Chaos Theory in a Wingbeat

Imagine this:

You go to buy a train ticket. You find that you are 1 cent short and didn't bother to bring your credit card. Bummer. So to paliate your annoyance at missing the professional conference you were going to where, little did you know, you would have met your future mentor and business partner where your combined skills would have created a revolutionary new product and made you both multi-millionaires... you go the station café for a coffee.

As chance would have it, a Jehovah's Witness is having a break from their 'ministering' but spots a potential victim and says hello.

Your guard is down and as you recount your unfortunate story he makes sympathetic noises and even jokes that if he had been behind you in the queue he would certainly have given you the extra cent you needed for your ticket.

You warm to him, and, to cut a long story short, accept an invitation to a meeting, meet a whole bunch of 'nice' people who welcome you in, you struggle for a long time before eventually 'accepting' their beliefs, spend years standing outside metro stations with religious magazines in your hand getting in everyone's way, renounce your ambition and drive, sink into misery and are overwhelmed by feelings of uselessness and pointlessness, and end up jumping in front of a train on the very same line the train you would have taken travelled, without you, all those unhappy years ago, for the want of a cent, and killing yourself.

This is an example of chaos theory. The idea that a very small change in the initial starting conditions can cause wildly differing results.

A simpler example is that you're in a traffic jam and you miss your train to the airport by 30 seconds. The next train is in 30 minutes. When you eventually get to the airport, you've missed your plane. The next available flight isn't until tomorrow. A 30 second lateness seems to be responsible, ultimately, for a whole day's delay.

Mathematicians call this a non-linear effect, where the initial small thing can lead on to much bigger things.

A classic example is the weather. Why can't we accurately predict what the weather will be more than a few days in advance, if that? Because the system is far too complex to be able to take into account all the starting conditions, and even less so all the possible consequences.

Despite the name, the use of chaos here doesn't imply totally random events. Indeed, chaos theory is about separating totally random phenomena, which are by definition totally unpredictable, from chaotic ones, which have some chance of being predicted correctly?

Still with me? Never mind. There's a fair bit of chaos gone into writing this little article (I predicted that right from the start), and so much the better, I reckon.

The final example I'll give is actually the most famous: the Butterfly Effect. This is a much-quoted and purposely over-the-top scenario where the gentle flapping of a butterfly's wings in one country causes, eventually, a horribly destructive tornado on another continent.

Chaos theory can be applied to many real-world 'systems', from a leaky tap to the orbit of Pluto. The prices of shares and organisations are also seen to experience chaotic behaviour, where small changes can set of a chain reaction of monumental importance, or a major upheaval can go relatively unnoticed.

So there you have it, more than you probably ever need to know about chaos theory; aren't you glad you didn't ask?

See you in the gene pool

FrabulousFrog

How Long is the Coast of Britain?

Benoît B. Mandelbrot, the mathmatician famous for, amongst other things, asking the above question, died in October of this year, 2010. Take note of that middle 'B.', by the way; it's vitally important, as we'll discover in a moment...

The question might seem a funny one, because obviously we know - we can probably just look it up on Wikipedia or something, right?

Well, no, not right, according to Mr.M. It all depends on what you measure. More specifically, what actually is the coast of Britain, or any other country for that matter.

Any map is obviously a simplified representation of reality. If we actually zoomed in on the grey map on the right, we'd discover lots of little nooks and crannies (bays, promontories etc.) which are definitely part of the coastline but which haven't been taken into account, and which would change the numbers.

And let's take an even closer look, insisted Mandelbrot. What if we have a perfectly straight stretch of beach. We can just draw a line from one end to the other and measure it, can't we? Or is that not a gross simplification and should we not be tracing the contours of every pebble, no, every grain of sand which makes up the line where that beach meets the water to have a better figure. And what about the atomic level... Oh dear!

So Mandelbrot worked on this and many other mathmatical problems, coining the terms fractal and his famous Mandelbrot set.

A fractal is a shape which can be split up into parts and those parts are more or less the same as the big shape, and so on, ad absurdum. Mountains and leaves are other real-world examples of this phenomenon.

A Mandelbrot Set is explained here, (what do you mean I haven't got a clue what I'm talking about, of course I do, look:) and this is a some-would-say rather beautiful pictoral representation of one...

Finally, he was well-known for applying his ideas to the real world where unclear boundaries exist, in the realms of economics and information theory. For example trying to explain price fluctuations in markets.

Just a couple more snippets before we go. He had French / American / Polish nationalities. President Sarkozy said that Mandelbrot had "a powerful, original mind that never shied away from innovating and shattering preconceived notions". Sarkozy also added, "His work, developed entirely outside mainstream research, led to modern information theory." He worked in the research department of IBM for 35 years. He has an asteroid (27500 Mandelbrot) named after him. He's got a heap of honours, including the French Legion of Honour (the big one!).

He was also closely linked to Chaos Theory and the Butterfly Effect, which I'll post about shortly, and... oh yes: apparently the 'B.' in his name doesn't mean anything at all; he invented it just for a laugh, the old dog! Haven't had enough of Mandelbrot yet? Watch this and you will have!

See you in the gene pool

FrabulousFrog