Von Koch Curves
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:
Also notice that the length of the polygonal line is doubled at
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).
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