Author Topic: Hidden Structures of the Mandelbrot and Julia Sets  (Read 306 times)

Hidden Structures of the Mandelbrot and Julia Sets
« on: August 11, 2018, 04:38:36 PM »
This is among the most interesting things I've ever seen - but then, I'm a geek.


Quote
An exploration of the correlations between the Mandelbrot Set and the Julia Sets for z^2 + c. Followed by some flybys of the full Mandelbrot/Julia structures. Finally, an attempt to show the complete 4D Mandelbrot Set..with partial success.


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Offline Baruch

Re: Hidden Structures of the Mandelbrot and Julia Sets
« Reply #1 on: August 11, 2018, 08:28:54 PM »
OK ... let me summarize from number theory.

In number theory, as I pointed out before, some numbers are algorithmic and others are not.  That was for the real number line.  If you make the move to complex numbers, then for a given algorithm plus a starting point in the complex plane ... the iteration converges or not.  We are naturally interested in the choices that converge in some sense (at least bounded to some finite region.  Other choices are unbounded.

In QFT (Quantum Field Theory) you have calculations that converge very slowly to a single value.  The most slowly converging real numbers are called "normal transcendental" numbers.  Once such number is Chaitin's constant.

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Which is related to the fundamental "halting" problem of computer science.  Normally we prefer real numbers (or complex numbers) that are not maximally slow to converge to a single value.  That is the maximum boundedness.  Most of the interior points (convergent in the sense of bounded) are not that nice.  They stay within a particular zone of numbers, rather than converging to a single value (however slowly).

So here is my idea ... that these "normal" transcendental numbers (in the complex plane) correspond to the infinitesimally-just-inside points of the Mandelbrot that corresponds to the particular algorithm used in the example (usually more complicated than the equation used in the video).  If they were infinitesimally-just-outside  points, they wouldn't converge at all, but it would be hard to determine which side they are on (they would move to complex infinity very slowly).

Notice the application to analog systems vs digital systems.  An infinity of real numbers (and complex numbers) are not computable.  An AI is a computation.  In so far as life/consciousness lies within that set of numbers (not the algorithmic ones) no AI can emulate it (get the correct value by the correct means).  It can approximate it arbitrarily (since any non-computable number is close to an infinite converging series of computable numbers) but here's the rub ...

Approximation doesn't allow emulation, only simulation.  It isn't differentiable.  For example, you can approximate the real number line with the computable numbers (or even the subset ... the rational numbers) but if you take the derivative of it, you are basically producing a series of positive/negative spikes, not a smooth curve at all ... aka what is called "shot noise" in electronics.  These spikes make up something that isn't a proper function, a "distribution" which is most often seen as a single positive spike (Dirac function) in quantum mechanics.

« Last Edit: August 11, 2018, 11:28:57 PM by Baruch »
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Offline Hakurei Reimu

Re: Hidden Structures of the Mandelbrot and Julia Sets
« Reply #2 on: August 12, 2018, 10:59:51 PM »
Baruch, the flaw in your argument is that a real intelligence, artificial or not, has to work in the real world. The reason we have intelligence is not to be intelligent, but rather we're intelligent because our intelligence helps us deal with problems in the real world, and help us survive (to certain values of "help" and "survive"). Said problems tend to be macroscopic in nature — it doesn't really matter much what the individual molecules or atoms or subatomic particles are doing in the system presenting us the problem, as such individual deviations average out over the untold bajilions of particles that constitutes the system.

Remember, people find QM baffling precisely because we just don't see it on the level we operate on day-to-day. Our intelligence simply isn't built around trying to make sense of it, because under normal circumstances QM simply doesn't enter into it — the quantum mechanical nature of the universe is at the large scale blurred by its thermodynamical nature.

In the real world, not the world of a theoretical undisturbed quantum system, there's thermal noise and outside interference, and as such any system that works in the real world has to be robust to these disturbances. An intelligent system that is too frequently prone to making inappropriate responses to consistent situations due to thermal noise or interference will find themselves quickly weeded out in favor of more robust systems.

In short, a system you describe that is so equisitely sensitive to the exact state of the system as it could not be acceptably emulated by a computer would have behavior that is on any practical level indistinguishable from true randomness, and wanton randomness in life tends to be disfavored and selected out.

We operate at a temperature of around 300K, and that's an appreciable amount of noise introduced into the quantum systems making up our thoughts. It translates into about ~25.8 meV, and any state that differs in energy by less across some scale of interest would be utterly lost in the noise, and each brain state that matters (macrostate) is represented by an ensemble of untold fucktons of actual quantum states that may be taken (microstates). Indeed, the degree of coarseness actually implemented in the brain is not single electrons, single microtubles, or even single neurons, but most likely a few tens of neurons, each composed of around 1014 atoms.

The statement that life/intelligence lies in the undescritized continuity is therefore very much premature, and given what else we know, highly suspect. Evolution would absolutely disfavor biochemistry that is so sensitive to thermal noise, where the border between life and death is so razor thin. Indeed, evolution would much favor biochemistry that works to stay as far the hell away from those borders as practically possible (homeostasis). Same with intelligence; evolution would disfavor intelligence so sensitive to practical randomness as to keep their behavior squarely within predictability, and thus respond reliably to similiar stimuli. This is also borne out by psychology experiments, which have revealed that we are actually remarkably predictable.

