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.

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:

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.

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 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.^{[4]}

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.

- Coastline problem
- 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
- 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.

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

- 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.

- "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

The American Falls is the second-largest of the three waterfalls that together are known as Niagara Falls on the Niagara River along the Canada–U.S. border. Unlike the much larger Horseshoe Falls, of which two-thirds is in Ontario, Canada and one-third in the U.S. state of New York, the American Falls is entirely within the United States.

Chaos gameIn mathematics, the term chaos game originally referred to a method of creating a fractal, using a polygon and an initial point selected at random inside it. The fractal is created by iteratively creating a sequence of points, starting with the initial random point, in which each point in the sequence is a given fraction of the distance between the previous point and one of the vertices of the polygon; the vertex is chosen at random in each iteration. Repeating this iterative process a large number of times, selecting the vertex at random on each iteration, and throwing out the first few points in the sequence, will often (but not always) produce a fractal shape. Using a regular triangle and the factor 1/2 will result in the Sierpinski triangle, while creating the proper arrangement with four points and a factor 1/2 will create a display of a "Sierpinski Tetrahedron", the three-dimensional analogue of the Sierpinski triangle. As the number of points is increased to a number N, the arrangement forms a corresponding (N-1)-dimensional Sierpinski Simplex.

The term has been generalized to refer to a method of generating the attractor, or the fixed point, of any iterated function system (IFS). Starting with any point x0, successive iterations are formed as xk+1 = fr(xk), where fr is a member of the given IFS randomly selected for each iteration. The iterations converge to the fixed point of the IFS. Whenever x0 belongs to the attractor of the IFS, all iterations xk stay inside the attractor and, with probability 1, form a dense set in the latter.

The "chaos game" method plots points in random order all over the attractor. This is in contrast to other methods of drawing fractals, which test each pixel on the screen to see whether it belongs to the fractal. The general shape of a fractal can be plotted quickly with the "chaos game" method, but it may be difficult to plot some areas of the fractal in detail.

The "chaos game" method is mentioned in Tom Stoppard's 1993 play Arcadia.With the aid of the "chaos game" a new fractal can be made and while making the new fractal some parameters can be obtained. These parameters are useful for applications of fractal theory such as classification and identification. The new fractal is self-similar to the original in some important features such as fractal dimension.

If in the "chaos game" you start at each vertex and go through all possible paths that the game can take, you will get the same image as with only taking one random path. However, taking more than one path is rarely done since the overhead for keeping track of every path makes it far slower to calculate. This method does have the advantages of illustrating how the fractal is formed more clearly than the standard method as well as being deterministic.

CoastThe coast, also known as the coastline or seashore, is the area where land meets the sea or ocean, or a line that forms the boundary between the land and the ocean or a lake. A precise line that can be called a coastline cannot be determined due to the Coastline paradox.

The term coastal zone is a region where interaction of the sea and land processes occurs. Both the terms coast and coastal are often used to describe a geographic location or region; for example, New Zealand's West Coast, or the East and West Coasts of the United States. Edinburgh for example is a city on the coast of Great Britain.

A pelagic coast refers to a coast which fronts the open ocean, as opposed to a more sheltered coast in a gulf or bay. A shore, on the other hand, can refer to parts of land adjoining any large body of water, including oceans (sea shore) and lakes (lake shore). Similarly, the somewhat related term "[stream bed/bank]" refers to the land alongside or sloping down to a river (riverbank) or body of water smaller than a lake. "Bank" is also used in some parts of the world to refer to an artificial ridge of earth intended to retain the water of a river or pond; in other places this may be called a levee.

While many scientific experts might agree on a common definition of the term "coast", the delineation of the extents of a coast differ according to jurisdiction, with many scientific and government authorities in various countries differing for economic and social policy reasons. According to the UN atlas, 44% of people live within 150 kilometres (93 miles) of the sea.

Coastline of BrazilThe coastline of Brazil measures 7,491 km, which makes it the 16th longest national coastline of the world. All the coast lies adjacent to geographical features can be found all through the coastal areas, like islands, reefs and bays. The beaches of Brazil (2095 in total) are famous in the world and receive a great number of tourists.A famous expression in Brazil is "from Oiapoque to Chuí", which means from the extreme south to the extreme north of the country. However, the actual northernmost point in Brazil is the Monte Caburaí in the far from the coast state of Roraima, whereas the southernmost point is in Santa Vitória do Palmar, the city from which Chuí emancipated.

