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, which
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.