Deep zooms need more iterations to show the edge.
Click to zoom in, shift-click or right-click to zoom out, drag to move, or use the wheel. Hover the Mandelbrot set to see the Julia set of each point. The address of the page keeps the view, to share it.
The simplest formula with the most detail
Take a number c on the complex plane. Start with z = 0, and repeat: z becomes z² + c. For some values of c, z stays small forever; for others it flies off to infinity. The Mandelbrot set is all the c that stay small, painted black; the colors outside count how fast the others escape.
Gaston Julia and Pierre Fatou studied these iterations during the First World War, by hand. In 1980 Benoit Mandelbrot, at IBM, had a computer draw the set that now carries his name. Its edge has infinite detail: zoom in anywhere along it and you find spirals, seahorses, elephants, and tiny copies of the whole set, each with its own decorations.
Each point c also has its Julia set: this time c is fixed, and the starting z varies. Pick c inside the Mandelbrot set and its Julia set is one connected piece; pick it outside and the Julia set crumbles into dust. The Mandelbrot set is a map of all the Julia sets, and near its edge, it looks like the Julia set of the point you are looking at. Move the pointer over it and watch the small view.