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The Mandelbrot set is the set of complex numbers ''c'' for which the sequence (''c'', ''c''² + ''c'', (''c''²+''c'')² + ''c'', ((''c''²+''c'')²+''c'')² + ''c'', (((''c''²+''c'')²+''c'')²+''c'')² + ''c'', …) does not approach infinity. The set is closely related to Julia sets (which include similarly complex shapes) and is named after the mathematician Benoit Mandelbrot, who studied and popularized it. Mandelbrot set images are made by sampling complex numbers and determining for each whether the result tends towards infinity when a particular mathematical operation is iterated on it. Treating the real and imaginary parts of each number as image coordinates, pixels are colored according to how rapidly the sequence diverges, if at all. More precisely, the Mandelbrot set is the set of values of ''c'' in the complex plane for which the orbit of 0 under iteration of the complex quadratic polynomial : remains bounded.〔(【引用サイトリンク】title=Mandelbrot Set Explorer: Mathematical Glossary )〕 That is, a complex number ''c'' is part of the Mandelbrot set if, when starting with ''z''0 = 0 and applying the iteration repeatedly, the absolute value of ''z''''n'' remains bounded however large ''n'' gets. This can also be represented as : :〔(【引用サイトリンク】title=Escape Radius, Mu-Ency at MROB )〕 For example, letting ''c'' = 1 gives the sequence 0, 1, 2, 5, 26,…, which tends to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set. On the other hand, ''c'' = −1 gives the sequence 0, −1, 0, −1, 0,…, which is bounded, and so −1 belongs to the Mandelbrot set. Images of the Mandelbrot set display an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The "style" of this repeating detail depends on the region of the set being examined. The set's boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts. The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules, and is one of the best-known examples of mathematical visualization. ==History== The Mandelbrot set has its place in complex dynamics, a field first investigated by the French mathematicians Pierre Fatou and Gaston Julia at the beginning of the 20th century. This fractal was first defined and drawn in 1978 by Robert W. Brooks and Peter Matelski as part of a study of Kleinian groups.〔Robert Brooks and Peter Matelski, ''The dynamics of 2-generator subgroups of PSL(2,C)'', in 〕 On 1 March 1980, at IBM's Thomas J. Watson Research Center in Yorktown, Heights, New York, Benoit Mandelbrot first saw a visualization of the set. Mandelbrot studied the parameter space of quadratic polynomials in an article that appeared in 1980.〔Benoit Mandelbrot, ''Fractal aspects of the iteration of for complex '', Annals NY Acad. Sci. 357, 249/259〕 The mathematical study of the Mandelbrot set really began with work by the mathematicians Adrien Douady and John H. Hubbard,〔Adrien Douady and John H. Hubbard, ''Etude dynamique des polynômes complexes'', Prépublications mathémathiques d'Orsay 2/4 (1984 / 1985)〕 who established many of its fundamental properties and named the set in honor of Mandelbrot. The mathematicians Heinz-Otto Peitgen and Peter Richter became well known for promoting the set with photographs, books, and an internationally touring exhibit of the German Goethe-Institut.〔Frontiers of Chaos, Exhibition of the Goethe-Institut by H.O. Peitgen, P. Richter, H. Jürgens, M. Prüfer, D.Saupe. since 1985 shown in over 40 countries.〕 The cover article of the August 1985 ''Scientific American'' introduced the algorithm for computing the Mandelbrot set to a wide audience. The cover featured an image created by Peitgen, et al.〔Fractals: The Patterns of Chaos. John Briggs. 1992. p. 80.〕 The Mandelbrot set became prominent in the mid-1980s as a computer graphics demo, when personal computers became powerful enough to plot and display the set in high resolution. The work of Douady and Hubbard coincided with a huge increase in interest in complex dynamics and abstract mathematics, and the study of the Mandelbrot set has been a centerpiece of this field ever since. An exhaustive list of all the mathematicians who have contributed to the understanding of this set since then is beyond the scope of this article, but such a list would notably include Mikhail Lyubich, Curt McMullen, John Milnor, Mitsuhiro Shishikura, and Jean-Christophe Yoccoz. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Mandelbrot set」の詳細全文を読む スポンサード リンク
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