal dimension
is more than a line but
less than a surface, since its fractal dimension lies between
and
.
As depicted by the Peano curve the Hausdorff-Besicovitch dimension of a
fractal doesn't have to be noninteger. The curve has an integer dimension
. Actually it is area-filling.
The fractal dimension of the Koch curve in sect. 2.5
can be determined in the same way. In each step new line segments are
generated which are rescaled by a factor
(see fig. 4).
Hence the fractal dimension is
.
There is another difference between Euclidean objects and fractals. The scaling factor of Euclidean objects can be choosen arbitrary (see fig. 6), while it is fixed for fractals (see fig. 7).