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Author Topic: Broken symmetry  (Read 768 times)
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FractalStefan
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« on: April 18, 2017, 07:41:46 PM »

Hello to all,

I'm new to this forum and exploring the Mandelbrot set for some weeks using JavaScript (HTML5 canvas element etc.).

One experiment I did last weekend was to record the path of the Z values which gives this kind of images:



If I restrict the recorded sequences of Z values to only those which diverge (i.e. for Cs which aren't part of the Mandelbrot set), I get pictures like this one:



The above picture has 1000 max. iterations and 100 million randomly chosen C starting points. However, if I increase the max. iterations to 1 million, things are getting weird:


(1 million max. iterations, 1 million random points)


(1 million max. iterations, 10 million random points)

Is there any explanation why the image suddenly loses its symmetry? Could it be because of the limited precision of the JavaScript floating-point numbers (which is 52 bits), or maybe due to the unperfect random number generation?

Here's the JavaScript programm I used.

(NB: Above pictures were histogram-optimized afterwards.)

Stefan
« Last Edit: May 03, 2017, 10:40:50 AM by FractalStefan, Reason: Updated link to JavaScript application » Logged
PieMan597
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« Reply #1 on: April 18, 2017, 09:19:38 PM »

If you don't know already, what you have rendered is called the Buddhabrot, which I think should get more recognition.

The glitches are probably just that-glitches. I have never seen the Buddhabrot do that before, but at least they do look pretty cool!
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Chillheimer
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« Reply #2 on: April 18, 2017, 09:32:05 PM »

Hi FractalStefan - and congratulations on discovering this all by yourself! seriously! smiley

pieman, what glitches do you mean? the asymetry?
that's not glitches but natural. as we know, the buddhabrot method uses random points - so the chance that you hit exactly the same random point of a 10million iteration chain on the oposite side, mirrored is pretty darn low.
i personally find that these 'glitches' are indeed what makes the buddhabrot special - as they really shows single iterationchains of one starting-value each in context with the mandelbrot set.
« Last Edit: April 18, 2017, 09:40:05 PM by Chillheimer » Logged

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FractalStefan
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« Reply #3 on: April 18, 2017, 10:19:47 PM »

Thanks @PieMan597 and @Chillheimer for your feedback. Yes, I was thinking about methods to colorize the "inner" part of the "apple", and while playing around with the Visualization of the Mandelbrot iterations and looking at the "wild leaps" of the Z points in the Gaussian number plane, I thought that it would be interesting to visualize the "path of Z".

I too thought that the random numbers might be the cause for the asymmetry, but in this case I had expected more "randomness" in the form of "noise" etc., but not entire macroscopic "objects" like for instance the two big "rings" in the 3rd picture. The whole thing looks to me a bit like this ancient machine - what's it called? (let's google) - the "Antikythera mechanism". cheesy

So the question ramains how these big asymmetric structures are being formed. The random numbers are separate from each other (at least they should be), so a current Z number doesn't know if previous Z numbers have already gone a certain path, so it cannot "decide" to go there too in order to enforce a certain object...
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claude
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« Reply #4 on: April 18, 2017, 10:36:44 PM »

The asymmetries are in my opinion negative artifacts of not sampling enough points to get a good approximation to the true "infinite iterations" limit set of the Buddhabrot.  However, I don't even know if this limit is well-defined, see: https://math.stackexchange.com/questions/1393110/is-the-buddhabrot-well-defined

I do conjecture that the limit of the anti-Buddhabrot (only plotting iterates of points in the Mandelbrot set) is likely to be well defined, but perhaps dependent on the area of the boundary of the Mandelbrot set being 0 or otherwise, see: https://mathr.co.uk/blog/2013-12-30_ultimate_anti-buddhabrot.html
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Chillheimer
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« Reply #5 on: April 18, 2017, 10:51:25 PM »

I too thought that the random numbers might be the cause for the asymmetry, but in this case I had expected more "randomness" in the form of "noise" etc., but not entire macroscopic "objects" like for instance the two big "rings" in the 3rd picture. The whole thing looks to me a bit like this ancient machine - what's it called? (let's google) - the "Antikythera mechanism". cheesy

So the question ramains how these big asymmetric structures are being formed. The random numbers are separate from each other (at least they should be), so a current Z number doesn't know if previous Z numbers have already gone a certain path, so it cannot "decide" to go there too in order to enforce a certain object...

