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] }

- Mathematical aspects
- Discovery
- Measuring a coastline
- See also
- References
- Citations
- Sources
- External links

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 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 a maximum 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.

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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 infinitesimally small distances. The following animation illustrates how a smooth curve can be meaningfully assigned a precise length:

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.

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.^{ [6] } 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).

Coastlines are 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.^{ [7] }

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, but the Spanish reported it as 1214 km. This was the beginning of the coastline problem, which is a mathematical uncertainty inherent in the measurement of boundaries that are irregular.^{ [8] }

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 dividers 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 is monotonically increasing when the common length of the segments is decreased. In effect, the shorter the ruler, the longer the measured border; the Spanish and Portuguese geographers were simply using different-length rulers.

The result most astounding to Richardson is that, under certain circumstances, as *ℓ* approaches zero, the length of the coastline approaches infinity. 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 perimeter of a regular polygon inscribed in a circle approaches the circumference with increasing numbers of sides (and decrease in the length of one side). In geometric measure theory such a smooth curve as the circle that can be approximated by small straight segments with a definite limit is termed a rectifiable curve.

More than a decade after Richardson completed his work, Benoit Mandelbrot developed a new branch of mathematics, fractal geometry, to describe just such non-rectifiable complexes in nature as the infinite coastline.^{ [9] } His own definition of the new figure serving as the basis for his study is:^{ [10] }

I coined

fractalfrom the Latin adjectivefractus. The corresponding Latin verbfrangeremeans "to break:" to create irregular fragments. It is therefore sensible ... that, in addition to "fragmented" ...fractusshould also mean "irregular".

A key property of some fractals is self-similarity; that is, at any scale 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 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 stochastic arrangement of bays and promontories formed from the small objects at hand. In such an environment (as opposed to smooth curves) Mandelbrot asserts^{ [9] } "coastline length turns out to be an elusive notion that slips between the fingers of those who want to grasp it".

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:^{ [11] }

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 2*ε*. *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.^{ [12] } Mandelbrot showed that *D* is independent of *ε*.

- Alaska boundary dispute – Alaskan and Canadian claims to the Alaskan Panhandle 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 fjord-dense region.
- Fractal dimension
- Gabriel's Horn, a geometric figure with infinite surface area but finite volume
- "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension", a paper by Benoît Mandelbrot
- Paradox of the heap
- Zeno's paradoxes

**Benoit B.****Mandelbrot** was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life". He referred to himself as a "fractalist" and is recognized for his contribution to the field of fractal geometry, which included coining the word "fractal", as well as developing a theory of "roughness and self-similarity" in nature.

In mathematics, a **fractal** is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory.

In mathematics, **Hausdorff dimension** is a measure of *roughness*, or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the *Hausdorff–Besicovitch dimension.*

The **Mandelbrot set** is the set of complex numbers for which the function does not diverge to infinity when iterated from , i.e., for which the sequence , , etc., remains bounded in absolute value.

In mathematics, a **self-similar** object is exactly or approximately similar to a part of itself. Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed.

In mathematics, a **fractal dimension** is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the scale at which it is measured. It has also been mythologized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently and in a fractal dimension, i.e. one that does not have to be an integer.

**Lewis Fry Richardson**, FRS was an English mathematician, physicist, meteorologist, psychologist, and pacifist who pioneered modern mathematical techniques of weather forecasting, and the application of similar techniques to studying the causes of wars and how to prevent them. He is also noted for his pioneering work concerning fractals and a method for solving a system of linear equations known as modified Richardson iteration.

In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second along a path whose arclength is equal to that distance. The distance between two points of a metric space relative to the **intrinsic metric** is defined as the infimum of the lengths of all paths from the first point to the second. A metric space is a **length metric space** if the intrinsic metric agrees with the original metric of the space.

In mathematics, **Hausdorff measure** is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in or, more generally, in any metric space.

This is a **list of fractal topics**, by Wikipedia page, See also list of dynamical systems and differential equations topics.

In fractal geometry, the **Minkowski–Bouligand dimension**, also known as **Minkowski dimension** or **box-counting dimension**, is a way of determining the fractal dimension of a set in a Euclidean space , or more generally in a metric space . It is named after the Polish mathematician Hermann Minkowski and the French mathematician Georges Bouligand.

"**How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension**" is a paper by mathematician Benoit Mandelbrot, first published in *Science* on 5 May 1967. In this paper, 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.

