Coastline paradox
The coastline paradox is the counterintuitive observation that the
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.
Discovery
Shortly before 1951, Lewis Fry Richardson, in researching the possible effect of border lengths on the probability of war, noticed that the Portuguese reported their measured border with Spain to be 987 km (613 mi), but the Spanish reported it as 1,214 km (754 mi). This was the beginning of the coastline problem, which is a mathematical uncertainty inherent in the measurement of boundaries that are irregular.[6]
The prevailing method of estimating the length of a border (or coastline) was to lay out n equal straight-line segments of length l with
The result most astounding to Richardson is that, under certain circumstances, as l approaches zero, the length of the coastline approaches
Mathematical aspects
This section needs additional citations for verification. (February 2015) |
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
Not all curves can be measured in this way. A fractal is, by definition, a curve whose perceived 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.
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.
Coastlines are less definite in their construction than idealized fractals such as the
Measuring a coastline
More than a decade after Richardson completed his work,
I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means "to break:" to create irregular fragments. It is therefore sensible ... that, in addition to "fragmented" ... fractus should also mean "irregular".
In "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension", published on 5 May 1967,[12] 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.[13]
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
A key property of some fractals is
There are different kinds of fractals. A coastline with the stated property is in "a first category of fractals, namely curves whose fractal dimension is greater than 1". That last statement represents an extension by Mandelbrot of Richardson's thought. Mandelbrot's statement of the Richardson effect is:[15]
where L, coastline length, a function of the measurement unit ε, is approximated by the expression. F is a constant, and D is a parameter that Richardson found depended on the coastline approximated by L. He gave no theoretical explanation, but Mandelbrot identified D with a non-integer form of the Hausdorff dimension, later the fractal dimension. Rearranging the expression yields
where Fε−D must be the number of units ε required to obtain L. The broken line measuring a coast does not extend in one direction nor does it represent an area, but is intermediate between the two and can be thought of as a band of width 2ε. D is its fractal dimension, ranging between 1 and 2 (and typically less than 1.5). More broken coastlines have greater D, and therefore L is longer for the same ε. D is approximately 1.02 for the coastline of South Africa, and approximately 1.25 for the west coast of Great Britain.[5] For lake shorelines, the typical value of D is 1.28.[16]
Solutions
The coastline paradox describes a problem with real-world applications. To resolve this problem, several solutions have been proposed.[17] These solutions resolve the practical problems around the problem by setting the definition of "coastline," establishing the practical physical limits of a coastline, and using mathematical integers within these practical limitations to calculate the length to a meaningful level of precision.[17] These practical solutions to the problem can resolve the problem for all practical applications while it persists as a theoretical/mathematical concept within our models.[18]
Criticisms and misunderstandings
The coastline paradox is often criticized because coastlines are inherently finite, real features in space, and therefore, there is a quantifiable answer to their length.[17][19] The comparison to fractals, while useful as a metaphor to explain the problem, is criticized as not fully accurate as coastlines are not self-repeating and are fundamentally finite.[17]
The source of the paradox is based on the way we measure reality and is most relevant when attempting to use those measurements to create cartographic models of coasts.
See also
- Staircase paradox, similar paradox where a straight segment approximation converges to a different value
- Alaskan Panhandlediffered greatly, based on competing interpretations of the ambiguous phrase setting the border at "a line parallel to the windings of the coast", applied to the fjord-dense region.
- Fractal dimension
- Gabriel's horn, a geometric figure with infinite surface area but finite volume
- List of countries by length of coastline
- Scale (geography)
- Paradox of the heap
- Zeno's paradoxes
References
Citations
- .
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.
- S2CID 128975381.
- ^ Richardson, L. F. (1961). "The problem of contiguity: An appendix to statistics of deadly quarrels". General Systems Yearbook. Vol. 6. pp. 139–187.
- S2CID 15662830. Archived from the originalon 2021-10-19. Retrieved 2021-05-21.
- ^ ISBN 978-0-7167-1186-5.
- ISBN 0-521-38297-1.
- S2CID 235508504.
- ^ Post & Eisen, p. 550 (see below).
- ^ Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, Chaos and Fractals: New Frontiers of Science; Spring, 2004; p. 424.
- ^ a b Mandelbrot 1982, p. 28.
- ^ Mandelbrot 1982, p. 1.
- S2CID 15662830.
- ^ "Dr. Mandelbrot traced his work on fractals to a question he first encountered as a young researcher: how long is the coast of Britain?": Benoit Mandelbrot (1967). "Benoît Mandelbrot, Novel Mathematician, Dies at 85", The New York Times.
- ISBN 978-0747404132.
- ^ Mandelbrot 1982, pp. 29–31.
- S2CID 235508504.
- ^ S2CID 255441171.
- S2CID 214198004.
- ^ a b Sirdeshmukh, Neeraj. "Mapping Monday: The Coastline Paradox". National Geographic. Retrieved 25 November 2023.
Sources
- ". Journal of Legal Studies XXIX(1), January 2000.
- ISBN 978-0-7167-1186-5.
External links
- "Coastlines" at Fractal Geometry (ed. Michael Frame, Benoit Mandelbrot, and Nial Neger; maintained for Math 190a at Yale University)
- The Atlas of Canada – Coastline and Shoreline
- NOAA GeoZone Blog on Digital Coast
- What Is The Coastline Paradox? – Veritasium