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FIGURE 2.1
The first few steps in the construction of the Koch curve.

2.3 An Iterative Geometric Construct: The Koch Curve

In your previous studies of 2-D geometry, you encountered classical geomet-
ric objects such as the circle, the triangle, the square, different polygons, etc.
These shapes only approximate the shapes that you observe in nature (e.g.,
the shapes of clouds, mountain ranges, rivers, coastlines, etc.). In a successful
effort to address the limitations of classical geometry, mathematicians have
developed, over the last century and more intensely over the last three
decades, a new geometry called fractal geometry. This geometry defines the
geometrical object through an iterative transformation applied an infinite
number of times on an initial simple geometrical object. We illustrate this
new concept in geometry by considering the Koch curve (see Figure 2.1).
The Koch curve has the following simple geometrical construction. Begin
with a straight line of length L. This initial object is called the initiator. Now
partition it into three equal parts. Then replace the middle line segment by an
equilateral triangle (the segment you removed is its base). This completes the
basic construction, which transformed the line segment into four non-colin-
ear smaller parts. This constructional prescription is called the generator. We
now repeat the transformation, taking each of the resulting line segments,
partitioning them into three equal parts, removing the middle section, etc.

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