Today I start posting the Storm series.

In the previous series, Sequence Fractals V, I declared that any mathematical significance was gone. I continued to post formulas and images with "if you do this then that happens" type of commentary. Leaning towards the explore side of the explore/construct scale.

Now I move to the construction side of the scale and embrace the chaos. "Storm" seems like the perfect theme.

$ 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.

$ 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+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=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, 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:-0.493-0.255; 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:-0.597-0.183; 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; 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+c)^{s_i}$
s₁=1, s_i+1=(0.2+0.5i)*s+(0.5+0.2i) (repeat after 10 steps)
Center:5.681-4.541i; Zoom = 328

After writing the words for yesterday's post, Sequence Fractals Part V #30, I zoomed into the right side. There is still a lot going on, but now there is also some consistency. The fractal like repeating of gently morphing shapes invites you to explore the details, while the overall white/blue/brown background anchors the frame.

$ (z+c)^{s_i}$
s₁=1, s_i+1=(0.2+0.5i)*s+(0.5+0.2i) (repeat after 10 steps)
Center:5.188-4.49i; Zoom = 1.6

When I created this, I was still trying to figure out the math. The location is a relatively long distance from the origin. Composition wise, there is too much going on here. No overall cohesion. But many areas are screaming out for magnification.

$ (z+c)^{s_i}$
s₁=1, s_i+1=(0.9+0.1i)*s_i (repeat after 22 steps).
Center:-0.5+.2i; Zoom = 0.05

Yesterday's picture, Sequence Fractals Part V #28, had an escape radius of 100. That is a good number for polynomial fractals, but not here. The mathematical reason for having an escape reason is invalid when you leave the realm of polynomials.

With this formula, the escape radius is just another parameter to tweak to create different effects. Compared to yesterday's image, this one is zoomed out and given a much larger escape radius, and larger max iterations. The palette was altered to create the white background.

$ (z+c)^{s_i}$
s₁=1, s_i+1=(0.9+0.1i)*s_i (repeat after 22 steps).
Center:-0.2-0.2i; Zoom = 0.25

Now let's switch to non-real exponents. Back in Sequence Fractals Part V #19, I said that if either the real or imaginary part of w is irrational, then z^w has infinitely many values. Actually z^w has infinitely many values if the angle is irrational (as in today's formula). I suspect this is also true anytime the imaginary part of w is non-zero. (But I am too lazy to verify that right now.) So today's equation is one of those where at each step we pick out one of an infinite number of choices for the exponentiation.

I do not wake up each morning, create a picture, write these words and make a blog post. That is only an illusion I create by scheduling the posts to come out at midnight each day. The next few pictures where created very early during the sequence fractal series. This one was created back in May. I am writing these words just a few days (about a week) before publishing the post.

I spent the last few months on the polynomial (positive integer exponent) variations. I put together a rough draft for this non-integer exponent set during that time. Of course I threw out the rough draft and have just been winging it.

Why do I bother to mention this? At the time I created these I was still trying to figure out and demonstrate some mathematical truth in the formula. So my mindset today is substantially different now than it was when I made the picture.

Still, it is part of the story, they take a long time to render, and I do not want to waste it.

$ z^{s_i}+c$
s_i = 0.5+0.2i
s_i+1=-0.5(s_i²-s_i+2.75)
Center:-1.680+0.8466i; Zoom = 1600

The sequence is defined by a set of simple operations. Three affine transformations surrounding s_i². When combined, assuming I can read the code correctly and simplify an expression, the result is s_i+1=-0.5(s_i²-s_i+2.75).

This setup, adding affine transformations (az+b) to a base sequence results in many knobs (parameters) to tune. Each producing continuous changes.

Here, a variety of spirals are caught in a spider's web.

$ z^{s_i}+c$
s_i = 0.5, 1.5, 2.5 … (three step repeating)
Center:-0.291+0.109i; Zoom = 50

Another simple sequence, 1/2, 3/2, 5/2. Most of these, especially the ones with positive exponents have a stable area near 0. There is a high-gradient area near the edge of the stability. Compare with Sequence Fractals Part V #20.

Here the colors suggest a kind of duality between the two areas. Pan and zoom, as well as setting the escape threshold, can be used to find a desired range of cut size and density. Both sides of the chasm have areas the scream out for deeper exploration.

$ z^{s_i}+c$
s_i = 2.5, 1.5, -0.5 … (three step repeating)
Center:0.097006+0.123293i; Zoom = 1600

Adding a third, negative exponent. This looks like the Minimalist Geometric Abstractions series I posted in 2019. The previous series is on the construct end of the discover/construct axis. It would be interesting to merge the two approaches.

