HOW LONG IS THE COAST OF BRITAIN?
How long is the coast of Britain?
Statistical self-similarity and fractional dimension
Science: 156, 1967, 636-638
B. B. Mandelbrot
Geographical curves are so involved in their detail that their lengths are often infinite or more
accurately, 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. In particular, it exceeds the
value unity associated with ordinary curves.
1. Introduction
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. I propose to reexamine in this light, some empirical
observations in Richardson 1961 and interpret them as implying, for example, that the dimension of the
west coast of Great Britain is
D = 1.25.
Thus, the so far esoteric concept of a ‘‘random figure of fractional
dimension’’ is shown to have simple and concrete applications of great usefulness.
Figure 1 :fig width=page frame=none depth='4.25i' place=bottom. :figcap. :figdesc. Data from Richardson
1961, Fig. 17, reporting on measurements of lengths of geographical curves by way of polygons which have
equal sides and have their corners on the curve. For the circle, the total length tends to a limit as the side
goes to zero. In all other cases, it increases as the side becomes shorter, the slope of the doubly logarithmic
graph having an absolute value equal to
D - 1 .
(Reproduced by permission.) :efig.
Self-similarity methods are a potent tool in the study of chance phenomena, wherever they appear,
from geostatistics to economics (M 1963b{E }), and physics (M 1967i{N9}).
Very similar considerations apply
in the study of turbulence, where the characteristic sizes of the ‘‘features’’ (which are the eddies) are also
very widely scattered, a fact first pointed out by Richardson himself in the 1920’s. In fact, many noises have
dimensions
D
between 0 and 1; therefore, scientists ought to consider dimension as a continuous quantity
ranging from 0 to infinity.