I'm neither mathematician nor programmer, so my explanation comes solely from observation and my own conclusions after years of playing around with this - and may lack mathematical clarity:
First of all, you do get noise. but most noise will escape pretty fast and happens only in the lower ranges. when iteration chains escape fast, they doesn't have time to "colorize many points brightly"
But if you by chance get an iteration chain that is stable for 1 million iterations, it will colorize one million points - and most of the time, like with the hige circles, these iteration bands will stay very close to each other, adding to the brightness of that feature (compared to the points that escape fast).
You can now go and also set a lower limit for iterations, say only displaying chains (or orbits as they are often called) with between at least 10000 and a max of 1million. you will get far less noise that way.

I often do these - and watch the process closely - i find it fascinating to see these single orbit-chains drawn. and there is a sweet spot - if you keep rendering an image with even billion of iterations for months, you will get the same cloudy image as with 100 max. (except you can zoom deeper in).
Here's one of my images, a closeup of the main bulb, with only high iteration orbits. you can see some "nets" forming - each net "layer" is ONE single orbit chain, each pixel is one single iteration of this long chain. if you overlay hundreds of these nets you will get "noise".

I suggest you take a look at this http://www.fractalforums.com/announcements-and-news/buddhabrot-mag(nifier)-a-realtime-buddhabrot-zoomer/
« Last Edit: April 18, 2017, 10:59:39 PM by Chillheimer » Logged

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« Reply #6 on: April 18, 2017, 10:55:58 PM »

The asymmetries are in my opinion negative artifacts of not sampling enough points to get a good approximation to the true "infinite iterations" limit set of the Buddhabrot.  

It hink this is where the buddhabrot is totally misunderstood imho.
This are NOT artifacts -this is what the mandelbrot-set is actually "made of".
We observe single iteration chains, we observ the pattern chain of one starting value you enter into z-->z▓+c - and the closer it is to the border, the longer the chain will be - visually!
and i find that absolutely stunning and incredible.

Of course - you don't get a clear image of the whole set this way - but it opens up another perspective on the mandelbrot-set that is far more interesting than I expected when I first came across this.
It's like the "quantum physics" of the mandelbrot set - you watch it's elementary particles and how they behave...
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FractalStefan
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« Reply #7 on: April 18, 2017, 11:49:48 PM »

But if you by chance get an iteration chain that is stable for 1 million iterations, it will colorize one million points - and most of the time, like with the hige circles, these iteration bands will stay very close to each other, adding to the brightness of that feature (compared to the points that escape fast).

This is indeed a very good explanation. I think that's it! smiley

So these "objects" are probably coming from very long sequences whose paths are very close to each other.

You can now go and also set a lower limit for iterations, say only displaying chains (or orbits as they are often called) with between at least 10000 and a max of 1million. you will get far less noise that way.

Good thought, I will try this!

I often do these - and watch the process closely - i find it fascinating to see these single orbit-chains drawn.

Unfortunately this is not possible with my little JavaScript. Perhaps I will try to achieve this. Perhaps by rendering and updating the image after each long sequence with - let's say - 50% of the maximum iterations count (this value should be configurable). Another option would be to save each such of these updated images; however, I'm afraid this is not possible in JavaScript (since accessing the file system is a potential security risk for a scripting language running in a web browser)...

and there is a sweet spot - if you keep rendering an image with even billion of iterations for months, you will get the same cloudy image as with 100 max.

Interesting thought. The more interesting would be such an automatical image saving feature...
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Chillheimer
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« Reply #8 on: April 19, 2017, 12:04:21 AM »

glad I could help clarify (and I hope I'm actually correct.. wink

don't know if you missed the link i posted directly under my image, that tool is really worth a look!

So these "objects" are probably coming from very long sequences whose paths are very close to each other.

I'd go as far as to say that these objects are actually one SINGLE sequence, coming from just one set of very deep (as in many .digits) coordinates.
I whish there was a possibility to know exactly what these coordinates are, but no tool I've tried yet has this capabilities to find out after/while rendering.
I'd love to explore them and shape them, find similar ones and just play around with single orbits.
like this: https://www.geogebra.org/m/Npd3kBKn
but for muuuch deeper values..
maybe some day someone will come up with such thing. but it's... well kind of a niche, only few fractal fanatics care about.. very uncharted territory..

here is one single orbit. If I remember correctly I set a minimum of 10 million iteration for this and a max of 50 million. so this chains length is somewhere in between.
I love it that this image is nothing but one pair of coordinates put into z->z▓+c  showing the path that number takes until it finally after millions of iterations is "allowed to escape to infinity"... that incredible complexity, not just in the whole set, but in a single escape orbits - and there are many other forms and shapes to explore.


