The length of a coastline can be measured by using a map and a pair of
compasses with an opening of and counting the number of steps
needed for one roundtrip (fig. 8). The length of the
coastline is given as
.
Figure 8: Measuring the coastlength of Iceland
Tab. 1 shows the number of steps and the length
of the coastline for different openings
. It is obvious that the measured
length increases as the opening of the compasses is reduced.
Table 1: Length of Iceland's coastline
If the coastline has a well-defined length, then should approach
a constant value
as
. However this is not the case.
The double logarithmic plot in fig. 9 shows that
does not reach a fixed value as
is reduced.
Figure 9: versus
Thus 'How long is the coast of iceland?' is not a good question. The length depends on what someone wants to do. If somebody wants to build a fence around Iceland with fenceposts every ten meters, for him the coast is longer than for someone who wants to place lighthouses every fifty kilometers. Since the values in fig. 9 are lying nicely on a straight line a power law
is a proper fit. is the compass, divider or ruler dimension
[8][3]. It can be estimated from the slope
of the regression line in fig. 9. The slope is:
With a constant of for the man with the fence the coastline
would have a length of
,
for the man with the lighthouses
.
The coast length differs by a factor of
.
The compass dimension is a measure to compute the fractal dimension of natural objects, since it is an estimator of the Hausdorff-Besicovitch dimension. For example the Koch curve (see sect. 2.4) has a compass dimension equal to its Hausdorff-Besicovitch dimension.