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The Mandelbrot set (/ ˈ m æ n d əl b r oʊ t,-b r ɒ t /) [1] [2] is a two-dimensional set with a relatively simple definition that exhibits great complexity, especially as it is magnified. It is popular for its aesthetic appeal and fractal structures.
Every pixel that contains a point of the Mandelbrot set is colored black. Every pixel that is colored black is close to the Mandelbrot set. Exterior distance estimate may be used to color whole complement of Mandelbrot set. The upper bound b for the distance estimate of a pixel c (a complex number) from the Mandelbrot set is given by [6] [7] [8]
Without doubt, the most famous connectedness locus is the Mandelbrot set, which arises from the family of complex quadratic polynomials : = + The connectedness loci of the higher-degree unicritical families, +
Mandelbrot set rendered using a combination of cross and point shaped orbit traps. In mathematics, an orbit trap is a method of colouring fractal images based upon how close an iterative function, used to create the fractal, approaches a geometric shape, called a "trap". Typical traps are points, lines, circles, flower shapes and even raster ...
gives rise to the Mandelbrot set: Complex squaring map: discrete: complex: 1: 0: acts on the Julia set for the squaring map. Complex cubic map: discrete: complex: 1: 2: Clifford fractal map [13] discrete: real: 2: 4: Degenerate Double Rotor map: De Jong fractal map [14] discrete: real: 2: 4: Delayed-Logistic system [15] discrete: real: 2: 1 ...
An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set. [1] Although this curve is only rarely a half-line (ray) it is called a ray because it is an image of a ray. External rays are used in complex analysis , particularly in complex dynamics and geometric function theory .
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The famous Mandelbrot set is a subset of this parameter space, consisting of the points in the complex plane which give a bounded set of numbers when a particular iterated function is repeatedly applied from that starting point. The remaining points, which are not in the set, give an unbounded set of numbers (they tend to infinity) when this ...