How Long is the Coastline of Britain?

The question "How long is the coastline of Britain?" posed by Benoit Mandelbrot, the father of modern fractal theory, in his book The Fractal Geometry of Nature is not as simple as it appears. The problem is that one's answer to this question depends on the length of the ruler one uses. Unlike circles and the other shapes from classical geometry, coastlines are very irregular. They're full of inlets, bays, and rocky shores. A shorter measuring stick will fit more snugly in these nooks and crannies and increase the estimated length of the coastline. Hence, if we measure the length of Britain's coastline using a mile-long ruler, we will get one value. If we use a shorter ruler, say a yardstick, we will get a larger value because a yardstick can more closely approximate Britain's convoluted boundary. In fact, as the scale of measurement decreases, the estimated length increases without limit. Thus, as the length of the ruler approaches zero, the estimated length of the coastline approaches infinity. This difficulty in measuring due to the irregularity of the object being measured is characteristic of fractal curves and surfaces.