This applet demonstrates the beauty of the Mandelbrot set
The iteration is done for each point (pixel) on the screen. With z0 being the complex number represented by the point, and c = 0. Technically the set is the set of all points whose iterative sequences converge to zero.
The colors are an arbitrary 'coding' of whether the sequence for each point converges to zero or diverges. In the middle (the central white bits) the sequences normally converge quite quickly to zero. At the outer edges the sequences quickly diverge. But the boundary between them is fractal and exhibits chaotic behaviour. The colours represent 'contours' or regions that
diverge at approximately the same rate (in other words the sequences will be at approximately the same distance from 0 after the iterations have finished).
Feel free to play around with this fascinating piece of maths. But please note that it can be rather slow, especially on a slow computer.
Pressing 'Iterations +' or 'Iterations -' changes the number of iterations per point. A large number ot iterations gives a better picture but is much slower.
Pressing 'Start' draws the picture. Once you have a picture on the screen you can enlarge
small regions of it by clicking those regions (once only) with your mouse. Please note that some regions, especially near the white centre, can be very slow.
Clicking 'Reset' returns you to the 'Start' screen. Be patient, and prepare to be stunned by some of the images!