By Daniel J. Velleman
Read Online or Download American Mathematical Monthly, volume 117, number 2, February 2010 PDF
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Extra info for American Mathematical Monthly, volume 117, number 2, February 2010
All other children of curve AC are strictly separated from curve MC by line M S. Furthermore, all primary children of curve AC are pairwise disjoint, and thus if an intersection occurs, it must be between curve M S and curve MC (or between the left counterparts of curve M S and MC, but by the natural left-right symmetry of curve AC, it is unnecessary to search both the left and right sides for intersection). 1 can be bettered. Instead of checking the whole wedge, our focus can be restricted to a properly chosen trapezoid.
Likewise, let the lower strip be the minimum constant-radius tubular neighborhood of segment MC so that the union of the strip, ξk , and βk contains the intersection of curve MC with the critical trapezoid. Let μ be the radius of the upper strip, and let λ be the radius of the lower strip. 2 collects the importance of the preceding definitions. 2. , α, ζ , and the upper strip are disjoint from ξk , βk , the lower strip) then curve AC does not self-intersect. Proof. Because the lines lk and lk+1 exist, the critical trapezoid, denoted T , exists.
In the very paper in which he introduced the curve, von Koch provided a family of curves generalizing his original construction. He allowed that the deleted section might vary in size and position at every stage of the process, so long as the curve remained bounded. This is a greatly flexible definition that allows the final curve to have many different properties; here we will focus on a more restrictive and more symmetric generalization. In this vein, we vary the length of the middle segment that is removed.
American Mathematical Monthly, volume 117, number 2, February 2010 by Daniel J. Velleman