How long is the coast of Britain? Statistical self-similarity and fractional dimension

Abstract

Geographical curves are so involved in their detail that their lengths are often infinite or, rather, undefinable. However, many are statistically "self-similar," meaning that each portion can be considered a reduced-scale image of the whole. In that case, the degree of complication can be described by a quantity D that has many properties of a "dimension, " though it is fractional; that is, it exceeds the value unity associated with the ordinary, rectifiable, curves.