$ z^{s_i}+c$
s_i = 2, 3, … (two step repeating)
Center:0+0i; Zoom = 0.4

Sequence Fractals Part V will feature formulas where the exponent of z contains the sequence variable. As before, we will start gently, then go crazy.

Here is the simplest such formula, the sequence variable replaced the exponent 2 in the normal quadratic formula. Ironically, this form did not occur to me until I tried several others, and then started to prepare posts for the blog. So the blog will not be following the same path as my personal journey through the exponents.

Today's sequence consists of two repeating integers. The result is similar to the previous Sequence Fractal sets. As long as we stick with positive integers in the sequence, each iteration is a polynomial function.

$ z^2+s_i*z+c$
s₀=0.5, s_i+1=-2.0*s_i²+0.7007
Center:-0.891+0i; Zoom = 256

The sequence adder is increased by 0.0001 compared to yesterday's image, Sequence Fractals Part IV #35. The capture set is gone. Gone at least in this area, and as far as I can tell, everywhere else.

The high density in both of these images is probably a result of choosing a sequence so close to the capture/no capture boundary.

$ z^2+s_i*z+c$
s₀=0.5, s_i+1=-2.0*s_i²+0.7006
Center:-0.891+0i; Zoom = 256

Another sequence fractal formula variation. A z term is added and it is multiplied by the sequence value. The constant sequence 0 would give rise to the usual quadratic / Mandelbrot situation.

I tried several different sequences, and a few zooms on each. I found nothing that stood out as distinctly different from the other formula variations presented so far.

This is the first of two zooms along the negative real axis. Here, the sequence slowly converges to a 32 cycle. Tomorrow's image, Sequence Fractals Part IV #36 is the same location with a small increase in the adder.

$ s_i*z^2+c$
s₀=0.1i, s_i+1=-3.0*s_i²+0.1
Center:-15.61563-0.37619i; Zoom = 13107

Here is a deep zoom on the left side of yesterday's picture, Sequence Fractals Part IV #33. Just a little above the center line, the real axis.

$ s_i*z^2+c$
s₀=0.1i, s_i+1=-3.0*s_i²+0.1
Center:-2+0i; Zoom = 0.04

This one uses the a*s_i²+b sequence generator with a = -3, b = 0.1. The sequence converges quickly to a small value. The result is a large fractal. Large in the sense that the capture area extends well beyond the |c|<=2.0 that is typical with quadratic polynomials. The horizontal range of this image is (-25,+25).

$ s_i*z^2+c$
s₀=0.1i, s_i+1=(0.5+0.1i)*s_i+(0.5-0.1i)
Center:2.7243-0.0262i; Zoom = 3276

Another zoom, same setup as yesterday. Sequence Fractals Part IV #31

$ s_i*z^2+c$
s₀=0.1i, s_i+1=(0.5+0.1i)*s_i+(0.5-0.1i)
Center:-1.312+0.069i; Zoom = 256

This sequence starts at 0.1i and converges to a value close to 1.0. The convergence is quick, but not a straight line.

Again, since the sequence converges quickly, the image looks almost like a regular fractal.

$ s_i*z^2+c$
s₀=1.5, s_i+1=0.9*s_i+0.1
Center:-0.7348+0.180i; Zoom = 128

The sequence multiplies z² again. The sequence starts at 1.5 and converges to 1.0 along the real axis.

Since the sequence converges quickly to the multiplicative identity, 1.0, the resulting images look like Mandelbrot set zooms.

$ s_i*z^2+c$
s_i=1.0, 0.001 repeating
Center:-12.2+0.0i; Zoom = 3

Introducing a new variation on the sequence fractal formula. The z² part is mulplied by the sequence value s_i.

To get my footing with the new formula I started out with a simple two step repeating sequence. The "do-nothing" value of one alternates with the a very small value, 0.001. If an orbit tries to escape, every other step pulls it back. The result is that a large area around the origin is captured. Today's view is far out to the left (-12 units) on the negative real axis.

$ z^2+c*s_i$
s₀=0.3, s_i+1=(-0.5+0.25i)*s_i²+(1.2+0.5i)
Center:-0.8244795+0.27693854i; Zoom = 102400

Here is a zoom into the top of the mountain peak in the lower center of Sequence Fractals Part IV #27.

I feel compelled to try to describe this when I should just let the image speak for itself. This has some characteristic of the jigsaw puzzle fractals. See Sequence Fractals Part III #27 and Sequence Fractals Part IV #16 for a description of "jigsaw puzzle fractals". The areas that I image as the puzzle pieces have a fractal spirals along the edges. The "drop cloth" fractals, see Sequence Fractals Part IV #22 and Sequence Fractals Part IV #23 also have fractal shapes on the edges of puzzle pieces. This one however seems much more organized than the splashed paint appearance of the others.

