$ 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.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.716
Center: 0; Zoom = 0.25

Zooming out to see the big picture. C1.00 and C1.01 points are the larger distorted brots above and below. C2.02 and C2.01 are the small white area to the left, above and below the central chaos. See Sequence Fractals Part II #18 for a list of named point values for this fractal. The nearby mini slightly to the right appears to host a M1,3 point.

Here are the first few iterates of -0.074+0.384i near the center of the right side mini, which gives evidence that an attracting M1,3 point is nearby.

0 :  0.0000000000,  0.0000000000
1 :  1.1186973098, -0.2226672276
2 :  0.7986249557,  0.2897826579
3 : -0.0192503327, -0.0160305150
4 :  1.1180435564, -0.2235550021
5 :  0.7967073694,  0.2888181202
6 : -0.0135847570, -0.0190021871
7 :  1.1185795766, -0.2236186754
8 :  0.7975798288,  0.2882324026
9 : -0.0136806798, -0.0158793598
10:  1.1184661988, -0.2234036595
11:  0.7975737434,  0.2887061709
12: -0.0148163399, -0.0167138452
13:  1.1184113007, -0.2234957519
14:  0.7973969097,  0.2885950187
15: -0.0142411724, -0.0169491521
16:  1.1184598547, -0.2234949011
17:  0.7974818960,  0.2885534577
18: -0.0142878044, -0.0166716323
19:  1.1184478907, -0.2234766718

My M point calculator refuses to find this point. This is probably a shortcoming in my program. A M1,3 point is a root of a 128 degree polynomial, the program is only finds half of them before giving up. There are things I could do to improve the program, and I would like that additional confirmation. I choose not to head down that rabbit hole right now. So let me get by with "appears to be".

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.716
Center: M1.1.00 (for critical point 0); Zoom = 2
Start: z₀ = $ \pm \sqrt{0.716-c}$

Wondering about the other critical point? Here is a look at the same area, starting the iteration with one of the other critical points. Zoom is reduced by a factor of 1000, otherwise the picture would be solid white. The previous two pictures are fully contained in the period two bulb. If you go back to Sequence Fractals Part II #19, you can see small portions of the valley between period two and the cardioid in this picture on the right edges of the main shape in the earlier picture.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.716
Center: M1,1.00; Zoom = 2000

Let's get a lot closer. Zoom is 50x the previous picture. Sequence Fractals Part II #20, center is back at M1,1.00. M1,1.01 on the other side looks similar. It is a little bigger and somewhat squished horizontally. The distortions can be seen in the previous picture.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.716
Center: -1.6 + 0.0i; Zoom = 40

Zooming in 4x from yesterday, Sequence Fractals Part II #19, and picking a center point midway between M1,1.00 and M1,1,01. The two M points are at the tip of the nubs on the left and the right.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.716
Center: M1,1.00; Zoom = 10

M1,1.00 at the same zoom as Sequence Fractals Part II #14.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.716
Center: M1,1.00; Zoom = 1

Another small, 0.001 increase for a to -0.716, and we have colliding M points. Boom an explosion of white.

Named Points:
 2 fixed points
C1.00:   0.216000,  1.087198
C1.01:   0.216000, -1.087198
  6 two cycles
C2.00:  -0.302421,  0.452026
C2.01:  -0.302421, -0.452026
C2.02:   0.412158,  1.039344
C2.03:   0.412158, -1.039344
C2.04:   0.538264,  1.634787
C2.05:   0.538264, -1.634787
  4 1/1 preperiodic points
M1,1.00:  -0.176018,  0.000000
M1,1.01:  -0.146881, -0.000000
M1,1.02:   0.593450,  1.627621
M1,1.03:   0.593450, -1.627621

Notice what happened to the M1,1 points. M1,1.00 and M1,1.01 are no longer a conjugate pair. Now both lie on the real axis. If you imagine a as time you can picture the M1,1 points moving as a increases. They have been moving nearly vertically down and up towards each other. Sometime between the last frame and this one, they collided and are now moving away from each other along the real axis.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.717
Center: M1,1.00; Zoom = 1

Normal view of M1.1.00 again. Zoom is 1/10 of Sequence Fractals Part II #14. Notice that we are getting dangerously close to the reflected (complex conjugate) M1,1.01 point directly below. (This is of course just a set up for tomorrow's picture.)

