$ 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 = -0.680
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 = -0.65
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 = -0.62
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 = -0.61
Center: -0.20+0.25; Zoom = 2

Landed

$ 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 = -0.62
Center: -0.20+0.25; Zoom = 2

Absorbed

$ 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 = -0.50
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 = -0.25
Center: 0; Zoom = 0.3

The C1 points are now fully absorbed.

$ 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 = -0.05
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.