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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:
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 ...
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.
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,
A preperiodic orbit. In mathematics, a Misiurewicz point is a parameter value in the Mandelbrot set (the parameter space of complex quadratic maps) and also in real quadratic maps of the interval [1] for which the critical point is strictly pre-periodic (i.e., it becomes periodic after finitely many iterations but is not periodic itself).
A "scale-2" Mandelbox A "scale-3" Mandelbox A "scale -1.5" Mandelbox. In mathematics, the mandelbox is a fractal with a boxlike shape found by Tom Lowe in 2010. It is defined in a similar way to the famous Mandelbrot set as the values of a parameter such that the origin does not escape to infinity under iteration of certain geometrical transformations.
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 ...
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