Out of the 26 Brazilian states, 9 are landlocked, as well as the Distrito Federal. Most of the 17 coastal states have their capitals lying near the coast, exceptions being Porto Alegre (Rio Grande do Sul), Curitiba (Paraná), São Paulo (São Paulo), Teresina (Piauí), Belém (Pará) and Macapá (Amapá). Porto Alegre, Belém, Teresina and Macapá lie all near large navigable rivers, though.

Coastline of the United KingdomThe 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 a large number of 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.

Great LakesThe Great Lakes (French: les Grands-Lacs), also called the Laurentian Great Lakes and the Great Lakes of North America, are a series of interconnected freshwater lakes primarily in the upper mid-east region of North America, on the Canada–United States border, which connect to the Atlantic Ocean through the Saint Lawrence River. They consist of Lakes Superior, Michigan, Huron, Erie, and Ontario, although hydrologically, there are four lakes, Superior, Erie, Ontario, and Michigan-Huron. The connected lakes form the Great Lakes Waterway.

The Great Lakes are the largest group of freshwater lakes on Earth by total area, and second-largest by total volume, containing 21% of the world's surface fresh water by volume. The total surface is 94,250 square miles (244,106 km2), and the total volume (measured at the low water datum) is 5,439 cubic miles (22,671 km3), slightly less than the volume of Lake Baikal (5,666 cu mi or 23,615 km3, 22–23% of the world's surface fresh water). Due to their sea-like characteristics (rolling waves, sustained winds, strong currents, great depths, and distant horizons) the five Great Lakes have also long been referred to as inland seas. Lake Superior is the second largest lake in the world by area, and the largest freshwater lake by area. Lake Michigan is the largest lake that is entirely within one country.The Great Lakes began to form at the end of the last glacial period around 14,000 years ago, as retreating ice sheets exposed the basins they had carved into the land which then filled with meltwater. The lakes have been a major source for transportation, migration, trade, and fishing, serving as a habitat to a large number of aquatic species in a region with much biodiversity.

The surrounding region is called the Great Lakes region, which includes the Great Lakes Megalopolis.

How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension"How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" is a paper by mathematician Benoît Mandelbrot, first published in Science in 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.

Julia setIn the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values.

Thus the behavior of the function on the Fatou set is "regular", while on the Julia set its behavior is "chaotic".

The Julia set of a function f is commonly denoted J(f), and the Fatou set is denoted F(f). These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century.

LandmassLandmass is a contiguous region of land surrounded by ocean.It may be a continent or an island:

Continent#Definition

Island#Differentiation from continents

List of U.S. states and territories by coastlineThis is a list of U.S. states and territories ranked by their coastline length. Thirty states have a coastline: twenty-three with a coastline on the Arctic, Atlantic, and/or Pacific Ocean, and eight with a Great Lakes coastline; New York has coasts on both. States with no coastline are not included; smaller border lakes such as Lake Champlain or Lake of the Woods are not counted. Five U.S. territories have coastlines — three U.S. territories have a coastline on the Pacific Ocean, and two U.S. territories have a coastline on the Atlantic Ocean (Caribbean sea). The U.S. Minor Outlying Islands have coastlines, but their coastlines are not counted.

Two separate measurements are used: method 1 only includes states with ocean coastline and excludes tidal inlets; method 2 includes Great Lake coastline and the extra length from tidal inlets. For example, method 2 counts the Great Bay as part of New Hampshire's coastline, but method 1 does not. The resulting figures differ significantly due to the ambiguity inherent in all attempts at measuring coastlines, as expressed in the coastline paradox.

Method 1 does not include the coastlines of the territories of the United States, while method 2 does.

The data for method 1 were retrieved from a CRS Report for Congress using data from U.S. Department of Commerce, National Oceanic and Atmospheric Administration, The Coastline of the United States, 1975. This is based on measurements made using large-scale nautical charts. The figure for Connecticut was arrived at separately and may not reflect the correct comparative distance. These numbers exclude the Great Lakes coastlines.

The data for method 2 are from a list maintained by the Office of Ocean and Coastal Resource Management of the National Oceanic and Atmospheric Administration (NOAA). The state coastline lengths were computed by an unspecified method that includes tidal areas not included in the first method. These numbers also include the Great Lakes coastlines, which do not have similar tidal areas.

