Sequence Fractals Part V #31

412.jpg

(z+c)^{s_i}
s1=1, si+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.

Sequence Fractals Part V #29

411.jpg

(z+c)^{s_i}
s1=1, si+1=(0.9+0.1i)*si (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.

Sequence Fractals Part V #28

409.jpg

(z+c)^{s_i}
s1=1, si+1=(0.9+0.1i)*si (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 zw has infinitely many values. Actually zw 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.

Sequence Fractals Part V #27

408.jpg

z^{s_i}+c
si = 0.5+0.2i
si+1=-0.5(si2-si+2.75)
Center:-1.680+0.8466i; Zoom = 1600

The sequence is defined by a set of simple operations. Three affine transformations surrounding si2. When combined, assuming I can read the code correctly and simplify an expression, the result is si+1=-0.5(si2-si+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.

Sequence Fractals Part V #26

407.jpg

z^{s_i}+c
si = 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.

Sequence Fractals Part V #25

406.jpg

z^{s_i}+c
si = 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

Sequence Fractals Part V #24

405.jpg

z^{s_i}+c
si = 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.

Sequence Fractals Part V #23

404.jpg

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

Yesterday |si| = 1, today |si| = 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.

Sequence Fractals Part V #22

403.jpg

(z^2)^{s_i}+c
s0=1.0, si+1=(0.8+0.6i)*si
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.