Mandelbrot set - definition. What is Mandelbrot set
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%ما هو (من)٪ 1 - تعريف

FRACTAL NAMED AFTER MATHEMATICIAN BENOIT MANDELBROT
Mandelbrot Set; Mandlebrot set; Mandlebrot fractal; Mandelbrot spiral; Mandelbrot fractal; The mandelbrot set; Mandel Set; Mandelbrot sequence; MLC conjecture; Z^2+c; Minibrot
  • Attracting cycle in 2/5-bulb plotted over [[Julia set]] (animation)
  • A mosaic made by matching Julia sets to their values of c on the complex plane. Using this, one may see that the shape of the Mandelbrot set is formed. This is because the Mandelbrot set is itself a map of the connected Julia sets.
  • Attracting cycles and [[Julia set]]s for parameters in the 1/2, 3/7, 2/5, 1/3, 1/4, and 1/5 bulbs
  • With <math>z_{n}</math> iterates plotted on the vertical axis, the Mandelbrot set can be seen to bifurcate where the set is finite.
  • The first published picture of the Mandelbrot set, by [[Robert W. Brooks]] and Peter Matelski in 1978
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  • Image of the Tricorn / Mandelbar fractal
  • Periods of hyperbolic components
  • Zooming into the boundary of the Mandelbrot set
  • A 4D Julia set may be projected or cross-sectioned into 3D, and because of this a 4D Mandelbrot is also possible.
  • Feigenbaum ratio]] <math>\delta</math>.
  • Correspondence between the Mandelbrot set and the [[bifurcation diagram]] of the [[logistic map]]
  • External rays of wakes near the period 1 continent in the Mandelbrot set

Mandelbrot set         
¦ noun Mathematics a particular set of complex numbers which has a highly convoluted fractal boundary when plotted.
Origin
1980s: named after the Polish-born mathematician Benoit B. Mandelbrot.
Mandelbrot set         
The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e.
Mandelbrot set         
<mathematics, graphics> (After its discoverer, {Benoit Mandelbrot}) The set of all complex numbers c such that | z[N] | < 2 for arbitrarily large values of N, where z[0] = 0 z[n+1] = z[n]^2 + c The Mandelbrot set is usually displayed as an {Argand diagram}, giving each point a colour which depends on the largest N for which | z[N] | < 2, up to some maximum N which is used for the points in the set (for which N is infinite). These points are traditionally coloured black. The Mandelbrot set is the best known example of a fractal - it includes smaller versions of itself which can be explored to arbitrary levels of detail. {The Fractal Microscope (http://ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html/)}. (1995-02-08)

ويكيبيديا

Mandelbrot set


The Mandelbrot set () is the set of complex numbers c {\displaystyle c} for which the function f c ( z ) = z 2 + c {\displaystyle f_{c}(z)=z^{2}+c} does not diverge to infinity when iterated from z = 0 {\displaystyle z=0} , i.e., for which the sequence f c ( 0 ) {\displaystyle f_{c}(0)} , f c ( f c ( 0 ) ) {\displaystyle f_{c}(f_{c}(0))} , etc., remains bounded in absolute value.

This set was first defined and drawn by Robert W. Brooks and Peter Matelski in 1978, as part of a study of Kleinian groups. Afterwards, in 1980, Benoit Mandelbrot obtained high-quality visualizations of the set while working at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York.

Images of the Mandelbrot set exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications; mathematically, one would say that the boundary of the Mandelbrot set is a fractal curve. The "style" of this recursive detail depends on the region of the set boundary being examined. Mandelbrot set images may be created by sampling the complex numbers and testing, for each sample point c , {\displaystyle c,} whether the sequence f c ( 0 ) , f c ( f c ( 0 ) ) , {\displaystyle f_{c}(0),f_{c}(f_{c}(0)),\dotsc } goes to infinity. Treating the real and imaginary parts of c {\displaystyle c} as image coordinates on the complex plane, pixels may then be coloured according to how soon the sequence | f c ( 0 ) | , | f c ( f c ( 0 ) ) | , {\displaystyle |f_{c}(0)|,|f_{c}(f_{c}(0))|,\dotsc } crosses an arbitrarily chosen threshold (the threshold has to be at least 2, as -2 is the complex number with the largest magnitude within the set, but otherwise the threshold is arbitrary). If c {\displaystyle c} is held constant and the initial value of z {\displaystyle z} is varied instead, one obtains the corresponding Julia set for the point c {\displaystyle c} .

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. It is one of the best-known examples of mathematical visualization, mathematical beauty, and motif.