Sequence Fractals Part I #17
$ z^2+c+s_i$
si = 1, -1, 1, -1, …
Center:-3.631717+0i , Zoom = 6000
This is a zoom into the far left side of #1, also see #9, #10, #11, for more views of the neighborhood. The center is another M1,1 point. This point is similar to the point -2, a M2,1 point in the Mandelbrot set. (The Mandelbrot set has no M1,1 points.)
Self-similarity is in full force here. Every zoom looks the same. I offer the following "proof by recursive pictures": For the parameter space non-escape set, M, for our function, -3.631717… is an accumulation point of M, a boundary point of M, and not connected to any other point of M.
In the Mandelbrot set, -2 has the first two properties, but not the third.
If you liked the bit about derivatives in yesterday's post, for today's picture
f'(z1) = 24.1509+0i |f'(z1)| = 24.1509
So there is self-similarity with a 24x zoom and no rotation.