posted on Oct, 24 2003 @ 06:03 PM
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.