why mandelbrot's boundary is 2? [closed]

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Closed 7 years ago.

I'm trying to understand why we're iterating through Mandelbrot points until |z| < 4. why 4? is there somekind of a law? or is it based on statistical measurements?

thanks, igal

回答1:

Consider the Mandelbrot set with along y=0, which would would be z(i) = z(i-1)^2 + c.

Consider when c = (x=-2, y=0)

z(0) = 0
z(1) = 0^2 + -2 = -2
z(2) = (-2)^2 + -2 = 4 - 2 = 2
z(3) = 2^2 + -2 = 4 - 2 = 2
z(...) = 2^2 + -2 = 4 - 2 = 2

This example (x=-2,y=0) is the point with the greatest magnitude that will never blow up. Thus when z^2 > 4, there is no point in further iteration since you already know it will blow up.

All other points where the magnitude of the point >= 2 will blow up. 

来源：https://stackoverflow.com/questions/10690224/why-mandelbrots-boundary-is-2