Mathematics? It's all just life.

I am a writer and looking for ideas based on the 'life is just maths' theorem, e.g. how a beautiful face is just a measure of symmetry, and butterfly-flaps-wings-world-war-three-breaks-out type formulae. Any ideas?

Nature follows some pretty basic but very elegant mathematical patterns.I would check out Fibonacci numbers , the golden rectangle and logarithmic spirals.

I'm not a big fan of chaos theory and, but for the beauty of the metaphor, the "butterfly effect" idea with its stretched-to-the-breaking-point logic would have never made it out of the research paper it originally appeared in.

You can also get some really cool ideas from scientists who have tried to explain physics in lay terms like Richard Feynman and Stephen Hawking but that might take a little longer than what you had in mind.

Good luck - sounds like a cool project. If possible, why not post the story here when you are through with it. I'd love to read it as I'm sure many here would.

A classic.

Thanks Beest, I may well do that, although it is some way off completion...

That Fibonacci thing is interesting, and has presented a somewhat animated discussion point for at least two dinner parties. Cheers :beer:

Originally posted by Slartibartfast
Another Classic

Wow. :thumb:

I read a story years ago...cant for the life of me recaol either title or author. It was part of a collection of short stories. The basic premise was a time travel holiday firm. You could buy a trip back in time but it was all very tramlined. You had to stay on the "path" which existed sperately from its surroundings. At no point were you to step fro the path....The central character steps from the path....a minor adventure ensues whilst he attempts not to alter history....By the time he gets back to the path it seems he has succeeded in not damaging or affecting anything major. Just a butterfly dies thats all.
When he is returned to his own time he steps into the travelcentre lobby and the signs are all written in a strange format, more like old english than modern. As he leaves the travel station he sees more and more differences. he had chanegd the past when he killed the butterfly....Ok it wasmuch better the waythe wrter did it...I am recalling from 12 years ago and am a tad stoned.

(edited to say i think the writer was JG Ballard )

It's a famous Rob't Heinlein story but I can't remember the title.

There is a Heinlein story that is similar but the one i am thinking about I am pretty sure is a JG Ballard story. I just went looking at some bookcovers online and i think it was a short story in the collection "Myths of the Near Future"....

I used to love Heinlein's stories though. The Cat Who Walked Through Walls, Job and Time Enough for Love are my favourites.

It's a Ray Bradbury, isn't it?

edit to add: Yes, it is.

Yes! thats the one.

Originally posted by Undertoad
It's a famous Rob't Heinlein story but I can't remember the title.

I've read quite a number of Heinlein's books and don't recall a premise like that.

Could you work on it? I'd like to add it to my collection.

It has already been established that the story in question was by Ray Bradbury, and is called "A Sound of Thunder."

(That's the story that I linked to in my post above.)

Thanks for the Ray Bradbury guys... good foundations, kind of thing I'm looking for. Anything else you can think of would be great :)

Larry Niven wrote several short stories using time travel, which remind me of the Bradbury premise. Niven's books are more lighthearted, though. His premise is that since time travel is fantasy, a time machine would send you to mythological times, with dragons, unicorns, etc. They center around a big game hunter collecting specimens for the royal zoo.

"Despite the fact that J.S. Bach was not widely appreciated during his lifetime, among those who knew music well, his reputation had spread. In the year 1747 he was given the honor of visiting the court of Frederick the Great at Potsdam (May 7th and 8th) where he brilliantly improvised a fugue on a difficult theme proposed (and composed) by the King (himself, an amateur flautist). Stories of that amazing improvisation (a six-part fugue) on a theme conceived almost to defy the possibility of such improvisation, have become legendary and have been likened to a man playing a hundred games of chess simultaneously and winning. It was a feat of unparalleled skill and musical genius."

That fugue has been described as almost mathematically impossible for a human mind to create.

The keyword is "almost".

Let's see how badly I can mangle physical theory based on my memory of high school.

So you've got one body floating in space. It's simple to demonstrate it's position and behaviour; if nothing acts upon it, it just sits there.

Two bodies in space. When they have a gravitational effect on each other, they trace out an ellipse. Their motions can be predicted and explained analytically using mathematics. Newton solved the 2 body problem.

Add a third body? The math goes to hell. It is generally thought that the three body problem lends itself to chaos theory, and that there is no analytical way to describe the movements of three bodies insofar as how they act on each other. It was originally postulated that one could predict the future of an object's motion assuming you knew the forces acting upon it, but the three body problem proved this to be false; and essentially gave birth to the idea of chaos theory.

In 1912, Karl Sundman developed a convergent infinite series as a solution to the restricted three body problem. Problem is, getting it to any level of precision required something on the order of 10^8,000,000 terms, and his solution is of little practical use.


I always was fascinated by the fact that science cannot explain something as "seemingly" simple as how three objects affect one another.

Brains perhaps this could be applied to human relationships as well (e.g. the age old three's-a-crowd) or why social groups of three or more are often surprisingly complex, and if you extend this to society how chaos inevitably ensues. Does it's complexity render it inexplicable? Or is this just our inability to comprehend it?

I always was fascinated by the fact that science cannot explain something as "seemingly" simple as how three objects affect one another.

