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Coastline paradox

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 curve–like properties of coastlines; i.e., the fact that a coastline typically has a fractal dimension. Although the "paradox of length" was previously noted by Hugo Steinhaus,[1] the first systematic study of this phenomenon was by Lewis Fry Richardson,[2][3] and it was expanded upon by Benoit Mandelbrot.[4][5]

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 precise the measurement device, the closer results will be to the true length of the edge. When measuring a coastline, however, the closer measurement does not result in an increase in accuracy—the measurement only increases in length; unlike with the metal bar, there is no way to obtain an exact value for the length of the coastline.

In three-dimensional space, the coastline paradox is readily extended to the concept of fractal surfaces, whereby the area of a surface varies depending on the measurement resolution.

  1. ^ Steinhaus, Hugo (1954). "Length, shape and area". Colloquium Mathematicum. 3 (1): 1–13. doi:10.4064/cm-3-1-1-13. The left bank of the Vistula, when measured with increased precision would furnish lengths ten, hundred and even thousand times as great as the length read off the school map. A statement nearly adequate to reality would be to call most arcs encountered in nature not rectifiable.
  2. ^ Vulpiani, Angelo (2014). "Lewis Fry Richardson: scientist, visionary and pacifist". Lettera Matematica. 2 (3): 121–128. doi:10.1007/s40329-014-0063-z. MR 3344519. S2CID 128975381.
  3. ^ Richardson, L. F. (1961). "The problem of contiguity: An appendix to statistics of deadly quarrels". General Systems Yearbook. Vol. 6. pp. 139–187.
  4. ^ Mandelbrot, B. (1967). "How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". Science. 156 (3775): 636–638. Bibcode:1967Sci...156..636M. doi:10.1126/science.156.3775.636. PMID 17837158. S2CID 15662830. Archived from the original on 2021-10-19. Retrieved 2021-05-21.
  5. ^ Mandelbrot, Benoit (1983). The Fractal Geometry of Nature. W. H. Freeman and Co. pp. 25–33. ISBN 978-0-7167-1186-5.

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