long coast britain de mandelbrot benoît (1 resultados)

First edition. THE FIRST PAPER ON FRACTALS "FAMOUS IN THE HISTORY OF MATHEMATICS". First edition, journal issue in original printed wrappers, of Mandelbrot's first paper on fractals (a term he coined in 1975). "Today Mandelbrot's paper on the coast of Britain is famous in the history of mathematics" (). "Mandelbrot had come across the coastline question in an obscure posthumous article by an English scientist, Lewis F. Richardson, who groped with a surprising number of the issues that later became part of chaos [theory] . Wondering about coastlines, Richardson checked encyclopaedias in Spain and Portugal, Belgium and the Netherlands, and discovered discrepancies of 20% in the estimated lengths of their common frontiers . [Mandelbrot] argued [that] . the answer depends on the length of your ruler. Consider one plausible method of measuring. A surveyor takes a set of dividers, opens them to a length of one yard, and walks them along the coastline. The resulting number of yards is just an approximation of the true length, because the dividers skip over twists and turns smaller than one yard, but the surveyor writes the number down anyway. Then he sets the dividers to a smaller length - say, one foot - and repeats the process. He arrives at a somewhat greater length, because the dividers will capture more of the detail and it will take more than three one-foot steps to cover the distance previously covered by a one-yard step. He writes this new number down, sets the dividers at four inches, and starts again . Common sense suggests that, although these estimates will continue to get larger, they will approach some particular final value, the true length of the coastline . if a coastline were some Euclidean shape, such as a circle, this method of summing finer and finer straight-line distances would indeed converge. But Mandelbrot found that as the scale of measurement becomes smaller, the measured length of a coastline rises without limit" (Gleick, pp. 94-96). A copy of Richardson's 'obscure' article, posthumously published in 1961 although written in the 1920s, accompanies Mandelbrot's article here. Richardson proposed, in section 7 ('Lengths of land frontiers or seacoasts') of his article, that the measured length of the coastline should be proportional to G1 - D, where G is the length of the ruler and D is a number, possibly fractional, greater than or equal to 1. On p. 636 of his article, Mandelbrot notes that: "Such a formula, of an entirely empirical character, was proposed by Lewis F. Richardson [in the offered paper] but unfortunately it attracted no attention." Mandelbrot suggests that D should be regarded as the dimension of the coastline - it is now known as the 'fractal dimension'. "Although the key concepts associated with fractals had been studied for years by mathematicians, and many examples, such as the Koch 'snowflake' curve were long known, Mandelbrot was the first to point out that fractals could be an ideal tool in applied mathematics for modeling a variety of phenomena from physical objects to the behavior of the stock market. Since its introduction in 1975, the concept of the fractal has given rise to a new system of geometry that has had a significant impact on such diverse fields as physical chemistry, physiology, and fluid mechanics. Many fractals possess the property of self-similarity, at least approximately, if not exactly. A self-similar object is one whose component parts resemble the whole. This reiteration of details or patterns occurs at progressively smaller scales and can, in the case of purely abstract entities, continue indefinitely, so that each part of each part, when magnified, will look basically like a fixed part of the whole object . This fractal phenomenon can often be detected in such objects as snowflakes and tree barks. All natural fractals of this kind, as well as some mathematical self-similar ones, are stochastic, or random; they thus scale in a statistical sense" (Britannica). "The paper examines the coastline paradox: the property that the measured length of a stretch of coastline depends on the scale of measurement . Th[e] 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 (1881-1953), who observed that the measured length L(G) of various geographic borders was a 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) = MG1 - D where M is a positive constant and D is a constant, called the dimension, greater than or equal to 1 [now known as the 'fractal dimension']. 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 to 1.25 for the West coast of Britain. "Mandelbrot then describes various mathematical curves, related to the 'Koch snowflake,' 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 G1 - D.The 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.