The idea of something that appears simple hiding a multitude of complexities is a promising theme as is its reversal - chaos that is actually incredibly simple.

Did I mention that the idea of chaos theory was first proposed by Henri Poincare, a French mathematician, while he was trying to solve the 3-body problem?

I used to have a chaos pendulum to remind me that life was crazy

There's a billion neato mathematical curiosities if we're talking about space ;-)

All could have literary analogies, easily.

Ever heard of the Kirkwood gaps? So there's this belt of asteroids between Mars and Jupiter, the Main Belt, but they're not evenly distributed. There are certain gaps, where no astroids exist, and "enhancements", where the astroids are clustered in concentration. The location of the gaps is due to 3-body physics, and therefore directly related to chaos theory.

If you take the region of the Main Belt where the mean motion of the asteroids is half the mean motion of Jupiter, a region known as the 2:1 Resonance Region, there are no astroids. If you somehow "placed" an astroid in this region, the eccentricity, e, would vastly increase until the asteroid was in the same orbit as Mars, and either collided or was thrown out.
Def'n of eccentricity
But, if you take the 3:2 Resonance Region, the eccentricity remains stable, and asteroids can survive here for long periods of time, and therefore they cluster.

Bottom line, part of chaos theory says that when you change the initial conditions no matter how little (like, the initial conditions for how the earth reacts with the sun might be hinged on what side of your fridge you keep the jam), the results vary wildly, exponentially. BUT, there are FAMILIES of solutions! It's not RANDOM, but it's chaotic.

And guess what this discovery gave us?
Fractals. Mandlebrot figured that the dimensions in a chaotic system had to be fractionated. So you can represent many complex chaotic systems with a fractal.

And how does this relate to a butterfly flapping its wings? So you'll always be informed at parties, a butterfly flaps its wings in New York, which changes Earth's "initial conditions" to such a small degree, but chaos theory takes over and "magnifies" this change in conditions exponentially, and it rains in Japan. Something big begat by something miniscule.

So you'll always be informed at parties, a butterfly flaps its wings in New York, which changes Earth's "initial conditions" to such a small degree, but chaos theory takes over and "magnifies" this change in conditions exponentially, and it rains in Japan. Something big begat by something miniscule.

And therein lies my difficulty with chaos theory (or at least this famous example). There are probably over 100 trillion variables that, taken together, determine whether or not it will rain in Japan. To connect the rain in Japan to a single one of them with the implicit assumption of causality is preposterous. To suggest that all of the 100 trillion (or whatever) variables were in perfect balance and the lone butterfly tilted the scale just enough is to remove one more shovel full of dirt from a (nearly) bottomless hole.

Unlike the last voice in Horton Hears a Who, all the variables that contribute to an event are not all pointing in the same direction. Some amplify each other, some cancel each other out, some diffract others and the behaviour of others depends on the state of yet still others (if there's no rain in the clouds, then its not going to rain no matter what else happens). To connect events on either side of a stochastic system is in itself a contradiction.

I think your difficulty with that analogy lies in a slight misinterpretation of it's meaning.

Now, don't get me wrong, I'm not trying to insult you, I've never even met you ;)

But the way you describe it...That's not quite how it's meant to be interpreted.

I'll start at the beginning and summarize...I apologize if you already know some or all of this, but at least I wont be backtracking later.

Scientists developed a model of a deterministic universe, widely pushed by Newton, who developed laws which, in the span of a few sentences, accurately described thousands of complicated universal processes. He stated that the universe is rather like a game of billiards; if you hit the ball the same way each time, it'll roll the same direction. Hence, if your initial conditions are precisely the same, the outcome will always be the same.

Makes logical sense, right? If the golfball is on the same blade of grass, all the weather is identical, and you hit it with identical speed and force, it should land precisely the same place every time 400 yards down the course.

Now here's the issue; you can never measure your initial conditions to infinite accuracy. To do this, you would need, say, a thermometer with an infinite number of decimal places, otherwise there's a tiny bit of room for error, no matter how small, maybe a millionth of a degree.

For a while, people believe that if your measurement of initial conditions was more accurate, your prediction of the outcome of an event would be more accurate. If you could measure the force you hit the golf-ball to 50000 decimal places, you could more accurately determine where it would land.

Scientists went on believing this until that arse Poincare came along and tried to look at 3 planetary bodies, all interacting with one another. He realized that even if he took insanely precise measurements of the planets and all the initial conditions, he could not accurately predict how they would move in relation to each other. The reason was that, if his initial conditions were off by a hundredth of a decimal place, the resulting outcome was ENORMOUSLY different than he predicted. If he made his measurements of the initial conditions 100,000x more accurate, the end result would STILL be way, way different if you went a billion decimal places down the line and changed a 2 to a 3.

He figured out that some systems, minute changes in initial conditions grow to have an enormous effect in a tiny amount of time. The uncertainties (and there will always be uncertainties, no instrument can measure to infinite decimal places) will always overwhelm any calculation and defeat the accuracy of your prediction. No matter how tiny these uncertainties are!

This is where the butterfly effect comes in. It's just what I said it was, an analogy. "Whether a butterfly does or does not flap its wings can determine whether a storm will arise, a year later, on the other side of the world".

