Mathematics? It's all just life.

[Carbonated_Brains]

"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.

[wolf]

The keyword is "almost".

[Carbonated_Brains]

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.

[Catwoman]

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.

[Carbonated_Brains]

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?

[jaguar]

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

[Carbonated_Brains]

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.

[Beestie]

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.

[Carbonated_Brains]

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?

[Troubleshooter]

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

[Carbonated_Brains]

Beestie just said that.

[Troubleshooter]

Originally posted by Carbonated_Brains
Beestie just said that.

Actually he said that it didn't.

[Carbonated_Brains]

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

[Troubleshooter]

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.

[Carbonated_Brains]

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"