Month: August 2021

s_i = -1.27i, 1, (repeating)
Center:-0.351+0.546i , Zoom = 25.6

Here is a zoom into the upper left area of yesterday’s picture Sequence Fractals Part IV #5 .

Continuing yesterday thoughts…

The ‘*s_i‘ and ‘+s_i‘ variations of two-step sequence fractals are actually different parameterizations of the universe of (affine conjugate equivalence classes of) quartic polynomials. This is fundamentally different than the pan and zoom views of the Mandelbrot set.

Anyway, I think (at least for today, I change my mind daily) that it would be better to dig into the different parameterizations of quartic polynomials as a separate topic, which I may or may not do someday. Even a small understanding of the general situation will help inform the exploration of specific parameterizations. Without that even minimal understanding, it feels like wandering in the dark.

Besides I am now over four months of posts on sequence fractals, with at least another month and a half to go. A lighter touch on the remaining topics is called for.

The next few posts are random views of z²+c*s_i two step sequence fractals.

s_i = -1.27i, 1, (repeating)
Center:1.0+0.0i , Zoom = 0.5

I have to confess that I am undecided on where to go next. The ‘*s_i‘ and ‘+s_i‘ cases are different. These deserve the full treatment as the +s_i case in Part I and Part II. That includes looking at the Julia sets. Finding Cycles and Misiurewicz points. Exploring how the C and M points move around as the sequence changes. Look at pictures of the other critical points. (These all start at 0 which is always a critical point, the other critical points depend on c and move around. That means extra programming for me.)

On the other hand, both of these classes, ‘*s_i‘ and ‘+s_i‘ are part of the universe of slices of the connectivity locus of quartic polynomials. In some sense these two classes are two parts of some much larger whole. I want to avoid “been there, done that”.

To be clear, there are an infinite number of genuinely unique ways to parameterize the space of quartic polynomials. And in each of them, there are an infinite number of genuinely unique 2D cross sections, or slices. And any one of those infinite times infinite number of slices can be selected to explore with pan and zoom.

Contrast this the quadratic case. There are only a small number (afaik) of useful parameterizations, z²+c being ubiquitous. There, the whole universe exists in a single 2 dimensional slice of the parameter space, which we explore with pan and zoom.

To be continued.

s_i = 1, -1.598, (repeating)
Center:0.9978+0.0i , Zoom = 160

A zoom near the center of yesterday’s image, Sequence Fractals Part IV #3 where the tips of the two ‘brots collide.

s_i = 1, -1.598, (repeating)
Center:1.0+0.0i , Zoom = 0.5

Another repeating two step sequence. Again, if we look at every other iteration, we are basically looking at quartic polynomials.

Back in Sequence Fractals Part III #6 I was musing about the different ways to set up the parameter spaces for quartic polynomials. The full parameter space always has three complex (six real) dimensions. I am going to skip the details for now, but if you write down any degree four polynomial in z with parameters a,b,c sprinkled into the coefficients, you will almost always define a parameterization that captures almost all of the affine-conjugate classes.

If we choose to parameterize the quartic space with . (Not that anyone would do this unless they were playing with two-step sequence fractals based on .) Then today’s picture arises from taking a c-slice of the parameter space, fixing a = -1.598 and b = 0.

In general a c-slice with b=0 is equivalent to the sequence fractal with two-step sequence 1,a.

s_i = 1, 2, … (two steps, repeating)
Center:-0.4617+0.2148i , Zoom = 512

Here is a zoom into the previous picture, Sequence Fractals Part IV #1.

s_i = 1, 2, … (two step repeating)
Center:-0.5+0i; Zoom = 1

Now, on to Sequence Fractals, Part IV. Next up, I want to experiment with different ways to incorporate the sequence into the basic fractal formula. Here, the parameter c is multiplied by the sequence step, s_i.

As before, let’s start with baby steps, simple two step sequences.

If two steps of the iteration are combined we get a quartic polynomial. This are some similarities to the earlier case Sequence Fractals Part I #1 of z²+c+s_i with a two-step sequence, which also gave rise to quartic polynomials.

As in the earlier case, 0, our starting z value, is a critical point.