Coastline

Under coastline means the length of a coastline . Because of the irregular shape of every natural coast, the length determined depends heavily on the accuracy of the map base used and the accuracy of the measurement. Finer measurements lead to a longer coastline. The mathematician Benoît Mandelbrot compared the determination of the length of a coast with that of self-similar curves . The same applies to the determination of the bank length of inland waters.

Coast length as country information

Coast length, along with other data such as area and longitude and latitude, is sometimes used to describe a country or region geographically. Both the absolute length of the coast and the relationship to other variables such as the length of the land borders of the respective country can be of interest.

Alexander von Humboldt determined the relationship between the length of the coast and the area of ​​the continents as a measure of the horizontal division of the land masses. In a greater contact with the sea, he saw a better way of opening up a country for world trade. This ratio is particularly favorable for Europe because of the long coastline and particularly unfavorable for Africa.

Coastline of Germany

There are different specifications for the length of Germany's coastline, but it is seldom stated which exact coastline they refer to and how they were determined. The northern German coastal states estimate the length of the mainland coast to be around 1200 km. However, this information does not include the coast lengths of the islands.

In the World Factbook of the CIA , the coastline is given as 2389 km, with no information on how this value was determined.

The individual German federal states sometimes state several or no coast lengths in their statistical reports. In Schleswig-Holstein a distinction is made between the length of the coast on the Baltic Sea (328 km, including Fehmarn : 402 km) and on the North Sea (202 km, including islands and Halligen : 468 km). The Schlei , a water arm reaching deep inland, is not taken into account. In Mecklenburg-Vorpommern the length of the outer coast (377 km) and the length is Bodden - and Lagoon coast (1568 km) specified.

Selected coastline

The total length of the world's coastlines is given in the World Factbook as 356,000 km. This includes the coastlines of all continents and islands.

Some states have extremely short coastlines in relation to the area of ​​their national territory. Some states with particularly short coastlines are listed in the following table:

Country	National territory	Coastline	Coast length per km² of state area
Congo	2,345,410 km²	40 km	0.017 m
Jordan	89,342 km²	27 km	0.30 m
Bosnia Herzegovina	51,129 km²	24 km	0.47 m
Togo	56,785 km²	56 km	0.99 m
Belgium	30,528 km²	72.3 km	2.3 m
Slovenia	20,273 km²	46.6 km	2.3 m

In comparison, there are around 6.3 meters of coastline in France per km², around 65 meters in Norway and 2,081 meters in the miniature state of Monaco and 2,161 meters in the island state of the Maldives. The ratio of the length of the coast to the area of ​​the state is, however, only partially suitable for describing the maritime nature of a state, since in larger states the area has a stronger effect for purely mathematical reasons. Other factors also play a role, such as the nature of the coast for natural harbors.

See also : List of Countries by Coast Length

Measurement of coastal lengths

Measuring the length of irregular lines such as coastlines is based on the principle that they are aligned by a measurable approximation curve. A possible approximation for determining length is to use a pair of dividers to determine points at a certain distance on the coastline. An approximation for the length of the coast can be given from the number of coastal sections found in this way and a remnant . If is small enough, this coastline is independent of which end point of the coastline the measurement is started from. ${\displaystyle G}$ ${\displaystyle L(G)}$ ${\displaystyle G}$

• Increasingly finer measurements of the length of the coast of Great Britain with straight measuring sections of different lengths
• 

  Straight measuring sections of 200 km in length.
  Total length about 2350 km.

• 

  Straight measuring sections of 100 km in length.
  Total length about 2775 km.

• 

  Straight measuring sections of 50 km in length.
  Total length about 3425 km.

Since not every detail of the coast can be displayed in the maps used, depending on the scale, and the coastline is approximated by an approximate curve during the measurement, the result depends on the map scale and the point spacing . Unlike in the case of smooth, mathematical curves, the estimated coastline length does not converge towards a limit value due to the very irregular coast shape with decreasing size , but becomes arbitrarily large within the limits of the comparison with finer measurements . ${\displaystyle G}$ ${\displaystyle L(G)}$ ${\displaystyle G}$

Lewis Fry Richardson discovered this characteristic when he wanted to investigate how the length of the border between two states relates to the probability that these states will be at war with one another. It struck him that the information on the limit length in various sources differed considerably from one another. In empirical studies he found between the dot pitch and the coastline thus determined the relationship with the positive factor and the constant whose value is at least 1 and which is characteristic of a border or coastline. For a straight line is such that the measured length is independent of . The more irregular the coast, the bigger it is . For the very irregular west coast of England, Richardson found the value ; H. halving will be about the factor larger. ${\displaystyle G}$ ${\displaystyle L(G)}$ ${\displaystyle L(G)=F\,G^{1-D}}$ ${\displaystyle F}$ ${\displaystyle D}$ ${\displaystyle D=1}$ ${\displaystyle G}$ ${\displaystyle D}$ ${\displaystyle D=1{,}25}$ ${\displaystyle G}$ ${\displaystyle L(G)}$ ${\displaystyle 1{,}19}$

Comparison with fractals

Benoît Mandelbrot dealt with self-similarity and fractal curves in the 1960s . Such curves are also assigned a non-integer dimension such as the Hausdorff dimension . In a paper by Lewis Fry Richardson on measuring coastlines, Mandelbrot discovered similarities to self-similar curves. Mandelbrot found another mention of this fact in Jean-Baptiste Perrin .

