$ 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.