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Coastline paradox

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An example of the coastline paradox. If the coastline of Great Britain is measured using units 100 km (62 mi) long, then the length of the coastline is approximately 2,800 km (1,700 mi). With 50 km (31 mi) units, the total length is approximately 3,400 km (2,100 mi), approximately 600 km (370 mi) longer.

The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal-like properties of coastlines, i.e., the fact that a coastline typically has a fractal dimension (which in fact makes the notion of length inapplicable). The first recorded observation of this phenomenon was by Lewis Fry Richardson[1] and it was expanded upon by Benoit Mandelbrot.[2]

The measured length of the coastline depends on the method used to measure it and the degree of cartographic generalization. Since a landmass has features at all scales, from hundreds of kilometers in size to tiny fractions of a millimeter and below, there is no obvious size of the smallest feature that should be taken into consideration when measuring, and hence no single well-defined perimeter to the landmass. Various approximations exist when specific assumptions are made about minimum feature size.

The problem is fundamentally different from the measurement of other, simpler edges. It is possible, for example, to accurately measure the length of a straight, idealized metal bar by using a measurement device to determine that the length is less than a certain amount and greater than another amount—that is, to measure it within a certain degree of uncertainty. The more accurate the measurement device, the closer results will be to the true length of the edge. When measuring a coastline, however, the issue is that the result does not increase in accuracy for an increase in measurement —it only increases; unlike with the metal bar, there is no way to obtain a maximum value for the length of the coastline.

Mathematical aspects

The basic concept of length originates from Euclidean distance. In Euclidean geometry, a straight line represents the shortest distance between two points. This line has only one length. On the surface of a sphere, this is replaced by the geodesic length (also called the great circle length), which is measured along the surface curve that exists in the plane containing both endpoints and the center of the sphere. The length of basic curves is more complicated but can also be calculated. Measuring with rulers, one can approximate the length of a curve by adding the sum of the straight lines which connect the points:

Using a few straight lines to approximate the length of a curve will produce an estimate lower than the true length; when increasingly short (and thus more numerous) lines are used, the sum approaches the curve's true length. A precise value for this length can be found using calculus, the branch of mathematics enabling the calculation of infinitely small distances. The following animation illustrates how a smooth curve can be meaningfully assigned a precise length:

However, not all curves can be measured in this way. A fractal is, by definition, a curve whose complexity changes with measurement scale. Whereas approximations of a smooth curve tend to a single value as measurement precision increases, the measured value for a fractal does not converge.

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This Sierpiński curve (a type of Space-filling curve), which repeats the same pattern on a smaller and smaller scale, continues to increase in length. If understood to iterate within an infinitely subdivisible geometric space, its length tends to infinity. At the same time, the area enclosed by the curve does converge to a precise figure—just as, analogously, the land mass of an island can be calculated more easily than the length of its coastline.

As the length of a fractal curve always diverges to infinity, if one were to measure a coastline with infinite or near-infinite resolution, the length of the infinitely short kinks in the coastline would add up to infinity.[3] However, this figure relies on the assumption that space can be subdivided into infinitesimal sections. The truth value of this assumption—which underlies Euclidean geometry and serves as a useful model in everyday measurement—is a matter of philosophical speculation, and may or may not reflect the changing realities of "space" and "distance" on the atomic level (approximately the scale of a nanometer). For instance, the Planck length, many orders of magnitude smaller than an atom, is proposed as the smallest measurable unit possible in the universe.

Coastlines less definite in their construction than idealized fractals such as the Mandelbrot set because they are formed by various natural events that create patterns in statistically random ways, whereas idealized fractals are formed through repeated iterations of simple, formulaic sequences.[4]

Practicality

In actuality, the concept of an infinite fractal is not applicable to a coastline; as progressively more accurate measurement devices are used (with the measurement result growing every time), other, practical problems with measurement emerge.

  • The sea is in constant motion, meaning there is no fixed "coastline".
  • Even if the sea's movement could be halted while measurement took place, there would be no way (beyond arbitrary decisions) to define where the coastline lies in terms of river outflow. A measurement including the entire banks of every river that extends into a landmass would conflict with many interpretations of a coastline, yet no established technique exists for selecting an arbitrary line where river becomes sea.
  • Even if the issue of rivers were overcome, it would still be impossible to decide which the boundary between land and water is, as land may be wet but not submerged.
  • Even if such a definition could be agreed upon, as increasingly accurate measurement is performed, the problem of measuring the boundary of an atom—which ultimately does not have a defined boundary—arises. Even in the classical model of an atom, which assumes atoms are composed of individual solid particles, most of an atom is made up of the space between these particles. This problem is further exacerbated when other, more modern models of the atom are considered.

See also

Notes and references

  1. ^ Weisstein, Eric W. "Coastline Paradox". MathWorld.
  2. ^ Mandelbrot, Benoit (1983). The Fractal Geometry of Nature. W.H. Freeman and Co. 25–33. ISBN 978-0-7167-1186-5.
  3. ^ Post & Eisen, p. 550.
  4. ^ Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, Chaos and Fractals: New Frontiers of Science; Spring, 2004; p. 424.

Bibliography

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