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.717
Center -0.167756+0.065847:; Zoom = 200
Start: z₀ = $ \pm \sqrt{0.717-c}$

Here is the view from the other critical point of the same area, same zoom and colors. The Mandelbrot surface is complete. Except for the chunks knocked loose in the previous picture Sequence Fractals Part II #15, the white areas are the same.

Something similar can be seen with the quadratic family z²+c. Within this family you can prove "If an attracting fixed point exists, then the critical point is attracted to it." This is again a consequence of the fact that there is only one critical point. And basically the reason we always want to start with the critical point. If you try to draw a Mandelbrot set starting with any other point, you will find pieces missing. Always less, never more. If you start far enough away, none of it remains.

I have not worked through the details, but I am pretty sure that what we see here is closely related.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.717
Center -0.167756+0.065847:; Zoom = 200

Here is a zoom into the area a little north of M1,1.00. We are trying to answer the question, "Does this look like the surface of the Mandelbrot set?". Any Star Wars fans? The center bulb reminds me of the death star 2.0. Is the Empire building a Mandelbrot death star? Or perhaps our M1,1 point is attacking and destroying the Mandelbrot set.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.717
Center: M1,1.00; Zoom = 10

Another 0.001 increase in the value of a. The zoom factor is 1/12 of the similar picture for a = -0.718. Sequence Fractals Part II #11. The self-similarity makes it hard to eyeball the zoom. At the same zoom as the other image, you cannot see the blue and orange swirlies.

Named Points:
  2 fixed points
C1.00:   0.217000,  1.088118
C1.01:   0.217000, -1.088118
  6 two cycles
C2.00:  -0.301061,  0.453422
C2.01:  -0.301061, -0.453422
C2.02:   0.412829,  1.040308
C2.03:   0.412829, -1.040308
C2.04:   0.539231,  1.635462
C2.05:   0.539231, -1.635462
  4 1/1 preperiodic points
M1,1.00:  -0.160367,  0.037868
M1,1.01:  -0.160367, -0.037866
M1,1.02:   0.594367,  1.628305
M1,1.03:   0.594367, -1.628305

Notice the transition between white and colors about a quarter of the way down from the top. Does it almost look like a section of the surface of the Mandelbrot set?

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.718
Center: M1,1.00; Zoom = 5
Start: 0
Julia / dynamic / variable view.

Going back to critical point 0 as the start point, and switching to Julia view. Compare to Sequence Fractals Part II #9. This one appears connected, but it is not. Under deep zoom there is a small gap on the center where the two shapes come together. This is expected as M1,1.00 is a repelling fixed point after one iteration. Note however there are large connected components.

Compare the Julia pictures and also compare the parameter space pictures M1,1.00 in Sequence Fractals Part II #8 (a=-0.179) and Sequence Fractals Part II #11 (a=-0.718). In the first picture of each set (a=-0.719) imagine M1,1.00 moving through a field of settled dust, in its wake it kicks up the dust produces the colorful swirlies. Yes, water would be a better analogy, but I need dust for the next part. Now imagine in the second picture (a=-0.718) there is a vacuum cleaner nearby. It sucks up most of the dust that gets disturbed, exposing the clean white table underneath.

I think the analogy comes pretty close to the actual dynamics. Points near a repelling cycle or pre-cycle will tend to dance around a lot before heading off to infinity. This is exactly the colorful spirals that make fractal pictures so interesting. If there is an attracting basin nearby, such as the attracting fixed point associated with some other critical point, some will get sucked in during the dance.