Von Koch Curves

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 star-like 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 self-similarity.

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. .

The final curve can be strongly modified when the iteration process is changed. Mandelbrot gives the following example, where the symmetry of the original Von Koch curve is broken: courbes

But one can think of quite more dramatic changes. The following pattern leads to the famous Peano curve, the continuous curve which completely fills a square. Only the three first steps are displayed below peano

but one easily guess that the whole blue square will be filled if the iteration is continued. Please notice that there should not be any rounded angles in the curves; they were placed only for the reader convenience, so as to follow the continuity of the curve more easily.

Also notice that the length of the polygonal line is doubled at every step.

Lastly, the iteration can be made even more complex by introducing periodical reversals or symmetries, by making the lines invisible from time to time and by introducing colour changes. One thus can get quite a lot of different pictures. Several experimental programs were written to explore the possibilities in this way. The pictures below were derived with Fracgen, by Doug Houck (1988, for the Amiga).

$fracgen$ Obviously, these pictures pertain to curiosity more than to Art, but there was no progress in this way, for lack of more advanced software.

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Discovering the fractal world:: Introduction - Mandelbrot Exploration - Lyapounov Exploration - Von Koch Curves - IFS Fractals - Fractal Dimension - Mandelbrot Relatives - Finest Fractal Pictures - Software - Biblio and Links
Fractals and mysticism:: Introduction - The Mysticism of Infinite - Non-Euclidean Art?
A new Art?: Introduction - Fascination of Fractals - Fractals and Photography - Definitions of Art - The Colour Choice - Other Colour Choices - Fractalists Painters - Compositions with Mandelbrot - Put a pretty girl - Algorithmic Art - Beyond the "Fractal" Art

Von Koch Curves

Generalized Von Koch curves