$ 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.)

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.5i.
Center:0; Zoom = 0.3

I have just a few more two-step pictures to post. First a few with a approaching 0 from the negative pure imaginary direction. As I post these I am going to dial way back on the math and write about other things. Feel free to ignore the words if I get too introspective.

I have some foresight into future posts, but not much.

I try to make a post every day. That does not mean that I start each day start with a blank page. There is a process of sorts. I explore a topic, create several low resolution pictures with rough notes on each. Then sort and group them, select a subset, put them in order and generate a plan for the next few days or the next week. Then I generate the high resolution images and turn the notes into a post and push it to the website. I try queue up a few days of posts in advance so there is time for a final proof read and link testing. The webpage is set to automatically publish the posts at 0000 hour on the scheduled day.

$ z^2+c+s_i$
s_i = a,-a,…; a = -i.
Center: i; Zoom = 0.3

This one starts part III. Mainly because Part II was getting too long, and I called the last post the "grand finale". However the basic formula has not changed. I have not tired of the desert color scheme yet. But I will convert to the more traditional rainbow colors with black insides so part III looks different compared to part II.

I had planned to base part III on the general two-step sequence, s_i = a,b,a,b,… However some of the first few picture I generated looked very familiar. For example 0,1… is the same as -.5,.5,…. At first I thought I got the picture files mixed up. It turns out all that matters is the difference a-b. The picture for general a,b,… is the same as 0,b-a,0,b-a… translated a units. It is easy to see that c'=c-a is a conjugation of both steps in the sequence fractal.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.05
Center: 0; Zoom = 0.3

The big finale, when all the plot points come together (pun intended). Surely you have seen this coming and were eager to get to this point.

Here is the list of named points for this function, a = -0.05, and for the Mandelbrot set. The => symbol points to the equivalent point in the MSet. The first set looks at the usual two-step function, quartic polynomial. When we get to a=0, the two steps are the same, the good old z²+c. (Picture not included, but you may have seen it elsewhere.) The Mandelbrot numbers are based on the normal single step, so the cycle numbers may double when comparing the two.

Named Points for a=(-0.050000, 0.000000)
  2 fixed points
C1.00:  -0.837298,  0.000000 => C2.00
C1.01:  -0.062702,  0.000000 => C1.00
  6 two cycles
C2.00:  -1.846047,  0.000000 => C4.00
C2.01:  -1.210164,  0.000000 => C4.01
C2.02:  -0.107257,  1.085808 => C4.02
C2.03:  -0.107257, -1.085808 => C4.03
C2.04:   0.285362,  0.540972 => C4.04
C2.05:   0.285362, -0.540972 => C4.05
  4 1/1 preperiodic points
M1,1.00:  -1.909295,  0.000000 => M2,1.00
M1,1.01:   0.031926,  1.058185 => M2,2.00
M1,1.02:   0.031926, -1.058185 => M2,2.01
M1,1.03:   0.045443,  0.000000 => C1.00

Named Points for Mandelbrot set
  1 fixed points
C1.00:   0.000000,  0.000000
  1 two cycles
C2.00:  -1.000000,  0.000000
  3 three cycles
C3.00:  -1.754878,  0.000000
C3.01:  -0.122561,  0.744862
C3.02:  -0.122561, -0.744862
  1 2/1 preperiodic points
M2,1.00:  -2.000000,  0.000000
  2 2/2 preperiodic point
M2,2.00:   0.000000,  1.000000
M2,2.01:   0.000000, -1.000000
  6 four cycles
C4.00:  -1.940800,  0.000000
C4.01:  -1.310702,  0.000000
C4.02:  -0.156520,  1.032247
C4.03:  -0.156520, -1.032247
C4.04:   0.282271,  0.530061
C4.05:   0.282271, -0.530061

Note that M1,1.03 in this list is M1,1.01 in all the previous lists.

tl;dr: Feel free to skip the rest.

Over explaining the naming convention: These named points are a single snapshot for a particular formula / parameter pair. In each list, the names are arbitrarily assigned by left to right (< real) order. In this sense there is no connection between one set of values and another for a different value of a. However the super function, considering all parameters as variables f(z,c,a) is just a polynomial and so continuous in all variables. That makes the set of named points a continuous function of the parameter a, and so it makes sense to track how these points evolve individually as a changes, as we have been doing all along.

Sometime since the last C/M list, the M1,01 point, on the real axis, moved to right of the former M1,1.02 and M1,1.03. Hence that name change.

