790:. To resolve this problem, several solutions have been proposed. 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. 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.
527:. 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 measured 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 then describes various mathematical curves, related to the
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study of fractal-like objects in nature that look random rather than regular. For this he defines statistically self-similar figures and says that these are encountered in nature. The paper is important because it is a "turning point" in
Mandelbrot's early thinking on fractals. It is an example of the linking of mathematical objects with natural forms that was a theme of much of his later work.
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117:. 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.
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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
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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. The comparison to fractals, while useful as a metaphor to explain the problem, is criticized as not fully accurate, as coastlines
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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
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on a map or aerial photograph. Each end of the segment must be on the boundary. Investigating the discrepancies in border estimation, Richardson discovered what is now termed the "Richardson effect": the sum of the segments monotonically increases when the common length of the segments decreases. In
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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. However, this figure relies on the assumption that space can be subdivided
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the same general configuration appears. A coastline is perceived as bays alternating with promontories. In the hypothetical situation that a given coastline has this property of self-similarity, then no matter how great any one small section of coastline is magnified, a similar pattern of smaller
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fractional dimension. Instead, it notes that
Richardson's empirical law is compatible with the idea that geographic curves, such as coastlines, can be modelled by random self-similar figures of fractional dimension. Near the end of the paper Mandelbrot briefly discusses how one might approach the
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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
105:. 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.
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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)
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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
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arrangement of bays and promontories formed from the small objects at hand. In such an environment (as opposed to smooth curves) Mandelbrot asserts "coastline length turns out to be an elusive notion that slips between the fingers of those who want to grasp it".
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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
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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.
814:, has made addressing the paradox much easier; however, the limitations of survey measurements and the vector software persist. Critics argue that these problems are more theoretical and not practical considerations for planners.
400:), 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
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bays and promontories superimposed on larger bays and promontories appears, right down to the grains of sand. At that scale the coastline appears as a momentarily shifting, potentially infinitely long thread with a
833:. Thus wide number of "shorelines" may be constructed for varied analytical purposes using different data sources and methodologies, each with a different length. This may complicate the quantification of
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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. Modern technology, such as
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1348:"What is the essence of a coastline, for example? Mandelbrot asked this question in a paper that became a turning point for his thinking: 'How Long Is the Coast of Britain'": James Gleick (1988)
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531:, which are defined in such a way that they are strictly self-similar. Mandelbrot shows how to calculate the Hausdorff dimension of each of these curves, each of which has a dimension
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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:
715:. 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
172:. Richardson had believed, based on Euclidean geometry, that a coastline would approach a fixed length, as do similar estimations of regular geometric figures. For example, the
516:, 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.
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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
1331:"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). "
462:, to describe just such non-rectifiable complexes in nature as the infinite coastline. His own definition of the new figure serving as the basis for his study is:
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differed 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
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effect, the shorter the ruler, the longer the measured border; the
Spanish and Portuguese geographers were simply using different-length rulers.
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converge to a precise figure—just as, analogously, the area of an island can be calculated more easily than the length of its coastline.
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length), which is measured along the surface curve that exists in the plane containing both endpoints and the center of the sphere. The
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317:, the branch of mathematics enabling the calculation of infinitesimally small distances. The following animation illustrates how a
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is, by definition, a curve whose perceived complexity changes with measurement scale. Whereas approximations of a smooth curve
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such a smooth curve as the circle that can be approximated by small straight segments with a definite limit is termed a
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is the longest, with the former two records a matter of fierce debate; furthermore, the problem extends to demarcating
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is its fractal dimension, ranging between 1 and 2 (and typically less than 1.5). More broken coastlines have greater
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means "to break:" to create irregular fragments. It is therefore sensible ... that, in addition to "fragmented" ...
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1138:
Richardson, Lewis Fry (1993). "Fractals". In
Ashford, Oliver M.; Charnock, H.; Drazin, P. G.; et al. (eds.).
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The coastline paradox describes a problem with real-world applications, including trivial matters such as which
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539:, which has a dimension exactly 2). The paper does not claim that any coastline or geographic border actually
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Richardson, L. F. (1961). "The problem of contiguity: An appendix to statistics of deadly quarrels".
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to a single value as measurement precision increases, the measured value for a fractal does not converge.
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Alternately, the concept of a coast "line" is in itself a human construct that depends on assignment of
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ways, whereas idealized fractals are formed through repeated iterations of simple, formulaic sequences.
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1647:(ed. Michael Frame, Benoit Mandelbrot, and Nial Neger; maintained for Math 190a at Yale University)
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McNamara, Gerard; Vieira da Silva, Guilherme (2023). "The
Coastline Paradox: A New Perspective".
