Professor Stewart's Hoard of Mathematical Treasures (42 page)

Lagrangian points, and associated energy contours.

More precisely, around 1750, Leonhard Euler proved that the points
L
₁
,
L
₂
and
L
₃
exist, and Lagrange discovered the other two. Lagrange did this calculation as part of an attack on a more general question, the motion of three bodies under gravity. Isaac Newton had shown that, for two bodies, the orbits are ellipses, and it was natural to ask what happens with three bodies. This
turned out to be a very difficult problem, and we now know why: the typical motion is chaotic (Cabinet, page 117).

The
L
₄
and
L
₅
points are stable, provided the mass of the Sun is at least
times that of the planet. That is, a mass located at such a point will remain nearby even if it is disturbed a little. The other three points are unstable. No natural occurrences of bodies orbiting at these points were known until astronomers noticed that unusually many asteroids are located near the Sun-Jupiter
L
₄
and
L
₅
points. They are spread out along Jupiter’s orbit in the same ‘banana’ shape as the energy contours near those points. Since then, other instances have been found:

Although the other three Lagrangian points are unstable, they are surrounded by stable orbits, called halo orbits, so a space probe or other artefact can be maintained near those points with very little expenditure of fuel. The James Webb Space Telescope, successor to the Hubble Telescope, will be positioned at the Sun- Earth
L
₂
point when it is launched in or after 2013. This location keeps the Earth and Sun in the same direction, as seen from the telescope, so that a single fixed shield can stop radiation from those two bodies warming it up and disturbing the delicate instruments. The only Lagrangian point that has not yet featured
in an actual or planned space mission is
L
₃
. All five of them have been exploited in numerous science fiction stories.

A wealth of further information can be found at:

A good pub puzzle. Start with six coins, numbered 1-6 and arranged as in the left-hand picture. Slide them one at a time, without disturbing the others, to rearrange them into the right-hand picture in the number order shown.

How can you achieve this by moving as few coins as possible?

 

Start here . . .

. . . and end here.

. . . and
then
what?

Chapter 94 of Snorri Sturluson’s Heimskringla: History of the Kings of Norway - which I’m sure you’re familiar with - tells of a game
of chance between King Olaf I of Norway
⁴¹
and the King of Sweden,
⁴²
to decide which country owned the island of Hísing.

According to Thorstein the Learned, the two kings agreed to throw a pair of dice, and whoever got the highest score also got the island.

The King of Sweden, who had won the right to go first by drawing lots, threw the dice, and scored a double six. ‘There is no use in you throwing,’ he said. ‘I cannot lose.’

‘There remain two sixes on the dice, my Lord,’ replied Olaf, as he shook the dice in his hand, ‘and it is a trifling matter for God to make the dice land that way.’ Then he rolled the dice ...

What do you think happened next?

 

Legend has it that the great geometer Euclid composed a puzzle which went as follows.

A mule and a donkey were stumbling along the road, each carrying several identical heavy sacks. The donkey started complaining, making a horrible groaning noise, and eventually the mule got fed up.

‘What are you complaining for? If you gave me one sack, I’d have twice as many as you! And if I gave you one sack, we’d be carrying the same load.’

How many sacks were the donkey and the mule carrying?

 

It is said that if a monkey sat at a typewriter and kept hitting keys at random, then eventually it would type the complete works of Shakespeare. This statement dramatises two things about random sequences: anything can turn up, and, therefore, the result need not appear random. The infinite monkey theorem goes further, and states that, if the monkey keeps typing for ever, then the probability that it will eventually type any given text is 1.

To test this proposition, all you need is two dice, of different colours or otherwise distinguishable, and a table of symbols. The one at bottom right is a space.

Simulated monkey.

Throw the two dice, choose the corresponding symbol, and write it down. For instance, if you throw 4 /1, then you get the letter D. Keep going, and see how long it takes to get a sensible word with, say, three or more letters. Your experience should be confirmed by two calculations:

In 2003, lecturers and students from the University of Plymouth MediaLab tried the experiment with real monkeys -
six Celebes crested macaques - and a computer keyboard. The experimental subjects produced five pages of typing, mainly looking like this:
and then trashed the keyboard comprehensively.

The mathematical statement goes back to Émile Borel, in a 1913 paper ‘Statistical mechanics and irreversibility’, and his 1914 book Le Hasard (Chance). The Argentine writer Jorge Luis Borges traced the underlying idea back to Aristotle’s Metaphysics. The Roman orator Cicero, unimpressed by Aristotle’s views, compared the statement to believing that ‘if a great quantity of the one-and-twenty letters, composed either of gold or any other matter, were thrown upon the ground, they would fall into such order as legibly to form the Annals of Ennius. I doubt whether fortune could make a single verse of them.’

Well, no . . . unless you used a really great quantity.

The monkey on the typewriter has been used to attack the theory of evolution.
⁴³
Random mutations in DNA are like the monkey. And while it is true that eventually the monkey can type anything, it is also true that it won’t type anything remotely interesting during the lifetime of the universe. Now, a key protein like haemoglobin, which carries oxygen in our blood, is specified by more than 1,700 DNA ‘letters’ A, C, T, G. The chance of this molecule arising by random mutations is so tiny that it might as well be zero. Therefore haemoglobin cannot have evolved, Darwin was wrong, God must have created it, QED.

This criticism turns out to be facile, and rests on several
misconceptions. One is that the haemoglobin molecule is a ‘target’ at which evolution must aim. However, haemoglobin is not the only molecule that could carry oxygen and deliver it where required. Haemoglobin does that job because it has two similar but distinct forms. In one of them, oxygen atoms bind to the four iron atoms in the molecule; in the other, they don’t. The molecule ‘flexes’ slightly from one form to the other. Most of the haemoglobin molecule plays no essential role in this process, although it does provide a suitably flexible scaffolding for the bits that matter. So a huge variety of other molecules could in principle do the same job. Nature evolved one, and that was all it needed. Well, actually it evolved several variants, which if anything helps to support the point I’m making.