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Coastline of Britain

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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.
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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/
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```
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?

```

```
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

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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Associated Topics:
High School Fractals

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