19 Facts About Mandelbrot set


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.

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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.

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Mandelbrot set has its origin in complex dynamics, a field first investigated by the French mathematicians Pierre Fatou and Gaston Julia at the beginning of the 20th century.

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Mandelbrot set studied the parameter space of quadratic polynomials in an article that appeared in 1980.

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The mathematical study of the Mandelbrot set really began with work by the mathematicians Adrien Douady and John H Hubbard, who established many of its fundamental properties and named the set in honor of Mandelbrot for his influential work in fractal geometry.

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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.

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Mandelbrot set is the set of values of c in the complex plane for which the orbit of the critical point under iteration of the quadratic map.

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Mandelbrot set can be defined as the connectedness locus of the family of quadratic polynomials, while its boundary can be defined as the bifurcation locus of this quadratic family.

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Mandelbrot set is a compact set, since it is closed and contained in the closed disk of radius 2 around the origin.

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Dynamical formula for the uniformisation of the complement of the Mandelbrot set, arising from Douady and Hubbard's proof of the connectedness of, gives rise to external rays of the Mandelbrot set.

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In other words, the boundary of the Mandelbrot set is the set of all parameters for which the dynamics of the quadratic map exhibits sensitive dependence on i e, changes abruptly under arbitrarily small changes of It can be constructed as the limit set of a sequence of plane algebraic curves, the Mandelbrot curves, of the general type known as polynomial lemniscates.

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Hypothetical non-hyperbolic components of the Mandelbrot set are often referred to as "queer" or ghost components.

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Mandelbrot set is self-similar under magnification in the neighborhoods of the Misiurewicz points.

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Shishikura's result intuitively states that the Mandelbrot set boundary is so "wiggly" that it manages to locally fill up space as efficiently as a two-dimensional planar region.

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At present, it is unknown whether the Mandelbrot set is computable in models of real computation based on computable analysis, which correspond more closely to the intuitive notion of "plotting the set by a computer".

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The value of for the corresponding is not that of the image center but, relative to the main body of the Mandelbrot set, has the same position as the center of this image relative to the satellite shown in the 6th zoom step.

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For general families of holomorphic functions, the boundary of the Mandelbrot set generalizes to the bifurcation locus, which is a natural object to study even when the connectedness locus is not useful.

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However, the quaternion Mandelbrot set is simply a solid of revolution of the 2-dimensional Mandelbrot set, and is therefore largely uninteresting to look at, as it does not have any of the properties that would be expected from a 4-dimensional fractal.

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Mandelbrot set is widely considered the most popular fractal, and has been referenced several times in popular culture.

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