More Mandelbrot

The Mandelbrot set holds endless fascination for me.  Who would have thought that repeated iteration of a simple mathematical formula could show such intricate, repeating patterns?  For the first picture I used 5,000 iterations as the maximum necessary for determining whether a given point in the complex plane is a member of the set (shown black) or not.  This means that every tiny pixel in the black areas is black because the formula has been iterated 5,000 times to make it so.  It takes 3 or 4 seconds for my laptop to generate this picture, and roughly, say, a quarter of the picture is black and there are nearly a million pixels in the picture. So that alone necessitates 1,250,000,000 iterations, and each iteration means a lot of work for the computer.  What an amazing thing a modern laptop is!  Without such devices at our disposal, we never could have guessed at the intricacies depicted here.

The parts which are not black are points which have been rejected before 5,000 iterations have been performed for them, and their colour is dependent on the number of iterations necessary to decide they should be rejected.


By the way, I read somewhere that it has been proved there are no isolated pieces of the set, every little black bit is connected to the main chunk, even though we may not be able to see the connection because screen resolution is not fine enough or perhaps because we have not used enough iterations, resulting in a pixel being shown black when it shouldn't be.