It would seem that it is difficult to measure the coastline. Well, yes, it is complex, twisted. But this is not a miniature bacterium. Walked and measured everything along the border. However, as you understand, everything is not so simple here.
Shortly before 1951, Lewis Fry Richardson, while studying the alleged influence of the length of state borders on the likelihood of the outbreak of military conflicts, noted the following: Portugal stated that its land border with Spain was 987 km, and Spain determined it to be 1214 km.
This fact served as a starting point for studying the problem of the coastline and to an unusual conclusion: the length of the coastline turns out to be an unattainable concept, sliding between the fingers of those who try to understand it.
The main method for estimating the length of a border or coastline was to overlay N equal segments of length l on a map or aerial photograph using a compass. Each end of the line must belong to the boundary being measured. Investigating the discrepancies in the bounds estimates, Richardson discovered what is now called the Richardson effect: the scale of measurements is inversely proportional to the total length of all segments. That is, the shorter the ruler used, the longer the measured border. Thus, Spanish and Portuguese geographers were simply guided by measurements of different scales.
The most striking thing for Richardson was that when the value of the ruler goes to zero, the length of the coast goes to infinity. Initially, Richardson believed, relying on Euclidean geometry, that this length would reach a fixed value, as is the case with regular geometric shapes. For example, the perimeter of a regular polygon inscribed in a circle approaches the length of the circle itself with an increase in the number of sides (and a decrease in the length of each side). In the theory of geometric measurements, such a smooth curve as a circle, which can be approximately represented as small segments with a given limit, is called a rectifiable curve.
More than ten years after Richardson completed his work, Mandelbrot developed a new branch of mathematics - fractal geometry - to describe such non-rectifiable complexes that exist in nature, such as an endless coastline
The key property of fractals is self-similarity, which consists in the manifestation of the same general figure at any scale. The coastline is perceived as an alternation of bays and capes. Hypothetically, if a given coastline has the property of self-similarity, then no matter how much one or another part is scaled, a similar picture of smaller bays and capes still appears, superimposed on larger bays and capes, down to grains of sand. At this scale, the coastline appears to be an instantaneous, potentially endless thread with a stochastic location of bays and headlands. In such conditions (as opposed to smooth curves) Mandelbrot states: "The length of the coastline turns out to be an unattainable concept, sliding between the fingers of those who try to understand it."
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In reality, the coastlines lack details less than 1 cm [source unspecified 918 days]. This is due to erosion and other marine phenomena. In most places, the minimum size is much larger. Therefore, the infinite fractal model is not suitable for coastlines.
For practical reasons, choose the minimum size of parts equal to the order of units of measurement. So, if the coastline is measured in kilometers, then small line changes, much less than one kilometer, are simply not taken into account. To measure the coastline in centimeters, all small variations of about one centimeter must be considered. However, on scales of the order of centimeters, various arbitrary non-fractal assumptions have to be made, for example, where an estuary joins the sea, or where measurements are to be made at wide watts. In addition, the use of different measurement methods for different units of measurement does not allow converting these units using simple multiplication.
To determine the state territorial waters, so-called straight baselines are built connecting the officially established points of the coast. The length of such an official coastline is also easy to measure.
Extreme cases of the coastline paradox include coasts with a large number of fjords: these are the coasts of Norway, Chile, the northwest coast of North America, and others. From the southern tip of Vancouver Island in the northern direction to the southern tip of Southeast Alaska, the bends of the coast of the Canadian province of British Columbia make up more than 10% of the length of the Canadian coastline (including all the islands of the Canadian Arctic Archipelago) - 25,725 km out of 243,042 km at linear distance, equal to only 965 km.