$ 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.

$ z^2+c*s_i$
s₀=1.0, s_i+1=(-0.96+0.28i)*s_i
Center:0.138+0.0.402i; Zoom = 64

A small zoom into the previous picture, Sequence Fractals Part IV #16.

$ z^2+c*s_i$
s₀=1.0, s_i+1=(-0.96+0.28i)*s_i
Center:0.121+0.419i; Zoom = 32

Here is a zoom into the top edge of the circle in the previous picture, Sequence Fractals Part IV #15.

I call these jigsaw puzzle fractals. They does not have the normal behavior where just a few points around the edge of the capture set escape each step, resulting in smooth colors and swirls. Instead large chunks escape on distinct iteration steps. The result looks like fitting together distinctly colored puzzle pieces.

Here are some earlier examples of "jigsaw puzzle" fractals: Sequence Fractals Part III #27, Sequence Fractals Part III #38

$ z^2+c*s_i$
s₀=1.0, s_i+1=(-0.96+0.28i)*s_i
Center: 0+0i; Zoom = 0.5

Next up: Non-periodic sequences. This one is based on the 7-24-25 Pythagorean triple. The Pythagorean triples provide an easy way to find rational coordinates that generate an irrational angle. The sequence moves around the circle with radius 1.0 by a rotation of an irrational angle.

I used a similar sequence with $ z^2+c+s_i$ earlier, Sequence Fractals Part III #27 for example.

With the '*s_i' formula, almost all of these look like a circle. That makes some sense. The sequence s_i is dense in the unit circle. Informally that means it is everywhere. Formally, pick any point on the circle and your favorite small $ \epsilon $ and there is a point in the sequence within $ \epsilon $ of your point.

All c values (pixels) with a common radius will behave similarly. |*s_i| = 1.0. If r > 0, |c| = r, then the adder at each step, +c*s_i has |c*s_i| = r. The c*s_i are dense in the circle with radius r. For the most part, only |c| matters; all c's on a given circle have the same eventual fate, capture or escape. It is only near this circular capture/escape border the angle of c makes a difference.

If the colors were adjusted they would look like the sun with solar flares.

$ z^2+s_i*c$
s_i = 1, -.5, 1 (repeating)
Center:0.58873-0.26113i , Zoom = 4096

Zoom into the left side of yesterday's image, Sequence Fractals Part IV #13.

$ z^2+s_i*c$
s_i = 1, -.5, 1 (repeating)
Center: 0.58956-0.26120i , Zoom = 1024

Here is a look at the right side of the main body, Sequence Fractals Part IV #10. If you want to navigate from that image, locate the largest broken bulb attached to the lower right of the main object. This image is just above the top surface of that bulb.

$ z^2+s_i*c$
s_i = 1, -.5, 1 (repeating)
Center:-0.9769+0.0566i , Zoom = 256

A look at the area just above the object in the upper left of yesterday's picture Sequence Fractals Part IV #11

$ z^2+s_i*c$
s_i = 1, -.5, 1 (repeating)
Center:-0.916+0i , Zoom = 8

Zoom into the left side of yesterday's image, Sequence Fractals Part IV #10.

$ z^2+s_i*c$
s_i = 1, -.5, 1 (repeating)
Center:0+0i , Zoom = 0.5

Now a three step repeating sequence. Combining three steps results in a degree 8 polynomial. Don't worry, I am not going to go there. This specific formula and the small handful of zooms to follow will have to serve as a quick introduction

$ z^2+s_i*c$
s_i = 1, 1.2650895i, (repeating)
Center:-0.035238+0.0.816426i , Zoom = 1600

Almost the same setup as yesterday's picture Sequence Fractals Part IV #8, with a very small value change to the second value in the sequence.

My first attempt after changing the sequence value resulted in an almost totally white image. I had to increase the escape iteration count 10 fold to start seeing the details.

With the high resolution and large iteration count, these take over a day to render. I had to turn off anti-alias to have something ready to publish today.

$ z^2+s_i*c$
s_i = 1, 1.26509i, (repeating)
Center:-0.035238+0.0.816426i , Zoom = 1600

Zoom into the bridge near the center of yesterday's picture, Sequence Fractals Part IV #7.

Note: Yesterday's picture is a bit misleading. There appears to be a white bridge (points in the capture set) between the two bodies. Upon increasing the iteration count, I discovered that the two bodies were not actually connected. Most of the points in this area escape, but they take a very long time before deciding to leave.

$ z^2+s_i*c$
s_i = 1, 1.26509i, (repeating)
Center:0.0+0.5i , Zoom = 0.5

Another sequence, again alternating between a real number and pure imaginary.