Variations on a theme by gerrit

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Offline gerrit

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« Reply #45 on: May 25, 2019, 07:17:55 AM »
Here's another minibrot with filled Julia sets inside. This is for the "other" formula.

Coloring here is not orbit trap but this https://fractalforums.org/fractal-mathematics-and-new-theories/28/smooth-1d-coloring/2753/msg13961#msg13961.
Interesting that it also works nicely for fractals like this where no orbits ever escape.

Offline Spyke

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« Reply #46 on: May 25, 2019, 06:22:55 PM »
If you do this with the other critical orbit it looks sort of the same but the minbrot has vanished.

I have been quiet for a couple of days. (I think) I understand what is happening in Gerrit's observation. But I spent a whole day trying to prove it, only to discover my math skills are quite rusty. It has been 40 years since topology and complex functional analysis. I just wasted another hour this morning. So, I will present "conjecture". Please help fill in the details if you are so inclined.

The "official" definition of the Mandelbrot set is \( M = \{ c | J_c \) is connected \( \} \). That "is connected" part is difficult to work with, so we use the equivalent definition \( M_z = \{ c | z \in K_c \} \) where \( K_c \) is the filled in Julia set. \( M=M_0 \), and we have a natural generalization to alternate starting points. However \( K_c \) has hard coded assumptions that the other critical point is \( \infty \) and it is attractive. This generalization can be extended to polynomials, but it breaks down for rational functions.

The "connected Julia" definition may generalize nicely, but that one hurts my head. Let's agree that any generalization should include \( M_z = \{ c | z \in J_c \} \). (For polynomials,  \( K_c \supseteq J_c \), and the "extra" points are usually uninteresting.)

For \( z^2 + c \). If\(  |c| > 2 \), then then \( J_c \) is Fatou dust. If \( |z| > 2 \) and \( |c| < 2 \) then \( J_c \) is contained in the \( |z|<=2 \) disk. Now I want to swap planes, I keep tripping up on this part. Fix a starting point z, with \( |z| > 2 \), for the parameter plane calculation. Then pick a "pixel" value c. Does z converge for c? Only if z is in the 0-measure set \( J_c \). So unless you engineered the choice of z and c, the answer is no. The parameter plane image is empty, the brot disappears. Yes, this is a long way to say what everyone already knows. However, the \( |c| < 2 \) condition is often ignored in escape-checks. And this sets up a contrast with the rational function case. based on the nature of the Julia set.

Now suppose f is a rational function. Any of them discussed in the thread will work. Although degree numerator <= degree denominator may be required. For these rational functions we know there exists c where \( J_c = \bar{C} \). Now f(z,c) is continuous, so there is an open set around the c with this property. (Yes, this needs proof, but it makes intuitive sense and all images so far support the statement). So the parameter space image, will have an open set (measure > 0) around c with this property. The filled-in Julia set definition of M-set fails because \( \infty \) does not attract. Whatever we do to replace it should include\(  \{ c | z \in J_c \} \), in particular this c is included. Any of the alternate, non-escape time, coloring methods in this thread seem to do a good job of exposing these points. You might be suspicious that the starting point z has not been mentioned yet. For which starting values of z is this c an interesting point? When is \( z \in J_c \)? Always because \( J_c = \bar{C} \). The starting point does not matter!\(  \)

These images show distinct brot and antibrot components. I think it makes sense that the brot disappears without careful selection of the start value, while the antibrot seems unchanged with any start value.
Earl Hinrichs, offering free opinions on everything.

Offline Spyke

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« Reply #47 on: May 25, 2019, 06:45:10 PM »
Gerson: Can you try bailout = 10000? (Or more if you software is actually testing |z|2)

I suspect the pancakes are the result of the low bailout. Since infinity does not mean escape in this case, orbits are stopped prematurely. In this case, small bailout is like orbit trap, and it hides the fractal.

Of course if you like the pancakes, and consider it an artistic effect, then cool.

Offline Spyke

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« Reply #48 on: May 25, 2019, 07:12:13 PM »
a = 6, center = -0.885 + 0.244, width = 0.031

Gerrit: I like your min-Z colors, even when it is too bright. (Except for when it looks too much like orbit trap.)

This one is about color selection, and not any further insight into the formula.

