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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 examine. Seller Inventory # 5675
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