List of lakes of SwitzerlandThis page contains a sortable table listing all major lakes of Switzerland. The table includes all still water bodies, natural or artificial, that have a surface area of at least 0.30 square kilometres (0.12 sq mi), regardless of water volume, maximum depth or other merit. These lakes are ranked by area, the table including also the elevation above sea level and maximum depth. They are either natural (type N), natural but used as reservoirs (NR) or fully artificial (A). For a list of artificial lakes only, see List of dams and reservoirs in Switzerland. For a list of lakes above 800 metres that includes smaller water bodies, see List of mountain lakes of Switzerland.

Along with the mountains, lakes constitute a major natural feature of Switzerland, with over 1000 km of shores within the country. Lakes, large and small, can be found in almost all cantons and provide an important source of water as well as leisure opportunities. The two most extensive, Lake Geneva and Lake Constance, are amongst the largest in Europe and mark the border of the Swiss Plateau, along with the Alps and the Jura Mountains. The largest wholly Swiss lake is Lake Neuchâtel. Next in size comes Lake Maggiore, followed by Lake Lucerne and Lake Zurich. In total 103 lakes exist that are more than 30 hectares in surface area, and a considerable number of smaller lakes. All these lakes are found in the four major river basins of Switzerland: Rhine, Rhone, Po and Danube, at almost all elevations below the snow line.

List of rivers by lengthThis is a list of the longest rivers on Earth. It includes river systems over 1,000 kilometres (620 mi).

Menger spongeIn mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension.

North Sea Continental Shelf casesGermany v Denmark and the Netherlands [1969] ICJ 1 (also known as The North Sea Continental Shelf cases) were a series of disputes that came to the International Court of Justice in 1969. They involved agreements among Denmark, Germany, and the Netherlands regarding the "delimitation" of areas—rich in oil and gas—of the continental shelf in the North Sea.

Norwegian coastlineThe Norwegian coastline is the coastline of Norway along the Skagerrak, North Sea, Norwegian Sea, and Barents Sea. This considers only the mainland coastline and excludes Svalbard.

A straight line along Norway's sea borders (the coastal perimeter) is 2,650 kilometers (1,650 mi) long. Along the coast there are many fjords, islands, and bays, resulting in a low-resolution coastline of over 25,000 kilometers (16,000 mi). At 30-meter (98 ft) linear intercepts, this length increases to 83,281 kilometers (51,748 mi) (see the coastline paradox). Much of Norway's wealth is linked to its long coastline; for example, the petroleum industry, maritime transport, fishing, and fish farming.

The Norwegian landscape was formed by glaciers that eroded the basement rock and formed countless valleys and fjords, as well as the characteristic skerries that protect the land from the ocean along most of the mainland coastline. There are only a few shorter or longer stretches where the mainland is exposed to the open sea along the coast: at Lindesnes, Lista, Jæren, Stad, Hustadvika, and Folda in Trøndelag, and along the Varanger Peninsula.

PerimeterA perimeter is a path that surrounds a two-dimensional shape. The term may be used either for the path itself or its length—it can be thought of as the length of the outline of a shape. The perimeter of a circle or ellipse is called its circumference.

Calculating the perimeter has several practical applications. A calculated perimeter is the length of fence required to surround a yard or garden. The perimeter of a wheel (its circumference) describes how far it will roll in one revolution. Similarly, the amount of string wound around a spool is related to the spool's perimeter.

Shape factor (image analysis and microscopy)Shape factors are dimensionless quantities used in image analysis and microscopy that numerically describe the shape of a particle, independent of its size. Shape factors are calculated from measured dimensions, such as diameter, chord lengths, area, perimeter, centroid, moments, etc. The dimensions of the particles are usually measured from two-dimensional cross-sections or projections, as in a microscope field, but shape factors also apply to three-dimensional objects. The particles could be the grains in a metallurgical or ceramic microstructure, or the microorganisms in a culture, for example. The dimensionless quantities often represent the degree of deviation from an ideal shape, such as a circle, sphere or equilateral polyhedron. Shape factors are often normalized, that is, the value ranges from zero to one. A shape factor equal to one usually represents an ideal case or maximum symmetry, such as a circle, sphere, square or cube.

SinuositySinuosity, sinuosity index, or sinuosity coefficient of a continuously differentiable curve having at least one inflection point is the ratio of the curvilinear length (along the curve) and the Euclidean distance (straight line) between the end points of the curve. This dimensionless quantity can also be rephrased as the "actual path length" divided by the "shortest path length" of a curve.

The value ranges from 1 (case of straight line) to infinity (case of a closed loop, where the shortest path length is zero) or for an infinitely-long actual path.

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