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https://www.youtube.com/watch?v=gB-BP9HWiMo&feature=youtu.be

A rather old video. With the basics of how the mandelbrot set is generated. And a bit on how it relates to other things.

I like the passionate way he talks about the mandelbrot set.

great, thank you for linking :D

oh dr moriarty   ;D
its always quite charming to see noobs presenting about the mandelbrot set.  :D
they raise the maxiters from 100 to 1000 and they are like OMG do you even understand how many calculations are happening right now!?
and they zoom in 2 or 3 times and they are like bro do you even understand how mind-bogglingly deep we are right now!?
and of course the obligatory not-enough-maxiters where half the image is black (pretty much every single mandelbrot image ever posted, outside of fractalforums)
speaking of maxiters, is dr moriarty's "number of points" his maxiters?   :nerd:

lol, right, and i am quite disappointing that no one is referring to my "what happens in the mandelbrot calculation" they always explain it for the single point, but this does not show how i myself and me visualised the transformation, hrhrhr need to self promote this because this is what should at least be reviewed from some more than just 160 viewers :D

https://www.youtube.com/watch?v=ce0lms78nt4

hehe and as referenced in the bottom comment, the mandelbrot song just because

https://www.youtube.com/watch?v=ES-yKOYaXq0

Yes it is amusing to watch.

I am curious what his reaction would be if we let him use a perturbation based renderer. :laugh: Julia morphing would be amazing for him.

"That's an interesting approach, but how can it be made to automatically and reliably produce equal-quality images as the normal way?"

:/

That was in 2012. Nowadays it is much better. That is of course if you know how to do it.

I too look forward to the day we can simply say "the perturbation method is better", but for now it is solving a different problem / rendering different images (with no way to tell when you're looking at glitches or not).

It's possible Knighty will get there (I have complete faith in his mathemagics), but the principle of apples to apples comparisons should not be forgotten :)

We already can. It is better than classic rendering, in my opinion. But it requires practice.

I am happy Knighty and the others are having a look at it. But perturbation is good as it is.

So much for apples to apples... oh well :)

lycium, i know you think it is totally different, pertubation theory is an approximation, that is correct, glitches can easily be detected visually alone, for once, why not write a small program that calculates the 50 million iters per pixel by "hand" using a big numbers library, i wonder how long that would take to calculate, perhaps the comparation image would be rendering a year or so (at least with ultrafractal ) for comparison just the "inner" pixels need to be compared, when using exact same areas and iteration set up the 2 images should be exactly the same, or with some differencies the farther one is away from a reference point

nevertheless the results are mandelbrots (an apple) it is not that pertubation makes a pear out of an apple, it just grows the apple in the laboratory but at the end it is an apple :D my five pence on the "we cannot advertise pertubation theory because we dont know if its correct"

Please don't straw man (https://en.wikipedia.org/wiki/Straw_man), I wouldn't make such a ridiculous claim :)

As you agreed, I'm only saying it's an approximation, and well, saying something like "glitches can easily be detected visually alone" in the face of experience/evidence here on FF where people often miss glitches (besides the more general point that machines are made to automate, and requiring human checking is a fundamental step backwards) seems like a much stronger claim.

All I'm advocating is scientific rigour.

In any case, in my opinion the point isn't really worth discussing deeply anyway, since anyone could post an image saying it's a zoom to E49576495764957 and it's difficult to truly know if it is or not, since there is nothing new going on in these ultra deep zooms that doesn't happen earlier.

In some way you did.

Anyway.. It will never be possible to completely automate things and you will always need a person to act if the machine can't detect something. Evaluation is crucial if you want to improve something.

Remember, At this moment Knighty and the others are evaluating to improve the programs.


Edit: come on Lycium, don't hide your text.

Please explain, how did I say you can't advertise perturbation methods?

Circular argument here: you can't completely automate if the machine can't detect something authomatically, therefore you can never completely automate.

The original algorithm doesn't need checking, which is the goal, which is why Knighty and others are working on it. The UltraFractal author claimed to have a bulletproof method at the 2014 Fractal Symposium, and explained to us how it works, but I haven't seen any results yet (and it's obviously not something I can share).

Thanks for the reminder, it's almost as if I didn't say this in my 2nd post :-X

Scared of the Mandel-mob...

