Professor Stewart's Cabinet of Mathematical Curiosities
Triangle of Cards
Turnip for the Books
The Four-Colour Theorem
Shaggy Dog Story
Shaggy Cat Story
Rabbits in the Hat
River Crossing 1 – Farm Produce
More Curious Calculations
Extracting the Cherry
Make Me a Pentagon
What is π?
Legislating the Value of π
If They Had Passed It …
Much Undo About Knotting
To Find Fake Coin
Mathematical Jokes 1
An Age-Old Old-Age Problem
Why Does Minus Times Minus Make Plus?
How to Unmake a Greek Cross
How to Remember a Round Number
The Bridges of Königsberg
How to do Lots of Mathematics
Euler's Pentagonal Holiday
Who Was Pythagoras?
Proofs of Pythagoras
A Constant Bore
Fermat's Last Theorem
A Little-Known Pythagorean Curiosity
Squaring the Square
Squares of Squares
Ring a-Ring a-Ringroad
Pure v. Applied
How Old Was Diophantus?
If You Thought Mathematicians Were Good at Arithmetic …
The Sphinx is a Reptile
Six Degrees of Separation
Duplicating the Cube
Curves of Constant Width
The Stolen Car
The Square Wheel
Why Can't I Divide by Zero?
River Crossing 2 – Marital Mistrust
Wherefore Art Thou Borromeo?
Kinds of People
The Sausage Conjecture
Tom Fool's Knot
What is the Golden Number?
What are the Fibonacci Numbers?
The Plastic Number
Don't Let Go!
The Most Likely Digit
Why Call It a Witch?
Möbius and His Band
Three More Quickies
Miles of Tiles
Why No Nobel for Maths?
Is There a Perfect Cuboid?
When Will My MP3 Player Repeat?
The Poincaré Conjecture
Pig on a Rope
The Surprise Examination
Mathematical Jokes 2
Why Gauss Became a Mathematician
What Shape is a Crescent Moon?
What is a Mersenne Prime?
The Goldbach Conjecture
Turtles All the Way Down
A Puzzling Dissection
A Really Puzzling Dissection
Nothing Up My Sleeve …
Nothing Down My Leg …
Can You Hear the Shape of a Drum?
What is e, and Why?
May Husband and Ay …
Many Knees, Many Seats
What Day is It?
Logical or Not?
A Question of Breeding
The Sixth Deadly Sin
How Deep is the Well?
What is the Square Root of Minus One?
The Most Beautiful Formula
Why is Euler's Beautiful Formula True?
Your Call May be Monitored for Training Purposes
Archimedes, You Old Fraud!
Fractals – The Geometry of Nature
The Missing Symbol
Where There's a Wall, There's a Way
Constants to 50 Places
Are Hard Problems Easy? or How to Win a Million Dollars by Proving the Obvious
Don't Get the Goat
All Triangles are Isosceles
If π isn't a Fraction, How Can You Calculate It?
Let Fate Decide
What Shape is a Rainbow?
The Riemann Hypothesis
Disproof of the Riemann Hypothesis
Murder in the Park
The Cube of Cheese
The Game of Life
Drawing an Ellipse – and More?
Mathematical Jokes 3
The Kepler Problem
The Milk Crate Problem
Spherical Sliced Bread
Professor Stewart's Cunning Crib Sheet
Chaos Theory // 117
lattices, but they can occur naturally. So Penrose's discovery changed our ideas about natural arrangements of atoms in crystal-like structures.
The detailed mathematics and crystallography are too complicated to describe here. To find out more, go to: en.wikipedia.org/wiki/Penrose_tiling
Answers on page 278
Chaos Theory If you want your friends to accept you as a `rocket scientist', you have to be able to spout about chaos theory. You will casually mention the butterfly effect, and then you get to talk about where Pluto (no longer a planet but a mere dwarf planet) will be in 200 million years' time, and how really good dishwashers work.
Chaos theory is the name given by the media to an important new discovery in dynamical systems theory the mathematics of systems that change over time according to specific rules. The name refers to a surprising and rather counterintuitive type of behaviour known as deterministic chaos. A system is called deterministic if its present state completely determines its future behaviour; if not, the system is called stochastic or random. Deterministic chaos universally shortened to `chaos' is apparently random behaviour in a deterministic dynamical system. At first sight, this seems to be a contradiction in terms, but the issues are quite subtle, and it turns out that some features of deterministic systems can behave randomly.
