Take a number, square it, add a constant, and repeat. Some starting points stay put forever and others fly off to infinity. The border between them never ends, however far you zoom.
Drag to move, scroll or pinch to zoom, double-click to dive in. In the Mandelbrot set, the small picture in the corner shows the Julia set for the point under your pointer; shift-click, or press Open the Julia set here, to explore it. The address of the page always holds your view, so Copy a link lets you send someone exactly where you are. Ordinary computer numbers run out of precision at about a ten-trillion-fold zoom, so that's the floor.
In 1918 and 1919, during and just after the First World War, Gaston Julia and Pierre Fatou studied what happens when you apply z → z² + c over and over. They worked entirely by hand; nobody could see the shapes. Sixty years later, at IBM, Benoit Mandelbrot used a computer to draw the set of c values whose Julia sets hold together, and in 1980 the picture now called the Mandelbrot set appeared. Every point of it is a doorway to a Julia set: points inside give connected shapes, points outside give scattered dust.
The others are close relatives. The Multibrot sets use a higher power. The Burning Ship (Michael Michelitsch and Otto Rössler, 1992) folds z into one quarter of the plane before squaring. The Tricorn, named by W. D. Crowe and colleagues, flips z first. Newton's method is Isaac Newton's way of homing in on a root of an equation: colour each start by which of the three cube roots of 1 it ends up at, and the borders between them turn out to be fractal too.
