A familiar method of measuring the length of a curve is to 
approximate the curve
by straight line segments; add the lengths
of the line segments. 
Smaller line
segments should give a better approximation, and 
we look for a limiting value of the lengths of the line segments, as we
use smaller and smaller line segments. 

In more detail, 
*  we begin by selecting points on the curve and
connecting successive points wth straight line segments. 
*  The length of the portion of
the curve between successive points cannot be less than the length of the straight line
segment, so the collection of line segments has length less than (or equal to) the length
of the curve. 
*  Between successive points in the orginal collection,
insert more points of the curve and replace the original line segments with segments between
successive points of this new collection. 
*  Repeating this process, the lengths of the
collections of segments converges to the length of the curve, we hope. 

Our eyes tell the story: as
the segments are replaced with smaller segments, distinguishing the curve from the collection
of segments becomes more difficult. 
First, we apply this to a situation where it works: finding the length (circumference)
of a circle. 
Heartened by this success, we try the same approach to compute the length of the
Koch curve. 
This result may appear confusing, but some curves wiggle so much they
fill up an area. 
Perhaps the Koch curve is one of these. Let's compute its
area. 

Length is a onedimensional measure; area a twodimensional measure.
Neither is a useful measure for the Koch curve, so
the Koch curve is somehow more than onedimensional and less than twodimensional.
What is it? 