, with four cycle sequence.
, with convergent sequence.
The sequence is defined recursively by My rusty mental math says this converges to −10i.
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).
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, zi is close to the Sierpinski iteration, si. If it is, draw a dot there, the color of the dot depends on the iteration step, opacity is determined by how close.
If my mental math is correct the sequence spirals into the point
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.
The sequence is bounded, I know . 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.
twenty cycle sequence.
Getting into unfamiliar territory
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.