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Counterintuitive observation that the coastline of a landmass does not have a well-defined length
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The measured length of the coastline depends on the method used to measure it and the degree of
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between 1 and 2 (he also mentions but does not give a construction for the space-filling
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The prevailing method of estimating the length of a border (or coastline) was to lay out
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1276:"How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension"
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1035:"How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension"
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In three-dimensional space, the coastline paradox is readily extended to the concept of
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Coastlines are less definite in their construction than idealized fractals such as the
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How Long Is the Coast of
Britain? Statistical Self-Similarity and Fractional Dimension
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The result most astounding to Richardson is that, under certain circumstances, as
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How Long is the Coastline of Law? Thoughts on the Fractal Nature of Legal Systems
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The Collected Papers of Lewis Fry Richardson: Meteorology and numerical analysis
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82:–like properties of coastlines; i.e., the fact that a coastline typically has a
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is a parameter that Richardson found depended on the coastline approximated by
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with increasing numbers of sides (and decrease in the length of one side). In
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An animation showing the increasing length of the coastline of the island of
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because they are formed by various natural events that create patterns in
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1549:"Estimation of Typical High Intertidal Beach-Face Slope in Puget Sound"
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246: in this section. Unsourced material may be challenged and removed.
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exist when specific assumptions are made about minimum feature size.
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653:, later the fractal dimension. Rearranging the expression yields
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645:. He gave no theoretical explanation, but Mandelbrot identified
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86:. Although the "paradox of length" was previously noted by
1386:"The Fractal Scaling Relationship for River Inlets to Lakes"
1165:"The Fractal Scaling Relationship for River Inlets to Lakes"
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does not have a well-defined length. This results from the
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1142:. Vol. 1. Cambridge University Press. pp. 45–46.
908:
List of countries and territories by number of land borders
1384:
Seekell, D.; Cael, B.; Lindmark, E.; Byström, P. (2021).
1163:
Seekell, D.; Cael, B.; Lindmark, E.; Byström, P. (2021).
971:"Lewis Fry Richardson: scientist, visionary and pacifist"
55:
An example of the coastline paradox. If the coastline of
615:{\displaystyle L(\varepsilon )\sim F\varepsilon ^{1-D},}
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More than a decade after Richardson completed his work,
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with decreasing measuring units (coarse-graining length)
282:. In Euclidean geometry, a straight line represents the
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approaches zero, the length of the coastline approaches
629:, coastline length, a function of the measurement unit
90:, the first systematic study of this phenomenon was by
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are not self-repeating and are fundamentally finite.
691:{\displaystyle F\varepsilon ^{-D}\cdot \varepsilon ,}
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curve can be meaningfully assigned a precise length:
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Heinz-Otto Peitgen, Hartmut JĂĽrgens, Dietmar Saupe,
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1983:
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1483:Stoa, Ryan (15 Jun 2020). "The Coastline Paradox".
1333:Benoît Mandelbrot, Novel Mathematician, Dies at 85
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502:", published on 5 May 1967, Mandelbrot discusses
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837:using methods that depend on shoreline length.
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113:amount—that is, to measure it within a certain
510:between 1 and 2. These curves are examples of
331:Not all curves can be measured in this way. A
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1651:The Atlas of Canada – Coastline and Shoreline
70:is the counterintuitive observation that the
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1228:Chaos and Fractals: New Frontiers of Science
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1446:. Journal of Coastal Resources (1): 45–54.
742:is approximately 1.02 for the coastline of
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262:Learn how and when to remove this message
1605:"II.5 How long is the coast of Britain?"
872:List of countries by length of coastline
437:
2250:List of fractals by Hausdorff dimension
1523:"Mapping Monday: The Coastline Paradox"
1521:Sirdeshmukh, Neeraj (28 January 2013).
924:
152:equal straight-line segments of length
1217:Post & Eisen, p. 550 (see below).
849:– Alaskan and Canadian claims to the
633:, is approximated by the expression.
7:
284:shortest distance between two points
244:adding citations to reliable sources
140:reported their measured border with
547:A key property of some fractals is
1656:NOAA GeoZone Blog on Digital Coast
1547:Cereghino, P; et al. (2023).
821:which is not flat relative to any
25:
2232:How Long Is the Coast of Britain?
1113:. W. H. Freeman and Co. pp.
1022:. Vol. 6. pp. 139–187.
794:Criticisms and misunderstandings
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45:
36:
903:List of river systems by length
649:with a non-integer form of the
482:. The corresponding Latin verb
231:needs additional citations for
2256:The Fractal Geometry of Nature
1661:What Is The Coastline Paradox?
1609:The Fractal Geometry of Nature
1452:10.2112/JCOASTRES-D-22-00034.1
1109:The Fractal Geometry of Nature
812:Geographic Information Systems
584:
578:
94:, and it was expanded upon by
1:
1485:Rutgers University Law Review
494:should also mean "irregular".
