How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension
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How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension is a paper by mathematician Benoît Mandelbrot, first published in Science in 1967. In this paper Mandelbrot discusses self-similar curves that have Hausdorff dimension between 1 and 2. These curves are examples of fractals, although Mandelbrot does not use this term in the paper, as he did not coin it until 1975.
The paper examines the surprising property that the measured length of a stretch of coastline depends on the scale of measurement. Empirical evidence suggests that the smaller the increment of measurement, the longer the measured length becomes. If you were to measure a stretch of coastline with a yardstick, you would get a shorter result than if you were to measure the same stretch with a small ruler. This is because you would be laying the ruler along a more curvilinear route than that followed by the yardstick. The empirical evidence suggests a rule which, if extrapolated, shows that the measured length increases without limit as the measurement scale decreases towards zero.
In the first part of the paper Mandelbrot discusses research published by Lewis Fry Richardson on how the measured lengths of coastlines and other natural geographic borders are dependent on the scale of measurement. Richardson had observed that the measured length L(G) of various country borders was a function of the measurement scale G. Collecting data from several different examples, he conjectured that L(G) could be closely approximated by a function of the form
- L(G) = MG1 − D
Mandelbrot interprets this result as showing that coastlines and other geographic borders can have a property of statistical self-similarity, with the exponent D measuring the Hausdorff dimension of the border. With this interpretation, the examples in Richardson's research have dimensions ranging from 1.02 for the coastline of South Africa to 1.25 for the West coast of Britain.
Note that the paper does not claim that any coastline or geographic border actually is a fractal - that would be a physical impossibility. It simply states that the measured length of a coastline or border may be empirically observed to behave like a fractal over a range of measurement scales.
In the second part of the paper Mandelbrot describes various curves, related to the Koch snowflake, which are defined in such a way that they are strictly self-similar. Mandelbrot shows how to calculate the Hausdorff dimension of each of these curves, each of which has a dimension between 1 and 2. He also mentions (but does not give a construction for) the space-filling Peano curve, which has a dimension of 2.
The paper is important because it shows Mandelbrot's early thinking on fractals, and is an example of the linking of mathematical objects with natural forms that was a theme of much of his later work.
[edit] See also
[edit] References
- Benoît Mandelbrot, 1967, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. Science, New Series, Vol. 156, No. 3775. (May 5, 1967), pp. 636-638. doi:10.1126/science.156.3775.636
- PDF version from Mandelbrot's home page