The butterfly is the billionth decimal place of uncertainty. The weather system is the chaotic system.

So I see what you're saying, but if you have a MILLION tiny variables, each SINGLE one of them has the power to change your prediction into something completely different. The butterfly effect states that if you have EVERYTHING the same, and ONE butterfly flaps its wings, that's enough uncertainty to throw off your calculation by orders of magnitude.

Goddamn this is a long post.

Does it make more sense now?

Or, the butterfly is the straw that broke the camel's back.

Beestie just said that.

Originally posted by Carbonated_Brains
Beestie just said that.

Actually he said that it didn't.

He didn't say that it did, didn't he?

Originally posted by Carbonated_Brains
He didn't say that it did, didn't he?

To clarify:

It is my interpretation of Beastie's post that the impact of the flapping of said butterfly's wings was negligible, counterbalanced by too many other factors, and thus not of moment.

Ah.

That's what I was trying to dispell; it's not a matter of one butterfly out-weighing all the other variables, it's a matter of "something as small as one butterfly can vastly change the outcome of the calculation"

What I tried to say is that I have a hard time establishing any cause-and-effect relationship between an independent (air movement from a butterfly flapping its wings) and a dependent variable (weather in Japan) in a chaotic system.

Let me put it another way: is anyone saying that had the butterfly not flapped its wings that it would not have rained in Japan? It sounds to me like that is what the example is suggesting and I'm saying that the idea is ridiculous.

I was saying that a scenario whereby the butterfly is the straw that broke the camel's back is (to put it mildly) not plausible because of the number and frequency of interactions that occur between the butterfly and the rainstorm. To adopt the butterfly theory is to ascribe an equal liklihood to each of the variables which is to render each of them essentially negligible.

To assume that one variable out of a trillion can propogate through the entire system - reinforcing everything in its path or its successor's path so as to rival the magnitude of a weather pattern defies any application of common sense.

The 3-body problem, imho is a different problem - the variables are discreet in number and, therefore, traceable. In a chaotic system, the variables are not traceable therefore, the notion of cause and effect has no application. While the butterfly may have flapped its wings and it may have rained in Japan, wether the two events are dependent or independent is unknowable.

In Zen, we can speak about chaos but never of chaos.

What I tried to say is that I have a hard time establishing any cause-and-effect relationship between an independent (air movement from a butterfly flapping its wings) and a dependent variable (weather in Japan) in a chaotic system.

But how is this different than you pushing a line of dominos, resulting in the last domino falling? It seems like you're arbitrarily "deciding" that a butterfly cannot possibly have an affect a year later around the world. Chaos theory highlights the dependency of massive outcomes on tiny events, that's the entire basis behind it.

Let me put it another way: is anyone saying that had the butterfly not flapped its wings that it would not have rained in Japan? It sounds to me like that is what the example is suggesting and I'm saying that the idea is ridiculous.

That's exactly what the theory states. Allegorically of course. Always keep in mind this is an allegory, the butterfly just REPRESENTS an infinitismally small alteration of the initial conditions.

One initial condition, A, is the butterfly does flap its wings. It rains in Japan a year later.

Initial condition B, is the butterfly does NOT flap its wings. Japan goes through drought.

The initial conditions, tiny and almost identical, have enormous impact on the final outcome: That little deviation, between flapping and not flapping, exponentially propogates into a massive change.

I was saying that a scenario whereby the butterfly is the straw that broke the camel's back is (to put it mildly) not plausible because of the number and frequency of interactions that occur between the butterfly and the rainstorm.

That may be true, but it's not the point. Whether it's the butterfly that actually causes the storm is completely irrelevant. The allegory is demonstrating that something (whatever it is) as small as a butterfly flapping its wings, can determine whether a huge event does or does not happen.

A tiny event results in a massive difference in outcome.


To adopt the butterfly theory is to ascribe an equal liklihood to each of the variables which is to render each of them essentially negligible.

Again, there is no butterfly theory. Chaos theory does not ascribe equal likelyhood to all variables, but states that all variables are capable of enormous impact on the end result, even tiny ones.

To assume that one variable out of a trillion can propogate through the entire system - reinforcing everything in its path or its successor's path so as to rival the magnitude of a weather pattern defies any application of common sense.

Why does it defy common sense? Just because it seems unlikely? I think common sense would say that if you knock over the first domino, the last one eventually falls. There's nothing that should suggest a major event can't start from a tiny, tiny key event.

The 3-body problem, imho is a different problem - the variables are discreet in number and, therefore, traceable.

They are no more traceable than the butterfly scenario. The variables in a 3 body problem could be as numerous as the atoms in each celestial body.

In a chaotic system, the variables are not traceable therefore, the notion of cause and effect has no application. While the butterfly may have flapped its wings and it may have rained in Japan, wether the two events are dependent or independent is unknowable.

It's only unknowable due to the fact that you can't be infinitely accurate. If you had a set of infinitely accurate measuring tools, you could definitely spend an infinite number of years doing calculations that eventually trace back to a butterfly flapping its wings.


In Zen, we can speak about chaos but never of chaos.