« Last Edit: April 19, 2017, 12:16:22 AM by Chillheimer » Logged

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FractalStefan
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« Reply #9 on: April 19, 2017, 12:30:54 AM »

glad I could help clarify (and I hope I'm actually correct.. wink

I'm sure you are. Sounds at least very plausible to me.

don't know if you missed the link i posted directly under my image, that tool is really worth a look!

Indeed, I missed it. However, I like coding my own programs and discovering which amazing results will come out of simple algorithms. cheesy

I'd go as far as to say that these objects are actually one SINGLE sequence, coming from just one set of very deep (as in many .digits) coordinates.

Do you really think, all objects are one single sequence, or is each object one sequence? I'd think the latter, but I have to examine this further.

I whish there was a possibility to know exactly what these coordinates are, but no tool I've tried yet has this capabilities to find out after/while rendering.

It should not be very difficult to identify extra-long sequences and log them during the process. At the end of the process, the program could print a list of coordinates (C points) which could then be tested separately which path they will draw. I'll try coding this in the next time.

here is one single orbit. If I remember correctly I set a minimum of 10 million iteration for this and a max of 50 million. so this chains length is somewhere in between.

The shape looks similar to the above mentioned "rings" in my picture number 3.

I love it that this image is nothing but one pair of coordinates put into z->z▓+c  showing the path that number takes until it finally after millions of iterations is "allowed to escape to infinity"... that incredible complexity, not just in the whole set, but in a single escape orbits - and there are many other forms and shapes to explore.

Yeah, a very fascinating object, this so-called "buddhabrot"...! (What a funny name, sounds like the German word "Butterbrot" (buttered bread)... cheesy)
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FractalStefan
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« Reply #10 on: April 19, 2017, 01:27:53 AM »

@Chillheimer: To prove your theory I modified my little JavaScript app which displays the "path" of the Z values for a given C point so that it can now display dots instead of lines. The pattern of some Z value sequences near the rim of the Mandelbrot set are looking very similar to the "ring" objects which appear in the "Buddhabrot" pictures:



So I think, this, in combination with the random values, is the explanation for the visible asymmetries and the differences in each picture (despite equal parameters). I just rendered a second image, using the same parameters as image number 3 in the first post of this thread - and as expected, it looks different. For comparison purposes, here are the two images - the old one and the new one - each of them rendered with 1 million max. iterations and 1 million random points:





It's really cool that - other than the normal Mandelbrot set - each image is a new surprise due to the usage of random values...  afro

Regards & good night,
Stefan

(P.S., for those who try the JavaScript program: I mistakenly stated that the image was colored using the "sinus shape" linearity. In fact, it was the logarithmic gradient. Using other linearity methods will give only a very dark picture.)
« Last Edit: April 20, 2017, 09:26:14 PM by FractalStefan, Reason: Updated links and images » Logged
FractalStefan
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« Reply #11 on: April 22, 2017, 03:52:43 AM »

Hi Chillheimer & all,

the Script I used to create these images has now been updated so it can log the C numbers which are the starting point for the Z sequences, and then allows to browse through the list and to select one, several or all of these C numbers and display the corresponding Z orbits.

This is the Scipt:

http://www.stefanbion.de/fraktal-generator/mandelbrot-z-orbits.htm

And here are two screenshots:





And finally some cropped screenshots of Z orbits:



My favourite one is the last one - doesn't it look a litte like 2 happy goldfish...? cheesy

One observation I made was that the most orbits don't result from very long sequences of Z numbers. Most sequences of this example with 1 million maximum iterations were only a couple of thousands of Z numbers long.

Stefan
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Chillheimer
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« Reply #12 on: April 22, 2017, 09:38:11 PM »

I'm terribly busy right now and the whole next week, but this is now top on my list afterwards! Exciting! Thx fur sharing!
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FractalStefan
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« Reply #13 on: April 23, 2017, 04:15:33 AM »

I'm terribly busy right now and the whole next week, but this is now top on my list afterwards! Exciting! Thx fur sharing!

No problem, don't worry... wink

If anybody wants to see some more images of "Z orbits", look into this gallery:
http://www.stefanbion.de/fraktal-generator/z-orbits/

Stefan
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FractalStefan
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« Reply #14 on: May 03, 2017, 11:02:26 AM »

Hi,

I added new colorization options (gradient linearity, histogram settings) and some presets to the JavaScript application [Link] and also created a web page about "Mandelbrot Z Orbits" [Link] which explains a bit of the background and shows the images created using the presets.

Here's a screenshot of the application after rendering the image for the preset which I called "Alien Radar" (sorry, I found no sillier name... grin):



Stefan
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