Coastline of Britain

Date: 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/