I am trying a variation, of sorts, of min-z. I wanted a 2D version. Using the z that gives the minimum is discontinuous. It looks similar to Claude's i of min |zi| coloring. I wanted something continuous.

I take a starting value, v, large and arbitrary. On each iteration if |z| < |v| then replace v with a weighted average of z and v. For example, if b = |z|/|v| then v = b * v + (1-b) * z provides a smooth transition from v to z. v keeps getting smaller, so it is sort of like min-z. The final v is used for selecting the color.

Full disclosure: That is not the whole story, I added parameters for non-linear weighting, and to speed up / slow down the rate of change. Much parameter tweaking took place before deciding on the final version.

Offline gerrit

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« Reply #49 on: May 25, 2019, 09:00:50 PM »
This is for \( (z^2+i)/(z^2+1)+c \) using the smooth iteration coloring with a power of 3 and radius R=2.
The sharp boundaries are finite R effects, increasing R makes them go away.

This is my UF code for the smooth iteration coloring:
Code: [Select]
SmoothIters {
init:
  float func g(float x, float R, float p)
     float y = (1 - x/R)
     if y<0
        y = 0
     endif
     return y^p
  endfunc
  float acc = 0
  int it = 1
loop:
  float az = cabs(#z)
  if az <= @R
     acc = (it*acc + g(az,@R,@p))/(it+1)
     it = it + 1
  endif
final:
  #index = acc * @cs
default:
  title = "SmoothIters"
  float param p
    caption = "Exponent"
    default = 2
  endparam
  float param R
    caption = "Radius"
    default = 2.0
  endparam
  float param cs
    caption = "Color speed"
    default = 1
  endparam
}

Offline gerrit

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« Reply #50 on: May 25, 2019, 09:05:04 PM »
Same smooth iteration method reveals structure in the "totally random" looking (with the min(z) coloring) Julia sets. 2 examples.

Offline gerrit

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« Reply #51 on: May 27, 2019, 05:50:48 AM »
Determined to beat the subject to death, here's two more Julia sets with the smooth iterations bringing out out the structure.

Offline gerrit

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« Reply #52 on: May 27, 2019, 05:52:09 AM »
and two more smooth iteration show-offs.

Offline gerrit

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« Reply #53 on: May 27, 2019, 05:54:23 AM »
These two look best with the min(|z|) coloring. I do not object too much to the fact that it looks like an orbit trap, because it is an orbit trap.

Offline gerson

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« Reply #54 on: May 28, 2019, 01:04:54 AM »
@Spyke « Reply #47  image 01 below is the same that Reply #43 without orbit traps.
image 02 is the same that Reply #43 with orbit traps but with bailout 10000 (without orbit traps only got a black image)
Why don't I get minibrot as all others got? Is the color method?

Offline gerrit

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« Reply #55 on: May 28, 2019, 01:14:24 AM »
@gerson: I think your bailout 5 is too low, I use \( \infty \).
I think you'll see minis if you zoom.

Offline gerrit

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« Reply #56 on: May 28, 2019, 08:13:13 AM »
\( \frac{z^2+1+i}{z^3+1} + c \).

Offline mrrudewords

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« Reply #57 on: May 28, 2019, 09:37:55 AM »
@Spyke « Reply #47  image 01 below is the same that Reply #43 without orbit traps.
image 02 is the same that Reply #43 with orbit traps but with bailout 10000 (without orbit traps only got a black image)
Why don't I get minibrot as all others got? Is the color method?

@Gerson, with a low bailout you won't get the detail but you will not see anything colouring by iteration with a high bailout as nothing will escape. You need to colour using an orbit trap or similar as all points will be 'inside'.
Z = Z2 + C (obvs)

Offline Spyke

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« Reply #58 on: May 28, 2019, 08:48:44 PM »
Gerson: What are you using to color the first one? Is in min |z| similar to Gerrit, or perhaps min |zlast|? As Mr Rude pointed out, escape count coloring is boring since nothing escapes. Inside color or orbit trap variations are good candidates.

As for the mini, there is too much going on in this thread for my old tired brain. Can you repost the formula you are using, and the image screen coordinates? I will try to generate a matching image and look for the mini.

Offline gerrit

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« Reply #59 on: May 29, 2019, 09:46:17 AM »
Pattern of minibrots in \( \frac{z^2+c}{z^2 + 1} \) parameter space (c=pixel).


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