Chaos is the supreme ideal of Taoism. Chaos is wholeness, oneness, and Nature. Chaos represents the natural state of the world. Digging holes in the head of Chaos means destroying the natural state of the cosmos.

Originally posted by Carbonated_Brains

They are no more traceable than the butterfly scenario. The variables in a 3 body problem could be as numerous as the atoms in each celestial body.

Nyet. There are eighteen variables in the three body problem (position and velocity, in three dimensions, of the three bodies, with respect to the center of mass of the three). There are also three constants (mass of the three bodies). If you start worrying about the atoms in the bodies, it's not a three-body problem anymore.

I worded that wrong, methinks.

What I mean to say is, the main variables you mentioned are a direct result of things like atom configurations and on which side of the fridge you store the marmalade, etc.

Each major variable is an amalagamation of the exact positions and states of trillions of tiny components, an error in measurement of any one of those components can propogate and exponentially increase.

So if some football player in Kansas jumps into the air right when you're trying to do your calculation, the infinitesimally small amount of force he exerts to jump, and the resultant gravitational pull which brings the two bodies (him and the earth) together again, resulting in a tiny change in acceleration of the Earth, resulting in a temporary change of initial conditions...that could be enough to throw off the calculation.

I doubt I'm explaining this any better than before.

I am admitedly hard-headed but I still do not think that your examples demonstrate chaos theory or explain why my position is flawed.

Neither the domino example nor the 3-body problem are analogous to the butterfly example.

The domino example falls short for several reasons: a) there is only one element (the dominos), there is no intereference from other elements (reinforcing or dampening) and there is no provision for amplifying the magnitude of the effect as it ripples through the system.

The three body problem is also different than the butterfly example. The 3-body problem is one of sensitivity to initial assumptions. For example, how the universe formed (galaxies with space between them) is entirely dependent on (among other things) the mass of electrons and anti-electrons. Change either of those masses in a nearly infinitely small amount and the post- Big Bang universe is very different. Why? Because there are so many electrons and anti-electrons. Change the mass of a single electron (a single butterfly) and nothing happens.

It seems to me that what you are argueing as a chaotic system is more like the following example: a germ microbe infects a flea that bites a mouse that then goes nuts and chews through some wires in a nuclear submarine causing a meltdown that brings down the sub. So, in this example, a butterfly flapping its wings which blew the microbe left instead of right thereby infecting the flea did, in fact, lead directly and traceably to a nuclear explosion. This example, however is not an example of chaos theory in action.

Ultimately, a theory has to aid in our understanding of an event. My problem with the butterfly example is and remains that it really doesn't do or say anything nor does it prove or even allege anything that we didn't already know. When we see it raining in Japan, it is not illuminating to postulate that the rain might have been caused by a minute event that happened over a year ago. Obviously something caused the rain - but chaos theory as explained in the butterfly example brings us not one inch closer to understanding what or why. Nor does it eliminate any false notions of what caused the rain. Basically, its useless -we are no closer to the truth nor are we any further away from a lack of understanding than we were before the "theory" was introduced.

That is why I have such an academic contempt for this example - it sounds really smart but its as empty as outer space.

If you're game, I wouldn't mind taking another crack at this argument, but I feel some might think it's getting old or redundant.

Wanna keep it up, or should I respectfully drop the topic and we'll agree to disagree?

I'm interested.

If your email link is active (mine is), we can take it off line so as not to bore the rest of the Cellar to sleep :)

Just because we aren't all commenting doesn't mean we aren't entertained.

Great. Let's keep it here, CB.

I thought of two other problems with your examples that I touched on but did not address directly. In the butterfly example, something small (a butterfly sneeze or whatever) is deemed to have caused something large (a hurricane). In the domino example, the first domino falling causes the last domino to fall but the events are of equal magnitude. Connecting the first and last domino is much more realistic than connecting the sneeze with the hurricane. No one has addressed my concern that the initial small event continues to increase in magnitude (conveniently without interruption or interference) throught the system from beginning to end so as to terminate in something as big as a hurricane. I throw a rock in a lake and the way of the world is that the waves get smaller as it propogates. Why? Because the elements it encounters require energy to change their state - the energy of the rock hitting the water is slowly dissapated by creating the waves until enough water is encountered to have used up all the energy the rock had and the waves disappear completely. The sneeze causing a hurricane argues nearly the opposite. The sneeze actually has to pick up steam as it ripples through the world. Now, waves can be reinforced as they travel through a medium by other activity also in the medium. But, that is really stretching it - hence my invocation of common sense to put an end to the endless chain of increasingly preposterous suppositions necessary to connect a sneeze to a hurricane.

Additionally, in the three body problem, another way to demonstrate my unwillingness to accept it as an example of chaos can be illustrated by the difference between temperature and heat. Raise the termperature of one atom by one degree and the increase in heat is negligible as are the effects of the increase in heat. However, raise the temperature of every atom in the universe by one degree and the heat gain is nearly immeasurable. So, the "small change produces a big event" idea seems to hold but what is being called a small change is really a big change. A butterfly sneeze is a small change - one atom's temperature increase is a small change. In addition, an error in an initial setting is a factual error - the reality of the situation did not vary - only our understanding of it. The effect of the error has a direct and instantaneous effect on the outcome. The sneeze however, is but one event out of a trillion that help determine the weather.

