Mandelbrot [6] offers the following definition of a fractal: A fractal is a set for which the Hausdorff-Besicovitch dimension exceeds its topological dimension. This definition, although correct and precise, is too restrictive, since it excludes many fractals that are useful in physics [3].
An alternative definition is given by Mandelbrot in [7]: A fractal is a shape made of parts similar to the whole. This definition uses the concept of self-similarity. A set is called strictly self-similar if it can be broken into arbitrary small pieces, each of which is a small replica of the entire set.
Fig. 4 shows the construction of the Koch curve. It begins with a line. In the first step the middle third is replaced by an equilateral triangle and the baseline is removed. This procedure is applied repeatedly to the remaining lines. In the limit there is a strictly self-similar structure. Each fourth of this structure is a rescaled copy of the entire structure.
Figure 4: The Koch Curve
Let the original line () have a length of
.
After
steps the number of segments is
with length
. Thus one can find a measure according to
equ. (7):
The measure remains finite and equals
if and only if
. This critical value is the
Hausdorff-Besicovitch dimension
(comp. sect. 2.4).
Each stage of the construction is a line. In the limit
there is also a line with infinite length.
Thus its topological dimension is
(comp. sect. 2.2).
Since the Hausdorff-Besicovitch dimension exceeds
the topological dimension
, the Koch curve is a fractal
by definition.
Natural objects like coastlines or roots (see fig. 5) [2] do not show exactly the same shape but look quite similar when they are scaled down. Due to their statistical scaling invariance they are called statistical self-similar [9].
Figure 5: Root system of Cannabis Sativa
The miniature copy of a structure may be distorted, e.g. skewed (see fig. 12). For this case there is the notion of self-affinity [8].