Let me explain why.
You may remember the bit in Douglas Adams's The Hitch Hiker's Guide to the Galaxy that parodies the concept of determinism. No, really, you do remember the supercomputer Deep Thought? When asked for the answer to the Great Question of Life, the Universe and Everything, it ruminates for five million 118 // Chaos Theory
years, and finally delivers the answer as 42. The philosophers then realise that they didn't actually understand the question, and an even greater computer is given the task of finding it.
Deep Thought is the literary embodiment of a `vast intellect' envisaged by one of the great French mathematicians of the eighteenth century, the Marquis de Laplace. He observed that the laws of nature, as expressed mathematically by Isaac Newton and his successors, are deterministic, saying that: `An intellect which at a certain moment knew all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.'
In effect, Laplace was telling us that any deterministic system is inherently predictable in principle, at least. In practice, however, we have no access to a Vast Intellect of the kind he had in mind, so we can't carry out the calculations that are needed to predict the system's future. Well, maybe for a short period, if we're lucky. For example, modern weather forecasts are fairly accurate for about two days, but a ten-day forecast is often badly wrong. (When it isn't, they've got lucky.)
Chaos raises another objection to Laplace's vision: even if his Vast Intellect existed, it would have to know `all positions of all items' with perfect accuracy. In a chaotic system, any uncertainty about the present state grows very rapidly as time passes. So we quickly lose track of what the system will be doing. Even if this initial uncertainty first shows up in the millionth decimal place of some measurement with the previous 999,999 decimal places absolutely correct the predicted future based on one value for that millionth decimal place will be utterly different from a prediction based on some other value.
In a non-chaotic system such uncertainties grow fairly slowly, and very long-term predictions can be made. In a chaotic Chaos Theory // 119
system, inevitable errors in measuring its state now mean that its state a short time ahead may be completely uncertain.
A (slightly artificial) example may help to clarify this effect. Suppose that the state of some system is represented by a real number an infinite decimal between 0 and 10. Perhaps its current value is 5.430 874, say. To keep the maths simple, suppose that time passes in discrete intervals 1, 2, 3, and so on. Let's call these intervals `seconds' for definiteness. Further, suppose that the rule for the future behaviour is this: to find the `next' state the state one second into the future you take the current state, multiply by 10, and ignore any initial digit that would make the result bigger than 10. So the current value 5.430 874 becomes 54.308 74, and you ignore the initial digit 5 to get the next state, 4.308 74. Then, as time ticks on, successive states are:
and so on.
Now suppose that the initial measurement was slightly inaccurate, and should have been 5.430 824 differing in the fifth decimal place. In most practical circumstances, this is a very tiny error. Now the predicted behaviour would be:
See how that 2 moves one step to the left at each step making the error ten times as big each time. After a mere 5 seconds, the 120 // Chaos Theory
first prediction of 7.4 has changed to 2.4 a significant difference.
If we had started with a million-digit number, and changed the final digit, it would have taken a million seconds for the change to affect the predicted first digit. But a million seconds is only 111 days. And most mathematical schemes for predicting
2 the future behaviour of a system work with much smaller intervals of time thousandths or millionths of seconds.
If the rule for moving one time-step into the future is different, this kind of error may not grow as quickly. For example, if the rule is `divide the number by 2', then the effect of such a change dies away as we move further and further into the future. So what makes a system chaotic, or not, is the rule for forecasting its next state. Some rules exaggerate errors, some filter them out.
The first person to realise that sometimes the error can grow rapidly that the system may be chaotic, despite being
´ deterministic was Henri Poincare, in 1887. He was competing for a major mathematical prize. King Oscar II of Norway and Sweden offered 2,500 crowns to anyone who could calculate whether the solar system is stable. If we wait long enough, will the planets continue to follow roughly their present orbits, or could something dramatic happen such as two of them colliding, or one being flung away into the depths of interstellar space?
