Mandelbrots

The Mandelbrot set, often denoted M, is the set of complex numbers c for which the quadratic recurrence z_{n+1} = z_n^2 + c, starting from z_0 = 0, produces a bounded orbit. In practical terms, one asks whether the iterates of zero remain within a finite region when repeatedly applying f_c(z) = z^2 + c. The set is plotted in the complex plane and is famous for its distinctive shape, with a large cardioid and a cascade of circular bulbs.

M is contained in the disk |c| <= 2; if |c| > 2, the sequence diverges. The boundary of M is infinite and fractal, exhibiting self-similarity at many scales. The set is connected and is compact in the complex plane; its boundary has elaborate structure that has made it a central object in complex dynamics and fractal geometry.

Relation to Julia sets: For a fixed c, the Julia set J_c is connected if and only if c lies in M; when c is outside M, J_c is a Cantor set. Thus M serves as a parameter space for Julia sets of the family z^2 + c. The Mandelbrot set's boundary encodes the various dynamical behaviors of the quadratic family.

History and computation: The term Mandelbrot set honors Benoit Mandelbrot, who popularized it in the late 20th century after earlier work by Fatou and Julia on complex dynamics. It is commonly visualized by the escape-time algorithm, iterating z_{n+1} = z_n^2 + c and coloring by the iteration count at which |z_n| exceeds a threshold, usually 2. The images have influenced mathematics, computer graphics, and art.