coastline paradox

coastline paradox, a mathematical concept showing that the measured length of a coastline is not a definite quantity but varies based on the scale of measurement used and the level of detail that scale reveals. The smaller the measuring scale, the more details are revealed and the longer the measured length will be. This paradox occurs because coasts do not have straight lines; they have turns, curves, and promontories shaped by features such as bays, coves, inlets, and peninsulas. When a coastline is measured using smaller scales, more turns and curves are measured.

The phenomenon was first observed in the 1920s by English mathematician, physicist, meteorologist, and psychologist Lewis Fry Richardson, who hypothesized that the likelihood of war between neighboring countries would be proportional to the length of their shared border. During his research he noticed a discrepancy in the recorded length of the border between Spain and Portugal. The Spanish reported it to be 987 km (613.3 miles), whereas the Portuguese reported it to be 1,214 km (754.3 miles). Richardson then found similar discrepancies in the reported border lengths between other countries and realized that they were due to differences in the scale of measurement. His findings were published posthumously in “The Problem of Contiguity: An Appendix of Statistics of Deadly Quarrels” (1961).

Polish-born French American mathematician Benoit Mandelbrot, known as the father of fractals, expanded on Richardson’s work with the paper “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension” (1967). He noted that coastlines were self-similar shapes; that is, the curve of a coastline looks roughly the same no matter what scale at which it is viewed.

Because the curve of a coastline is fractal and self-similar, when it is measured on smaller scales, the coastline has as many curves and inlets as it does on larger scales. Therefore the length of the coastline depends on the scale of measurement.

Mandelbrot suggested a parameter, now known as the fractal dimension, to describe the degree of complication in coastline details. The fractal dimension is defined as the logarithm of how many pieces a self-similar shape may be broken into divided by the logarithm of how many times that piece may be magnified to produce the original figure.

For example, a square can be broken into n² pieces, each of which must be magnified n times to produce the original square. (If a square is divided into 25 squares (25 = 5 ⨉ 5), each smaller square must be multiplied by 5 to reproduce the original square.) The fractal dimension of the square is log(n²)/log(n) = 2 log(n)/log(n) = 2.

Coastlines have a fractal dimension between 1 and 2, with 1 being that of a straight line and 2 being that of a plane. For example, a fractal dimension of 1.47 represents a form midway between a line and a plane—thus it is twistier than a straight line but not spread out enough to fill a complete two-dimensional plane. The more completely a fractal fills a plane, the closer its dimensional value is to 2.

In addition to coastlines, fractals are used to describe many of the self-similar characteristic patterns found in nature, such as cloud formations, trees, rivers, mountain ranges, and seashells.