Topological dimension is always a nonnegative integer. To obtain a finer measurement of the complexity of a space, it is essential to have a way to ``measure'' things. Therefore, we will assume that our space X is a metric space with a distance function d. Thus, we cross the boundary from topology to geometry.
Measuring things plays a large role in calculus; we learn there how to
compute the lengths of curves, the areas of regions, and the volumes
of solids. It's worth reviewing how these are done before tackling the
question of measuring fractals. We start with simple geometric
figures that we know how to measure. The length of the line segment
form 
to 
is
according to Pythagoras' formula. The area of a rectangle is just its length times its width. For a more general set S, we try to approximate S by a finite union of small copies of these simple objects. See this figure for an example of a curve approximated by a sequence of line segments, which is usually called a polygonal curve.
Figure 39: Polygonal approximation of a curve
Then the approximate length of the curve is the sum of the lengths of the line segments. The tricky part is to choose finer and finer approximations in such a way that the length of the largest segment tends to 0. If the same limiting value L of the sum of the lengths of the line segments always occurs, we say the curve is rectifiable and that the length of the curve is L. For areas, we use the same sort of reasoning approximating the region by a union of small rectangles.
These achievements of calculus meet utter failure when trying to
measure something like the Koch snowflake curve. The generations
produced by the L-system associated to the Koch curve (see
this figure) are in fact polygonal
approximations to the limit curve. The n-th generation has 
line segments each of which measures 
, assuming we started with
generation zero being a segment of length 1. So the total length of the
polygonal curve is 
, which rapidly tends to 
as 
. (For instance, 
.)
It can be shown that no matter how we approximate the Koch curve, the
lengths of the approximations grow to at a comparable rate.
This feature of fractal curves was an important motivation for
Mandelbrot to connect fractal geometry with nature. Mandelbrot's
article ``How long is the coast of Britain?'' [Man67]
explores exactly how geographical boundaries exhibit the same
phenomenon. The numbers 4 and 3 give a better insight into the
complexity of the Koch curve than just its infinite length, and that
is exactly where fractal dimension originates.
Like topological dimension, fractal dimension begins with a covering
of the given set S by open balls 
. We call the
covering an 
-covering if the radius r of any ball
in the covering satisfies 
. We suppose our set S is
bounded, and the covering has the smallest number of balls
possible. Let this number of balls be 
. For the Koch
curve, the minimal number of -balls with 
is
roughly 
. With the aid of the magical logarithm, we
can work out the relationship between and in
this case. First, take logarithms:
The ratio of these two logarithms is independent of :
This constant is the fractal dimension of the Koch curve.
Definition of Fractal Dimension: Let S be a compact subset of a metric space. For each, let
necessary to cover S. Suppose
exists. Then
is called the fractal dimension of S.
Let's explain why this corresponds to the usual notion of topological
dimension. First, consider a rectifiable curve S. In that case, a
disk of radius covers a piece approximately 
long. So the number of disks necessary is roughly 
where L is the length of the curve. Then 
. The dimension occurs as the
negative of the exponent of in the growth law for
. Thus, we see that the fractal dimension of a
rectifiable curve is 1. Similarly, for a plane region of area A, a
disk of radius covers an area of 
. Hence,
disks
are needed. The fractal dimension of a plane region is 2. It can be
proved that the fractal dimension 
is always greater than or
equal to the topological dimension 
. Also, if the set S is
contained in 
, the fractal dimension 
is always
. Therefore, the fractal dimension serves as an interpolation
of the topological dimension.
There are several defects and subtleties in the various definitions of
fractional dimensions. One severe problem is that for certain compact
subsets the limit of 
may not exist.
Secondly, one might think that we need consider only balls of radius
exactly equal to , instead of less than or equal to
. However, there are compact sets for which the two
different definitions of dimension lead to different results. The
pathological aspects of dimension have required deep analytical
studies to delineate. In applications to natural science, one usually
takes the point of view that the fractals that occur in nature are
well-behaved with respect to the calculation of their dimension. This
is somewhat ironic since the genesis of the ``fractal geometry of
nature'' was the rejection that nature should be described by the
smooth objects of classical geometry.