$ 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

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.680
Center:0.03193+0.08066i; Zoom = 150

Just a quick digression. There is a baby Mandelbrot connecting each of the tufts. Picture only, just an observation. I am not going to try to analyze it. I have been on this topic for almost three months. Really, I am trying to pick up the pace.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.680
Center: -0.16; Zoom = 2.4

Notice the tufts connected to the main body on the right side. We have seen similar things before, see Sequence Fractals Part II #11. In those cases the decoration was either not connected, or connected at a single point. Here, there seems to be something else going on. (Building up anticipation for tomorrow's post…)

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.690
Center: -0.16; Zoom = 2.4

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.700
Center: -0.16; Zoom = 2.4

I did not generate a list of named points today. The M1,1.00 point that we were tracking is still the left tip of the center image. Notice that if you were to extend the white area to a full Mandelbrot set, as the other critical point does, it has crossed over into the period four bulb.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.705
Center: -0.16; Zoom = 2.4

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.712
Center: -0.16; Zoom = 2.4

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.713
Center: -0.16; Zoom = 2.4

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.714
Center: -0.16; Zoom = 2.4

Another 0.001 bump to the parameter a, and dramatic changes to the central shape. Yes, I realize at +0.001 a day this will take two years to get to 0. Let's look at a few more with this small change, then I will pick up the pace.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.715
Center: -0.16,0; Zoom = 2.5

Bump the a value by another 0.001. Center is midway between two M1,1 points. Holy Chaos Batman! Is that a Bat-brot?

Named Points for a=(-0.715000, 0.000000)
  2 fixed points
C1.00:   0.215000,  1.086278
C1.01:   0.215000, -1.086278
  6 two cycles
C2.00:  -0.303783,  0.450626
C2.01:  -0.303783, -0.450626
C2.02:   0.411487,  1.038380
C2.03:   0.411487, -1.038380
C2.04:   0.537296,  1.634112
C2.05:   0.537296, -1.634112
  4 1/1 preperiodic points
M1,1.00:  -0.205637,  0.000000
M1,1.01:  -0.119427,  0.000000
M1,1.02:   0.592533,  1.626936
M1,1.03:   0.592533, -1.626936

M1,1.00 and M1,1.01 are again the left and right tips along the centerline. The view from the other critical point is very similar to the a = -0.716 case, Sequence Fractals Part II #22. (So I am not going to waste a post on it.) M1,1.00 is very close to the center of the period 2 bulb. M1,1.01 has crossed over into the main cardioid. That accounts for the thin slice missing just left of M1,1.01.

It would be interesting to look at the picture for the a values where M1,1.00 (this critical point) = C2.0x (2-bulb center, other critical point), and when M1,1.01 (this) = 2-bulb, main cardioid bifurcation (join) point. Also from a few posts ago, the a value where M1,1.00 = M1,1.01 would be interesting. I need to add some software to calculate these collisions, but I am also trying to keep up a post-a-day pace. This is just one more thing in a long list of things I would do differently if things were different. I will save the whining for later.