The Mandelbrot set is the set of complex numbers {\displaystyle c} c for which the function {\displaystyle f_{c}(z)=z^{2}+c} {\displaystyle f_{c}(z)=z^{2}+c} does not diverge when iterated from {\displaystyle z=0} z=0, i.e., for which the sequence {\displaystyle f_{c}(0)} {\displaystyle f_{c}(0)}, {\displaystyle f_{c}(f_{c}(0))} {\displaystyle f_{c}(f_{c}(0))}, etc., remains bounded in absolute value.
A zoom sequence illustrating the set of complex numbers termed the Mandelbrot set.
Its definition and name are due to Adrien Douady, in tribute to the mathematician Benoit Mandelbrot.[1] The set is connected to a Julia set, and related Julia sets produce similarly complex fractal shapes.
Mandel_zoom_12_satellite_spirally_wheel_with_julia_islands by Wolfgangbeyer