As with the previous image, Sequence Fractals Part V #24, it is possible to dial in the density here. However, it never looks as clean as yesterday's image

$ z^{s_i}+c$
s_i = 0.5, 2.5, … (repeating)
Center:-1.3322+0.1473i; Zoom = 128

Now back to the repeating two-step square root sequences. See Sequence Fractals Part V #10

The story so far: Several sequence fractal formulas with the sequence as an exponent have been examined. There are examples of positive integer, integer, rational, and irrational exponents. The pictures get messier and math value become more doubtful with this progression.

In the last few days I have been dissecting my imagined dichotomy of discovery vs construction. Rather there is a process: exploration, discovery, knowledge, tool, construction.

So I am just going to embrace it, and spend the rest of this series at various points along that process. Can these equations be used to create interesting abstract art? Can something be extracted that I can add to my toolbox? I am going to explore, with more interest in finding future potential, than producing a finished masterpiece.

For the simple case, two simple fractions. 1/2 and 5/2, repeating. It is possible to get some control over the formula. The result exhibits the mosaic appearance from the non-integer exponents, while retaining the fractal self-similarity.

$ (z^2)^{s_i}+c$
s₀=2.0, s_i+1=(0.8+0.6i)*s_i
Center: -0.102+0.502i; Zoom = 32

Yesterday |s_i| = 1, today |s_i| = 2.

So yesterday Sequence Fractals Part V #22 I proposed the thesis fractal art = discovery and abstract art = construction. Today I have not a full refutation, rather that these are two steps in both cases, and generally in creating art.

While everyone starts with z²+c, the fractal artist soon moves beyond that and acquires a "palette" of formulas. Initially there is a period of exploration and discovery with each formula. Eventually the fractalist learns what to expect from each. After this point, the process becomes using that knowledge to select a formula and feature that most closely matches the artist's mood and intent.

Addition constructive steps follow. There is framing and scaling. Do I want the main feature dead center, or should I put is off to the side or corner? How large should it be? What colors should I use? Do I want a lot of colors or just a few?

Now consider the paint-splash variety of abstract art, (just as an example and because it is fun). Suppose I decide to make paint splash art. My first attempt will disappoint. It does not look like what I had in my mind. I try some experiments, more or less paint, thinner or thicker paint, different splash angles. I learn (discover) the nuances of splashing paint. The process is not a direct path from mental image, original intent, to final product. It detours though experimentation and discovery, and usually results in several variants, all better than the original intent. Along the way I have added paint splashing skills and knowledge to my toolbox. (The more traditionally minded can of course replace paint splashing with study of and practice with different brush types and sizes, and reach the same conclusion.)

Again, today's picture lies somewhere between discover and construction. My mind was blank, no specific intent other than to explore. Eventually I found these thorny objects and decided to work with them.

$ (z^2)^{s_i}+c$
s₀=1.0, s_i+1=(0.8+0.6i)*s_i
Center: -0.102+0.502i; Zoom = 32

I usually think of fractal art as discovery and abstract art as construction.

With fractals, I start with a well-known mathematical formula, explore it, and discover interesting landscapes. With abstract art I try to construct an image that is already in my mind. Put a line here, a circle over there, a squiggle between them, the splash paint on top.

The medium, computer or canvas does not matter. In theory, one could do the fractal calculations by hand and use the results to paint on a canvas. A lot of abstract art, in particular generative art is created with a computer. It is probably easier to draw lines and circles with a pencil or a brush than a computer. It is certainly easier to splash real paint than virtual paint. I choose to use a computer because I am a computer programmer, because my fractal program already had all the basics, and so that I do not have to clean all that stray paint off the floor and walls.

Discovery and construction.

What does any of this have to do with today's picture? There is no mathematical object behind it. It does not resemble my original intent. It lies somewhere in-between discovery and construction.

$ (z^2)^{s_i}+c$
s₀=1.0, s_i+1=(0.8+0.6i)*s_i
Center: 0+0i; Zoom = 0.004

Here is the original starting point for this series. The previous formula, $ z^{2^{s_i}}+c$, was my intent. I messed up the order of operations in the program code. I have a few images with this formula, and I felt compelled to first show the big picture view.

You may be familiar with the exponent rule (x^a)^b = x^(ab). The rule works for real numbers, but not for complex numbers. The multi-value nature of complex exponentiation gets in the way. Try x=-i, a=2, b=1/2 as an example.

The image is here only to introduce the formula. It is not very pretty, and I am not going to talk about mathematical significance, so there is nothing more to say.