$ 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.

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$ 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.

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

Many of these have a familiar / not familiar feel. At first glance this looks like it could be found in the Mandelbrot set. Looking closer you see the small separated features that hover above each terminal point.

The focus in this series is on the math, not the art.

As usual, I need to immediately qualify that statement. The math is not deep. I mix up sequences and fractals in various ways and ask "what does this looks like". A picture is generated which provides the answer "it looks like this". No deep proofs or theory. Just pictures of mathematics.

On the other side of the coin, the pictures still have some artistic value. This is more discovery and less constructive than when I set out to create art. Even on the discovery side I spend less than the usual amount of time framing and coloring the fractal. But still, I hope you find these more interesting than a graph of a parabola.

$ z^2+c+s_i$ where $ s_i$ is a two cycle sequence.

There are many ways to combine sequences and fractals. When I started on this adventure I made a list of several methods to investigate. $ z^2+c+s_i$ probably qualifies as the simplest way to incorporate a sequence. Also a two-cycle sequence, (two repeating numbers a,b,a,b,a,b...) is perhaps the simplest non-trivial sequence.

Already with this simple setup, it is clear that we are not looking at your father's Mandelbrot set.

I think it is obvious, but just in case. $ s_i$ is the i^th number in the sequence. The i^th step in the fractal calculation is $ z \leftarrow z^2+c+s_i$

Sequence Fractals

In mathematics a sequence is an infinite ordered list of numbers. https://en.wikipedia.org/wiki/Sequence

Sequences are cool, I decided to look for ways to combine sequences and fractals.

Of course I can take any topic and wonder how to combine it with fractals so this may not be a brilliant inspiration, rather just the normal inner workings of a fractal addled brain. And actually my short definition of sequence is wrong for multiple reasons. Technically sequences can be infinite or finite and do not have to be numbers. You could have a sequence of turtles. Let's keep it simple and stick with infinite sequences of complex numbers.

Oops, still not quite there. For practical time and space reasons, we won't be using the whole infinite sequence, just the first million or so values.

Today's picture? This is a picture of a particularly messy sequence. The sequence is bounded but does not converge. The picture is a portion of the complex plane with a fuzzy dot for each number in the sequence.

Fractal 198

And another

Fractal 197

Yet another fractal spiral

Fractal 196

At this point I gave up on my software testing and just embraced the fractal creation. The smooth colors are back.

Fractal 195

Another experiment. This time intentionally creating color bands between the fractal arms.

Fractal 194

Another one, In the same neighborhood.

Fractal 193

Same image with monochrome palette.

Fractal 192

This fractal uses iteration count coloring. If you look closely in the big blue areas you can see some color banding.

I was testing software and not in "artist mode". I prefer smooth coloring techniques for fractals, but did not want the where the extra smooth color code getting in the way.

As often happens, I am easily distracted and forgot I was in test mode and wandered off exploring fractal space.

Shattered

I really have no idea what to say about this one. Perhaps it should be titled "Artist Goes Off the Rails"

Spray Splatter Squares 2

Here I added some chaos to the splatter. The splatter is not uniformly random. There are blank spots and clumping and brushing. Interest is generated by screaming "fix me".

Spray Splatter Squares

As before, an experiment that produced a nice piece of abstract art. The idea and technique call for more study and development. Just not this week.

Displaced Lines / Faded Bluejeans

A variation of the previous experiment. It is interesting enough to stand on its own. I think it might work better as a background for something else. Again, that is something that I might look into as some unspecified future date.

Displaced Lines

This one was a test pattern for a distraction that I was letting distract me. Despite the origin I think it makes a nice piece of abstract art.

Fractal Color Experiment

This one is getting close to a "Complexity Limit". The fun decorations around the fractal work well here. But the fractal detail is starting to get lost. When I try this technique with a deeper, more complex fractal, the fractal details become random pixel dust. There are ways to fix that, ways to smooth out the transition from the background to the foreground. I will explore that rabbit hole later.

Fractal Color Experiment

Trying out ways to color the dead space "behind" and escape time fractal. I will probably pursue this further in the future. I was in the middle of something else. So for now, it is a placeholder for a future work.