**Arc length** is the distance between two points along a section of a curve.

In mathematics, **Lebesgue's density theorem** states that for any Lebesgue measurable set , the "density" of *A* is 0 or 1 at almost every point in . Additionally, the "density" of *A* is 1 at almost every point in *A*. Intuitively, this means that the "edge" of *A*, the set of points in *A* whose "neighborhood" is partially in *A* and partially outside of *A*, is negligible.

The **coastline of the United Kingdom** is formed by a variety of natural features including islands, bays, headlands and peninsulas. It consists of the coastline of the island of Great Britain and the north-east coast of the island of Ireland, as well as many much smaller islands. Much of the coastline is accessible and quite varied in geography and habitats. Large stretches have been designated areas of natural beauty, notably the Jurassic Coast and various stretches referred to as heritage coast. They are both very long, spreading through the mainland.

A **fractal curve** is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rectifiable curves — that is, they do not have finite length — and every subarc longer than a single point has infinite length.

In mathematics, the **curvature of a measure** defined on the Euclidean plane **R**^{2} is a quantification of how much the measure's "distribution of mass" is "curved". It is related to notions of curvature in geometry. In the form presented below, the concept was introduced in 1995 by the mathematician Mark S. Melnikov; accordingly, it may be referred to as the **Melnikov curvature** or **Menger-Melnikov curvature**. Melnikov and Verdera (1995) established a powerful connection between the curvature of measures and the Cauchy kernel.

**Lacunarity**, from the Latin lacuna, meaning "gap" or "lake", is a specialized term in geometry referring to a measure of how patterns, especially fractals, fill space, where patterns having more or larger gaps generally have higher lacunarity. Beyond being an intuitive measure of gappiness, lacunarity can quantify additional features of patterns such as "rotational invariance" and more generally, heterogeneity. This is illustrated in Figure 1 showing three fractal patterns. When rotated 90°, the first two fairly homogeneous patterns do not appear to change, but the third more heterogeneous figure does change and has correspondingly higher lacunarity. The earliest reference to the term in geometry is usually attributed to Benoit Mandelbrot, who, in 1983 or perhaps as early as 1977, introduced it as, in essence, an adjunct to fractal analysis. Lacunarity analysis is now used to characterize patterns in a wide variety of fields and has application in multifractal analysis in particular.

The **Minkowski sausage** or **Minkowski curve** is a fractal first proposed by and named for Hermann Minkowski as well as its casual resemblance to a sausage or sausage links. The initiator is a line segment and the generator is a broken line of eight parts one fourth the length.

- ↑ 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.

- ↑ 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. - ↑ Richardson, L. F. (1961). "The problem of contiguity: An appendix to statistics of deadly quarrels".
*General Systems Yearbook*. Vol. 6. pp. 139–187. - ↑ 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. - 1 2 Mandelbrot, Benoit (1983).
*The Fractal Geometry of Nature*. W. H. Freeman and Co. pp. 25–33. ISBN 978-0-7167-1186-5. - ↑ Post & Eisen, p. 550 (see below).
- ↑ Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe,
*Chaos and Fractals: New Frontiers of Science*; Spring, 2004; p. 424. - ↑ Richardson, Lewis Fry (1993). "Fractals". In Ashford, Oliver M.; Charnock, H.; Drazin, P. G.; et al. (eds.).
*The Collected Papers of Lewis Fry Richardson: Meteorology and numerical analysis*. Vol. 1. Cambridge University Press. pp. 45–46. ISBN 0-521-38297-1. - 1 2 Mandelbrot 1982, p. 28.
- ↑ Mandelbrot 1982, p. 1.
- ↑ Mandelbrot 1982, pp. 29–31.
- ↑ Seekell, D.; Cael, B.; Lindmark, E.; Byström, P. (2021). "The Fractal Scaling Relationship for River Inlets to Lakes".
*Geophysical Research Letters*.**48**(9): e2021GL093366. Bibcode:2021GeoRL..4893366S. doi:10.1029/2021GL093366. ISSN 1944-8007. S2CID 235508504.

- Post, David G., and Michael Eisen. "How Long is the Coastline of Law? Thoughts on the Fractal Nature of Legal Systems".
*Journal of Legal Studies*XXIX(1), January 2000. - Mandelbrot, Benoit B. (1982). "II.5 How long is the coast of Britain?".
*The Fractal Geometry of Nature*. Macmillan. pp. 25–33. ISBN 978-0-7167-1186-5.

- "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? – YouTube video by Veritasium
- The Coastline Paradox Explained – YouTube video by RealLifeLore

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