$ 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.

$ z^{s_i}+c$
s_i = 0.5, 2.5, … (two step repeating)
Center:-1.363-0.885i; Zoom = 16

Here is a zoom into the hot spot a little left and below center in Sequence Fractals Part V #8. Yesterday's picture Sequence Fractals Part V #9 demonstrated that the cut-density varies from very sparse to sub-pixel, the latter resulting in colored sand.

As the saying goes, this is a good thing and a bad thing. The good thing is that you can find whatever density that suits your mood. The bad thing is it takes a lot of searching and discarded images before finding something good.

$ z^{s_i}+c$
s_i = 0.5, 2.5, … (two step repeating)
Center:0.006+0.145i; Zoom = 25.6

In yesterday's picture, Sequence Fractals Part V #8, notice three hot spots near the center. One in the exact center and two a little to the left above and below the horizontal center line. Today's picture is a zoom into the top of the central hot spot.

The image suggests an abstraction of waves or trees. The white on the bottom suggests a fresh snowfall in the forest. That is serendipitous. I created this only to demonstration the cutting and chopping that results from non-integer exponents.

At the top the cuts are sparse. They get more dense in the middle to the point where the individual cuts are indistinguishable. Eventually the cuts are so dense, they are sub-pixel resulting in gray sand. I turned capture coloring back on, the white at the bottom is part of the capture set.

$ z^{s_i}+c$
s_i = 0.5, 2.5, … (two step repeating)
Center: 0.0+0.0i; Zoom = 0.1

Much more could be said about the rational functions in the previous posts. Rational Functions deserve their own topic, so I will not go there now. Let's switch to non-integer exponents, starting with half steps (square roots).

If you are familiar with complex numbers you know that the square root function has a discontinuity along the negative real axis. At every iteration step, for those orbits that are close to the negative real axis we put them into two groups. If you are above (or on) the axis you go here, if you are below then you go there. Of the points near -4, half will be sent close to 2i and the other half close to -2i. This creates discontinuities (tears) in our pictures.

Choosing the negative axis is a convention. The cut could be made along any line from 0 to ∞. (For that matter it could be any simple curve from 0 to ∞.) There is no way to avoid the discontinuity. It is always there, and always starts at 0. This is true of all non-integer exponents.

Today's picture is only here to demonstrate the cracks and tears that will be present for the rest of the pictures in this series.

$ z^{s_i}+c$
s_i = 2,-2, … (two step repeating)
Center:-1.23909+0.87230i; Zoom = 4096

Combining two steps of this sequence fractal give the rational function $ \frac{cx^4+2c^2x^2+c^3+1}{x^4+2cx^2+c^2}$. As with polynomial sequence fractals with a two-step sequence this is actually a picture of a degree 4 rational function.

These take a long time to render. Every point runs to the full iteration limit. My code is very inefficient, each step uses complex exponentiation. At higher iteration levels, most of the artifacts will disappear and the image will look like a sea of colored sand, containing a few smooth islands. (I was going to optimize the code and create such an image, but I ran out of time.)

$ z^{s_i}+c$
s_i = 2,-2, … (two step repeating)
Center:-1.059+0.792i; Zoom = 4

As mentioned yesterday, (non-polynomial) rational functions do not have points that "escape to infinity". That forces some changes when making pictures of the capture set. You cannot set a large radius and say when the orbit leaves that radius that it has escaped. In fact, the orbit is almost guaranteed to return. Perhaps another way to look at it is that since infinity, ∞, is just another point, nothing ever escapes.

To be sure rational functions still have critical points, and fixed and periodic points which may be attracting, repelling or indifferent. All the rich dynamics of polynomials are still present.

These images are generated without an "escape test", and also without an "inside color". There is of course an iteration limit. All orbits are run up to the full iteration limit. It does not make sense to color by iteration count, or to paint all the points that reach the iteration limit the same color. The coloring is based on a scheme I described in the series Smooth Colors in 2019. This scheme works well even when the concept of escape and capture are absent.

There are still attracting fixed points and cycles. With this coloring, they show up as smooth areas with gradually changing color.