How Long Is the Coast of Britain
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"How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" is a paper by
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
Benoit Mandelbrot, first published in ''Science'' on 5 May 1967. In this paper, Mandelbrot discusses self-similar curves that have
Hausdorff dimension 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 ...
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.


Overview

The paper examines the
coastline paradox 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 fr ...
: the property that the measured length of a stretch of coastline depends on the scale of measurement. Empirical evidence suggests that the smaller the increment of measurement, the longer the measured length becomes. If one were to measure a stretch of coastline with a
yardstick A meterstick, metrestick, or yardstick is either a straightedge or foldable ruler used to measure length, and is especially common in the construction industry. They are often made of wood or plastic, and often have metal or plastic joints so ...
, one would get a shorter result than if the same stretch were measured with a
ruler A ruler, sometimes called a rule, line gauge, or scale, is a device used in geometry and technical drawing, as well as the engineering and construction industries, to measure distances or draw straight lines. Variants Rulers have long ...
. This is because one would be laying the ruler along a more curvilinear route than that followed by the yardstick. The empirical evidence suggests a rule which, if extrapolated, shows that the measured length increases without limit as the measurement scale decreases towards zero. This discussion implies that it is meaningless to talk about the length of a coastline; some other means of quantifying coastlines are needed. Mandelbrot discusses an empirical law discovered by Lewis Fry Richardson, who observed that the measured length ''L''(''G'') of various geographic borders was a
fractal curve 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 rectif ...
of the measurement scale ''G''. Collecting data from several different examples, Richardson conjectured that ''L''(''G'') could be closely approximated by a function of the form :L(G)=MG^ where ''M'' is a positive constant and ''D'' is a constant, called the dimension, greater than or equal to 1. Intuitively, if a coastline looks smooth it should have dimension close to 1; and the more irregular the coastline looks the closer its dimension should be to 2. The examples in Richardson's research have dimensions ranging from 1.02 for the coastline of
South Africa South Africa, officially the Republic of South Africa (RSA), is the Southern Africa, southernmost country in Africa. It is bounded to the south by of coastline that stretch along the Atlantic Ocean, South Atlantic and Indian Oceans; to the ...
to 1.25 for the West coast of
Britain Britain most often refers to: * The United Kingdom, a sovereign state in Europe comprising the island of Great Britain, the north-eastern part of the island of Ireland and many smaller islands * Great Britain, the largest island in the United King ...
. Mandelbrot then describes various mathematical curves, related to the
Koch snowflake The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curv ...
, which are defined in such a way that they are strictly self-similar. Mandelbrot shows how to calculate the Hausdorff dimension of each of these curves, each of which has a dimension ''D'' between 1 and 2 (he also mentions but does not give a construction for the space-filling Peano curve, which has a dimension exactly 2). He notes that the approximation of these curves with segments of length ''G'' have lengths of the form G^. The resemblance with Richardson's law is striking. The paper does not claim that any coastline or geographic border actually ''has'' fractional dimension. Instead, it notes that Richardson's empirical law is compatible with the idea that geographic curves, such as coastlines, can be modelled by random self-similar figures of fractional dimension. Near the end of the paper Mandelbrot briefly discusses how one might approach the study of fractal-like objects in nature that look random rather than regular. For this he defines statistically self-similar figures and says that these are encountered in nature. The paper is important because it is a "turning point" in Mandelbrot's early thinking on fractals."What is the essence of a coastline, for example ? Mandelbrot asked this question in a paper that became a turning point for his thinking: 'How Long Is the Coast of Britain'": James Gleick (1988) '' Chaos: Making a New Science'', p.94. . It is an example of the linking of mathematical objects with natural forms that was a theme of much of his later work.


See also

*
Coastline paradox 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 fr ...
*
List of countries by length of coastline This article contains a list of countries by length of coastline, in kilometers. A coastline of zero indicates that the country is landlocked. Overview The coastline paradox states that a coastline does not have a well-defined length. Measurem ...


References

{{Fractals Mathematics papers Works originally published in Science (journal) Fractals