Month: June 2021

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.

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

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.

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…)

s_i = a,-a,…; a = -0.690
Center: -0.16; Zoom = 2.4

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.

s_i = a,-a,…; a = -0.705
Center: -0.16; Zoom = 2.4

s_i = a,-a,…; a = -0.712
Center: -0.16; Zoom = 2.4

s_i = a,-a,…; a = -0.713
Center: -0.16; Zoom = 2.4

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.