Coastline paradox
Under coast length means the length of a coastline. Because of the very irregular shape of many coasts that length strongly depends on the accuracy of the source map used and the accuracy of the measurement. It is shown that finer measurements lead to a greater length of coastline. The mathematician Benoît Mandelbrot has compared the length of a coastline with the provision of self-similar curves. The same applies to the determination of the shoreline inland waters.
Coastline as a national information
The coastline is sometimes given together with other data such as the face and the latitude and longitude of the geographical description of a country or a region. In this case, both the absolute length of the coast are of interest as well as the relationship with other variables such as the length of the land borders of the country.
Alexander von Humboldt certain the ratio of coastline to the surface of the continents as a measure of the horizontal layout of the land masses. In a larger contact with the sea he saw a better way of development of a country in world trade. This ratio is for Europe because of the long coastline particularly favorable and unfavorable particularly for Africa.
Coastline of Germany
There are different specifications for the coastline of Germany, but for which there is rarely indicated how accurate coastline they relate and how they were determined. The North German coastal states estimate the length of the mainland coast at about 1200 km. This figure, however, lack the coastal lengths of the islands.
The World Factbook of the CIA the coast length is given as 2389 km, without information about how this value was determined.
The individual German states indicate in their statistical reports in some cases several or no coastal lengths. In Schleswig -Holstein between the coastline on the Baltic Sea (328 km, including Fehmarn: 402 km ) and on the North Sea (202 km, including islands and islets 468 miles) distinguished. The Schlei, a deep inland reaching water arm is not included. In Mecklenburg- Western Pomerania, the length of the outer coast ( 377 km ) and the length of the bay and Haffküste (1568 miles) is specified.
Selected coastal lengths
The total length of coastlines worldwide is specified in the World Factbook with 356,000 km. This includes the coastlines of all continents and islands.
Some states have very short coastlines in proportion to the area of its territory. In the following table, some states are listed with particularly short coasts:
In comparison, in France come to an area approximately 6.3 km ² state meters, in Norway, about 65 meters and in the mini-state of Monaco even 2,081 meters, and the island state of the Maldives 2,161 meters of coastline. However, the ratio of the length of coastline as the state space is of limited use to describe the Maritimität of a state, as in the larger states, the area for purely mathematical reasons effect stronger. In addition, other factors play a role, such as the nature of the coast to natural harbors.
Measurement of coastal lengths
Measuring the length of irregular lines as coastal based on the principle that they are adjusted by a measurable approximation curve. A possible approximation for determining the length is to be determined with a pair of dividers points at a certain distance on the shoreline. From the number of coastal areas and thus found a scrap piece an approximation of the coastline can be specified. If is small enough, this coastline is regardless of which endpoint of the coastline, the measurement is started.
- Increasingly finer measurements of the coastline of Britain with different long straight sections measuring
Straight sections measuring 100 km in length. Total length is about 2775 km.
Straight sections measuring 50 km in length. Total length is about 3425 km.
Since the cards used depending on the scale, not every detail of the coast can be represented and the coastline is approximated by an approximate curve for the measurement, the result of the map scale and the dot pitch depends. The estimated length of coastline converges unlike smooth mathematical curves because of the very irregular coastline shape with decreasing not to a limit, but becomes arbitrarily large with finer measurements within the limits of the comparison.
This property has been found, as he wanted to investigate how the length of the border of two states with the probability that these states war with one another, is related Lewis Fry Richardson. He noticed that the information differed significantly between the boundary length in different sources from each other. In empirical studies, he found between the point and the distance so determined coastline the context of the positive factor and the constant whose value is at least 1 and is characteristic of a border or coast. In a straight line so that the length measured is independent of. Is more irregular is the coast, the greater. For the very irregular west coast of England Richardson found the value, that is, at a halving of increases by approximately a factor.
Comparison with fractals
Benoît Mandelbrot in the 1960s dealt with self-similarity and fractal curves. Such a curve is assigned to a non-integer dimension, such as the Hausdorff dimension. In an essay Lewis Fry Richardson on measurement of coastal lengths Mandelbrot discovered similarities to self-similar curves. Another mention of this fact was Mandelbrot 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 empirically found by Richardson in the determination of coast lengths comparable to the dimensions of self-similar curve, and thus described a possible use of fractals. As for coastlines is not the strict self-similarity of constructed curves such as the boiling between Snowflake, Mandelbrot called this geographical curve a statistically self-similar or random self-similar figure.
In the article published in 1967 Mandelbrot uses the term fractal yet, he speaks only of fractional dimensions ( fractional dimensions).
A relationship between the applied accuracy in the measurement of lengths very irregular curves and the determined length Hugo Steinhaus had been made in 1954 for the length of the west bank of the Vistula. These considerations, however, were less sensitive.
Limits of the comparison
Mandelbrot used the problem of determining lengths of coast only as a starting point to show a possible application of fractals. However, many non-scientists saw in the article a proof that the coastal length is arbitrarily large if it is determined accurately enough.
The empirical formula found by Richardson applies for the investigated range of him. In this scale range, coastlines behave like fractals. However, the formula can not be extrapolated to arbitrarily small spacings and fine measurements without further verification. An application of the formula to arbitrarily high accuracy has been, therefore, in the real world does not make sense, since the definition of the coastline due to the varying water level is not arbitrary estimated.
In nature, the self-similarity of structures for only a limited number of stages and not up into infinitely small structures is considered. Even so it can not be concluded from Richardson's empirical formula that coastlines are infinitely long.