The Mandelbrot set is defined as the collection of points

in the complex plane for which the orbit of the iterated relation

, known as the Julia set for

, remains in a finite, simply connected region as

. The Mandelbrot set, which has a fractal boundary, is shown in black on the graphic. Two dominant features are a cardioid and a circular disc tangent to one another. If

for any value of

, then the point cannot belong to the Mandelbrot set. Such points fall in a colored region, with different colors determined by the number of steps it takes for an orbit to reach a value

. Magnification of various regions of the complex plane reveals an incredible variety of fractal structures. The classic computer-generated images were published by Peitgen and Richter (

*The Beauty of Fractals*: Heidelberg, Springer-Verlag, 1986). This Demonstration cannot reproduce their computational power, but a general idea of the richness of the Mandelbrot set can be obtained with magnifications up to

with 300×300-pixel resolution. The 2D slider centers the image on a desired region for magnification. The magnification should be increased stepwise for optimal control.