$ z^2+c+s_i$
s₀=1.0, s_i+1=-0.5*s_i
Center-1.416536+-1.07538i; Zoom =8384

The first of two zooms.

The zooms are not much different from zooms for the quadratic case. As s_i+1 approaches 0, the iteration step get close to the usual $ z^2+c$ iteration.

$ z^2+c+s_i$
s₀=1.0, s_i+1=0.5*s_i
Center:0.0+0.0i; Zoom =0.5

Yesterday's sequence reset after 30 steps. Without a reset, arithmetic sequences head off to infinity. Fractal pictures with them just have some ghostly shadows, no chaos. The first few orbit steps for small numbers bounce around, but soon everything heads off to infinity. Only light shadows from the early iteration remain.

The same holds for geometric sequences that go to infinity. But not all geometric sequences blow up. Here is one that converges to 0. Usually the fractals built on these sequences, such as this one, have a large capture set, and no need to reset after a few steps.

$ z^2+c+s_i$
s₀=0.0, s_i+1=s_i+0.1. Reset after 30 steps.
Center:0.0+0.0i; Zoom =0.5

At 30 steps between resets the white capture set is gone.

I found some cracks that host minis, similar to yesterday's post. Sequence Fractals Part III #19 Much smaller than yesterday's, but after adjusting the scale they were so similar they do not deserve a separate post.

$ z^2+c+s_i$
s₀=0.0, s_i+1=s_i+0.1. Reset after 25 steps.
Center:-1.2101-0.41833i; Zoom = 32

Found a mini in a crack.

$ z^2+c+s_i$
s₀=0.0, s_i+1=s_i+0.1. Reset after 25 steps.
Center:0.0+0.0i; Zoom =0.5

At 25 steps between resets there is a small capture area in the nose.

One zoom coming up for this one, a peek at a brown spot in the wings.

$ z^2+c+s_i$
s₀=0.0, s_i+1=s_i+0.1. Reset after 20 steps.
Center:-0.98714+0.296305i; Zoom = 2048

Another zoom. The color entropy dial is turned down a little.

$ z^2+c+s_i$
s₀=0.0, s_i+1=s_i+0.1. Reset after 20 steps.
Center:-1.92826+0.00031i; Zoom = 32768

Near the stable areas there is interesting chaos. Here is the first of two zooms

$ z^2+c+s_i$
s₀=0.0, s_i+1=s_i+0.1. Reset after 20 steps.
Center:0.0+0.0i; Zoom = 0.5

Change the sequence to reset to after ever 20 steps.

The capture set is very small. That is not surprising. Each step we are taking the usual z²+c iteration and kicking it a little to the right. On every 20th step the orbit gets pushed 1.9 units to the right. It may be more surprising that anything remains in the capture set.

$ z^2+c+s_i$
s₀=0.0, s_i+1=s_i+0.1. Reset after 15 steps.
Center:-1.854112+0.001739i; Zoom = 13421

Third zoom of Sequence Fractals Part III #11. Close to yesterday's image Sequence Fractals Part III #13, a little above the real axis

$ z^2+c+s_i$
s₀=0.0, s_i+1=s_i+0.1. Reset after 15 steps.
Center:-1.8625+0.0i; Zoom = 52

Second of three zooms into Sequence Fractals Part III #11

This one is near the tip of the Pinocchio nose.

$ z^2+c+s_i$
s₀=0.0, s_i+1=s_i+0.1. Reset after 15 steps.
Center:-1.7396+0.0i; Zoom = 164

Here is the first of a few random zooms into the image posted yesterday, Sequence Fractals Part III #11

If you are keeping score, this is equivalent to a degree 2^15 = 32,768 polynomial.

$ z^2+c+s_i$
s₀=0.0, s_i+1=s_i+0.1. Reset after 15 steps.
Center:0.0+0.0i; Zoom = 0.5

Moving up to 15 steps before resetting the sequence.

The capture set is getting smaller, but there is still plenty of chaos.

$ z^2+c+s_i$
s₀=0.0, s_i+1=s_i+0.1. Reset after 10 steps.
Center:0.0+0.0i; Zoom = 0.5

Zoom out, full view of previous picture.

The sequence is shifted to start at 0 instead of -1. Changing the start point does not change the fractal shape, only shifts the location on the complex plane.

I am going to lighten up on the math. But I still feel compelled to show the "big picture" along with the zooms.

$ z^2+c+s_i$
s₀=-1.0, s_i+1=s_i+0.1, reset after 10 steps.
Center:0.549+0.373i; Zoom = 64

Let's continue with $ z^2+c+s_i$, and move beyond the two step sequence. This one is based on a simple arithmetical sequence that repeats after 10 steps.

This image is similar to the pictures of the quadratic case, yet different and strange. The kind of thing I was hoping to discover.

As with the two step sequences, since the sequence repeats after 10 steps, we could combine 10 steps, get a polynomial and then claim that we are iterating a polynomial. The polynomial has degree 2^10 = 1024. At this point, I am not sure being a polynomial matters much.

The starting point for the iteration is 0. That huge polynomial has only even powers of z, so 0 is still a critical point. Please do not expect me to calculate any of the other 1022 critical points.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.4-0.2i
Center: 0; Zoom = 0.4

Different values for a generate collisions at different angles. Let d be the angle, defined as a complex number with |d|=1. Let t be a real number. Set a = dt. For large t, typically |t|>1 there are two separated mini brots. The line between the two is roughly perpendicular to d. As t runs from 1 to 0, the two minis come together, collide and combine and eventually become the Mandelbrot set when t = 0. Different angles, d, produce different intermediate shapes.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.02i
Center: 0; Zoom = 0.4

Recap: The quartic polynomials affine-conjugate equivalence classes can be parameterized in many different, yet reasonable, ways. All of these are mathematically equivalent. However when drawing fractals, they generate different pictures. (It is not surprising, mathematicians think a torus and a coffee cup are the same thing.)

