How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension

"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 is one of Mandelbrot's first publications on the topic of fractals.

The paper examines the coastline paradox: the 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 one were to measure a stretch of coastline with a yardstick, one would get a shorter result than if the same stretch were measured with a one-foot (30.48 cm) ruler. This is because one 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 measure length increases without limit as the measurement scale decreases towards zero.

This discussion implies that it is meaningless to talk about the length of a coastline; some other means of quantifying coastlines are needed. Mandelbrot discusses an empirical law discovered by Lewis Fry Richardson, who observed that the measured length L(G) of various geographic borders was a function of the measurement scale G. Collecting data from several different examples, Richardson conjectured that L(G) could be closely approximated by a function of the form

where M is a positive constant and D is a constant, called the dimension, greater than or equal to 1. Intuitively, if a coastline looks smooth it should have dimension close to 1; and the more irregular the coastline looks the closer its dimension should be to 2. 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.

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