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Los Frupanishads

Started by 1Yacatismic1, August 07, 2007, 06:19:25 PM

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Bebek Sincap Ratatosk

Quote from: LMNO on September 04, 2007, 08:36:34 PM
Actually, some of the more Theological around here have decided that when talking metaphysics, Chaos* was the combination of Order and Disorder.


That is, disorder is the opposite of order, and only one half of the Chaos that is Universe.
*Chaos as Eris, as primordial, as Universe.

I've often considered that model and it seems quite useful. Of course, I like the metaphor too as a reminder of how much the interpretation of the data imapcts what we see (is it order, is it chaos? It depends on how you're programmed....)
- I don't see race. I just see cars going around in a circle.

"Back in my day, crazy meant something. Now everyone is crazy" - Charlie Manson

Triple Zero

Quote from: LMNO on September 04, 2007, 08:26:29 PM
Quote from: PeregrineBF on September 04, 2007, 08:21:37 PMAnd order does not eliminate chaos. Chaos can come from seemingly ordered systems.
For example: The quadratic map of X-> X^2+c where c is a real parameter. It's even more fun if you do it on the complex plane (z -> z^2+c). Highly chaotic, but a very simple, ordered beginning. And yet the chaos seems to contain ordered parts as well...

Hold on a second.  How are you defining chaos here?
American heritage dictionary says, "A condition or place of great disorder or confusion. "  Which seems to imply the inability to predict what will happen next.  But you seem to have a formula there which, if you do the math, will tell you exactly what will happen next.

What am I missing here?

this is mathematical chaos. sort of.

the formula given is the one of the mandelbrot fractal.

given different values of C (Z is always initialized to 0+0i), the "orbit" of the value of Z when iteratively applied this formula is either stable, periodic, goes off to infinity or chaotic. which makes up (respectively) the inside, inside, outside and edge of the mandelbrot set.

the thing is, a lot of iterative systems are often highly chaotic. even when they're governed by very simple rules. but because the output of the system gets fed back into it, it becomes very hard to predict the output of the system after a high number of iterations.

the mathematical definition of chaos is, something like, a system that changes over time, for which a certain pertubation in initial condition yields an exponential (relating to time) difference in output.
this means, if you have a chaotic system, say a point which moves in a way governed by very simple rules. say you have two points, but they're only 0.01 units apart, if you iterate this system, with 10 timesteps they may be 0.1 units apart, but with 100 timesteps they may be 10000 units apart, and with even more they'll be completely uncorrelated with eachother.

i could give some simple examples another time if you like. (not that i'm the expert on chaos theory, but i know a bit about it)
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e-prime disclaimer: let it seem fairly unclear I understand the apparent subjectivity of the above statements. maybe.

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LMNO

No need.

Mathemetical Chaos Theory is, to me, much different than what most people call "Chaos".  Much like the word "particle" means a much different thing when you're talking about electrons.

And yes, after a high number of iterations it can be unpredictable, but if your calculate each one, it certainly isn't.

So, it sounds to me like Pere is conflating two separate definitions here.

Triple Zero

no.

because it is only predictable in theory

if you even try to do it in practice, in a sufficiently chaotic system there will be precision errors, and they will, after only a few iterations explode exponentially to make the entire system unpredictable.
and even if you are going to try and calculate it all completely symbolic (= no loss of precision), after a number of iterations you will need a piece of paper larger than the known universe to write down all the symbols.

these are very real limits to the predictability of chaotic systems, even when they're based on very simple deterministic rules. often, the best you can do is to try and describe the large-scale behaviour of the system on a higher level, but (as we can see so clearly in the social sciences) you're going to have to accept a large margin of error with such descriptions.

the problem with chaotic systems is, that because of its exponential error-propagating properties, the old saying "but one day surely we'll have the technology to calculate this" simply has turned out to be wrong. some things cannot be solved with moar computing power, no matter how simple they seem.
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LMNO

If such things cannot be calculated, how is the Mandelbrot pattern achieved?

