The Mandelbrot set is the set of complex numbers c for which the sequence produced by repeatedly applying the function f(z) = z^2 + c, starting from z = 0, does not diverge to infinity. It is named for Benoit Mandelbrot, who obtained the first high-quality computer visualizations of it in 1980 while working at IBM's Thomas J. Watson Research Center, though the underlying iteration had been studied by Pierre Fatou and Gaston Julia decades earlier. The boundary of the Mandelbrot set is a fractal curve of unbounded complexity, revealing new detail at every level of magnification, and a given value of c belongs to the Mandelbrot set exactly when its corresponding Julia set is connected, making the Mandelbrot set a map of every connected Julia set at once.
Facts
Origin YearYear of Mandelbrot's first published high-quality visualizations at IBM; the underlying iterated function had already been studied by Fatou and Julia since the 1910s. FAQs
Is the Mandelbrot set connected?
Yes; Hubbard and Douady proved it is connected.
Yes. J. Hubbard and A. Douady proved that the Mandelbrot set is connected, so it is a single piece rather than a scattering of separate parts.
Connections
Associated With
A parameter c produces a connected Julia set exactly when c belongs to the Mandelbrot set.
Source Julia Set (Wikipedia)
Attributed To
Source PlanetMath: Mandelbrot Set - PlanetMath
In Branch
Source Mandelbrot Set (Wikipedia)
Is Kind Of Object
Sources
1. Mandelbrot Set (Wikipedia)
Wikimedia FoundationLead section
The boundary of the Mandelbrot set is a fractal curve
- In Branch: Dynamical Systems
View the Source Julia Set (Wikipedia)
PlanetMath: Mandelbrot Set - PlanetMath
PlanetMathAttributed To: Benoît Mandelbrot, Page textQuote, Attributed To: Benoît Mandelbrot, Page text
Discovered by Benoît Mandelbrot, the first print-out of it was in black and white.
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