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A mosaic made by matching Julia sets to their values of c on the complex plane. The Mandelbrot set is a map of connected Julia sets. As a consequence of the definition of the Mandelbrot set, there is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding Julia set. For instance ...
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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]
English: Collection of Julia sets laid out in a grid such that the centre of each image corresponds to the same position in the complex plane as the value of the set. When laid out like this the overall image resembles the Mandelbrot set. Here there are 300 sets in each direction for a total of 90,000 mini Julia sets. The dimensions are:
Boundary of Mandelbrot set as an image of unit circle under Uniformization of complement (exterior) of Mandelbrot set Let Ψ M {\displaystyle \Psi _{M}\,} be the mapping from the complement (exterior) of the closed unit disk D ¯ {\displaystyle {\overline {\mathbb {D} }}} to the complement of the Mandelbrot set M {\displaystyle \ M} .
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 ...
The Douady-Hubbard-Sullivan theorem (DHS) states that the multiplier map of an attracting periodic orbit is a conformal isomorphism from a hyperbolic component of the Mandelbrot set onto the unit disk and it extends homeomorpically to the boundaries. [2]
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