Coastline of BritainDate: 02/13/2002 at 20:45:40 From: Jonathan Vivian Subject: The Coastline of britain IS DEFINITELY finite! Dr Math, do you agree with my sloppy proof that it is false that Britain's coastline is infinite? This is what I want to disprove: _____________ A long, long time ago, fractal god Benoit Mandelbrot posed a simple question: How long is the coastline of Britain? His mathematical colleagues were miffed, to say the least, at such an annoying waste of their time on such insignifigant problems. They told him to look it up. Of course, Madelbrot had a reason for his peculiar question. Quite an interesting reason. Look up the coastline of Britain yourself, in some encyclopedia. Whatever figure you get, it is wrong. Quite simply, the coastline of Briutain is infinite. You protest that this is impossible. Well, consider this. Consider looking at Britain on a very large-scale map. Draw the simplest two- dimensional shape possible, a triangle, that circumscribes Britain as closely as possible. The perimeter of this shape approximates the perimeter of Britain. However, this area is of course highly inaccurate. Increasing the amount of vertices of the shape going around the coastline, and the area will become closer. The more vertices there are, the closer the circumscribing line will be able to conform to the dips and the protrusions of Britain's rugged coast. There is one problem, however. Each time the number of vertices increases, the perimeter increases. It must increase, because of the triangle inequality. Moreover, the number of vertices never reaches a maximum. There is no point at which one can say that a shape defines the coastline of Britain. After all, exactly circumscribing the coast of Britain would entail encircling every rock, every tide pool, every pebble which happens to lie on the edge of Britain. Thus, the coastline of Britian is infinite. ___________________ Here is my proof: The coastline of Britain theory says that since the perimeter increases every time you add another vertex, and since you have to have an infinite number of vertices to be 100% accurate, then the length of Britain's coastline is infinite. This is why that is wrong: In order to be accurate you must add vertices as the distance between them approaches zero. The closer the distance is to zero between each pair of vertices, the more accurate the estimation becomes. However, since an infinite number of vertices is needed to be 100% accurate, the distance between each pair of vertices must be _ZERO_, making the perimeter of the shape undefined. The limit of the perimeter of the shape as the number of vertices approaches infinity and the distance between each pair of vertices approaches zero is equal to the actual real perimeter of Britain and is most definitely a finite answer. Date: 02/14/2002 at 11:39:42 From: Doctor Peterson Subject: Re: The Coastline of britain IS DEFINITELY finite! Hi, Jonathan. There are two issues here. First, the coastline is really being used only as a metaphor; genuine fractals can't exist in the real world, because the real world is not infinitely divisible (as far as we know). Eventually you get down to the point where you are measuring around individual atoms; if you go farther, you will have to go around each proton or something, and we can't say whether protons are smooth or not. The whole thing doesn't really make sense at that point. What Mandelbrot is really talking about is a genuine fractal in an ideal Euclidean space where actual points exist and we can have line segments as small as we like. Second, let's suppose there are no physical limitations, and the irregularity of the coastline really does continue forever at smaller and smaller scales. The actual length of the coastline has to be taken as the limit of an infinite sequence of measurements at finer and finer scales. The mere fact that each measurement is greater than the previous one does not really imply that the sequence has no limit; many familiar infinite sequences, like 1/2, 3/4, 7/8, 15/16, ..., are always increasing yet never pass some set value (1 in this case). So the argument you are trying to answer is incomplete. However, your counterargument is just as faulty, because it is also possible for the product of a number growing toward infinity (the number of vertices) and a number decreasing toward zero (the average length of an edge) to be infinite. The fact is, we simply can't say whether a particular fractal-like line has finite or infinite length without knowing the details about it. Here is an example of such an infinitely long fractal, the Koch snowflake: An Amazing Phenomenon: Infinite Perimeter - Cynthia Lanius http://math.rice.edu/~lanius/frac/koch2.html You will see that the length of each segment decreases to zero (each 1/3 as long as the previous), while the number of sides increases infinitely (4 times as many each time), and the perimeter is infinite. Here are some previous discussions I found in our archives by searching for the words "fractal coastline": Is the Coastline of Britain Infinite? http://mathforum.org/dr.math/problems/puzzled.8.17.99.html Fractal Dimension of a Coastline http://mathforum.org/dr.math/problems/brotzman1.16.97.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 02/16/2002 at 05:41:43 From: Jonathan Vivian Subject: The Coastline of britain IS DEFINITELY finite! Dr Peterson, thank you very much for your reply. You have helped me see more light on the situation. However, I have just one more thing to clear up. From what I gather now, I was wrong and the coastline of Britain isn't _definitely_ finite. But, couldn't it be finite or infinite? I mean, as humans we don't know whether the irregularities of the coastline go on forever as we zoom in (infinite perimeter), or if there is a point where the irregularities become uniform (atomic level example) and finite. I mean for all we know, an island could be a perfect circle that has a finite perimeter by definition. So my question is: Shouldn't Mandlebrot have said "The Perimeter of the coastline of Britain COULD be infinite," rather than saying that it IS infinite? Thanks again for your time. Date: 02/16/2002 at 21:06:05 From: Doctor Peterson Subject: Re: The Coastline of britain IS DEFINITELY finite! Hi, Jonathan. Yes, that's part of what I was trying to say; I don't believe Mandelbrot was really talking about the actual coastline, since you would have to know its behavior all the way down to the smallest scale to say it is infinite. However, he may have had in mind the particular kind of irregularity you see in a coastline, and assumed for the sake of argument that it is a true fractal and has the same irregularity at all scales. Unfortunately, I don't have his book to look it up in and see just how he stated this argument; but I think he was using the coastline to illustrate a concept that can be proved rigorously for mathematically defined fractals. Here are some links that show some of the details of this argument: Coastline Paradox - MathWorld, Eric Weisstein http://mathworld.wolfram.com/CoastlineParadox.html Fractals and the Fractal Dimension - Vanderbilt Univ. http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html The latter illustrates specifically what the former states generally; and if you assume that the graphs shown continue forever, then the length will indeed be infinite! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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