Julia Set

A Julia set, named for the French mathematician Gaston Julia, is the boundary between starting points whose orbits stay bounded under iteration of z \to z^2 + c and those that escape to infinity. Unlike the Mandelbrot set the constant c is fixed and the plane is scanned over starting values z. Whether the resulting Julia set is a connected continuum or a disconnected dust depends on whether the chosen c lies inside the Mandelbrot set — a deep theorem of complex dynamics due to Fatou and Julia in the 1910s.

Here the constant traces a circle of radius 0.7885 in the complex plane, c = 0.7885\,e^{ia}, while the angle a animates from 0 to 2\pi — so the view sweeps continuously through the whole family of Julia sets on that circle. Each frame samples the square -1.8 \le x, y \le 1.8 with an iteration limit of 256, and the plasma palette grades pixels by escape speed. As a advances the fractal deforms through connected blobs, dendritic branches, and spiral arms, pinching and reconnecting as the parameter sweeps near the boundary of the Mandelbrot set.