$ z^2+c+s_i$ two cycle sequence.

In this introduction I want to provide a qualitative overview of sequence fractals. I will not be providing much detail on the formulas. Later I will build up an atlas with details on formulas and parameters.

Yes, that sounds good. I will go with it. And it may not be far from the truth. When I started this exploration I found a big new universe to explore. I started out without a master plan, and without keeping good notes. I still have the source code archived for each image. So I could extract all the detailed information. At this point, I do not think the value of the details is worth the effort to extract it.

After getting lost many times, I decided that to start building a map. I will follow up the introduction with the results from a more systematic exploration.

Familiar / Not Familiar

This looks like something you could find in the Mandelbrot set. But on a closer look you see it is something different.

Sequence fractals are not new. I came up with this idea 30+ years ago when I first started playing with fractals. I worked with it a little bit, and then moved on. There are so many other ways to mess up the standard fractal formula. All those other ways were calling for my attention. Occasionally, as here, I return to one of these early ideas and for a deeper look.

$ z^2+c+s_i$ ten cycle sequence.

So far, these all look like the Mandelbrot set with a few small additional features. It appears that way only because of my selection of what to keep and what to show here. Initially I was disappointed in the results. Most results were uninteresting. The wrong sequence, or the wrong choice of formula wipes out all fractal like features. I might have given up, except that I had invested too much time in writing the software.

So, baby steps. Start with the simplest formula and the simplest sequences.

$ z^2+c+s_i$ twenty cycle sequence.

Getting into unfamiliar territory

$ z^2+c+s_i$ non-repeating sequence

The sequence is bounded, I know $ |s_i| < 6$. I do not know if the sequence has a limit or a limit cycle.

I could figure that out, just not today. The sequence is generated by a recursive function. Looking at the function I can tell that it is bounded. The software generates the sequence, uses it to generate the picture, but does not print out the values or do any kind of convergence analysis.

I strongly suspect that the sequence converges to a single value.

$ sin(z)+c+s_i$

How much is due to replacing z² with sin(z) and how much is due to adding the sequence? I do not know, it is on my list to investigate someday.

$ z^2+c+s_i$ convergent sequence.

If my mental math is correct the sequence spirals into the point $ (1-i)/4$

Sierpinski coloring

Up until now, with the exception of the first post in the series, I have been using the sequence part of 'sequence fractal' in the iteration formula. Here the iteration is based on the good old z^2+c, but the coloring method incorporates the sequence.

Wikipedia describes a sequence that converges to the Sierpinski Triangle. The link also defines the Sierpinski triangle. The generating sequence has a random component. Each time you generate it you get a different set of numbers, but it always converges to the same set of points.

One such sequence is generated, then the normal fractal iterations starts. At each iteration step a test is done to see if the iteration value, z_i is close to the Sierpinski iteration, s_i. If it is, draw a dot there, the color of the dot depends on the iteration step, opacity is determined by how close.

Sierpinski coloring

I suspect Sierpinski has little to do with the final results. We might get the similar results with any similar sequence (bounded with a large or infinite number of accumulation points).

Sierpinski coloring

$ z^2+c+s_i$, with convergent sequence.

The sequence is defined recursively by $ s_i = 0.9 * s_{i-1} - i$ My rusty mental math says this converges to −10i.

$ (z/c)^2+s_i$, with four cycle sequence.

$ (z/c)^2+s_i$, with two cycle sequence.

The repeating sequence is  i, -1, i, -1, ...

Function sequence $ f_i(z)+c$, with ten cycle sequence

Now we have a sequence of functions rather than a sequence of numbers. This is an easy generalization. Many of the previous post could have been described as a function sequence with $ f_i(z) = z^2 + s_i$

In this case the sequence is a repeating ten-cycle of affine transformations ($ z \leftarrow a * z + b$). The ten functions are pulled randomly, with replacement, from a set of three.

Since everything is linear here, no squares, no trig functions, the straight lines are expected.

Function sequence $ f_i(z)+c$

Similar setup as yesterday, but with a non-repeating infinite random sequence of affine transformations.

Function sequence $ f_i(z)$ (without +c)

Six cycle of functions chosen at random from four different functions. Two affine transformations, z = z², and z = z + c.

I think there may be a lot of potential with this scheme. However in a very quick investigation, I found that infinite random sequence and long (10 or more) cycles do not produce good results. Everything escapes quickly. Also different random seeds produce very different results. I need to investigate this with deterministic, or at least less-random, function selection.

I should point out that this is not entirely new territory. With a different coloring scheme, and without the "+c", this starts to look like IFS and Flame fractals.