How long is the coastline of Britain?

I found this in the course of my studies:

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.

Now, it's obvious that the coastline of Britian isn't infinate, but Mandelbrot is just using that coastline as an example. Fractals can't exist in the real world (but they can be represented. Also, it's not easy to define precisely where the coastline and water terminates.

Thats quite interesting, i suppose it just shows estimations of shape and size just dont cut it Wouldnt this also be true for any other type of island ?

It can work for just about any extremely large object (like a island) that has somewhat of a rocky coastline.

But as I mentioned in the first post, you can't have these lengths in real life.

Here's a mathenatical way of looking at it;

f(x): 1/x
0 < x < 1

1/0.1=10
1/0.01=100
1/0.001=1000
1/0.0001=10000
and so on...

You can see if x approaches zero, the value of the function approaches infinity.

[Edited on 24-10-2003 by Saucerat]

[Edited on 24-10-2003 by Saucerat]

[Edited on 24-10-2003 by Saucerat]

Well there is a set number but its a hard job to measure, i cant say i agree with that statement, the coastline is NOT infinite its just very very hard to measure.

Maybe he should try other islands such as Japan or Greenland and see they're all the same.

how big is a piece of string, how deep is half a hole...

is the glass half empty or half full... so many questions so little time....

ahh the philosophy of maths......(in repsonce to your comment)


on a side note it would actually be interesting to know the circumferance of the brittish isles...

Wow.
Seriously.

Smart man, I'll go with what he said because it uses less math.