page_title: Mandelbrot set
section_title: Basic properties
hierarchical_section_title: Mandelbrot set / Basic properties
caption_attribution_description: English: Wakes near the period 1 continent in the Mandelbrot set. Boundary of the Mandelbrot set rendered with distance estimation (exterior and interior). Labelled with periods (blue), internal angles and rays (green) and external angles and rays (red).
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 Mandelbrot set is a compact set, since it is closed and contained in the closed disk of radius 2 around the origin. More specifically, a point  belongs to the Mandelbrot set if and only if
   for all 
  In other words, if the absolute value of  ever becomes larger than 2, the sequence will escape to infinity.
  The intersection of  with the real axis is precisely the interval [−2, 1/4]. The parameters along this interval can be put in one-to-one correspondence with those of the real logistic family,
  The correspondence is given by
  In fact, this gives a correspondence between the entire parameter space of the logistic family and that of the Mandelbrot set.
  Douady and Hubbard have shown that the Mandelbrot set is connected. In fact, they constructed an explicit conformal isomorphism between the complement of the Mandelbrot set and the complement of the closed unit disk. Mandelbrot had originally conjectured that the Mandelbrot set is disconnected. This conjecture was based on computer pictures generated by programs that are unable to detect the thin filaments connecting different parts of . Upon further experiments, he revised his conjecture, deciding that  should be connected. There also exists a topological proof to the connectedness that was discovered in 2001 by Jeremy Kahn.
  The 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. These rays can be used to study the Mandelbrot set in combinatorial terms and form the backbone of the Yoccoz parapuzzle.
  The boundary of the Mandelbrot set is exactly the bifurcation locus of the quadratic family; that is, the set of parameters  for which the dynamics changes abruptly under 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. The Mandelbrot curves are defined by setting p₀ = z, pₙ₊₁ = pₙ² + z, and then interpreting the set of points |pₙ(z)| = 2 in the complex plane as a curve in the real Cartesian plane of degree 2ⁿ⁺¹ in x and y.  These algebraic curves appear in images of the Mandelbrot set computed using the "escape time algorithm" mentioned below.
caption: External rays of wakes near the period 1 continent in the Mandelbrot set
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