Furthermore, I do not accept the premise that the absence of the sneeze prevents the hurricane because it presupposes something that was never established - that the sneeze caused it in the first place. And the weakness of chaos theory, at least in this example, is that it is powerless to prove me wrong or even suggest why I might be. Not to be overly dramatic about it but chaos theory - at least what I have heard/read so far is little more than a repackaging of "cause and effect" but with enough hot air to float a blimp. Its predictive value is zero as far as I can tell and that is, after all, what theories are supposed to be for.

"It's predictive value is zero"

Damn right, that's the entire point behind the theory; chaotic systems cannot be predicted.

But let me form a better, more detailed response, I'm currently at work and the engineers around here are in a huff about something.

btw, how mathy can I get?

as mathy as you wanna be ...

(come on, you were ALL thinking it. )

Originally posted by Carbonated_Brains
...chaotic systems cannot be predicted.

So the butterfly really didn't cause the hurricane then. Because the same analytic tools that confirmed it could have predicted it.

And a model that cannot predict anything is a fiction - it exists outside the realm of the verifiable.

No one has addressed my concern that the initial small event continues to increase in magnitude (conveniently without interruption or interference) throught the system from beginning to end so as to terminate in something as big as a hurricane.

I really don't know shit about this but it struck me that the sneeze could tip the balance between two larger, virtually equal forces. The result of which tips the balance between two slightly larger forces. In that way the result of the sneeze gathers force. Just a thought. ;)

Originally posted by Beestie


Furthermore, I do not accept the premise that the absence of the sneeze prevents the hurricane because it presupposes something that was never established - that the sneeze caused it in the first place. And the weakness of chaos theory, at least in this example, is that it is powerless to prove me wrong or even suggest why I might be. Not to be overly dramatic about it but chaos theory - at least what I have heard/read so far is little more than a repackaging of "cause and effect" but with enough hot air to float a blimp. Its predictive value is zero as far as I can tell and that is, after all, what theories are supposed to be for.

Yes, it is impossible to say a butterfly caused a hurricane, but it is a thought experiment that demonstrates how chaotic systems work. Taking two systems with almost identical initial conditions, you will get vastly diverging results because over time differences magnify rather than dissipate. Comparing two 'systems', one in which a butterfly flaps its wings, and another in which it doesn't, is a cool way of saying these two systems are identical- with the exception of the tiniest infinitesimal difference.

And chaos theory has been shown to happen in weather systems, using less variables than the near infinite of the real weather system of the earth. Scientists can run identical simulations over and over with perfect accuracy using computers. It is real easy to show that changing one variable a tiny amount will result in a vastly different end result.

Weather is unstable. Bruce just explained it really well. Imagine a ball perfectly balanced on the top of a smooth hill. The slightest wind in any direction will force the ball to fall in any number of directions. All the kinetic energy of that ball is released by a tiny push.

All the kinetic energy of that ball is released by a tiny push.

Potential Energy, but let's not split hairs.

That was a damn good explanation, and had my 10 pages of rebuttal not been deleted when the stupid message board asked me for my password upon clicking "submit", I would have probably said something along those lines, with much more dilly-dallying and diagrams.

The computer thing is interesting though, the computer really wont repeat the same experiment and get the exact same answer if it's a truly chaotic model, because the tiny rounding errors made by computer processors are enough to propogate into the results over time.

The same chaotic "prediction" made by two computers will be hugely different due to minute changes in the way the processors are made, and handle math.

But you're right, bottom line is the butterfly does not CAUSE the situation, it merely represents a difference in initial conditions.

Originally posted by Carbonated_Brains


Potential Energy, but let's not split hairs.

Oops.

Originally posted by Carbonated_Brains


The computer thing is interesting though, the computer really wont repeat the same experiment and get the exact same answer if it's a truly chaotic model, because the tiny rounding errors made by computer processors are enough to propogate into the results over time.

Hmmmm, but if you feed the same computer the same variables on two seperate occasions, it is going to round them in exactly the same way each time it runs the simulation.


edit: added the word 'same' before the word 'computer'

Big, expensive problems yesterday.

Bad, bad day.

Found a screw in my tire when I got home from work, second time in 2 weeks. Also, somebody bent the shit out of my gas-nozzle cover on my car, and the door wont close properly.

After all that, I'm gonna take one huge crack at explaining how not only is chaos theory absolutely plausible, but it's probably the basic structure of our universe, so much so that "linear" systems are considered rare in comparison.

Oh, and I'm writing this shit in Notepad, and copying it over. I was ready to break this computer yesterday when everything got deleted. Here goes!

(oh, btw, the 'big expensive problems' were not gas-nozzle related, but $200,000-piece-of-equipment-stuck-10-metres-underground related)

Just caught up with this thread keep it going guys its fascinating! Don't be surprised if in a few years time you find yourselves quoted in a small-time (heh heh heh that's what you think) English novel.

I like the ball on the hill example.

