Mandelbrot set

The set of complex parameters c for which the sequence starting at zero under z ↦ z² + c remains bounded. Its boundary forms a fractal with intricate structure at every scale.

Quadratic polynomial: A polynomial whose highest-degree term has degree two. The map z ↦ z² + c is a quadratic polynomial for each fixed parameter c.
Complex number: A number expressed as a real part plus an imaginary part. The parameter c and every iterated value belong to the complex plane.
Benoit Mandelbrot: A mathematician who popularized fractal geometry and brought the set to broad attention. The set bears his name after his influential study of its geometry.
Complex quadratic map: The map z ↦ z² + c on the complex plane for a fixed complex parameter c. Each parameter in the set selects one member of this family of maps.
Fractal: A geometric structure with intricate detail across scales, often described using fractal dimension. The Mandelbrot set became one of the most recognizable examples of fractal geometry.
Complex dynamical system: A dynamical system in which states and evolution rules use complex numbers. Each point c defines a complex map whose repeated action determines membership.
Complex plane: A plane whose horizontal and vertical coordinates represent real and imaginary parts. The set is plotted as a region in this plane of possible parameters.
Adrien Douady: A French mathematician who helped establish the theory of polynomial Julia sets and the Mandelbrot set. His work with John Hubbard proved major structural results about the set.
Parameter space: A space whose points represent possible values of a system’s parameters. The Mandelbrot set is the locus of bounded behavior in this family’s parameter space.
Hausdorff dimension: A dimension defined by measuring how a set scales under covers of decreasing size. Dimension estimates quantify the complexity of the set’s boundary beyond ordinary length or area.

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