Therefore, he published in 1967 in the journal Science the article How Long Is the Coast of Britain? (German: How long is the coast of Britain? ), in which he compared coastlines with self-similar fractal curves. He showed that the constant found empirically by Richardson in the determination of coastline lengths is comparable to the dimension of self-similar curves, and thus described one possible application of fractals. Since the strict self-similarity of constructed curves such as Koch's snowflake does not apply to coastlines , Mandelbrot called this geographical curve a statistically self-similar or random self-similar figure. ${\displaystyle D}$

In the article published in 1967 Mandelbrot does not yet use the term fractal , he only speaks of fractional dimensions .

Hugo Steinhaus had already established a connection between the accuracy used when measuring lengths of very irregular curves and the determined length for the length of the western bank of the Vistula in 1954 . However, these considerations have received less attention.

Limits of Comparison

Mandelbrot only used the problem of determining the length of the coast as a starting point to show a possible application for fractals. However, many non-scientists saw the article as proof that the length of the coast can be arbitrarily large if it is determined precisely enough.

The empirical formula found by Richardson applies to the area he is investigating. At this scale range, coastlines behave like fractals. However, the formula cannot be extrapolated to any small point distances and fine measurements without further checking . Applying the formula to an arbitrarily high level of accuracy does not make sense in the real world because the definition of the coastline cannot be determined with arbitrary precision due to the changing water level.

In nature, the self-similarity of structures only applies to a limited number of levels and not down to infinitely small structures: In addition to theoretical considerations, this also applies from a purely technical point of view: At the latest in the area of ​​individual smaller rocks and stones, the term "coast" or . "Shores" are spoken at their waterline, and the geographic interest in such objects (geographically relevant scales) is lost overall. Therefore, from Richardson's empirical formula, it cannot be concluded that coastlines are infinitely long with respect to a defined normal water level.

In addition, there is the problem of erosion and redistribution of sand, which together make up a significant proportion of the length of the bank, so that accuracy is increasingly time-dependent. If the measurement accuracy is too high, the value would become obsolete faster than it could be recorded. This also limits the technical meaningfulness of the fractal model.

literature

• Benoît Mandelbrot: How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. In: Science 156, May 5, 1967, pp. 636-638 ( PDF; 32 kB; English ).
• Benoît Mandelbrot: Fractals: Form, chance, and dimension. WH Freeman and Company, San Francisco 1977, ISBN 978-0-7167-0473-7 .

Individual evidence

1. ↑ Benoît B. Mandelbrot: The fractal geometry of nature. Birkhäuser Verlag, Basel 1987, pp. 37-46.
2. ^ Earth (distribution of land and water, horizontal and vertical structure) . In: Meyers Konversations-Lexikon . 4th edition. Volume 5, Verlag des Bibliographisches Institut, Leipzig / Vienna 1885–1892, p. 747.
3. ↑ Landscape and Climate ( Memento from March 5, 2007 in the Internet Archive ) with information from the previous State Statistical Office Schleswig-Holstein
4. ↑ ^{a b} Overview of the coastline in the World Factbook ( Engl. )
5. ↑ Statistical Yearbook Schleswig-Holstein 2008/2009 (PDF; 2.2 MB) (17th chapter: Territory and geographical information )
6. ↑ Statistical Office Mecklenburg-Western Pomerania (see data> State data overview)
7. ^ Lewis Fry Richardson: The problem of contiguity: an appendix of statistics of deadly quarrels. General Systems Yearbook 6, 1961, pp. 139-187.
8. ↑ ^{a b c} Benoît Mandelbrot: Comments on How Long Is the Coast of Britain? ( Memento of the original from June 22, 2010 in the Internet Archive ) Info: The  archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF, English; 32 kB)
9. ↑ Benoît B. Mandelbrot: The fractal geometry of nature. Birkhäuser Verlag, Basel 1987, p. 39.
10. ^ Hugo Steinhaus: Length, shape and area. Colloquium Mathematicum 3, pp. 1-13.
11. ↑ Armin Bunde, Markus Porto, H. Eduardo Roman: Physics on fractal structures. In: Fractals in Class. Kiel 1998, ISBN 3-89088-130-0 , pp. 255–273 ( PDF, 5 MB  ( page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and remove it then this note. ).