Fractal Fundamentals

“How long is the coast of Britain?” Mandelbrot (1982), establishing the great work of THE FRACTAL GEOMETRY OF NATURE, asked this interesting question. His answer was that its length is uncertain but depends on the length of the ruler or yardstick used. It is evident that its length is at least equal to the distance measured along a straight line between its beginning and its end. However, the typical coastline is irregular and undoubtedly winding. It is much longer than the straight line between its endpoints. To measure its length, a divider is set to a prescribed opening ε that is called the ruler length or yardstick length or unit of measurement. This divider is “walked” along the coastline with each new step starting where the previous step leaves off. The number of steps multiplied by ε is an approximate length L(ε). As the divider opening becomes smaller and smaller, L(ε) is expected to settle rapidly to a well-defined value called the true length when the above operation is repeated. However, this does not happen. In the typical case, the observed L(ε) tends to increase without limit. When a bay or peninsula on a map scaled to 1/100,000 is re-examined on a map scaled to 1/10,000, sub-bays or sub-peninsulas become visible. On a 1/1,000 scale map, sub-sub-bays or sub-sub-peninsulas appear. Each adds to the measured length. This analysis leads to the conclusion that the coastline length is very large and is so ill-determined that it is best considered infinite.