Sequence Fractals Part II #16

279.jpg

z^2+c+s_i
si = a,-a,…; a = -0.717
Center -0.167756+0.065847:; Zoom = 200
Start: z0 = \pm \sqrt{0.717-c}

Here is the view from the other critical point of the same area, same zoom and colors. The Mandelbrot surface is complete. Except for the chunks knocked loose in the previous picture Sequence Fractals Part II #15, the white areas are the same.

Something similar can be seen with the quadratic family z2+c. Within this family you can prove “If an attracting fixed point exists, then the critical point is attracted to it.” This is again a consequence of the fact that there is only one critical point. And basically the reason we always want to start with the critical point. If you try to draw a Mandelbrot set starting with any other point, you will find pieces missing. Always less, never more. If you start far enough away, none of it remains.

I have not worked through the details, but I am pretty sure that what we see here is closely related.

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