$ (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.

$ z^{2^{s_i}}+c$
s₀=1.0, s_i+1=(0.8+0.6i)*s_i
Center: -0.207+0.211; Zoom = 16

Compared to yesterday's picture Sequence Fractals Part V #19, the sequence still moves around the unit circle, but at a slightly smaller step. The palette has been greatly reduced.

Continuing yesterday's discussion, this image is messy, and lacks mathematical significance. Well, I like messy, so I am going to push on. But no more searching for mathematical significance. I will however continue to include information on the underlying formula in case someone wants to try this at home.

The colors in the limited palettes create a mood. The smaller transitions are less jarring. A few places of sharp contrast tease the viewer's focus, but does not overwhelm as the last image does.

$ z^{2^{s_i}}+c$
s₀=1.0, s_i+1=(0.6+0.8i)*s_i
Center: -1.141+1.458; Zoom = 4

Shallow zoom into the trash heap in the upper left of yesterday's Sequence Fractals Part V #18 picture.

Complex exponentiation is a multi-valued function. Yes, by definition a function must have be single valued. But sometimes in complex analysis that requirement is ignored. (One work around is the consider the function that returns sets of complex numbers. The set, i.e. the function value, may consist of more than one element, but the set itself is a singular, specific, entity.)

You are already familiar with this. 1 and -1 are both reasonable values for 1^0.5. If w is a rational number, say a/b in lowest terms, then there are b candidates for z^w. If w is irrational, or if w is complex and either the real or imaginary part is irrational, then z^w has infinite values. Of course the computer does not deal very well with sets and infinity. So we just pick one value and ignore the rest.

In summary, we cannot find critical points, we cannot find a meaningful escape radius. On top of that each of thousands of iteration steps involves an arbitrarily pick of one number out of an infinity of choices for the of exponentiation calculation. I think one could say that even though the pictures are generated with numbers, they are mathematically worthless.

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

Here is the usual home base view, horizontal range from -2 to 2, of yesterday's setup, Sequence Fractals Part V #17. Inside / capture coloring is turned off. The right side is in the capture set. The usual complex exponent cutting and chopping has totally shredded the upper right.

The start point for iteration is 0. I have no idea whether this is a critical point. Does the concept of a critical point even make sense here? Each iteration step is a different function. Each of those functions have a different set of critical points. Anyway, different starting points do not seem to make much difference in the result.

I have the escape radius set very large, 10000. Many orbits never leave this radius. But of the ones that do leave, do they stay gone? Or do they return as with rational functions. I suspect that they eventually return.

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

The start of this series Sequence Fractals Part V #1 was actually the third starting point I considered. Here is the second candidate. The stacked exponents seemed like a natural place to add an exponent to the traditional z²+c. After queuing up a few, I decided that $ z^{s_i}+c$ was a simpler place to start.

While no longer the starting point, these are still part of the story. The most noticeable traits are shared by the different placement of the exponent. In fact given the sequence s_i, we could create a new sequence r_i = $ 2^{s_i}$, and then use the previous formula, $ z^{r_i}+c$. Like I said, I did not think about these things when I started.

One of the favorite sequences, where s_i jumps around the unit circle is back. There is a large area of interest. This image is zoomed way out. The horizontal range is -250,250.

$ z^{s_i}+c$
s_i = 2.5, 0.5, -1.5 … (three step repeating)
Center:1.359+0.409i; Zoom = 6.4

The same formula as the previous two pictures. But with experimental coloring.

Orbit-escape detection is turned off. The negative exponent creates some similarities to Rational Fractions. See the discussion in Sequence Fractals Part V #5 and Sequence Fractals Part V #6. For the same reasons, iteration count coloring does not work here.

The coloring algorithm counts how many times the orbit gets close to zero. "Count" is inaccurate, it actually generates a weighted average of all the values that get close. The average value is a complex number. Each of the real and imaginary components are used to generate a color, and then the two colors are blended together.

For the record, here "close to 0" is defined as "|z|<0.9". That parameter is subject to much trial and error tweaking. Finding two palettes that blend nicely also takes some effort. The bottom line, today's picture is more about my aesthetic choices than mathematics.

$ z^{s_i}+c$
s_i = 2.5, 0.5, -1.5 … (three step repeating)
Center:-1.11937+0.07167i; Zoom = 1600

Same formula as yesterday's Sequence Fractals Part V #14, different neighborhood.

The orange blob looks like a distorted cubic multibrot. But there is nothing cubic in the formula. The exponents are 5/2, 1/2, -3/2. Seems unlikely, but perhaps the 3 in the numerator in the third exponent is doing it. I think the area surrounding the orange spot looks like ice crystals forming on glass on a cold winter night.

$ z^{s_i}+c$
s_i = 2.5, 0.5, -1.5 … (three step repeating)
Center:-1.16588+0.14008i; Zoom = 410

Now three repeating exponents. Each involving a square root, one is negative.

I really like this one. It is something that I would put into my abstract art collection. There is a range of densities, and levels of complexity. The viewer is challenged to enter the more complex and chaotic blue regions. But when that gets uncomfortable, focus snaps back to the larger features.

$ z^{s_i}+c$
s_i = 0.5, 3.0, … (two step repeating)
Center:-0.6144+0.660i; Zoom = 256

When the exponents alternate between 0.5 and 3.0, the opposite order as yesterday. The cubic multibrots in yesterday's picture Sequence Fractals Part V #12 are gone.

$ z^{s_i}+c$
s_i = 3.0, 0.5, … (two step repeating)
Center:-1.4347-0.1034i; Zoom = 50

When the exponents alternate between 3.0 and 0.5, the cubic multibrot minis show up. In the big picture (not posted), the central object is surrounded by mud. It is only near the outer edge where interesting distinct features can be found.

$ z^{s_i}+c$
s_i = 0.5, 2.5, … (two step repeating)
Center:-1.469-0.845i; Zoom = 16

When you do find something you like, it is hard to stop at just one. This has the same color scheme, and coordinates close to yesterday's picture, Sequence Fractals Part V #10.

A good question to ask is how much of this is from the non-integer exponents, and how much is due to the sequence fractal iteration (alternating two different exponents). I was prepared to say it is mostly due to the non-integer exponent. But before saying that I decided to do some experiments with the normal fractal algorithm on $ z^{2.5}+c$. The result looked almost like a typical polynomial iteration. There were obvious cuts, but the cuts were not very dense. In-between the cuts, and dominating the image, were the usual fractal swirlies. Perhaps I did not look in the right place, or choose the wrong exponent. But I am convinced that the sequence fractal aspect is an equal contributor.