Sequence Fractals Part IV #33


s0=0.1i, si+1=-3.0*si2+0.1
Center:-2+0i; Zoom = 0.04

This one uses the a*si2+b sequence generator with a = -3, b = 0.1. The sequence converges quickly to a small value. The result is a large fractal. Large in the sense that the capture area extends well beyond the |c|<=2.0 that is typical with quadratic polynomials. The horizontal range of this image is (-25,+25).

Sequence Fractals Part IV #29


si=1.0, 0.001 repeating
Center:-12.2+0.0i; Zoom = 3

Introducing a new variation on the sequence fractal formula. The z2 part is mulplied by the sequence value si.

To get my footing with the new formula I started out with a simple two step repeating sequence. The “do-nothing” value of one alternates with the a very small value, 0.001. If an orbit tries to escape, every other step pulls it back. The result is that a large area around the origin is captured. Today’s view is far out to the left (-12 units) on the negative real axis.

Sequence Fractals Part IV #28


s0=0.3, si+1=(-0.5+0.25i)*si2+(1.2+0.5i)
Center:-0.8244795+0.27693854i; Zoom = 102400

Here is a zoom into the top of the mountain peak in the lower center of Sequence Fractals Part IV #27.

I feel compelled to try to describe this when I should just let the image speak for itself. This has some characteristic of the jigsaw puzzle fractals. See Sequence Fractals Part III #27 and Sequence Fractals Part IV #16 for a description of “jigsaw puzzle fractals”. The areas that I image as the puzzle pieces have a fractal spirals along the edges. The “drop cloth” fractals, see Sequence Fractals Part IV #22 and Sequence Fractals Part IV #23 also have fractal shapes on the edges of puzzle pieces. This one however seems much more organized than the splashed paint appearance of the others.

Sequence Fractals Part IV #26


s0=0.3, si+1=(-0.5+0.25i)*si2+(1.2+0.5i)
Center:0+0i; Zoom = 0.5

Some more playing around with complex parameters in the a*si2+b sequence generator.

I have not investigated the long term behavior of this sequence. Given the large capture set and Mandlebrot-like right side, I suspect it the sequence converges to very short cycle, most likely a single point.

A couple of zooms to follow.

Sequence Fractals Part IV #24


s0=0.4, si+1=-1.5*si2+0.9
Center:0.50789+1.055828; Zoom = 10240

I have generated and discarded many images in the last few days, trying to find something unique. Most are interesting, but feel too much like “variation on a theme”. Along the way I got distracted, that happens a lot. I pulled up a palette that I was working on earlier this year for non-fractal abstract art. I tweaked it some to get a better fit for fractals, and here is the result.

As for the main theme, this is another sequence with a which converges to a long cycle. The big difference is just the new colors.