There are many naming conventions for the special points in the Mandelbrot set. For example, here is a comprehensive system developed by Robert P. Munafo href=http://www.mrob.com/pub/muency/r2namingsystem.html. I do not know of any naming system for the general quartic case. To keep things simple, I have chosen a less precise but much simpler convention.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.125
Center: 0; Zoom = 0.3

C1 Collision

A while back, back when a < -0.7, the two C1 points hosted baby brots above and below the real axis. Around a = -0.5, they merged with the blob in the middle, however you could still recognize them as distinct attachments. At a=-0.25 that blob, which is becoming more and more familiar looking, has fully swallowed the C1 points leaving no visual clue of their existence.

You can't see it in the picture, but somewhere in the middle of all the white, the two C1 points have collided.

The C1 points can be easily solved exactly with algebra C1.00 = C1.01 = -0.375.

I have been using a program I wrote to solve non-linear equations and find the C and M points. It uses a iterative process loosely based on the Newton-Raphson method (but without a derivative). Like NR, sometimes the iteration does not converge, sometimes convergence is very slow, and as with any computer program, round off errors may propagate. Despite all of that, until today, the program has been very robust and highly accurate. Double roots are one trouble area. Here is the table to be consistent with the other posts in this series. The C1 points are only accurate to 0.001, and are showing too many decimal digits. The other points, as far as I know, are accurate to six decimal digits.

Named Points for a=(-0.125000, 0.000000)
  2 fixed points
C1.00:  -0.375016, -0.000014
C1.01:  -0.374462,  0.000469
  6 two cycles
C2.00:  -1.699718,  0.000000
C2.01:  -1.057328,  0.000000
C2.02:  -0.033836,  1.161541
C2.03:  -0.033836, -1.161541
C2.04:   0.287359,  0.562280
C2.05:   0.287359, -0.562280
  4 1/1 preperiodic points
M1,1.00:  -1.770337,  0.000000
M1,1.01:   0.085355,  1.139318
M1,1.02:   0.085355, -1.139318
M1,1.03:   0.099627,  0.000000

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.25
Center: 0; Zoom = 0.3

The C1 points are now fully absorbed.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.50
Center: -0.20+0.25; Zoom = 1

Docking complete

Named Points for a=(-0.500000, 0.000000)
  2 fixed points
C1.00:   0.000000,  0.866025
C1.01:   0.000000, -0.866025
  6 two cycles
C2.00:  -0.763554,  0.000000
C2.01:  -0.500000,  0.000000
C2.02:   0.302321,  0.820455
C2.03:   0.302321, -0.820455
C2.04:   0.329456,  1.480979
C2.05:   0.329456, -1.480979
  4 1/1 preperiodic points
M1,1.00:  -0.987258,  0.000000  f'= 10.260169,  0.000000, |f'|= 10.260
M1,1.01:   0.190788,  0.000000  f'=  0.972658,  0.000000, |f'|=  0.973
M1,1.02:   0.398235,  1.471098  f'=  6.383586,  7.865033, |f'|= 10.130
M1,1.03:   0.398235, -1.471098  f'=  6.383584, -7.865034, |f'|= 10.130

M1,1.01 is now attractive. This is rare, I do not think it happens on the Mandelbrot set. It should mean that M1,1.01 is in the white area. It is difficult to see graphically. But with deep zoom and numerical calculation of orbits, I found points on either side of the boundary.
left boundary = M1,1.00 < M1.1.01 < 0.19080 < right boundary < 0.19085.
So M1,1.01 is indeed in the interior of the capture component.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.55
Center: -0.20+0.25; Zoom = 1

Zoom out by a factor of two. Here is C1.00 coming into the central object.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.62
Center: -0.20+0.25; Zoom = 2

Absorbed

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.60
Center: -0.20+0.25; Zoom = 2

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.61
Center: -0.20+0.25; Zoom = 2

Landed

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.62
Center: -0.20+0.25; Zoom = 2

The a parameter is increased to -0.620. Let’s watch the C2.00 landing on the left and M1,3 on the right.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.65
Center: -0.16; Zoom = 1

Bumping up a to -0.650 to find another interesting central shape. C2.00 is near the top, slightly left of center. The suspected M1,3 point from Sequence Fractals Part II #23 is about to land.

Named Points for a=(-0.650000, 0.000000)
  2 fixed points
C1.00:   0.150000,  1.024695
C1.01:   0.150000, -1.024695
  6 two cycles
C2.00:  -0.394902,  0.353242
C2.01:  -0.394902, -0.353242
C2.02:   0.370475,  0.974363
C2.03:   0.370475, -0.974363
C2.04:   0.474427,  1.589533
C2.05:   0.474427, -1.589533
  4 1/1 preperiodic points
M1,1.00:  -0.561969,  0.000000
M1,1.01:   0.095643,  0.000000
M1,1.02:   0.533163,  1.581689
M1,1.03:   0.533163, -1.581689