You can also use the same idea to describe politics and international relations. Look at the 2000 elections. Those elections were very close. The country was split pretty much 50/50. Almost perfectly balanced. Some clerk in a little county in Florida designs a ballot one way, instead of another, and as a result, the entire world is sent on a drastically different course.

No-one knows what would have happend if the butterfly ballott was designed differently, and Gore was elected. Maybe 9/11 would have happened, maybe not. Iraq almost certainly wouldn't have happened. The oil prices might still be low, or something else could have happened to cause them to be even higher.

There's your "butterfly effect" or is it a "butterfly ballot effect?"

Stay tuned: I think that CB is setting me up for the kill. :)

Knives at the ready... should be a good'un ;)

Originally posted by Carbonated_Brains



Oh, and I'm writing this shit in Notepad, and copying it over. I was ready to break this computer yesterday when everything got deleted. Here goes!

you do know that the "back" button will bring back a dumped post if you fuck up like that, don;t you?

or right click, undo within the edit box if you accidentally delete text.

This is a work computer, and somehow the security is set high, and I guess I'm not accepting cookies, so if I do ANYTHING I have to re-enter my username and password
which is what happened when i clicked submit
so I clicked back, to see if the text would still be here, and again i was asked for the user/pass
typed it in, got this form, sans text.

Originally posted by lumberjim
you do know that the "back" button will bring back a dumped post ...

That's what I would expect but sometimes, when I hit the back button, it reloads. I remember trying to go back to find a post that someone had deleted (I read it before they deleted it) and to my dismay, it reloaded the page instead of fetching the one in cache. That was some good blackmail stuff too, damnit. :)

In the spirit of a real debate, I'm going to try to address your individual concerns, Beestie, one by one.

"I'm not a big fan of chaos theory and, but for the beauty of the metaphor, the "butterfly effect" idea with its stretched-to-the-breaking-point logic..."

Logic is subjective, and if I can't bury you in the logic, I'll put you six feet under with cold, hard math ;-)

"To suggest that all of the 100 trillion (or whatever) variables were in perfect balance and the lone butterfly tilted the scale just enough is to remove one more shovel full of dirt from a (nearly) bottomless hole."

It just SEEMS implausible. After all, the universe is completely about balance! What could have created the miracle of human life, if not a trillion carefully balanced elements? If things were NOT hinged and balanced upon one another, entropy would rule, and the universe would be an evenly spaced bunch of atomic goo! For any cellestial "house of cards" to be built, a lot of crazy precarious things have to happen. But I digress...

If you want an identifyable, mathematical equivalent to the "butterfly effect", you need to look at what Edward Lorenz bumped into while he was studying weather patterns for MIT. He took a bunch of fluid dynamics formulae, and applied them to an atmospheric model. He boiled 'em down to the following three equations:

dx/dt = :nuke: * (y - x)
dy/dt = r * x - y - x * z
dz/dt = x * y - b * z

:nuke: is the Prandtl Number, which is a constant, Lorenz used the number 10. "r" represents the temperature gradient between the top and bottom of the volume of atmosphere Lorenz wanted to study. "b" is the width to height ratio of this volume "box", he used 8/3. X relates to the rate of rotation, Y is another temperature gradient, and Z is the deviation of the line from a graphed vertical temperature plot. The upshot is he graphed the sucker and ended up with this:

Bam!

Look familiar? It should!

Zoom!

The reason chaos theory is NEW, is because you couldn't really study it without a computer. It requires billions of complex calculations, something no mathematician would spend his life doing. Lorenz graphed that butterfly shape, known as the Lorenz Attractor, and found that the system he was graphing never precisely repeats itself...the trajectory of that line never traces over a previous trajectory, it loops forever and ever.

Lorenz, being a good scientist, initially said "No flippin way, I must be wrong." So he did a futher experiment, he made a waterwheel with 8 evenly spaced buckets, each on a swivel, and having a hole in the bottom. He opened a spout and began to fill the buckets, causing the waterwheel to spin at a somewhat constant rate. When he dropped the hammer on that spout, let the water flow really fast, the waterwheel started to shudder in one direction, jerk to a stop, move backward, shudder forward again, and dance around randomly. VERY randomly. Lorenz sat there for hours and recorded the waterwheel, and it never repeated its motion. If he graphed the waterwheel, he'd get a version of the Lorenz attractor; the waterwheel, and the atmospheric model, were both chaotic systems.

What's even weirder is that when you graph a chaotic system, you have to use dimensions which we never really thought existed. We're familiar with the first 3 dimensions, but graphed chaotic systems use non-integral dimensions. Which means you can be in dimension 2.8, or 7.1. What happens when you graph a chaotic mathematical model using non-integer dimensions? You get a fractal.

In fact, that's how Mandlebrot and Julia invented fractals. They are graphical representations of chaos theory.

This guy named Cantor did some funky stuff, too. He postulated that you can take a regular line segment and infinitely rip pieces out of it...draw a line, then erase the middle third. Then erase the middle thirds of the two line segments you got after the first erasing, etc, etc...until you have hundreds of tiny dots. These tiny dots, though infinite in number, have a combined length of zero. This is called "Cantor Dust".

