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# How long is the coastline of Koch's Island?

Question #50756. Asked by peasypod. (Sep 03 04 3:58 PM)

Brainyblonde

Koch's island does have a (mathematically) well-defined shape, which Figure 10 approximates quite well as far as the human eye can distinguish. The mathematically precise coastline of Koch's island is the 'curve' which is the limit of the infinite sequence of approximations to it, of which Figure 9 gives the first three. At this point the mathematics takes over from the human cartographer. Mathematically, this limit curve is precisely determined, and like any other curve will consist of an infinitude of points strung together to form a 'line'. The process of arriving at the limit curve is analogous to arriving at the number 1/3 as the limit of the infinite sequence of decimals
0.3, 0.33, 0.333, 0.3333, 0.33333 .....
Since Koch's island is a mathematically defined region of the plane, it will have a definite area. The actual numerical value of its area will depend on the units of measurement being used, of course, but it will certainly be finite. (It may be calculated as a limit of a sequence of numbers, much like the 1/3 example above: it is, in fact, exactly 1.6 times the area of the triangle in Figure 9(i).) What of the length of the coastline surrounding this finite area? Well, each successive stage of the Koch process increases the length of the 'coastline' by a factor of 4/3. By the time the Koch curve (as the limiting coastline is called) is reached, this 4/3 increase will have occurred infinitely often, and so the length of the Koch curve will be infinite.
http://www.music-mind.com/resour3.htm

I of course understand every word of that!!! LOL!!!

 Sep 03 04, 4:18 PM

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