I’d like to know the opinions of the posters about Chaos Theory. Do you believe that there is hidden organization to seemingly random systems?
Sorry. By some mischance, I accidentally sent the same topic twice. I was trying to add my signature. I suppose I should not be suffering myself.
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And I think that THAT answered your question!
I’m transferring this thread to General Questions, where it belongs.
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A math guru will probably step up and be able to answer this better than me, but I’ll say a few words anyway.
I don’t think there is any question that chaos theory is valid, but the patterns that are found by chaos theory are not found in systems that appear random. They are found in systems that are too complex to model directly. Weather is probably the most popular example of a complex system, and I think we can agree that while weather is hard to predict, it is far from random.
Also, the patterns found by chaos theory don’t result in precise predictions. They show general tendencies. So to use the weather example again (which may be inapt because I don’t think chaos theory is really used currently in everyday weather prediction), chaos theory might be able to say that it is very likely a certain storm pattern will pop up in two weeks in a certain region. It won’t however predict things like the exact time or temperature of the storm.
I agree with undead dude in that chaos theory is valid.
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Another important aspect of chaos theory is that systems that you would not EXPECT to generate a pattern (because they are not deterministic or periodic in the sense we usually think of it) often do anyway. A famous fractal pattern of nested triangle shapes called Serpienski’s (sp?) gasket can be created by making a random dot on a piece of paper, then choosing a random position on the same piece of paper and placing a dot halfway between that position and your first dot; choosing another random position and placing a dot halfway between that position and your second dot; etc.
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I think this is implied in the discussion so far but I would like to make it explicit. Chaos theory, as a mathematical construct, is certainly valid, i.e., mathematically correct. I believe the intent of the OP, and this is the point I wanted to clarify, is to ask whether chaos theory as applied to physical systems produces valid results. I think the other posters have dealt with that.
I will add that I hope you’re not basing your question on what you learned from the irritating Jeff Goldblum character in Jurassic Park. That explanation was just movie hokum. Yahoo Serious did a better job of explaining relativity in Young Einstein.
“If you had manifested fatigue upon noticing that you had been an ass, that would have been logical, that would have been rational; whereas it seems to me that to manifest surprise was to be again an ass.”
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I may be misremembering, but I think to use your algorithm for generating the Sierpinski gasket, you need to start with 3 points at the vertices of an equilateral triangle (labeled to distinguish them). Then you make successive random choices of which vertex to move towards. The gasket, using this algorithm, is an attractor–any sequence of points will rapidly move toward it, and any point that is actually on the gasket will remain on it through successive iterations. Note that the pattern you generate with this algorithm is (unless you happen to land on the gasket with your first point–which is unlikely since it has measure zero on the plane) only a shadow of the actual gasket, although it is indistinguishable from the gasket at any finite resolution.
This is just in case somebody was getting ready to start coding this up :).
Rick, I did code that algorithm about 7 years ago and found it not to work. I assumed that it was due to lack of precision. All I had were double precision floats.
Also, I thought I saw the same algorithm used with an isosceles triangle, but I’m not sure.
Oh, and to clarify, I coded the triangle-based algorithm that Rick described. I hadn’t heard of one which wasn’t based on a triangle. The random choices were the starting point which needed to be within the triangle, and the random choice of which vertex to approach for each line.
RickG’s explanation is right about the three fixed points, but the triangle, as Undead points out, doesn’t need to be equilateral. (And, obsessive proofreader as I am, I’ll point out that the name is “Sierpinski”. Close, though.)
But even single-precision floats should work perfectly well, as long as you’re not going for extreme resolution. I’m sure a search on the Web would find working code.
See http://spanky.triumf.ca/ , http://sprott.physics.wisc.edu/fractals.htm , and many more for software and images.
Bob the Random Expert
“If we don’t have the answer, we’ll make one up.”
You are all correct that you don’t need an equilateral triangle. I was thinking particularly of the ‘classic’ Sierpinski gasket, which is equilateral. Using an arbitrary triangle (not necessarily isoceles), will, I think, give you a distorted version of the gasket. Note that the initial point need not be within the initial triangle–the attractor is global.
You don’t even need a triangle.
I had so much fun with that incredibly easy fractal, that I played with modifying it by changing the number of attractors, their distance to one another, and the resolution.
BTW, did anyone here know that a Sierpinski gasket has been used to design incredibly small high reception antennas?
One cell-phone I saw recently incorporates it.