You are being silly like you always do. I am out of this discussion.

unlocked it, it is a valid discussion here

in my opinion it is is way over my head to argue the theoretical foundations of pertubation theory, what lycium wants is a value for the error, i believe this is somewhere stated in the proofs of the theory, but in no way known to me .... at least physicians use pertubation theory to build atom bombs ... or to simulate atom bombs and then build these fluffing crap .... so that it is too wrong cant be possible, for me they are both apples, one grown naturally and the other grown in the artifical apple building lab ....

Thank you for unlocking the topic (it seems when he leaves the discussion, no one else may continue); it is frankly unbelievable that a "moderator" resorts to ad hominem when he runs out of substantial replies (after getting his last word, of course, not about the points raised, of course). As for being "silly", you can see why I was hesitant to write an aside with my opinion...

Again, all I was doing is advocating clear thinking / distinctions, and once again I regret doing so on this forum. I'm out of this "discussion" too, serves me right for trying to raise a point about furthering the field.

I'm pretty sure Dr. Moriarty actually knows quite a bit about perturbation theory since it's heavily applied in physics and particularly in those kinds of physics he studies. Indeed it's a rather large problem that you almost can't calculate anything meaningful in the full standard model of physics (as well as things like super symmetry or SUSY) without making use of perturbation theory, and to my knowledge (I might be wrong: It's not like I have worked with that stuff yet) it's a huge challenge to get all those calculations to meet all conditions for them to actually be valid.
I can only assume that the random artifacts you tend to see appear for the same reason. It's a challenge for rendering fractals because it's a challenge everywhere. It's a topic where you have to step forward with extreme caution, as per the motto: "Garbage in -> Garbage out", but the garbage in this case looks almost indistinguishable from the real thing.
And many people are actually calling for non-perturbative (i.e. exact) rewrites of those same theories in hopes to get rid of all the mess that it entails. Many of the tools we currently have available just aren't up to snuff for now. It's being worked on, at the very edge of today's math and as such it'll probably take years if not decades for the hopefully eventually successful tools to trickle down into niche communities such as fractal art creators.

Of course that doesn't mean people shouldn't be trying anyway. :)

lycium has taken this stance before about perturbation.  i think (?) he is more saying, how can we Prove that perturbation will always produce the exact same output as the standard method?  I don't think one can offer such a proof.  Anecdotally speaking after a few years of using perturbation rendering with automatic glitch detection and correction, it appears to be solid enough to use.  One would also have to decide what exactly they mean by "proof" and "exact."  Maybe if you used full arbitrary precision for all variables in the perturbation method it could be "proven" (?) to produce "exactly the same" results as the standard method, but the whole reason we use perturbation is to use less precision to gain speed and hope it all works out in the end.  How much does this affect the outcome and in what ways?  Are the numbers maybe a little bit off, but the iterations still come out the same, or maybe are the number of iterations sometimes off by a few in the end?  I dont think anyone has really investigated this much.

At least practically speaking, perturbation itself appears to produce identical or virtually identical images.  What is really left to nail down 100% is the utilization of series approximation to initialize points - more specifically, the stop condition, which is what is currently being further investigated, and looks to have a promising outcome thanks to the likes of knighty and claude  :beer:

at least i dont understand why the visual proof is not beeing accepted, stuff like the shape stacking and the obvious behaviour that is quite regularly e.g. doubling spirals, real minibrots and - in my point of view - are concrete hints to that the general approach seems to be intact using the method ... i just mean, if it looks like a mandelbrot, if it smells like a mandelbrot, sounds like a mandelbrot, feels like a mandelbrot and behaves like a mandelbrot ... it has to be THE mandelbrot

in my youth i really was thinking there is something new down there, today we now it is "just" endles repetition of previous behaviour, spiral doubling every n iterations and so on

recent competition nevertheless produced some stunning views from down there

exactness in this case is rather easy to define.
If you go into the limit of n->inf for the perturbation method, does it converge against the exact same thing as if you were to use a standard iteration method? - Rounding errors and the like not withstanding.
If the answer is yes, then it's exact.

It's kinda silly that things like "automatic error detection" are even necessary. If it really is correct, and there should be a way to make it fully correct, there will be no error (larger than what ever the error is for a given n) - certainly no additional structure that persists no matter how high you push n. And as a result there will be no need for any kind of error detection.

And that should be provable. - In a formal mathematical sense.

well, again, it seems the perturbation method itself is correct and will result in no errors if enough precision is used.  it is the utilization of not enough precision which introduces the errors.  hapf mentioned once conducting some experiments using more precision when necessary to avoid having to recalculate points.  i suspect this must be what the ultrafractal author meant if he in fact made some statement about "avoiding" glitches.  as it is a precision issue, there would really be no other way.