Triple Zero

the mandelbrot set is the set of points (represented as a complex number: x+iy) for which the iterative formula does not go off to infinity. this is the black "inside" of the mandelbrot set. the pretty colours on the outside are generated depending on which iteration the "orbit" of the point goes outside a certain treshold circle (usually the circles of radius 2 units), after which it is guaranteed to go to infinity, so the calculation can stop there. (and going off to infinity is not considered chaotic)

the points on the inside of the set will eventually converge to a point or a number of fixed points, so that's also not chaotic either.

only the points exactly on the edge between the inside and the outside of the set may display chaotic behaviour .. but then i would think not even all of them .. now you're making me wonder as well, which parts of the mandelbrot set are really chaotic in the "tiny pertubation of initial conditions causes exponential difference after some time" meaning.

possibly there are different kinds of chaotic behaviour. i know at least that you can calculate the "Lyapunov exponent" for a chaotic system, which is a measure of the speed of the exponential increase in error. maybe it's rather low for the mandelbrot fractal?

i know i've run into precision problems with the mandelbrot set, but nothing that couldn't be solved by just slapping a few extra bits of precision on.

on the other hand, if you look at the weather-prediction as a traditional chaotic system, i'm very sure about the computational limitations, because you need an exponential increase in computing power in order to gain a linear increase in prediction power (or possibly even less, because we actually *have had* an exponential increase in computing power since the 60s and is predicted to continue for at least some 10-15 more years).
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nurbldoff

I guess the boundary points of the Mandelbrot set are chaotic in the sense that exactly at the edge of the set, an "infinitesimal" movement on the complex plane results in a large change in long term behaviour. There's your sensitivity to initial conditions.
Nature is the great teacher. Who is the principal?

Triple Zero

yeah, but that would be the case for every "treshold" function. and not all of them are chaotic. i think. i guess i'll have to read more about chaos theory.
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bubz_the_troll

Quote from: Professor Cramulus on August 07, 2007, 09:53:51 PM
A lot of the longer prose was amusing, but very difficult to read due to size and resolution. Is there a full-page PDF available?
Out of boredom one day I started to "clean up" Los Fru and make it somewhat legible.  I think I only got the first page done and I got bored again decided to masturbate instead.

nurbldoff

I'm definitely no expert either, but this is what I'm thinking right now:

The Mandelbrot set is a "map" of complex polynomials. The polynomials themselves aren't chaotic; each one just either diverges, converges or it's periodic. The map, however, can behave "chaotically". On the edge of the set, a tiny movement in essentially any direction will bring "unpredictable" changes in behaviour of the polynomial. Or something.

It's easier to make sense out of the logistic map for example... it has one dimension less to worry about.
Nature is the great teacher. Who is the principal?

Darth Cupcake

I just want to throw in that this is really interesting.

I don't have a whole lot to add because I'm very slow with math (not bad, per se, just take a while), but I'll be damned if it's not one of the coolest freaking things out there.

So, you know, keep up the good work. That's all I really have to contribute to this. It's god damned fascinating, though.
Be the trouble you want to see in the world.

nurbldoff

Well we sure jacked this thread good. And what happened to my avatar?
Nature is the great teacher. Who is the principal?

Triple Zero

avatars got lost with the servermove.

but you're right (about what you said in your previous post), so what were we talking about again? it's late, i'll try to figure it out maybe tomorrow :)
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e-prime disclaimer: let it seem fairly unclear I understand the apparent subjectivity of the above statements. maybe.

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nurbldoff

I couldn't even upload a new one, apparently the attachment thingy is write-protected or somesuch. How do you people do it, host your own?

And I actually think the incessant threadjacking is one of the things that make this forum great! Let's discuss that!
Nature is the great teacher. Who is the principal?

Triple Zero

1) yes host it at imageshack, photobucket, etc
2) threadjacking is nice though i kinda like where this thread is at now, although the trend so far is that treadjacks only made it better
3) i also realize it is "tomorrow" right now you i owe you some complicated reply about chaos and the mandelbrot set (i thought i was going to pull Julia in there as well, you realize, poor Gaston never could have seen what beautiful things they look like?) , but to be honest it's not nearly "tomorrow" enough, i woke up halfway the night and didn't really get any proper sleep since. bweargh.
Ex-Soviet Bloc Sexual Attack Swede of Tomorrow™
e-prime disclaimer: let it seem fairly unclear I understand the apparent subjectivity of the above statements. maybe.

INFORMATION SO POWERFUL, YOU ACTUALLY NEED LESS.