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Journal Article

The Trouble with von Koch Curves Built from ngons

Tamás Keleti and Elliot Paquette
The American Mathematical Monthly
Vol. 117, No. 2 (February 2010), pp. 124-137
DOI: 10.4169/000298910x476040
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Page Count: 14
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The Trouble with von Koch Curves Built from ngons
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Abstract The von Koch curve, as Helge von Koch himself observes, can be quite naturally generalized. One such family of generalizations can be built using n-gons. To build these curves, fix some positive scalar c less than one. Draw a line segment with unit length. Replace the middle c portion of the segment with the sides of a regular n-gon whose own sides are length c. 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. The set of c for which the curve self-intersects is not necessarily an interval, a phenomenon that we explore here.

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