Department of Analysis, Eötvös Loránd University
Department of Mathematics, University of Washington
Abstract: The von Koch Curve, by Helge von Koch's own admission, can be quite naturally generalized. One such family of natural generalizations can be built using n-gons. To build these curves, fix some positive scalar c less than one. Draw a line segment with length L. Replace the middle cL portion of the segment with the sides of a regular n-gon whose own sides are length cL. This produces a total of n+1 line segments. Recursively apply this procedure to each line segment, and take the limit to produce the (n,c)-von Koch Curve. For fixed n and c sufficiently large, the curve intersects itself. Likewise, for c sufficiently small, the curve does not. The trouble with the (n,c)-von Koch curve is what happens in between. We show that the set of c for which the curve self-intersects is not necessarily an interval.
American Mathematical Monthly, February 2010 (vol. 117, no. 2, pp. 124-137).
The applet and materials below supplement the article.
The (n,c)-von Koch curve is constructed recursively. On a closed line segment of length L, glue a regular n-gon with side length cL to the middle of the line segment, and delete the interior of the overlapped middle-cL segment. This produces n+1 line segments from 1, and so it may be repeated, always gluing the polygons with a consistent orientation outward. The "edges" field in the applet specifies n; the value of c can be entered directly into the "c" field or changed using the "Increment c" and "Decrement c" buttons below the graph. The applet repeats the procedure for the number of times specified in the "generations" field, provided the length of the line segments added in the next generation is greater than the value in the "detail threshold" field. The limit of this procedure is the (n,c)-von Koch curve, so the applet renders an approximation to this curve, using the specified parameters.
For every n there is a c sufficiently small so that the von Koch curve embeds naturally into the plane. By the same token, there is a c sufficiently large so that the von Koch curve can be assured to intersect itself. To answer more exacting questions about the set of pairs (n,c) for which the (n,c)-von Koch curve self-intersects, a particular wedge of the curve is critical.
This wedge is located between the largest n-gon and the baseline of the curve. By studying this wedge, it becomes apparent that there is a meshing phenomenon that takes place, wherein the curve intersects itself at some c1 but does not at some c2 larger than c1. Enabling "only show wedge" in the applet focuses the graph on this wedge. A table of some parameters that demonstrate this property is given below.
To view these examples, enable "only show wedge" and set "generations" = 10 and "detail threshold" = 0.1.