si = 1, -1, 1, -1, …
Center:0+0i , Zoom = 0.25
Way back at the start Sequence Fractals Part I #1 we had these definitions:
f0(z,c) = z2+c+1
f1(z,c) = z2+c-1
fc(z) = f(z,c) =f1(f0(z,c),c) =
Derivative: fc‘(z) = 4z(z2+c+1)
Critical points are where fc‘(z) = 0. One critical point is 0, which we have studied in some detail. The other two are the roots of (z2+c+1),
The sign does not matter (it disappears with the first z2), both roots behave the same. So, for today only, let z0 be one of these roots and the starting point for the fractal iteration. Also forget about the two step function for now, and look at the original sequence fractal definition,
and write out the first few iterations (notice the alternating function subscript)
z0 was defined to be a root of (z2+c+1)=0 which is exactly f0(z,c).
How convenient! We can rewrite the orbit as
z0, 0, f1(0,c),
After the first step this is exactly the sequence fractal for si = -1, 1, -1, 1, … starting at 0. That sequence will be the subject of the next several posts.