### Variations on a theme by gerrit

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#### Spyke

• Strange Attractor
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#### Variations on a theme by gerrit

« on: April 28, 2019, 08:48:17 PM »
$$\frac{z^2+a}{z^2+1}+ c$$
a = 3 + 3i
center = -1.7 - 1.5i
horizontal center to edge width = 8

Earl Hinrichs, offering free opinions on everything.

#### Spyke

• Strange Attractor
• Posts: 98

#### Re: Variations on a theme by gerrit

« Reply #1 on: April 28, 2019, 09:31:23 PM »
After posting the previous image it occurred to me some explanation may be necessary.

First, gerrit was investigating the fractional part of the formula in the Fractal Randoms thread. I added the +c because I was not paying attention.

Second, have you ever set the escape radius too low? (Don’t lie, I know it happens.) You get holes in the image, you know the fractal extends beyond the edge, but nothing is there. In this formula intermediate orbit values can get very large (when preceding point is close to $$\pm i$$). And very large orbit points head near 1.0 on the next iteration. No escape radius is large enough here. I use 1e5, and crank it higher if I notice the holes.

Third, (now I am stuck counting paragraphs) If there is no escape radius then every iteration takes the full maximum iteration count. Very slow. I suppose one could try to detect attracting cycles to bail out early. That would catch most the pixels and help in this long view. But I did not bother. For many values of c, there are no attracting cycles, and with drill down into the interesting areas, cycle detection becomes just extra overhead, and error prone.

Next, (no more counting paragraphs) if every pixel goes the full max iteration, then iteration count coloring is quite boring. Distance estimation and similar also fail. One option is to count how many times the orbit lands in some small distance from the origin. In this image I use 5.0, with accumulator smoothing as I described in the recent 1D smooth coloring thread. If you try this at home, you will need some non-standard coloring algorithms.

And finally, these colors popped up the first time I got something reasonable after dealing with the above problems. Even though some detail is lost in the dark area, I kept this image for artistic reasons. The mini brot is content in a clean uniform universe when suddenly the fabric of space tears revealing another rich, complex, chaotic universe on the other side.

In case you are wondering, this is real. No post-processing. I did not superimpose the mini from another image.

#### Spyke

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#### Re: Variations on a theme by gerrit

« Reply #2 on: April 29, 2019, 05:28:07 PM »
Same parameters as before, but with the color entropy dialed back slightly to show more of the details in the triangle.

#### Spyke

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• Posts: 98

#### Re: Variations on a theme by gerrit

« Reply #3 on: May 19, 2019, 12:58:06 AM »
Each side of the chaos triangle in the previous image has distinctly different characteristics. Or at least that was my initial impression. I now think that if you tweak the parameters these "features" can move to any edge. But anyway, going with my initial impression, the right side looks somewhat like the boundary of a normal mandelbrot. One, highly speculative explanation is that the origin is off in that direction.

Sorry, I only have the image available right now. I have the source is checked into git. If anyone is interested I will dig out and post the usual the details.

#### Spyke

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• Posts: 98

#### Re: Variations on a theme by gerrit

« Reply #4 on: May 19, 2019, 03:22:08 PM »
On the left side of the chaos triangle, in the zoomed out view, you see bumps along the edge. Zooming in, these are minis.

#### Spyke

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#### Re: Variations on a theme by gerrit

« Reply #5 on: May 20, 2019, 04:42:53 PM »
Along the bottom of the chaos triangle there appear to be threads coming unraveled. These turn out to be filaments of minis swimming in the chaos.

#### Spyke

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#### Re: Variations on a theme by gerrit

« Reply #6 on: May 20, 2019, 08:46:00 PM »
If anyone is interesting in exploring this formula, I just dug out the source code, with the coordinates, for the previous three images.
Right Side: -0.289-0.484i, center to edge width = 0.25
Left Side: -1.951+0.232i, width = 0.125
Bottom: -0.886-1.359i, width = 0.125

These are all parameter space (M-like) views.

#### Spyke

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#### Re: Variations on a theme by gerrit

« Reply #7 on: May 20, 2019, 08:51:41 PM »
I found a few more when I restored the code.

This one if found a little south of the mini in yesterday's "left side" post. It is inside a bulb-like hole that pokes into the chaos region.
-2.1011-0.1724i, (half) width=.001

#### mrrudewords

• Fractal Fluff
• Posts: 375
• Dat Mandel!

#### Re: Variations on a theme by gerrit

« Reply #8 on: May 20, 2019, 10:20:22 PM »
Very nice!
Z = Z2 + C (obvs)

#### gerson

• Fractal Furball
• Posts: 290

#### Re: Variations on a theme by gerrit

« Reply #9 on: May 20, 2019, 11:48:39 PM »
what software are you using to do that?

