By Zdzislaw Alexander Melzak
Read or Download Companion to Concrete Mathematics, Vol. 1 PDF
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Extra resources for Companion to Concrete Mathematics, Vol. 1
Example text
Our first problem is to find a plane closed rectifiable curve C of fixed length L which encloses the largest possible area A. It is sometimes called the problem of Dido, or the problem of Hengist and Horsa, these semimythical personages having been concerned in shady land-grant deals involving as much land as can be enclosed by a hull's hide cut into thin strips and tied together. The official mathematical name of our problem is the isoperimetric problem of the circle. 8. ISOPERIMETRIC PROBLEMS FOR CONVEX HULLS 21 y b y i a FIGURE 1.
We have then for the average ii(L) ii(L) = J L,-,C,-#0 n(L)dL/J dL = 2l(C1 )/l(C). (2) LnC#0 If C1 is also a closed convex curve we have L(C1 ) < L(C) and we find the following probabilistic interpretation: if C and C 1 are closed convex curves, with C1 inside C, then a line cutting C will also cut C1 with the probability equal to the ratio of the lengths of the curves. If C 1 is arbitrary we have the following: some straight line must cut C 1 in at least 21( C1 )/l( C) points. From the foregoing we can also deduce a very simple solution of the celebrated needle-problem due to Buffon (1777): a plane is lined with parallel straight lines one unit apart and a thin needle N of length I < I is thrown at random on the plane, what is the probability p that N intersects one of the lines?
GEOMETRY 34 It is clear that the string can be moved along itself, remaining tight. We show that it can also be spread, so that the distance d changes, while still wrapping up the box tightly. For this we imagine the box cut open along its edges and spread out in the plane. Adding to the plane net certain reflections offaces in sides, we obtain the configuration of Figure 6b, in which the string appears as a straight segment. It is now clear that, within certain limits, the string can be moved parallel to itself, and each position corresponds to a different closed octagonal-shaped geodesic wrapping of the original parcel.