You can't rely on quantum mechanics to save us from the implications of AI. That is just Deepak Chopra crapola. There's no evidence or research that supports the notion that quantum mechanics or "analogue computing" is vital for our kind of intelligence.
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Offline Baruch

Re: Hidden Structures of the Mandelbrot and Julia Sets
« Reply #3 on: August 12, 2018, 11:01:40 PM »
OK, then build a sentient sex bot and go have his baby ...
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Offline Hakurei Reimu

Re: Hidden Structures of the Mandelbrot and Julia Sets
« Reply #4 on: August 13, 2018, 07:12:27 PM »
I'll take that as a concession of your loss.
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Offline Baruch

Re: Hidden Structures of the Mandelbrot and Julia Sets
« Reply #5 on: August 13, 2018, 08:57:39 PM »
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I'll take that as a concession of your loss.

I feel no loss.  You have never realized that life isn't a contest for king of the hill.  It simply isn't worth my time (but no reflection on you personally) to nit pick this with you.  Same as when you went off into calculating the various moments of continuous statistical distributions once upon a datum.
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Offline Hakurei Reimu

Re: Hidden Structures of the Mandelbrot and Julia Sets
« Reply #6 on: August 16, 2018, 02:21:56 PM »
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I feel no loss.  You have never realized that life isn't a contest for king of the hill.  It simply isn't worth my time (but no reflection on you personally) to nit pick this with you.  Same as when you went off into calculating the various moments of continuous statistical distributions once upon a datum.
Sounds like sour grapes to me. You don't get to say what I have "realized" because you're not me. You don't get me. You've never gotten me. I am a person who thrills in discovery, and I get annoyed at anyone who not only aren't actually interested in the thrill of discovery more than they are exercising their own preconceptions, but who are actively destructive to others engaged in the same thing.

I answered your notion that "brains are quantum therefore AI is theoretically impossible," and you responded with your ridiculous sex droid challenge, as if that were an answer to how we would know AI is theoretically impossible. Us not knowing how to construct AIs currently does not constitute knowledge that AI is theoretically impossible. That's just an appeal to ignorance.

So, I take this as a concession of your loss. Whether you feel the loss is up to you, and I don't particularly care, as long as you don't try to use it as grist in your drama queen mill.
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Offline Baruch

Re: Hidden Structures of the Mandelbrot and Julia Sets
« Reply #7 on: August 16, 2018, 07:15:58 PM »
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Sounds like sour grapes to me. You don't get to say what I have "realized" because you're not me. You don't get me. You've never gotten me. I am a person who thrills in discovery, and I get annoyed at anyone who not only aren't actually interested in the thrill of discovery more than they are exercising their own preconceptions, but who are actively destructive to others engaged in the same thing.

I answered your notion that "brains are quantum therefore AI is theoretically impossible," and you responded with your ridiculous sex droid challenge, as if that were an answer to how we would know AI is theoretically impossible. Us not knowing how to construct AIs currently does not constitute knowledge that AI is theoretically impossible. That's just an appeal to ignorance.

So, I take this as a concession of your loss. Whether you feel the loss is up to you, and I don't particularly care, as long as you don't try to use it as grist in your drama queen mill.

Yes, yes .. already.  I acknowledge that you are the god of atheism.  Better than being the god of toilets.  Being the god of atheism, also makes you the god of irony ;-)
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Offline trdsf

Re: Hidden Structures of the Mandelbrot and Julia Sets
« Reply #8 on: August 17, 2018, 12:53:06 PM »
By the bye, for those interested in exploring Mandelbrot and Julia sets, the interrelationship between them, as well as other fractal types, cellular automata and iterated function systems, let me recommend the granddaddy of them all, You are not allowed to view links. Register or Login.  I've been messing with it off and on since 1991.  It still has a few things that belie its origins in a DOS world, but it's fast, stable, feature-rich and free.  There are, of course, You are not allowed to view links. Register or Login.

I recall the first time I realized that at the depth I was exploring a Mandelbrot set, it was still showing fine detail while scale of the complete set was effectively the size of the orbit of Jupiter.  All you can do is walk away and get some coffee or ice water or something at that point.  And take a few deep breaths.

Edit: Speaking of iterated function sets - an online You are not allowed to view links. Register or Login
« Last Edit: August 17, 2018, 01:07:53 PM by trdsf »
Sir Terry Pratchett, on being told about the theory that the universe is a computer simulation: "If we all get out and in again, would it start to work properly this time?"

Offline Baruch

Re: Hidden Structures of the Mandelbrot and Julia Sets
« Reply #9 on: August 17, 2018, 01:18:38 PM »
Better to be 'iterate than non-'iterate ;-)
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Re: Hidden Structures of the Mandelbrot and Julia Sets
« Reply #10 on: August 17, 2018, 01:18:50 PM »
There's also You are not allowed to view links. Register or Login that's very interesting.


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Offline Hakurei Reimu

Re: Hidden Structures of the Mandelbrot and Julia Sets
« Reply #11 on: August 17, 2018, 09:29:45 PM »
There's a free app for MacOS X, Mandelbrot on Cocoa.

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Just in case you're that kind of person.
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Re: Hidden Structures of the Mandelbrot and Julia Sets
« Reply #12 on: August 18, 2018, 01:59:07 PM »
That looks pretty good. Can I get it for my Android phone?





[edit]
I looked at the google play store and didn't see it, so I guess I can't get it.
Oh, well, such is life!
[/edit]
« Last Edit: August 18, 2018, 03:15:48 PM by Unbeliever »
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"The Republicans went from Abraham Lincoln to Sarah Palin to Donald Trump. No wonder they don't believe in evolution."
Any Borowitz

 

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