100 Essential Things You Didn't Know You Didn't Know (2 page)

If we move on to three-dimensional shapes then the situation is quite different: Cauchy showed that
every
convex polyhedron (i.e. in which the faces all point outwards) with rigid faces, and hinged along its edges, is rigid. And, in fact, the same is true for convex polyhedra in spaces with four or more dimensions as well.

What about the non-convex polyhedra, where some of the faces can point inwards? They look much more squashable. Here, the question remained open until 1978 when Robert Connelly found an example with non-convex faces that is not rigid and then showed that in all such cases the possible flexible shifts keep the total volume of the polyhedron the same. However, the non-convex polyhedral examples that exist, or that may be found in the future, seem to be of no immediate practical interest to structural engineers
because
they are special in the sense that they require a perfectly accurate construction, like balancing a needle on its point. Any deviation from it at all just gives a rigid example, and so mathematicians say that ‘almost every’ polyhedron is rigid. This all seems to make structural stability easy to achieve – but pylons do buckle and fall down. I’m sure you can see why.

^fn1
http://www.drookitagain.co.uk/coppermine/thumbnails.php?album=34

2

A Sense of Balance

Despite my privileged upbringing, I’m quite well-balanced. I have a chip on both shoulders.

Russell Crowe in
A Beautiful Mind

Whatever you do in life, there will be times when you feel you are walking a tightrope between success and failure, trying to balance one thing against another or to avoid one activity gobbling up every free moment of your time. But what about the people who really are walking a tightrope. The other day I was watching some old newsreel film of a now familiar sight: a crazy tightrope walker making a death-defying walk high above a ravine and a rushing river. One slip and he would have become just another victim of Newton’s law of gravity.

We have all tried to balance on steps or planks of wood at times, and we know from experience that some things help to keep you balanced and upright: don’t lean away from the centre, stand up straight, keep your centre of gravity low. All the things they teach you in circus school. But those tightrope walkers always seem to carry very long poles in their hands. Sometimes the poles flop down at the ends because of their weight, sometimes they even have heavy buckets attached. Why do you think the funambulists do that?

The key idea you need to understand why the tightrope walker carries a long pole to aid balance is inertia. The larger your inertia,
the
slower you move when a force is applied. It has nothing to do with centre of gravity. The farther away from the centre that mass is distributed, the higher a body’s inertia is, and the harder it is to move it. Take two spheres of different materials that have the same diameter and mass, one solid and one hollow, and it will be the hollow one with all its mass far away at its surface that will be slower to move or to roll down a slope. Similarly, carrying the long pole increases the tightrope walker’s inertia by placing mass far away from the body’s centre line – inertia has units of mass times distance squared. As a result, any small wobbles about the equilibrium position happen more slowly. They have a longer time period of oscillation, and the walker has more time to respond to the wobbles and restore his balance. Compare how much easier it is to balance a one-metre stick on your finger compared with a 10-centimetre one.

3

Monkey Business

I have a spelling chequer

It came with my pee sea

It plainly marques four my revue

Miss takes I cannot see

I’ve run this poem threw it

I’m shore yaw pleased to no

It’s letter perfect in its weigh

My chequer told me sew . . .

Barri Haynes

The legendary image of an army of monkeys typing letters at random and eventually producing the works of Shakespeare seems to have emerged gradually over a long period of time. In
Gulliver’s Travels
, written in 1726, Jonathan Swift tells of a mythical Professor of the Grand Academy of Lagado who aims to generate a catalogue of all scientific knowledge by having his students continuously generate random strings of letters by means of a mechanical printing device. The first mechanical typewriter had been patented in 1714. After several eighteenth- and nineteenth-century French mathematicians used the example of a great book being composed by a random deluge of letters from a printing works as an example of extreme improbability, the monkeys appear first in 1909, when the French mathematician Émile Borel suggested that randomly
typing
monkeys would eventually produce every book in France’s Bibliothèque Nationale. Arthur Eddington took up the analogy in his famous book
The Nature of the Physical World
in 1928, where he anglicised the library: ‘If I let my fingers wander idly over the keys of a typewriter it
might
happen that my screed made an intelligible sentence. If an army of monkeys were strumming on typewriters they
might
write all the books in the British Museum.’

Eventually this oft-repeated example picked the ‘Complete Works of Shakespeare’ as the prime candidate for random recreation. Intriguingly, there was a website that once simulated an ongoing random striking of typewriter keys and then did pattern searches against the ‘Complete Works of Shakespeare’ to identify matching character strings. This simulation of the monkeys’ actions began on 1 July 2003 with 100 monkeys, and the population of monkeys was effectively doubled every few days until recently. In that time they produced more than 10
³⁵
pages, each requiring 2,000 keystrokes.

A running record was kept of daily and all-time record strings until the Monkey Shakespeare Simulator Project site stopped updating in 2007. The daily records are fairly stable, around the 18- or 19-character-string range, and the all-time record inches steadily upwards. For example, one of the 18-character strings that the monkeys have generated is contained in the snatch:

. . . Theseus. Now faire UWfIlaNWSK2d6L;wb . . .

