Nature abounds with intricate fragmented shapes and structures, including coastlines, clouds, lightning bolts, and snowflakes. In 1975 Benoit Mandelbrot coined the term fractal to describe such irregular shapes. The essential feature of a fractal is the existence of a similar structure at all length scales. That is, a fractal object has the property that a small part resembles a larger part, which in turn resembles the whole object. Technically, this property is called self-similarity and is theoretically described in terms of a scaling relation.
Chaotic dynamical systems almost inevitably give rise to fractals. And fractal analysis is often useful in describing the geometric structure of a chaotic dynamical system. In particular, fractal objects can be assigned one or more fractal dimensions, which are often fractional; that is, they are not integer dimensions .
To see how this works, consider a Cantor set , which is defined recursively as follows (Fig. 6).
Figure 6: Construction of Cantor's middle thirds set.
At the zeroth level the
construction of the Cantor set begins with the unit interval, that is,
all points on the line between 0 and 1. The first level is obtained from the
zeroth level by deleting all points that lie in the ``middle third,'' that is,
all points between 1/3 and 2/3. The second level is obtained from the first
level by deleting the middle third of each interval at the first level, that is,
all points from 1/9 to 2/9, and 7/9 to
8/9. In general, the next level is obtained from the
previous level by deleting the middle third of all intervals at the previous
level. This process continues forever, and the result is a collection of
points that are tenuously cut out from the unit interval.
At the nth level the set consists of 
segments, each of
which has length 
, so that the length of the Cantor set is
In the 1920s the mathematician Hausdorff
developed another way
to ``measure'' the size of a set. He suggested that we should examine the
number of small intervals, 
, needed to ``cover'' the set at
a scale 
. The measure of the set is calculated from
An example of a fractal dimension is obtained by inverting this equation,
Returning to the Cantor set, we see that at the nth level the length of the
covering intervals are
,
and
the number of intervals needed to cover all segments at the nth level
is 
. Taking the limits 
( 
), we find
The middle-thirds Cantor set has a simple scaling relation, because the factor 1/3 is all that goes into determining the successive levels. A further elementary discussion of the middle-thirds Cantor set is found in Devaney's Chaos, fractals, and dynamics. In general, fractals arising in a chaotic dynamical system have a far more complex scaling relation, usually involving a range of scales that can depend on their location within the set. Such fractals are called multifractals .