#### Spyke

• Strange Attractor
• Posts: 98

#### Re: Variations on a theme by gerrit

« Reply #10 on: May 21, 2019, 03:08:20 PM »
Mr. Rude: Thank you.

Gerson: I am using software that I wrote. I have been working on the program for over 30 years now.  The software does nothing special on the render (other than automatically add my signature). I have not used other software since Fractint. But I suspect that the commonly used commercial and shared fractal programs do more. My program is for general mathematical / algorithmic / computer generated art. Emphasis is on creating / storing / reusing (and debugging) code blocks. Still, fractals are my first choice and I spend most of my time on fractal algorithms.

So, sorry no parameter file for this. I have thought about modifying my program so it generates c# source code for each fractal. But I figure it would be a lot of effort to get a clean code file, with very little benefit.

For this image, the formula and coordinates are in the post. I am using a simple 2D coloring variant of the 1D coloring method we discussed here: https://fractalforums.org/fractal-mathematics-and-new-theories/28/smooth-1d-coloring/2753. The image is anti-aliased nine-to-one, equal weight.

As is typical for rational functions, every pixel gets iterated to the full iteration count, so iteration count coloring fails.

#### gerson

• Fractal Furball
• Posts: 290

#### Re: Variations on a theme by gerrit

« Reply #11 on: May 21, 2019, 04:34:43 PM »
Thanks for your explanation.
In past I used to play with Fractal Explorer to test formulas, now I am using Fractal Zoomer that let to user define formulas, bailout and so on, but I am not a programer and all that mathematic sometimes are difficult to understand.
This image was done with Fractal Zoomer with your parameter and using bailout 5 and 100 iterations. I understand your explanation about that holes (brown in my image) but I didn't get the mini brot maybe because color option.
Redered at 5000x5000 and resised to 1600x1200.

#### Spyke

• Strange Attractor
• Posts: 98

#### Re: Variations on a theme by gerrit

« Reply #12 on: May 21, 2019, 06:11:37 PM »
I started another render, still cooking right now. It will be zoomed out, with simplified color scheme. We should be able to find something similar in fractal zoomer. By eliminating the deep zoom and coloring, we should be able to figure out how to get the two programs to render something similar.

My iteration count was 768. Every pixel should hit that max. Remove the bailout entirely or set it to something large. I was using 10000 for the bail out radius. My program uses |z| in the test, not |z|2 as some programs do. You may need eight zeros.

I use R=1.0 for the coloring. (The coloring radius does not have to match the bail out radius.) I am not sure how your color palette works, if the full color range is the iteration count, there is not enough colors. If the distance between colors is 1.0, then there are too many. I multiple the value produce by the color calculation by 0.004 before passing it to the palette.

Zoom in my program is probably backwards of other programs. My zoom number is the horizontal distance from the center to the edge of the screen. Vertical distance (9/16) is adjusted to maintain 1 to 1 aspect ratio. So smaller numbers mean deeper zoom.

Your green triangle looks familiar. Try zooming in, however your program does zoom.

• 3f
• Posts: 1369

#### Re: Variations on a theme by gerrit

« Reply #13 on: May 21, 2019, 06:25:48 PM »
here's a fragm version of the first image, coloured by iteration index of minimum |z| with edge detection

#### Spyke

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• Posts: 98

#### Re: Variations on a theme by gerrit

« Reply #14 on: May 21, 2019, 07:01:54 PM »
That is wild, Claude. It reveals, or at least suggests something about the dynamics. The left point of the triangle does not behave like the other two. I imagine the mini as a ferocious pit bull guarding its territory. The chaos monster does not dare let its color bands get near the dog's territory.

Gerson: I looked at the docs for Fractal Zoomer. I did not see a coloring algorithm that I think will work. They all appear to use either the iteration count or the last z value. The coloring algorithm should some how take into account the whole orbit.

Claude's min |z| is checking every point. I am counting (actually weighted sum, adding 1.0-|z|) when the orbit point is within distance 1.0 of the origin. Anything that takes into account the full orbit should produce something interesting.

I see Fractal Zoomer comes with source code, do you modify the source code? I could give you some c# which should be easy to convert to Java.

This render has center -2.101-0.172, with "spyke-zoom" = 1.0. Simple black and white palette.

Near the top center you can see the mini poking at the chaos region (orange in my 5/19 post). Near the center there is a white brot like shape inside which you can see some minis. My second image post yesterday was from this area. That mini is too small to see in this image.

Anyway, if you can reproduce something like this, then we can working on zooming and coloring.

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