$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:0.044082+0.464238i , Zoom = 10

Here is a zoom of the top center of Sequence Fractals Part I #2, the little wing above the c=0 shape. This is also home to two-cycle 2.04 in the list in yesterday's post. 2.05 is a mirror image reflected across the real axis.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-2.21745+0i , Zoom = 800

Same location as yesterday Sequence Fractals Part I #5, with 10x zoom and 50% more color entropy. This is another two-cycle.

A two cycle means $ f(f(z_0,c),c)=z_0 \ and \ f_c(z_0) \ne z_0$. The second iteration comes back to the start point. The second part of the definition excludes fixed points. f is degree 4 polynomial in z and degree 2 in c. With the fixed start point z₀=0 the double iteration in the definition is a degree 8 polynomial in c. It has 8 roots, with 6 left over after removing the fixed points (0 and -3).

Here are the (c-parameter values) of the six two-cycles

2.00:  -3.604653,  0.000000
2.01:  -3.079229,  0.000000
2.02:  -2.217445,  0.000000
2.03:  -0.186836,  0.000000
2.04:   0.044082,  0.464238
2.05:   0.044082, -0.464238

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-2.21745+0i , Zoom = 8

If you look back at Sequence Fractals Part I #1, you will see there is something in the middle, near c=-2, that was cut off from the zooms to the right Sequence Fractals Part I #2 and left Sequence Fractals Part I #3. Here is a zoom of that thing in the middle. I am not sure whether this is a part of the feature as c=-3 or c=0, or if it is something by itself. I am not sure how to define "part of". I think I will call this a part of c = -3. Notice the satellites are nicely placed at 90 and 45 degrees.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-3.604665+0i , Zoom = 1024

Here is a zoom into the dark spot on the left side of Sequence Fractals Part I #3.

Recall from Sequence Fractals Part I #1 the two-step iteration formula, $ f(z,c) = (z^2+c+1)^2+c-1$. Call the center of this image c_this = -3.604665+0i. Then f(f(0,c_this),c_this) = 0. I may (will definitely) get sloppy at times and say that c_this is a 2-cycle. But these are parameter space (c) pictures. z is the variable that is iterated and it is the z orbit has properties like convergence, divergence and being a two cycle. Not the parameter c. I should say that the z₀=0 orbit with the parameter value c = c_this, is a two cycle. Or maybe that c hosts a two-cycle.

But that is too much of a mouthful, so I will from time to time use the technically incorrect expression. And as long as I am being technical, this is a two-cycle of the two-step function. So it is a four-cycle of the original sequence fractal definition.

And as long as I am dwelling in the minutiae… I have been trying to use the same palette for this set of pictures. The previous palette version of this picture was too dark in the area around the mini. I doubled the color entropy (colors change twice as rapidly) to expose more detail.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-3+0i , Zoom = 3.2x.

Here is the other fixed point. Again, a normal looking mset with satellites. The satellites are much smaller. They also seem to appear in a simple pattern, shooting out from each tip or terminal point.

$ 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