This problem turned out to be far too difficult, but Poincare ´ managed to make progress on a simpler question a hypothet- ical solar system with just three bodies. The mathematics, even in this simplified set-up, was still extraordinarily difficult. But
´ Poincare was up to the task, and he convinced himself that this `three-body' system sometimes behaved in an irregular, unpre- dictable manner. The equations were deterministic, but their solutions were erratic.
He wasn't sure what to do about that, but he knew it must be true. He wrote up his work, and won the prize. Chaos Theory // 121
And that was what everyone thought until recently. But in 1999 the historian June Barrow-Green discovered a skeleton in
´ Poincare's closet. The published version of his prizewinning paper was not the one he submitted, not the one that won the prize. The version he submitted which was printed in a major mathematical journal claimed that no irregular behaviour would occur. Which is the exact opposite of the standard story.
Barrow-Green discovered that shortly after winning the prize,
´ an embarrassed Poincare realised he had blundered. He withdrew the winning memoir and paid for the entire print run of the journal to be destroyed. Then he put his error right, and the official published version is the corrected one. No one knew that there had been a previous version until Barrow-Green discovered a copy tucked away among the archives of the Mittag-Leffler Institute in Stockholm.
Anyway, Poincare deserves full credit as the first person to appreciate that deterministic mathematical laws do not always imply predictable, regular behaviour. Another famous advance was made by the meteorologist Edward Lorenz in 1961. He was running a mathematical model of convection currents on his computer. The machines available in those days were very slow and cumbersome compared with what we have now your mobile phone is a far more powerful computer than the top research machine of the 1960s. Lorenz had to stop his computer in the middle of a long calculation, so he printed out all the 122 // Chaos Theory
numbers it had found. Then he went back several steps, input the numbers at that point, and restarted the calculation. The reason for backtracking was to check that the new calculation agreed with the old one, to eliminate errors when he fed the old figures back in.
At first the new numbers were the same as the old ones, but then they started to differ. What was wrong? Eventually Lorenz discovered that he hadn't typed in any wrong numbers. The difference arose because the computer stored numbers to a few more decimal places than it printed out. So what it stored as 2.371 45, say, was printed out as 2.371. When he typed that number in for the second run, the computer began calculating using 2.371 00, not 2.371 45. The difference grew chaotically and eventually became obvious.
When Lorenz published his results, he wrote: `One meteor- ologist remarked that if the theory were correct, one flap of a
(Left) Initial conditions for eight weather forecasts, apparently
identical but with tiny differences. (Right) The predicted
weather a week later the initial differences have grown
enormously. Italian weather is more predictable than British.
[Courtesy of the European Medium Range Weather Forecasting
Centre, Reading.] Chaos Theory // 123
seagull's wings could change the course of weather for ever.' The objection was intended as a put-down, but we now know that this is exactly what happens. Weather forecasters routinely make a whole `ensemble' of predictions, with slightly different initial conditions, and then take a majority vote on the future, so to speak.
Before you rush out with a shotgun, I must add that there are billions of seagulls, and we don't get to run the weather twice. What we end up with is a random selection from the range of possible weathers that might have happened instead.
Lorenz quickly replaced the seagull by a butterfly, because that sounded better. In 1972 he gave a lecture with the title `Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?' The title was invented by Philip Merilees when Lorenz failed to provide one. Thanks to this lecture, the mathematical point concerned became known as the butterfly effect. It is a characteristic feature of chaotic systems, and it is why they are unpredictable, despite being deterministic. The slightest change to the current state of the system can grow so rapidly that it changes the future behaviour. Beyond some relatively small `prediction horizon', the future must remain mysterious. It may be predetermined, but we can't find out what has been predetermined, except by waiting to see what happens. Even a big increase in computer speed makes little difference to this horizon, because the errors grow so fast.
For weather, the prediction horizon is about two days ahead. For the solar system as a whole, it is far longer. We can predict that in 200 million years' time, Pluto will still be in much the same orbit as it is today; however, we have no idea on which side of the Sun it will be by then. So some features are predictable, others are not.
Although chaos is unpredictable, it is not random. This is the whole point. There are hidden `patterns', but you have to know how to find them. If you plot the solutions of Lorenz's model in three dimensions, they form a beautiful, complicated shape
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