$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:0+0i , Zoom = 1.

$ f(z_0,0)=z_0\ and\ f(z_0,-3) = z_0$, in other words, z₀ is a fixed point for c=0, -3. These are the centers of the two brots in the previous picture.

Here is a closer look at the shape centered at 0. It looks like a well-formed mset. The little disconnected snake-like satellites are about the only clue that this is not a normal Mandelbrot set. In the Mandelbrot set, all of the minis are connected by thin threads. Also notice that the shapes at 0 and -3 (off screen to the left) are not connected.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center: -1.8+0i, Zoom = 0.4.

The description "combine fractal and sequence" is intentionally vague. Part I will look at the simplest way to combine fractals and sequences, namely "just add". The sequence in this case is simple, alternating 1 and -1.

Define the follow functions:
$ f_0(z,c) = z^2+c+1$.
$ f_1(z,c) = z^2+c-1$.
$ f_c(z) = f(z,c) = f_1(f_0(z,c),c) = (z^2+c+1)^2+c-1$.
f₀ is the first step, of the iteration, f₁ is the second step. The iteration alternates between the two, f₀, f₁, f₀, f₁ … .
f_c(z) = f(z,c) combines the first two steps, and the even orbits are found by iterating f.

For now, we will hide the odd orbit points, and the fact that we have two different functions, and just look into f. f is a degree 4 polynomial, we can leverage some of the experience from z²+c.

The derivative with respect to z is $ f'_c(z) = 4z(z^2+c+1)$, so z=0 is a critical point (f'=0) for all values of c. $ z_0=0$ was used as the starting point for this all similar these pictures unless otherwise stated. A degree 4 polynomial has three critical points, the other two are the subject of a later post.

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.

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)+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.

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

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

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

$ 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.

Sierpinski coloring

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

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.

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

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

$ 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$ 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.

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

Getting into unfamiliar territory

$ 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$