I agree with the statement above.  I can't speak to the perturbation method in general but when applied to fractals, as discussed here, there are no approximations beyond numerical precision.  When deriving the perturbation expressions (for example equation (1) in the K.I. Martin technical note) nowhere is an approximation made.  Rather the equations are recast in terms of quantities that, for a suitable reference point, can be calculated using lower precision.  These quantities are the orbit relative to a known reference orbit which is usually calculated using high precision math.  Glitches occur when the reference orbit is not close enough to the orbit of interest such that their difference can be adequately expressed in terms of standard precision.

This isn't the case when the perturbation method is used with series approximation.  When using series approximation you are truncating the expression for the relative orbit which is an approximation beyond just numerical precision.

Ok, that sounds fine. Though it ought to be possible to get bounds on when higher precision or a closer orbit are necessary without any sophisticated method to check that all is fine. What does this error correction actually entail? How does it decide whether an error has occurred and we need a boost? If that also is a simple check - akin to checking whether a point falls outside the circle of radius 2 to decide that it isn't in the Mset - then I really don't see a problem with the technique, based on what you guys are saying.

The celebrated method discovered by Pauldelbrot here is as follows:
http://www.fractalforums.com/announcements-and-news/pertubation-theory-glitches-improvement/msg73027/#msg73027 (http://www.fractalforums.com/announcements-and-news/pertubation-theory-glitches-improvement/msg73027/#msg73027)
Directly under his last formula, he briefly notes the implementation which is not difficult to code for.

Note that a simpler check of verifying the size of the relative orbit remains small is also needed.  Depending on how one codes the remaining details of the perturbation method, further checking may also be required.

Ok, I see... Well, the solution Pauldelbrot came up with is brilliant but it still has a slight ad-hoc-ness smell to it. It's probably fine but I bet there are some further improvements to be made. For instance, perhaps we could figure out why is a sensible threshold. Though the most convoluted part is where he starts hybridizing the two techniques.

Mind you, that's perfectly fine for image generation. If they look correct, that's probably good enough from an artistic standpoint. But there probably still are good improvements to be had. - And even if the numbers all turn out exactly correct (within the error bounds), the scheme could probably be simplified while yielding the same thing.

That post is over 2 years old, however. I didn't follow all the perturbation theory stuff too closely (and clearly missed out on a big deal) so I can only assume that improvements on that idea have been made since?

http://www.fractalforums.com/announcements-and-news/pertubation-theory-glitches-improvement/msg91934/#msg91934

In a previous post I said I left the topic. I regret saying that. I will join the discussion again as I have changed my mind and I have something to contribute.

Anyway, I have sent an e-mail to Sir Moriarity. And he replied, see the screenshots below:


Edit: Hey quasor!

I see, so this discussion is already happening elsewhere :)

Meanwhile, Dr. Moriarty's reply is more relevant to this thread. Btw, TheRedshiftRider, he was kind enough not to point it out but you might have an i to many in his name there ;)
Keep us posted if that correspondence develops any further though. He probably wouldn't have the time but perhaps he could have a look at the perturbation work and give a pointer towards improvements.
The perturbation theory link he gave is sure to be a good start:
http://fizika.unios.hr/~ilukacevic/dokumenti/materijali_za_studente/qm2/Lecture_2_Perturbation_theory.pdf (http://fizika.unios.hr/~ilukacevic/dokumenti/materijali_za_studente/qm2/Lecture_2_Perturbation_theory.pdf) for your convenience.

nice one redshiftrider,  maybe you can get him to join the fractalforums  :D
hes gotta take a break from doing more important things once in a while doesnt he?
he wrote a mandelbrot program after all

I believe he likes to play e-guitar and drum kit for fun

Thank you for the url. :) I had just enough time to post so I forgot it.
Oops... thanks again. I'll correct it and apologise. :)

I will give him a link to this thread so he can read this all. He could decide for himself to join.

Yes.

I got another reply. A pretty nice one actually. ^-^

Edit:

It reminds me of this:

https://youtu.be/HvgomEfQfKc

dont worry phil, we are just pretentious fractal enthusiasts, carry on  :evil1:

I asked him this as well. This is very interesting, absolutely amazing. :nerd:


Edit: link: http://www.nottingham.ac.uk/~ppzpjm/Frontiers-Nano-Ver2-1-May2015.pdf