Here we give a few hints on computing a few fractal dimensions.
Exercise V.1.1 of Barnsley [Bar93] asks the reader to
compute the fractal dimension of a point. For any 
, the
number 
of balls of radius 
necessary to
cover a point is (drum roll, please) just 1. So
and the dimension is just 0. Exercise V.1.3 then asks for the
dimension of a set of three points. What is in that
case? You may not be able to precisely determine , but
you can give a very precise upper bound 
that
allows you to solve the problem.
Now suppose S=[0,1], the unit line segment. Again we may not be able
to exactly determine for each 
, but that is
not necessary. We will show how to use the ``squeezing theorem'' of
calculus to evaluate the fractal dimension. The width of a disk of
radius is 
. Thus, if N disks of radius
cover S, we must have 
. On the
other hand, if 
, we can in fact produce a covering
by laying the disks down in a row with only a very slight overlap.
This proves that
Then
since 
is an increasing function. Dividing by
(which is positive for small ), we
get
Here we are giving lower and upper bounds for the quotient
in the middle. The upper and lower bounds are somewhat complicated;
however, we only need to apply the basic facts from calculus about
limits to evaluate the limits as 
. It's better
to set 
and consider the limits as 
.
The lower bound is
Since 
as , the limit of our lower
bound is 1 as 
. The reader may show by the same sort
of algebra that the upper bound has the same limit. The squeezing theorem
then says
There are several other worthwhile exercises of this type in Barnsley (see this list). For the numerical computations from pictures, it is very useful to have a piece of fine engineering graph paper that may be laid over the picture. Then you can count the boxes of various sizes of the graph paper that touch the fractal.
Finally, we shall sketch an example that shows a weakness in the definition of fractal dimension, which is the reason that for theoretical purposes its big brother Hausdorff-Besicovitch dimension is more commonly used. Consider the set
This is a sequence of points together with its one limit point 0. It is both closed and bounded, and therefore compact. We will show that the fractal dimension is at least 1/2, which contradicts our intuition that such a scant sequence of points should be 0-dimensional.
Choose a positive integer n>1 and choose to lie in the following
interval:
The outer expressions are the distances between 
and
and between 
and . Then there is
no way for a disk of radius to cover two of the points 1,
, 
, since any pair of these points
are more than apart. Thus,
for this choice of . Hence,
using properties of logarithms. The last expression tends to 1/2 as
, proving our claim that the fractal dimension is at
least 1/2. In fact, we leave it to the reader to find an upper bound that proves
that the fractal dimension is exactly 1/2.