3rd Order Evolution Of Tilings

[Dinkydau]

EDIT: I have changed the image to a new version. The previous version contained some glitches. Right now it should be completely free of glitches! I also changed the anti-aliasing algorithm to prevent some of the Moiré effects caused by the fine tilings.

fractal: Mandelbrot Set
Computed with Claude's GMP fork of Kalles Fraktaler
KFB map converted to MMIT with my KFB to MMIT converter
Colored in Mandel Machine with a custom gradient designed in Photoshop

Because of lack of inspiration for new renders, I'm currently dealing with my backlog of ideas that require extreme render times. I have been since early this year, actually. Most of my ideas since last November came within a few days.

This is my deepest render ever, made possible by the further development of programs that render the Mandelbrot set using perturbation and series approximation. The overall shape is the same as that of Dragon (Deep Version) except that this has 3 blobs with shapes connected to each "knot", instead of 4. Another, more crucial, difference is that the shapes here are custom, which requires more zoom depth. Applying evolution to 3 different shape types severely restricts the number of morphings that can be done per shape, unless you just zoom deeper. That's why this had to be so deep.

The first shape type is a Golden Ratio Tiling. The second shape type is a Hyperbolic tiling, which (like the fractal that the link leads to) also has golden ratio tiling "pistil". The third shape type is something different that I haven't made before. The surrounding tiled bands are highly rotational symmetrical copies of the initial repetitive zoom path that the tilings inside the main shape are also made out of.

Magnification:
2^27847.24
7.17440059980 E8382

Location:

Code:

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[Kalles Fraktaler]

Simply Amazing!
The surrounding is fantastic rather than the dense stripes that is common in such deep images!

I am not able to see the misformed parts in this view?
There is one issue that has been revealed lately that may occur when doing very large resolution, that when the number of terms used in approximation exceeds 100 it is possible that glitches are missed. When that occurs the result is that the end points of the pattern becomes blurry, but this seems not be the case with this image.
I think we should ask Claude to limit the number of automatic terms to something like 60.

Edit: the bold marked essential words were unfortunately missed...

[claude]

Beautiful and incredible!

I think we should ask Claude to limit the number of automatic terms to something like 60.

Not wanting to hijack this thread, let's discuss it on the new forums here:
https://fractalforums.org/kalles-fraktaler/glitches-with-high-order-series-approximation-used-for-large-images

[Dinkydau]

Thanks for your replies

The glitches are small and look like the result of a regular julia transformation. Here they are encircled:

Both should have been round, but they look morphed for some reason.
It almost looks intentional, but it wouldn't be possible to do this normally without affecting the larger embedded tilings such as the one in the center.

[Kalles Fraktaler]

I am sorry I haven't registered on the new forum yet.....  

In function CFraktalSFT::CalculateApproximation
I caluclate the number of terms to be used, between the thumb and index finger, to

Code:

if (m_bAutoTerms){
	int nT = sqrt((double)m_nTotal)*0.021;
	if (nT<5)
		nT = 5;
	SetTerms(nT);
}

So just a condition that it is not greater than 60 would help a lot:

Code:

if (m_bAutoTerms){
	int nT = sqrt((double)m_nTotal)*0.021;
	if (nT<5)
		nT = 5;
	if(nT>60)
		nT=60;
	SetTerms(nT);
}

And I do recognize that type of glitch as caused by too many approximation terms making the iterations where the glitch is detectable to be missed.
Which means, unfortuantely, that less iterations can be skipped by approximation, and that render time will increase...

[claude]

a condition that it is not greater than 60 would help a lot

Thanks, I implemented this change, but in my test glitches are still there (albeit reduced)

Will try increasing the probe point count when I have time next week.

[hapf]

What is the iteration range of the image? How many references?

[Dinkydau]

Lowest iteration count: 32 477 267
Highest iteration count: at least  32 543 000

There was only one reference.

I can't determine the highest iteration count exactly because some pixels are at the max iteration count, which I think is because they were not computed yet. I should explain. The render didn't even finish. I rendered it in a virtual machine and saved it every 10% and at 95% and 99%. After reaching 99%, the program crashed at some point (probably when it was at 100%), so I went back to the saved 99% state. I exported the partially finished KFB map and decided to work with that. I'm not entirely sure what would have happened if the program didn't crash. Normally it hides glitches from view, but I can see them in KF's window, so I don't think another reference would have been used to repair them.

