The Mandelbrot set is the set of all numbers for which the function fc(z) = z2 + c does not diverge when beginning with z0 = 0. Many of the points outside of the set go on long journeys to diverge, especially when close to the boundary with the Mandelbrot set. In order to estimate whether a point is on the Mandelbrot Set, we iterate over that point up to a certain limit, and if the point has not yet diverged (has a magnitude of over 2), we consider it in the set. The more iterations we consider, the more accurate the rendering of the set.
While it may at first seem unintuitive, many numbers that eventually diverge from the Mandelbrot set have steps that go through it. This is possible because the z0 component is added back in at every step of the way. It may be helpful to see the path that z takes as it attempts to diverge.
Click on any point in the rendered image to visualize the path the iterations take to escape from that point. Points close to the edge of the set tend to have the most interesting trajectories. Red lines indicate the path was non-escaping. Green lines indicate the path was escaping.