Seacoast shapes are examples of highly involved curves with the property that — in a statistical sense — each portion can be considered a reduced-scale image of the whole. This property will be referred to as ‘‘statistical self-similarity.’’ The concept of ‘‘length’’ is usually meaningless for geographical curves. They can be considered superpositions of features of widely scattered characteristic sizes; as even finer features are taken into account, the total measured length increases, and there is usually no clear-cut gap or crossover, between the realm of geography and details with which geography need not be concerned.
Quantities other than length are therefore needed to discriminate between various degrees of complication for a geographical curve. When a curve is self-similar, it is characterized by an exponent of
similarity, D, which possesses many properties of a dimension, though it is usually a fraction greater that the dimension 1 commonly attributed to curves.
B. Mandelbrot, “How long is the coast of Britain? Statistical self-similarity and fractional dimension”, Science: 156, 1967, pp. 636-638.
One cool consequence: the number of dimension(s) of the west coast of Britain is 1.25.