$ z^{2^s_i}+c$
s₀=1, s_i+1=(0.8+0.6i)*s_i
Center:0; Zoom = 1.6

The sequence stays on the unit circle, |s_i| = 1, but does not repeat or converge.

Another early experiment. There is too much color in the center. It needs a reduced palette. The left and right sides have lacy triangles that vaguely resemble a Sierpinski Triangle.

$ z^{2^s_i}+c$
s₀=1, s_i+1=(-0.8+0.6i)*s_i
Center:-0.597-0.183; Zoom = 25

I made yesterday's image five months ago. This one was made a few days ago, as I was writing the words for yesterday's post. I have six of the older pictures from back when I was more interested in the math than artistic composition. I am posting them not as finished product, but as something study and find potential.

Of course, with my short attention span, as soon as I mention something in a post, I find myself playing with those ideas rather than finishing the post. And hence the alternating last spring and last week pictures.

Here is a zoom into the lower right of yesterday's picture, Sequence Fractals Part V #32. After some searching, then careful framing and small tweaking of parameters, I came up with this interesting piece.

$ z^{2^s_i}+c$
s₀=1, s_i+1=(-0.8+0.6i)*s_i
Center:-0.493-0.255; Zoom = 51

Yesterday's, Sequence Fractals Part V #33, comments work just as well for today's picture, so I will not repeat them. Same limited palette. Same lacy patterns. But more drama and a feeling on conflict in today's picture

$ z^{2^s_i}+c$
s₀=1, s_i+1=(0.8+0.6i)*s_i
Center:0; Zoom = 0.4

Back to the older pictures. Here the sequence starts out small, then grows, following a logarithmic spiral outward.

$ z^{2^s_i}+c$
s₀=1, s_i+1=(0.8+0.6i)*s_i
Center:0; Zoom = 0.4

I tried to just post yesterday's image, Sequence Fractals Part V #35, and let it go. I tried and couldn't. So here is another, more artistic, interpretation of the same formula.

$ z^{2^s_i}+c$
s₀=1+2i, s_i+1=(0.71+0.71i)*s_i
Center0.310-0.023i; Zoom = 2

Like yesterday, Sequence Fractals Part V #26, the sequence is following a logarithmic spiral outward. The iteration formula has been modified, the stacked exponents are gone and now the sequence value is directly the exponent of z.

The white dots are an artifact of the coloring algorithm. As usual there is too much going on here. The left and right side are too different. Perhaps with some work, maybe a lot of work. the two sides could represent a contrasting duality. Dots vs rainbows. Of course to highlight that contrast, other aspects should be consistent throughout the frame. It is almost there, if I stare at it long enough.

$ z^{2^s_i}+c$
s₀=-0.01+i, s_i+1=(0.71+0.71i)*s_i
Center:473+.005i; Zoom = 1

This one is similar to yesterday's post, Sequence Fractals Part V #37. Same formula, a different starting point for the spiral sequence.

The upper left is a mess. Most of the rest of the image is populated with repeating, slightly morphing shapes. There seems to be a lot of potential. In fact, this is close to something that I would just say "done" and publish it.

This one is the last of the older pictures that were created with a "fractal mindset". That is, more or less raw pictures intended to give some insight into the math. I have several more queued up that I created with the "art mindset". The formulas create a background, that I use as inspiration to build a piece of abstract art. Many of those are derived from today's formula.

$ z^{2^{s_i}}+c$
s₀=1.0, s_i+1=(0.6+0.8i)*s_i
Center: 0+0i; Zoom = 0.5

I am going to close the Sequence Fractal series (plural) for now. I may add to it later, or start a part VI. I started the series on April 5, Sequence Fractals Introduction #1. It is time to take a break.

I have not followed up on few things in the introduction. I leave these behind with the intention to return someday. Although I have to confess that much of my "do someday" list will migrate to the "never got around to it" list. Here are a few of the things in the introduction that I will try to return to:

The above mentioned Sequence Fractals Introduction #1 directly draws a sequence.

Sequence Fractals Introduction #11 and Sequence Fractals Introduction #12 uses the sequence in the coloring algorithm on a normal fractal calculation.

Sequence Fractals Introduction #17 uses a sequence of functions rather the a sequence of numbers. (Yes in a mathematical sense there is an equivalence between numbers and functions. But the different starting points lead you down different paths.)

I said yesterday Sequence Fractals Part V #38 that I was going to continue with more artistic images derived from the formulas in this series. I will be posting them, but not calling them sequence fractals, or any other kind fractal. I am not sure what to call them, but they have certainly moved away from fractals to generative art or abstract art. I will not (usually) post the underlying formula. It is a distraction and a lot of manually copying number from one program to another for me. If you are every curious whether some was generating with a formula, and want information on it, just send me and email or ask in a comment.

To close, I reimagine Sequence Fractals Part V #18. Some small changes remove the trash in the upper right. Actually not removed, just painted over so it blends in. A nice transition from a raw fractal-like imaging to a more artistic interpretation.