
The genuine Von Koch curve, also called
snowflake curve, is derived as the limit of a polygonal contour.
At every step, as shown on the left, the middle third of
every side of the polygone is replaced with two linear segments at
angles 60° and 120°.
Starting from an equilateral triangle, the two first steps lead to
the starlike curves plotted on the left. If one goes on long enough
one finally gets the curve right below. Ideally the process should
go on indefinitely, but, in practice, the curve displayed on
the screen no longer changes when the elementary side becomes less than
the pitch, and then the iterations can be stopped.
What is thus obtained was long considered a mathematical monster, a
curve plotted in a bounded domain, but
with an infinite length (one easily sees that the
length is multiplied by 4/3 at every step), continuous but nowhere
differentiable (i.e. nowhere a tangent can be defined). It
is now regarded as an elementary example of fractal
–"elementary" because of the simplicity of the construction.
The pattern
can be seen everywhere along the curve, at every scale,
from visible to infinitesimal. This feature is called selfsimilarity.

For a while, it was thought that such a curve was a possible
starting point
for the design of tools for drawing complex natural curves (such as the
rocky coast of the celebrated example of Benoît Mandelbrot),
with a little number of control parameters. Here, the parameters are
just the 6 relative coordinates of the indentation points and the result
looks really complex when one goes out from the field of classical geometry
and its perfectly smooth curves.
Of course, the Von Koch curve does not look like a "natural" curve. It
is too regular in its irregularity to simulate the randomness of a
rocky coast. This regularity comes from the strictness of its
construction process. It can be loosened by introducing random fluctuations (for
instance random moves of the indentation points) but, according to
Mandelbrot, one gets better simulations tools with other recipes.
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