Cantor dust is used in electrical engineering. Engineers looking at electronic transmission powerlines observed periods of error-free transmission, then bursts of errors, then periods of calm, etc...analysing the bursts, they found that these bursts contained small periods of error free transmissions, then sub-bursts, etc...in fact, it follows Cantor's model. Cantor Dust is essential in modeling intermittency.

There's even chaos washing machines. Goldstar, in 1993, invented one which used a tiny pulsator which rose and fall randomly using a chaotic algorhithm, on the premise that it would produce cleaner clothes with fewer tangles...and it sort of worked, to an extent.

Guess what else is a chaotic system...the stock market. Billions of variables, no predictability, yet patterns can rise and fall with time. The stock market is completely useless in the short term, yet serious profit can be made by analysing trends and participating in long term trading.

These are examples of nonlinear systems, which are also dynamic. Chaos theory is the study of these systems.

"Ultimately, a theory has to aid in our understanding of an event. My problem with the butterfly example is and remains that it really doesn't do or say anything nor does it prove or even allege anything that we didn't already know. When we see it raining in Japan, it is not illuminating to postulate that the rain might have been caused by a minute event that happened over a year ago. Obviously something caused the rain - but chaos theory as explained in the butterfly example brings us not one inch closer to understanding what or why. Nor does it eliminate any false notions of what caused the rain. Basically, its useless -we are no closer to the truth nor are we any further away from a lack of understanding than we were before the "theory" was introduced."

Chaos theory alleges something that physicists have been terrified of since Newton. Mathematicians and physicists believed that if you knew the initial conditions with great accuracy, you could get formulas to describe all events in the future, and explain the ones in the past. Like playing a movie backward and foreward, everything is explainable and calculatable based on those conditions.

Chaos theory threw that out the window. Now, scientists cannot predict complex celestial motion, because they cannot ever know the initial conditions. To say that has small ramifications is like saying Hurricane Mitch was just a spatter of raindrops.

I think someone earlier said it best...the butterfly flapping its wings, is the minute difference between two sets of initial conditions. Assuming everything else is the same, you still have two systems where the initial conditions are infinitesmally different. And this difference is enough to cause a mathematical spiral of the Lorenz attractor...in a year, you're in 2 completely different places if you make a prediction based on the 2 sets of initial conditions, no matter how similar they are.

Scientists now believe the universe is itself a chaotic system, and linearity is rare. This is making scientists sweat, and religious types pay attention.

Hawking said it best: "If we find the answer to that (the universe), it would be the ultimate triumph of human reason-for then we would know the mind of God."

My old physics teacher (smartest man I've ever met) always said, if you want to explain something, use science. But if you want to know WHY, and you want a philosophical answer, you need to look to faith.

Not making this argument theological, it's just interesting.

Now wiggle your rebut!

Betcha didn't expect a 1300 word reply =)

If you have a system which you know exhibits chaotic behavior, and where you don't know most of the initial conditions, nor the possible interactions among the parts of the system, how meaningful are your long-term predictions of the behavior of the system?

...did I just get banned?

I can't post with the other account. I can't even email Undertoad.

What's going on?

why would you get banned......? there wasn;t even a poll.

i suspect your pesky work computer again.

Yet I can post with this account and not the other!

I think Adam Smith (an economist) said it long before these guys when he postulated his Invisible Hand theory.

Your post was very illuminating and I will be taking a closer look. The Smith peice pretty much sums up where I'm coming from - that chaotic systems behaviour is the sum of countless other small scale actions acting independently. No single action or lack of action dictates the outcome.

To use your stock market analogy, no single small investor moves the market - its the sum of ALL the small investors (forget the big boys - mutual fund managers, Warren Buffet, etc. for the moment).

Put another way, here is an experiment that I propose: Take a chaotic system and "bottle it." Now, put it through a process so as to generate an outcome. Now, replicate that process over and over again but each time, remove ONE variable and put back the previously removed variable until the process has been run without the contribution of each variable in the system at least once. That is to say if there are ten variables, remove one and run it. Then put it back, remove a different one and run it again. When you are done, you will have run the process 10 times: once with 10 variables and 9x with 9 but a different 9 each time. Obviously, the values of the variables must be held constant.

My hypothesis is that the outcomes would never change because no single variable can change the outcome. If a single variable could influence the outcome then the system wouldn't be chaotic. It if for this (hypothesized) reason that I cannot accept the butterfly example.

Even if I am wrong (very possible), at least I hope I have explained my objections clearly.

What you just described is a linear system.

You can't "bottle" a chaotic system and treat it as periodic; if you somehow did that, it wouldn't be a chaotic system.

A chaotic system is dynamical, in the sense that it's a model of an actual system whose behavior varies with time.

The idea of chaos theory doesn't mesh with the very linear scientific practice of putting stuff in a box and pretending it has no interactions with the outside world. Frictionless surfaces, perfect springs, perpetual pendulums and such are all fine and good for approximations of LINEAR systems, but you can't take a CHAOTIC system (which is based on the tiniest preturbations of the initial conditions) and pretend it's going to react in a similar way if you stick it in a box.

But now that I think of it, Lorenz stuck his model of the atmosphere in a box, and still got chaotic results. Check this out, I believe this diagram refutes your claim that ONE variable cannot affect the outcome.

haha, now that I look again, this is the best diagram yet. It perfectly rebuts the idea that small changes cannot individually have an impact.

