Mandelbrot explorer

Choose palette

The fractals shown here are all related to the famous Mandelbrot fractal set. The mapping is very simple:

z_n+1 = z_n²+ p

Any points in the complex plane that do not diverge are in the set. In this representation the algorithm is iterated up to 255 times. If the point has not diverged (|r|>2) after 255 iterations the point is considered to be in the set. If the point diverges before 255 iterations it is coloured according to the number of iterations at which it diverged. In the default palette the "colder" colours indicate a fast divergence and the "hotter" colours indicate a slower divergence.

The Mandelbrot³ set is the same as the Mandelbrot set except the algorithm uses a cube instead of a square:

z_n+1 = z_n³+ p

The Julia set is closely related to the Mandelbrot except a constant number (J_x+iJ_y) is added in each iteration. Changing the values of J_x and J_y can lead to dramatic changes in the fractal's shape.

z_n+1 = z_n²+ c

c = J_x + i J_y

This algorithm uses the "escape time" algorithm, where the colour palette is divided into discrete steps. This can lead to very obvious contours in the respresentation, which can be visually appealing or distracting, depending on the zoom and coordinates of the image, as well as your personal taste. The images tend to break as you zoom further in, because there are limits to how reasonably precise the coordinates can be specified.