I also tried convincing KF.exe that the map was the one and only frame of a zoom video, because there's a glitch reparation tool for zoom videos (file -> Examine zoom sequence), but that didn't work. The program crashes without a warning. Oh and by the way, Claude's versions always crash when I try to use the glitch reparation tool, with this error message:

Assertion failed!

Program: <path>/kf.exe
File: fraktal_sft/CDecNumber.h, Line 124

Expression: ! (b.m_dec == 0)

I also looked more closely at the glitches and it appears this is actually a case of the blurry glitch. It's just not very apparant with my coloring and 5×5 anti-aliasing. Here's a screenshot of the largest glitch without AA and with Mandel Machine's default settings (much denser gradient):

[hapf]

You need a couple references more to avoid the glitch. What skip did you use? Looks like a place where a lot can be skipped. At this depth you use extended (long) doubles for the iterations?

[Dinkydau]

I don't know. Kalles Fraktaler decides how many iterations to skip and which datatype to use.
I had "approximation low tolerance" unchecked to reduce the render time. In a test it seemed like it made such a drastic difference that I didn't consider leaving it checked (and in that test I didn't notice the glithes). If that's what caused the glitches, it means it was either no image or this faulty one.

So it appears that even today, rendering at a depth like this is extremely hard.
Finding the location was also a pain, even with Newton-Raphson zooming. It's easy to forget what to do next if you have to wait hours or even days, and it's easy to lose track of where you are while zooming manually (where that is still required) if rendering a small preview takes more than an hour (including reference). I kept the plan, along with the coordinates and depths of the minibrots found, in a text file so I wouldn't forget what I had to do and where I was. Also I kept screenshots of the test renders with annotations.

[Kalles Fraktaler]

Yes indeed it is hard to get there even today,  but it would be impossible some years ago
The unchecked "low tolerance" is also a factor that can cause glitches, together with the number of terms.
For this depth the datatype used is floatexp (a double for mantissa and int for exponent).
In principle it would be possible to use scaled long double in the same way as scaled double and that would make it possible to use long double up to e9800 instead of currently e4900. I think that would be easier to implement for Claude now when long double doesn't require a separate dll compiled in another compiler (gcc)

[claude]

I also tried convincing KF.exe that the map was the one and only frame of a zoom video, because there's a glitch reparation tool for zoom videos (file -> Examine zoom sequence), but that didn't work. The program crashes without a warning.

I add debugging this crash to my TODO list.

Claude's versions always crash when I try to use the glitch reparation tool, with this error message:

That is to prevent a division by zero, which would also crash with a possibly less informative error message. because GMP big floats don't have infinity/nan (unlike MPFR which is more IEEE-like).  I could make x/0=0 for all x, but that may simply mask errors that should be fixed elsewhere.  I'll try to debug this to find out why it wants to divide by zero.  Added to my TODO list.

In principle it would be possible to use scaled long double in the same way as scaled double and that would make it possible to use long double up to e9800 instead of currently e4900.

Added to TODO list too.

[claude]

Setting x/0 = 0 in CDecNumber seems to allow "examine zoom sequence" to work, provided there are at least 2 images.  There is a logic bug around https://code.mathr.co.uk/kalles-fraktaler-2/blob/refs/heads/formulas:/fraktal_sft/main.cpp#l2200 that reads out of bounds memory (->crash) when there is only 1 image, I need to try to understand that code and maybe rewrite a lot of it.  It's a bit too tricky for me to solve today, and I am busy in the coming days, so I hope you can wait a little longer.

[Dinkydau]

Of course. I expect nothing so everything is a bonus.

[Kalles Fraktaler]

Setting x/0 = 0 in CDecNumber seems to allow "examine zoom sequence" to work, provided there are at least 2 images.  There is a logic bug around https://code.mathr.co.uk/kalles-fraktaler-2/blob/refs/heads/formulas:/fraktal_sft/main.cpp#l2200 that reads out of bounds memory (->crash) when there is only 1 image, I need to try to understand that code and maybe rewrite a lot of it.  It's a bit too tricky for me to solve today, and I am busy in the coming days, so I hope you can wait a little longer.

I have assumed that there would be a full sequence and not a single image.
So I use the zoom of two images to calculate the zoom size.
Once I wanted to fix a single image, then I copied the kfb file and renamed it to what it would have been named if it would be a zoom sequence.
The CStringTable class will return an empty string and not NULL if out of bounds, however strrchr will return NULL.