I. The Escape-Time Algorithm
The escape-time algorithm is one of the earliest coloring algorithms, and in many programs it is still the only option available. Its simplicity makes it a favorite with those learning to create fractal software. From an artistic viewpoint it is becoming less important, because it produces discrete values; continuous coloring algorithms have generally superceded it.
The algorithm itself is based on the number of iterations necessary to determine whether the orbit sequence tends to infinity or not. It can be strictly demonstrated that when the orbit of any value of z0, z1, z2… zn exceeds a border region R, it always diverges towards infinity. The minimum size and shape of R are different for each fractal type, of course. If the orbit sequence is stopped as soon as zn is outside the border region R, then the coloring value for the escape-time algorithm is simply the length of the sequence, that is, n. (This value is readily available in actual implementations of fractal software, because a limit must be placed on iterations in order to prevent infinite iterating of a single orbit.)
Traditionally, R is set as a circle, centered at the origin, with radius 2. This is because for the Mandelbrot set, it can be proven that as soon as |z| > 2, the orbit will diverge. Interesting variations have been created by changing the shape and position of R; ellipses, triangles, stars, and so on may be used instead. And while mathematically, R must include the circle of radius 2 to be accurate in testing for divergence, this has not stopped some artists from experimenting with smaller radii.