Stay with me here:

If you take the Lorenz equations and use:

:nuke: = 10
b = 2.6667
r = 20
you get this:



This is the same graph with X plotted against time:


The BLACK line in the above picture is if you make your initial X = 5, instead of 1 (the red line assumes X = 1) See how the black line follows the red line, and near the end they're pretty much equal?

Seems to suggest a little change results in not much difference of outcome...
BUT WAIT!

What if we use the same conditions as in the first graph, except we change r = 28? So:

:nuke: = 10
b = 2.6667
r = 28

we get:


Wow, that's crazy different. The thing no longer spins around a single point in the x-y plane, but it flips randomly between 2 points. Shall we plot it on the X vs Time graph again?



NOW! Here's the kicker! Suppose we take the value of X, which is 1.0000, and change it to 1.0001! We'll superimpose the black line on the red line.



For a while they're the same, but then black diverges and takes an entirely different random path! In fact, the futher you run the experiment the weirder black is, in relation to red!

Absolute graphical mathematic proof that if you take the initial condition x = 1.0000 , and change it to x = 1.0001, you get an ENTIRELY different end result, given time!


Are you convinced yet? Please? My fingers are blistering!

btw, the change from x = 1.0000 to x = 1.0001 is analagous to using a thermometer which isn't accurate to the fourth decimal place. Changing that TEN THOUSANDTH little bit was enough to blow up the end result. If we went 12 more decimal places down the line (far down enough to be in the realm of measurement error!) we'd still see the same end result; crazy differences.

My old nickname works! Sweet!

Kneel. Kneel before Professor Chaos!


MWAHAHAHAHA!!!!!!

btw, is anybody reading this hooey besides the 5 or so people who have posted?

dude, i tried.

i just feel really stupid now.

somebody pull my finger!

I am, but I confess that I barely understand a lot of it. :)

I just remember reading all of it from those oatmeal packets I ate as a kid.

Those "daily facts" are detailed as hell.

Same here. I'm pretty sure it's very interesting though.

Don't stop just yet!

Carbonated, I don't think there is a problem in 'bottling' chaos. Computers can create chaotic results from very simple formulas. The problem is in predicting chaotic behavior, not creating it.


Okay, Beestie, one chart demonstration of the butterfly effect. This is the granddaddy example from Lorenz himself who stumbled onto these results.

Lorenz had a weather simulator on his computer. It would spit out a row of numbers for each day. It would take maybe a minute to simulate each day. One day, Lorenz was looking at old data, and he decided he wanted to run that particular simulation over again. To save time, he decided he wanted to start the simulation 'from the middle' rather than from the beginning. This seemed easy enough, he input as his initial conditions the numbers from some day in the middle of the data he had on the printout. Now here's the catch - the data on the printout were printed with three decimal places. The computer really kept track of things up to six decimals. In other words, the numbers that Lorenz input were just a tiny tiny bit off from what they had been in with the simulation first time around.

But this tiny difference shouldn't make a difference, right?

But these were his results:


Time is the X axis, and the results from both simulations are plotted on the same graph. Notice how on the left, the beginning of the simulation, there is one solid line. That means both simulations were spitting out almost identical results. But now look as how the two lines diverge over time. By the end of the graph, the two lines are totally apart. That means that with the weather simulation Lorenz was using, tiny differences in initial conditions will result in totally different outcomes. Maybe not at first, but after some time.

test your chaos theories:

HP linked this originally for the artistic potential, but i think it might apply


The Scribbler!!

Originally posted by Carbonated_Brains
My old nickname works! Sweet!

Old nickname, my ass. You went and drug some poor grad student out of the research lab at MIT and made him post all that stuff under your "so-called" alias :)

Actually, the Lorenz graphs you posted were new to me and quite persuasive - particularly the last one. I also liked Slarti's graph (which, seemed to be making the same point). In particular, I thought the idea of "starting in the middle" to be significant since, although we haven't talked about it, chaotic systems don't really have a discreet beginning or end. Its not like dropping a bowling ball and measuring the dent in the ground. Its more like jumping on a moving freight train, taking some measurements and jumping off again.

What I don't get is how such a small change can pick up strength as it ripples through the system over time. While Bruce provided a metaphorical scenario in which it was possible, it required a "parting of the Red Sea" w/r to all other variables - a state I considered so unlikely that it was hardly worth considering. At this point, I'll accept that my position was not correct -the graphs do a good job of explaining the increasing divergence. But now I'm really curious to understand how and why.

Oh, and thanks for taking the time to completely deconstruct my mental model for understanding the universe. Now that I understand what causes hurricanes, I'm going to go butterfly hunting - the world can thank me later. After that, I want to hunt down the cockroach that's going to cause the next earthquake - or is it prevent the next earthquake - I forget :)

Well, that was fun!:D

"Oh, and thanks for taking the time to completely deconstruct my mental model for understanding the universe."


:eek:


That is...

That is the best compliment I've ever received!

You damn well helped me get a much better understanding of this by forcing me to question myself about it, too!

All in all, a damn fine debate, one I'll be saving.