2330:Problems in spatial analysis
1390:Geophysical Research Letters
1303:10.1126/science.156.3775.636
1169:Geophysical Research Letters
1059:10.1126/science.156.3775.636
707:must be the number of units
2272:Chaos: Making a New Science
1351:Chaos: Making a New Science
103:cartographic generalization
2351:
808:Global Positioning Systems
454:developed a new branch of
988:10.1007/s40329-014-0063-z
969:Vulpiani, Angelo (2014).
829:is semi-arbitrary and in
1594:Journal of Legal Studies
1020:General Systems Yearbook
934:"Length, shape and area"
932:Steinhaus, Hugo (1954).
290:length (also called the
190:geometric measure theory
1033:Mandelbrot, B. (1967).
938:Colloquium Mathematicum
898:List of longest beaches
847:Alaska boundary dispute
734:is longer for the same
2264:The Beauty of Fractals
1611:. Macmillan. pp.
1596:XXIX(1), January 2000.
776:territorial boundaries
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404:enclosed by the curve
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296:length of basic curves
107:Various approximations
1601:Mandelbrot, Benoit B.
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434:Measuring a coastline
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304:
274:The basic concept of
132:Shortly before 1951,
115:degree of uncertainty
2210:Lewis Fry Richardson
2205:Hamid Naderi Yeganeh
1995:Burning Ship fractal
1927:Weierstrass function
1497:10.2139/ssrn.3445756
1410:10.1029/2021GL093366
1396:(9): e2021GL093366.
1189:10.1029/2021GL093366
1175:(9): e2021GL093366.
660:
572:
428:statistically random
240:improve this article
212:Mathematical aspects
134:Lewis Fry Richardson
92:Lewis Fry Richardson
1968:Space-filling curve
1945:Multifractal system
1828:Space-filling curve
1813:Sierpinski triangle
1527:National Geographic
1402:2021GeoRL..4893366S
1295:1967Sci...156..636M
1230:; Spring, 2004; p.
1181:2021GeoRL..4893366S
1051:1967Sci...156..636M
951:10.4064/cm-3-1-1-13
882:Paradox of the heap
788:geometric modelling
711:required to obtain
651:Hausdorff dimension
637:is a constant, and
508:Hausdorff dimension
398:space-filling curve
2195:Aleksandr Lyapunov
2175:Desmond Paul Henry
2139:Self-avoiding walk
2134:Percolation theory
1778:Iterated function
1719:Fractal dimensions
1557:10.25923/6ssh-tn86
1551:. NOAA Fisheries.
1337:The New York Times
1103:Mandelbrot, Benoit
975:Lettera Matematica
835:ecosystem services
784:erosion monitoring
688:
612:
551:; that is, at any
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329:
306:
280:Euclidean distance
204:is independent of
2310:Coastal geography
2292:
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2238:Coastline paradox
2215:Wacław Sierpiński
2200:Benoit Mandelbrot
2124:Fractal landscape
2032:Misiurewicz point
1937:Strange attractor
1818:Apollonian gasket
1808:Sierpinski carpet
1622:978-0-7167-1186-5
1374:, pp. 29–31.
1289:(3775): 636–638.
1124:978-0-7167-1186-5
1045:(3775): 636–638.
887:Staircase paradox
877:Scale (geography)
861:Fractal dimension
851:Alaskan Panhandle
506:curves that have
452:Benoit Mandelbrot
272:
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264:
198:Benoit Mandelbrot
194:rectifiable curve
96:Benoit Mandelbrot
84:fractal dimension
68:coastline paradox
16:(Redirected from
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2155:Michael Barnsley
2022:Lyapunov fractal
1880:Sierpiński curve
1833:Blancmange curve
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1644:Fractal Geometry
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921:
916:
843:
827:Intertidal zone
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780:property rights
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712:
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666:
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569:
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18:Length of coast
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2044:Newton fractal
2041:
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2027:Mandelbrot set
2024:
2019:
2018:
2017:
2012:
2010:Newton fractal
2007:
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1823:Fibonacci word
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1798:Koch snowflake
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1632:External links
1630:
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1582:Post, David G.
1578:
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1569:
1539:
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1423:
1376:
1364:
1360:978-0747404132
1341:
1324:
1272:Mandelbrot, B.
1263:
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1236:
1219:
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981:(3): 121–128.
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857:-dense region.
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823:Vertical datum
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529:Koch snowflake
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424:Mandelbrot set
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310:straight lines
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88:Hugo Steinhaus
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2039:Multibrot set
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2015:Douady rabbit
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1856:
1855:Hilbert curve
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1838:De Rham curve
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1803:Menger sponge
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1788:Barnsley fern
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1586:Michael Eisen
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1249:, p. 28.
