Question

according the following article : wolfram Mandelbrot set, I’m trying to understand how they exactly managed to calculate the Ln(C)=Zn=R(max) values.
i do understand that Rmax is a constant, equals 2,(|Zn| < 4 for all points that are inside the Mandelbrot set), and Ln(C) should be the amount of iterations i spent for each C(point), but how using these 2 i get to calculate

L1(C)   =   C   
L2(C)   =   C(C+1)
          ....
          ....

thanks for your help!

Answer 1

You start by setting z=C (or, basically equivalently as it happens, z=0) and then repeatedly setting z := z^2+C. Keep doing this until you get a z with |z|>Rmax.

If you never do — of course in practice you won’t go on literally for ever, but will stop after a certain maximum number of iterations — then your point is in the Mandelbrot set, and if you’re drawing a picture you typically colour it black.

If after N iterations you do get |z|>Rmax, then your point wasn’t in the Mandelbrot set, and N gives some indication of how thoroughly outside the set it is; if you’re drawing a picture, you typically plot the point in a colour determined by N.

The description of L_n on the Wolfram page is pretty bad. What they mean is: define L_n(C) to be the value of z after n iterations when you use the parameter C; then you can plot the curves defined by |L_n(c)|=Rmax. These are the boundaries between the different-coloured regions when you plot points as described above.