**Fractals Defined**

Self-similarity, which is a consequence of a power law relationship, and
non-integral dimension are the main characteristics of **fractals**.

A fractal (from the Latin *fractus*, derived from the verb
*frangere*, to break, to divide infinitely) is an object or a
geometrical construct (or set of such objects) whose parts are somewhat
identical or statistically similar with the whole through transformations
that include translations, rotations and zooming. A structure is fractal
if its finer shapes are indistinguishable or statistically similar to
coarser ones, and thence there is no dominating scale.

The Koch curve is such and object, and in order to identify it as fractal it is
customary to determine its fractal dimension. That
the fractal dimension of the Koch curve is greater than unity, which is the Euclidean
dimension of a line, should not be surprising; it simply indicates the
high degree of roughness of the Koch curve with respect to a
line.

The Koch curve tends to to fill out the two-dimesional plane, and thus it is
reasonable that its dimension should be more than one, although certainly
less than 2. Winding coastlines or the winding path
of river channels can be simulated with a Koch curve, since typical
coastlines have fractal dimensions around 1.25.

**Other examples of fractal objects...**

Consider the unit line segment but use the following rule: divide the
line segment into three equal segments and discrad the middle one.
Iterate. The result is the so-called Cantor's dust (named after German
mathematician Georg Cantor). This is a fractal object that consists of
just a line of disjoint points, hence the name dust. The fractal
dimension, i.e. the ration logN/log(1/R) is a constant, equal to 0.63093.
Intuitively it is reasonable to expect that since the rule simply pokes
holes into a line, the resulting fractal dimension should be less than
unity, as in fact it is.

Other fractal structures of similar nature can be obtained by poking holes
in a plane or in a three dimensional volume. For instance, to construct
the so-called Menger sponge (named after Austrian mathematician Karl
Menger) we start with a solid cube of unit side. The rule is: scale down
the side of the cube by; 1/3, to obtain 27 cubes of side 1/3, discard the
mid-face cubes and the one at the center. Iterate. The fractal dimension
of the Menger sponge is 2.72684..., reasonable less than the original
dimension of the cube because the sponge has voids at all scales.