How big is the quartic parameter space?

Quadratics have a single complex parameter. It is two dimensional. We can see the whole parameter space on a plane. Quartics have three complex parameters, and so live in a six dimensional space. We cannot visualize it.

In this six dimensional space resides a six dimensional solid called the "connectedness locus". (The Mandelbrot set is the 2 dimensional case.) We can cut out two dimensional slices and make quartic fractals. Typically the slices would be made perpendicular to the axis of one of the parameters. You might get fancy and tilt the slice at up three different angles (one for each parameter). But in all cases we only see very small part of the parameter space.

All of the different parameterization and all of the different slices will generate interesting pictures. But in this vast space of parameterization formulas and slices, you will never see the two-step sequence fractals.

You would either need to create a twisted curving non-linear slice, or specifically design the parameterization formula.

One small technicality, the connectedness locus consists of the points where the Julia set is connected, or eqivalently where the iteration of all critical points are bounded. Most of the pictures here examine the behavior of a single critical point.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.125i
Center: 0; Zoom = 0.4

I want to follow up on my comment in Sequence Fractals Part III #4 that sequence fractals are a special class, more than just a subset of the quartics. The comments are general, not specific to today's picture. Start by asking what does the quartic parameter space look like?

The general quartic polynomial is $ a_1z^4+a_2z^3+a_3z^2+a_4z+a_5$. But this gets reduced to three parameters.

Convention is to group the polynomials into affine-conjugate classes. Sounds complicated right, math words are so cool. An affine transformation is a simple function of the form $ b_1z+b_0$. Affine transformations are the only complex functions that are differentiable and invertible.

It turns out that $ f_1(z)=g^{-1}(f(g(z))$ behaves just like f. This, $ g^{-1}fg$, is affine conjugacy. It is why by convention we only look at $ z^2+c$ and not the general case for quadratics. Each belongs to a different affine-conjugate class, and collectively all classes are represented.

The first step is to select one representative function from each affine conjugate equivalence class. There are many ways to do this. But in all cases there are three complex parameters, a,b,c, not just c as with the quadratics.

For example we could drop one of the middle terms. (Like we do for quadratics.)

$ z^4+az^2+bz+c$
$ z^4+az^3+bz+c$
$ z^4+az^3+bz^2+c$

The first case is called monic (first coefficient is 1) and centered (second coefficient is 0). The critical points sum is 0 for monic, centered polynomials. In the third case, 0 is always a critical point. Calculating the other critical points is difficult.

Knowing the critical points is handy for generating fractals (as well as for complex dynamics in general). It is useful to work backwards from the critical points. Consider

$ 4(\int z(z-a)(z-b))+c$ critical points 0, a, b
$ 4(\int(z-a)(z-b)(z+a+b))+c$ critical points a, b, -a-b
$ 4\int(z-a)(z-b)(z-c)$ critical points a, b, c

Here ∫ does not indicate an actual iteration, I am using it to symbolize the anti-derivative as in a first calculus course. It makes is easy to see the critical points. The first one in the usual format is $ z^4+\frac{4}{3}(a+b)z^3+2abz^2+c$. Expanding the others are equally messy and just obscures the critical points. The second one is monic-centered.

I have a feeling that these are just the tip of the iceberg. Many other reasonable formulations possible.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.1453i
Center: -0.6335+0.3092i; Zoom = 0.4

I did a manual search, (meaning I created a lot of low resolution images) to find the point where the two shapes collide. It happens at some point between here and when a = -0.1450i. The interesting space between the two shapes is very narrow. I set the aspect ratio to 2 to stretch the image horizontally.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.15i
Center: 0; Zoom = 0.4

I started this series on sequence fractals just for fun. I was getting bored with the Mandelbrot set, and wanted something similar, but fresh. The idea has no mathematical significance, just random variations on the basic algorithm to generate different fractal pictures.

The simplest case, $ z^2+c+s_i$ with a two-step sequence certainly delivered. I managed to spend two months on the simple case, and found many new fractal beasts. Ironically it has been mostly guided by the mathematics.

It felt like the sequence fractals led me to the quartic polynomials, a serendipitous event to be sure, but perhaps I should have just started by studying quartic polynomials.

I no longer feel that way. I do not think that any study of quartic polynomials would have discovered these pictures. Basically the universe of quartic polynomials is so vast, this little speck would go unnoticed.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.25i.
Center: 0; Zoom = 0.4

I had no idea where this was headed with I started the Sequence Fractal series. Will actually I did have a good idea, it just that that idea was nothing like what actually happened. That is a good thing, I like the actual result much more than the original plan.

I am writing a blog, not a book. If I were writing a book I could embrace the new directions found while writing. I could reorder and retitle the chapters and present a finished product that is a nice unified whole. But with a blog, the earlier pages are already out there. It is too late to change the starting point or to hide a change in direction.

I suppose I could go back and retitle and reorder the blog posts. But that is too much work for me, and would generate more confusion than clarity for the reader.

You might suggest that I could have created the final draft of all posts in the series before posting the first. That sounds good in principle, but does not work well. Stuff gets misplaced, I get bogged down in details and lose sight of the big pictures. Staying organized becomes a bigger chore than creating content. And then soon the website has gone years without a new post. (I have been doing this for three decades, and there have been several multi-year hiatuses in that span.)