$ z^2+c*s_i$
s₀=0.3, s_i+1=(-0.5+0.25i)*s_i²+(1.2+0.5i)
Center:-0.82443+0.27707i; Zoom = 5120

Here is a zoom into the left side of the comb in the upper left of the previous image, Sequence Fractals Part IV #26

Are those puzzle pieces on the left?

$ z^2+c*s_i$
s₀=0.3, s_i+1=(-0.5+0.25i)*s_i²+(1.2+0.5i)
Center:0+0i; Zoom = 0.5

Some more playing around with complex parameters in the a*s_i²+b sequence generator.

I have not investigated the long term behavior of this sequence. Given the large capture set and Mandlebrot-like right side, I suspect it the sequence converges to very short cycle, most likely a single point.

A couple of zooms to follow.

$ z^2+c*s_i$
s₀=0.4, s_i+1=(-0.7-0.6i)*s_i²+(0.5+0.1i)
Center:-1.971+0.325; Zoom = 256

Complex numbers are used in the sequence generating formula for the sequence.

$ z^2+c*s_i$
s₀=0.4, s_i+1=-1.5*s_i²+0.9
Center:0.50789+1.055828; Zoom = 10240

I have generated and discarded many images in the last few days, trying to find something unique. Most are interesting, but feel too much like "variation on a theme". Along the way I got distracted, that happens a lot. I pulled up a palette that I was working on earlier this year for non-fractal abstract art. I tweaked it some to get a better fit for fractals, and here is the result.

As for the main theme, this is another sequence with a which converges to a long cycle. The big difference is just the new colors.

$ z^2+c*s_i$
s₀=0.0, s_i+1=-0.5*s_i²+2.8026
Center:-0.33725+0.00007i; Zoom = 5120

Another drop cloth fractal. The same generating formal as yesterday, but with a different seed. I have made minor tweaks to the location, zoom and color.

$ z^2+c*s_i$
s₀=0.4, s_i+1=-0.5*s_i²+2.8026
Center:-0.33742+0.00008i; Zoom = 2560

Now a very small change to the adder in the sequence generating formula. The capture set is gone. The big picture, zoom = 1, look very much like Sequence Fractals Part IV #18 except the main body is no longer white. It is a palette-dependent non-capturing color.

By my spreadsheet investigation, The sequence converges slowly to a 32-cycle. It may have bifurcated to a 64-cycle.

The result is on the boundary between normal swirly fractals and the jigsaw fractals. It reminds me of a paint drop cloth after a sloppy painter has used it for many projects.

$ z^2+c*s_i$
s₀=0.2.36, s_i+1=-0.5*s_i²+2.8
Center:-0.359+0i; Zoom = 320

Same generating sequence as the previous few pictures. With the starting point selected to speed up the convergence of the cycle. (The starting point is within 0.01 of one of the points on the 32-cycle.) As expected, the quicker convergence puts the sequence closer to a simple repeating (although somewhat long cycle) sequence, and the capture set is larger.

$ z^2+c*s_i$
s₀=0.4, s_i+1=-0.5*s_i²+2.8
Center:0.0305+0.2902i; Zoom = 410

Zoom of Sequence Fractals Part IV #18

$ z^2+c*s_i$
s₀=0.4, s_i+1=-0.5*s_i²+2.8
Center:0.0311229+0.2895398i; Zoom = 105000

A zoom into yesterday's picture Sequence Fractals Part IV #18

I had put this formula into a spreadsheet and generated the first forty values. There were no patterns, It appeared to be a chaotic sequence. In other situations a chaotic sequence leads to "jigsaw puzzle fractals". Long story short, the sequence is slowly converging to a long cycle. When I extended the spread sheet to 300 rows the sequence appears to be a freshly-bifurcated 32 cycle.

If one were to combine the 32 steps like we did with 2-cycle sequences, we would get a billion degree polynomial. As ridiculous as that number sounds, the pictures do seem normal, about what you would expect from a polynomial.

$ z^2+c*s_i$
s₀=0.4, s_i+1=-0.5*s_i²+2.8
Center:-0+0i; Zoom = 1

Now the sequence is generated by a*s_i²+b, a and b real numbers. You may be familiar with sequences like this, the Logistic Map and the real axis of the Mandelbrot set are two examples.

The parameters can be tweaked to produce a variety of behaviors in the sequence. For a < 0 and b > 0 and small the sequence will converges to a point. As b increases it goes through a period doubling phase where is converges to a 2 cycle, then 4, 8 as so on. Eventually after hitting every power of two, it becomes chaotic. The Wikipedia page linked above describes this phenomena in more detail.