The first 18 characters match part of an extract from
A Midsummer Night’s Dream
that reads

. . . us. Now faire Hippolita, our nuptiall houre . . .

For a while the record string was 21-characters long, with

. . . KING. Let fame, that wtIA’”yh!”VYONOvwsFOsbhzkLH . . .

which matches 21 letters from
Love’s Labour’s Lost

KING. Let fame, that all hunt after in their lives,

Live regist’red upon our brazen tombs,

And then grace us in the disgrace of death; . . .

In December 2004 the record reached 23 characters with

Poet. Good day Sir FhlOiX5a]OM,MLGtUGSxX4IfeHQbktQ . . .

which matched part of
Timon of Athens

Poet. Good day Sir

Pain. I am glad y’are well

Poet. I haue not seene you long, how goes the World?

Pain. It weares sir, as it growes . . .

By January 2005, after 2,737,850 million billion billion billion monkey-years of random typing, the record stretched to 24 characters, with

RUMOUR. Open your ears; 9r”5j5&?OWTY Z0d ‘B-nEoF.vjSqj[ . . .

which matches 24 letters from
Henry IV Part 2

RUMOUR. Open your ears; for which of you will stop

The vent of hearing when loud Rumour speaks?

Which all goes to show: it is just a matter of time!

4

Independence Day

I read that there’s about 1 chance in 1000 that someone will board an airplane carrying a bomb. So I started carrying a bomb with me on every flight I take; I figure the odds against two people having bombs are astronomical.

Anon.

Independence Day, 4 July 1977 is a date I remember well. Besides being one of the hottest days in England for many years, it was the day of my doctoral thesis examination in Oxford. Independence, albeit of a slightly different sort, turned out to be of some importance because the first question the examiners asked me wasn’t about cosmology, the subject of the thesis, at all. It was about statistics. One of the examiners had found 32 typographical errors in the thesis (these were the days before word-processors and schpel-chequers). The other had found 23. The question was: how many more might there be which neither of them had found? After a bit of checking pieces of paper, it turned out that 16 of the mistakes had been found by both of the examiners. Knowing this information, it is surprising that you can give an answer as long as you assume that the two examiners work independently of each other, so that the chance of one finding a mistake is not affected by whether or not the other examiner finds a mistake.

Let’s suppose the two examiners found A and B errors respectively and that they found C of them in common. Now assume
that
the first examiner has a probability a of detecting a mistake while the other has a probability b of detecting a mistake. If the total number of typographical errors in the thesis was T, then A = aT and B = bT. But if the two examiners are proofreading
independently
then we also know the key fact that C = abT. So AB = abT
²
= CT and so the total number of mistakes is T = AB/C, irrespective of the values of a and b. Since the total number of mistakes that the examiners found (noting that we mustn’t double-count the C mistakes that they both found) was A + B – C, this means that the total number that they didn’t spot is just T – (A + B – C) and this is (A – C)(B – C)/C. In other words, it’s the product of the number that each found that the other didn’t divided by the number they both found. This makes good sense. If both found lots of errors but none in common then they are not very good proofreaders and there are likely to be many more that neither of them found. In my thesis we had A = 32, B = 23, and C = 16, so the number of unfound errors was expected to be (16 × 7)/16 = 7.

This type of argument can be used in many situations. Suppose different oil prospectors search independently for oil pockets: how many might lie unfound? Or if ecologists want to know how many animal or bird species might there be in a region of forest if several observers do a 24-hour census.

A similar type of problem arose in literary analysis. In 1976 two Stanford statisticians used the same approach to estimate the size of William Shakespeare’s vocabulary by investigating the number of different words used in his works, taking into account multiple usages. Shakespeare wrote about 900,000 words in total. Of these, he uses 31,534 different words, of which 14,376 appear only once, 4,343 appear only twice and 2,292 appear only three times. They predict that Shakespeare knew at least 35,000 words that are not used in his works: he probably had a total vocabulary of about 66,500 words. Surprisingly, you know about the same number.

5

Rugby and Relativity

Rugby football is a game I can’t claim absolutely to understand in all its niceties, if you know what I mean. I can follow the broad, general principles, of course. I mean to say, I know that the main scheme is to work the ball down the field somehow and deposit it over the line at the other end and that, in order to squalch this programme, each side is allowed to put in a certain amount of assault and battery and do things to its fellow man which, if done elsewhere, would result in 14 days without the option, coupled with some strong remarks from the Bench.

P.G. Wodehouse,
Very Good, Jeeves

Relativity of motion need not be a problem only for Einstein. Who has not had the experience of sitting in a stationary railway carriage at a station, then suddenly getting the sensation of being in motion, only to recognise that a train on the parallel track has just moved off in the other direction and your train is not moving at all?

Here is another example. Five years ago I spent two weeks visiting the University of New South Wales in Sydney during the time that the Rugby World Cup was dominating the news media and public interest. Watching several of these games on television I noticed an interesting problem of relativity that was unnoticed by the celebrities in the studio. What is a forward pass relative to? The written rules are clear: a forward pass occurs when the ball
is
thrown towards the opposing goal line. But when the players are moving the situation becomes more subtle for an observer to judge due to relativity of motion.