2. A. Ineffective Ways to Measure

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 one-dimensional measure; area a two-dimensional measure. Neither is a useful measure for the Koch curve, so the Koch curve is somehow more than one-dimensional and less than two-dimensional. What is it?