A long, long time ago, fractal genius Benoit Mandelbrot posed a simple question: How long is the coastline of Britain? Mandelbrot's mathematical colleagues were miffed, to say the least, at what appeared to be such an annoying waste of their time. Mandelbrot's colleagues told him to look up the answer. Of course, Madelbrot had a reason for his peculiar question. Quite an interesting reason, actually.
Actually if you would look up this figure yourself, in some encyclopedia, whatever figure the book might give you would certainly be wrong. Quite simply, the coastline of Britain is infinite.
You may say that this is impossible. Instead of jumping to that conclusion however, 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.