page_title: Mandelbrot set
section_title: Further results
hierarchical_section_title: Mandelbrot set / Other properties / Further results
caption_attribution_description: English: An image illustrating the relationship between the Mandelbrot set and Julia sets. The upper image is the zoom in of the Mandelbrot set centered about (-0.743643887037151 + 0.131825904205330i) with a horizontal diameter of 0.000000000051299. The lower image is the corresponding Julia set centered about (0 + 0i) and with a horizontal diameter of 0.009860138819832269. 3x supersampling anti-aliasing, warm hue line, normalized iteration count and square root noise reduction have been used in both instances to increase the aesthetic worth and make the images and their structures easily visible.
context_page_description: The Mandelbrot set is the set of complex numbers for which the function does not diverge when iterated from, i.e., for which the sequence, etc., remains bounded in absolute value.
  Its definition is credited to Adrien Douady who named it in tribute to the mathematician Benoit Mandelbrot. The set is connected to a Julia set, and related Julia sets produce similarly complex fractal shapes.
  Mandelbrot set images may be created by sampling the complex numbers and testing, for each sample point, whether the sequence goes to infinity. Treating the real and imaginary parts of as image coordinates on the complex plane, pixels may then be coloured according to how soon the sequence crosses an arbitrarily chosen threshold, with a special color used for the values of for which the sequence has not crossed the threshold after the predetermined number of iterations. If is held constant and the initial value of —denoted by —is variable instead, one obtains the corresponding Julia set for each point in the parameter space of the simple function.
context_section_description: The Hausdorff dimension of the boundary of the Mandelbrot set equals 2 as determined by a result of Mitsuhiro Shishikura. It is not known whether the boundary of the Mandelbrot set has positive planar Lebesgue measure.
  
  In the Blum–Shub–Smale model of real computation, the Mandelbrot set is not computable, but its complement is computably enumerable. However, many simple objects (e.g., the graph of exponentiation) are also not computable in the BSS model. 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". Hertling has shown that the Mandelbrot set is computable in this model if the hyperbolicity conjecture is true.
caption: A zoom into the Mandelbrot set illustrating a Julia "island" and a similar Julia set.
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