and
The Mandelbrot set was created by Benoit B. Mandelbrot as an index to the Julia sets: each point in the complex plane corresponds to a different Julia set. Those points within the Mandelbrot set correspond precisely to the connected Julia sets, and those outside correspond to disconnected ones.
It can be shown that once the modulus of z_{n} is larger than 2 (in cartesian form[?], when x_{n}^{2} + y_{n}^{2} > 2^{2}) the sequence will tend to infinity, and c is therefore outside the Mandelbrot set. This value, known as the bailout value, allows the calculation to be terminated for points outside the Mandelbrot set. For points inside the Mandelbrot set, i.e. values of c for which z_{n} doesn't tend to infinity, the calculation never comes to such an end, so it must be terminated after some number of iterations determined by the program. This results in the displayed image being only an approximation to the true set.
Whilst it is of no mathematical importance, most fractal rendering programs display points outside of the Mandelbrot set in different colours depending on the number of iterations before it bailed out, creating concentric shapes, each a better approximation to the Mandelbrot set than the last.
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