1248:
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1149:0-521-38297-1
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1081:on 2021-10-19
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831:constant flux
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252:February 2015
245:
241:
235:
234:
229:This section
227:
223:
218:
217:
211:
209:
199:
195:
191:
187:
186:circumference
183:
179:
176:of a regular
175:
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135:
127:
125:
123:
118:
116:
110:
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104:
99:
97:
93:
89:
85:
81:
80:fractal curve
77:
73:
69:
58:
57:Great Britain
48:
39:
30:
19:
2284:Chaos theory
2279:Kaleidoscope
2270:
2262:
2254:
2237:
2180:Gaston Julia
2160:Georg Cantor
1985:Escape-time
1917:Gosper curve
1865:LĂ©vy C curve
1850:Dragon curve
1729:Box-counting
1642:
1608:
1593:
1560:. Retrieved
1542:
1530:. Retrieved
1526:
1488:
1484:
1478:
1461:10072/421013
1443:
1393:
1389:
1379:
1367:
1349:
1344:
1336:
1327:
1286:
1282:
1266:
1261:, p. 1.
1254:
1227:
1222:
1213:
1172:
1168:
1158:
1139:
1133:
1108:
1083:. Retrieved
1079:the original
1042:
1038:
1028:
1019:
1013:
978:
974:
964:
957:rectifiable.
955:
941:
937:
927:
816:
801:
797:
757:
744:South Africa
718:
703:
700:
624:
563:
546:
540:
518:
511:
504:self-similar
499:
497:
467:
465:
449:
421:
413:
405:
401:
330:
308:Using a few
307:
292:great circle
273:
258:
249:
238:Please help
233:verification
230:
200:showed that
163:
147:
131:
119:
111:
100:
67:
65:
29:
2305:Cartography
2275:(1987 book)
2267:(1986 book)
2259:(1982 book)
2245:Fractal art
2165:Bill Gosper
2129:LĂ©vy flight
1875:Peano curve
1870:Moore curve
1756:Topological
1741:Correlation
1532:25 November
944:(1): 1–13.
819:Tidal datum
537:Peano curve
456:mathematics
396:(a type of
2335:Topography
2299:Categories
2083:Orbit trap
2078:Buddhabrot
2071:techniques
2059:Mandelbulb
1860:Koch curve
1793:Cantor set
1669:Veritasium
1639:Coastlines
1085:2021-05-21
914:References
558:stochastic
476:adjective
138:Portuguese
2325:Paradoxes
2190:Paul LĂ©vy
2069:Rendering
2054:Mandelbox
2000:Julia set
1912:Hexaflake
1843:Minkowski
1763:Recursion
1746:Hausdorff
1667:video by
1562:29 August
1505:214198004
1470:255441171
1418:235508504
1205:235508504
1197:1944-8007
1005:128975381
919:Citations
772:coastline
754:Solutions
750:is 1.28.
683:ε
680:⋅
672:−
668:ε
602:−
595:ε
588:∼
582:ε
521:yardstick
472:from the
466:I coined
417:nanometer
174:perimeter
128:Discovery
72:coastline
2320:Fractals
2100:fractals
1987:fractals
1955:L-system
1897:T-square
1705:Fractals
1603:(1982).
1354:, p.94.
1319:15662830
1311:17837158
1274:(1967).
1105:(1983).
1075:15662830
1067:17837158
841:See also
513:fractals
485:frangere
315:calculus
288:geodesic
170:infinity
158:dividers
76:landmass
2049:Tricorn
1902:n-flake
1751:Packing
1734:Higuchi
1724:Assouad
1665:YouTube
1575:Sources
1398:Bibcode
1291:Bibcode
1283:Science
1177:Bibcode
1047:Bibcode
1039:Science
997:3344519
491:fractus
479:fractus
469:fractal
333:fractal
178:polygon
60:longer.
2315:Coasts
2148:People
2098:Random
2005:Filled
1973:H tree
1892:String
1780:system
1619:
1584:, and
1503:
1468:
1416:
1358:
1317:
1309:
1203:
1195:
1146:
1121:
1073:
1065:
1003:
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768:border
701:where
625:where
319:smooth
276:length
182:circle
2224:Other
1641:" at
1613:25–33
1501:S2CID
1491:(2).
1466:S2CID
1414:S2CID
1315:S2CID
1279:(PDF)
1201:S2CID
1115:25–33
1071:S2CID
1001:S2CID
855:fjord
804:LiDAR
764:beach
760:river
553:scale
525:ruler
474:Latin
392:This
156:with
142:Spain
74:of a
1617:ISBN
1564:2024
1534:2023
1356:ISBN
1307:PMID
1193:ISSN
1144:ISBN
1119:ISBN
1063:PMID
810:and
565:is:
498:In "
406:does
402:area
337:tend
66:The
1592:".
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1553:doi
1493:doi
1456:hdl
1448:doi
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541:has
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242:by
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206:ε
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