$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-2.216607+-2.216607i , Zoom = 2048

Three cycle host 3.12 (3.13). Located above the two cycle near -2.2. Top feature in Sequence Fractals Part I #5.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-3.053155+0.209961i , Zoom = 10000

Here is 3.06 (also reflected 3.07) in yesterday's three-cycle list. It is the little dust speck above the c=-3 brot in the world map Sequence Fractals Part I #1.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-3.630601+0.i , Zoom = 400000

Grab the electron microscope and crank up the zoom into the center of Sequence Fractals Part I #10. There is the mini brot that hosts a three-cycle.

Here is a list of the thirty c values that host three-cycles.

3.00:  -3.630601,  0.000000
3.01:  -3.613512,  0.000000
3.02:  -3.595016,  0.000000
3.03:  -3.568618,  0.000000
3.04:  -3.272072,  0.000000
3.05:  -3.136341,  0.000000
3.06:  -3.053155,  0.209961
3.07:  -3.053155, -0.209961
3.08:  -3.012519,  0.060576
3.09:  -3.012519, -0.060576
3.10:  -2.768406,  0.000000
3.11:  -2.308309,  0.000000
3.12:  -2.216607,  0.033713
3.13:  -2.216607, -0.033713
3.14:  -2.147552,  0.000000
3.15:  -0.324744,  0.181271
3.16:  -0.324744, -0.181271
3.17:  -0.320181,  0.000000
3.18:  -0.024650,  0.495512
3.19:  -0.024650, -0.495512
3.20:  -0.006748,  0.135925
3.21:  -0.006748, -0.135925
3.22:  -0.004470,  0.435048
3.23:  -0.004470, -0.435048
3.24:   0.072243,  0.489918
3.25:   0.072243, -0.489918
3.26:   0.104301,  0.479303
3.27:   0.104301, -0.479303
3.28:   0.146654,  0.152514
3.29:   0.146654, -0.152514

Several are on the real axis, they can be found by looking at the other axis-specks in Sequence Fractals Part I #1, they all look similar to today's picture. 3.22 and 3.23 are found in Sequence Fractals Part I #8. In the Mandelbrot set, the largest real-axis mini, and the top and bottom bulbs host three-cycles. 3.05, 3.08, 3.09, are attached to c=-3, and 3.17 3.20. 3.21 to c=0 in the same way.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-3.630601+0.i , Zoom = 1600

Now move to the left side of yesterday's picture Sequence Fractals Part I #9, and increase the zoom by 50x. The image is boring, but it is interesting how flat things look even at this magnification.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-3.604653+0.i , Zoom = 32

Go back to the picture that kicked off this series, Sequence Fractals Part I #1, and notice the small speck on the far left. Today we zoom into that area. At the center is the two-cycle visited in Sequence Fractals Part I #4. At the far left of this image is a three-cycle, and the subject of tomorrow's post.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-0.02465+0.495512i , Zoom = 1000

Now zoom into the feature in the upper left of yesterday's image Sequence Fractals Part I #7. This c value hosts a three-cycle. $ f(f(f(z_0,c),c),c)=z_0\ and\ f(z_0,c) \ne z_0$ . The left side of the first equation in the definition is a degree 32 polynomial. Take away the two fixed points and we expect to find 30 c values that host three cycles.

$ 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