$ 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.

$ z^2+s_i*c$
s_i = -1.27i, 1, (repeating)
Center:-0.351+0.546i , Zoom = 25.6

Here is a zoom into the upper left area of yesterday's picture Sequence Fractals Part IV #5 .

Continuing yesterday thoughts…

The '*s_i' and '+s_i' variations of two-step sequence fractals are actually different parameterizations of the universe of (affine conjugate equivalence classes of) quartic polynomials. This is fundamentally different than the pan and zoom views of the Mandelbrot set.

Anyway, I think (at least for today, I change my mind daily) that it would be better to dig into the different parameterizations of quartic polynomials as a separate topic, which I may or may not do someday. Even a small understanding of the general situation will help inform the exploration of specific parameterizations. Without that even minimal understanding, it feels like wandering in the dark.

Besides I am now over four months of posts on sequence fractals, with at least another month and a half to go. A lighter touch on the remaining topics is called for.

The next few posts are random views of z²+c*s_i two step sequence fractals.

$ z^2+s_i*c$
s_i = -1.27i, 1, (repeating)
Center:1.0+0.0i , Zoom = 0.5

I have to confess that I am undecided on where to go next. The '*s_i' and '+s_i' cases are different. These deserve the full treatment as the +s_i case in Part I and Part II. That includes looking at the Julia sets. Finding Cycles and Misiurewicz points. Exploring how the C and M points move around as the sequence changes. Look at pictures of the other critical points. (These all start at 0 which is always a critical point, the other critical points depend on c and move around. That means extra programming for me.)

On the other hand, both of these classes, '*s_i' and '+s_i' are part of the universe of slices of the connectivity locus of quartic polynomials. In some sense these two classes are two parts of some much larger whole. I want to avoid "been there, done that".

To be clear, there are an infinite number of genuinely unique ways to parameterize the space of quartic polynomials. And in each of them, there are an infinite number of genuinely unique 2D cross sections, or slices. And any one of those infinite times infinite number of slices can be selected to explore with pan and zoom.

Contrast this the quadratic case. There are only a small number (afaik) of useful parameterizations, z²+c being ubiquitous. There, the whole universe exists in a single 2 dimensional slice of the parameter space, which we explore with pan and zoom.

To be continued.

$ z^2+s_i*c$
s_i = 1, -1.598, (repeating)
Center:0.9978+0.0i , Zoom = 160

A zoom near the center of yesterday's image, Sequence Fractals Part IV #3 where the tips of the two 'brots collide.

$ z^2+s_i*c$
s_i = 1, -1.598, (repeating)
Center:1.0+0.0i , Zoom = 0.5

Another repeating two step sequence. Again, if we look at every other iteration, we are basically looking at quartic polynomials.

Back in Sequence Fractals Part III #6 I was musing about the different ways to set up the parameter spaces for quartic polynomials. The full parameter space always has three complex (six real) dimensions. I am going to skip the details for now, but if you write down any degree four polynomial in z with parameters a,b,c sprinkled into the coefficients, you will almost always define a parameterization that captures almost all of the affine-conjugate classes.

If we choose to parameterize the quartic space with $ z^4+2cz^2+bz+c^2+ac$. (Not that anyone would do this unless they were playing with two-step sequence fractals based on $ z^2+s_i*c$.) Then today's picture arises from taking a c-slice of the parameter space, fixing a = -1.598 and b = 0.

In general a c-slice with b=0 is equivalent to the sequence fractal with two-step sequence 1,a.

$ z^2+c+s_i$
s_i = 1, 2, … (two steps, repeating)
Center:-0.4617+0.2148i , Zoom = 512

Here is a zoom into the previous picture, Sequence Fractals Part IV #1.