100 Essential Things You Didn't Know You Didn't Know Page 18
Tall buildings funnel winds into narrow channels around them at street level (it’s just like putting your finger partly over the nozzle of a garden hose to make the jet reach further – the increased pressure brought about by the constriction results in a higher velocity of waterflow) and this can have a horrible effect on passersby and people using the building. They feel as if they are in a wind tunnel. The narrowing of the building at the base reduces these unwanted wind effects because there is less constriction of the airflows. The tapering of the top half also plays an important role. If you stand at ground level beside a conventional untapered tower block and look upwards, the building dwarfs you and blots out a large fraction of the sky. A tapering design opens up more of the sky and reduces the dominating effect of the structure because you can’t see the top from close-by on the ground.
The other striking feature of this building’s exterior is that it is round, not square or rectangular. Again, this is advantageous for smoothing and slowing the airflow around the building. It also assists in making the building unusually eco-friendly. Six huge triangular wedges are cut into each floor level from the outside in. They bring light and natural ventilation deep into the heart of the building, reducing the need for so much conventional air-conditioning and making the building twice as energy efficient as a typical building of the same scale. These wedges are not set immediately below each other from floor to floor, but are slightly rotated with respect to those on the floors above and below. This helps increase the efficiency of the air suction into the interior. It is this slight offsetting of the six wedges from floor to floor that creates the exterior spiral pattern that is so noticeable.
Looking at the rounded exterior from a distance, you might have thought that the individual surface panels are curved – a complicated and expensive manufacturing prospect – but in fact they are not. The panels are small enough, compared with the distance over which the curvature is significant, that a mosaic of flat, four-sided panels is quite sufficient to do the job. The smaller you make them, the better they will be able to approximate the covering of the curved outside surface. All the changes of direction are made at the angles joining different panels.
85
Being Mean with the Price Index
The average human has one breast and one testicle.
Des McHale
All economically developed countries have some measure of the change of the average person’s cost of living that is derived from the price of standard units of some collection of representative goods, including staple foods, milk, heat and light. They have names like the Retail Price Index (RPI) or the Consumer Price Index (CPI), and are a traditional measure of inflation, which can then be used to index salaries and benefits to allow for it. Citizens therefore want these measures to come out on the high side, whereas governments want them to be on the low side.
One way to work out a price index is to take a simple average of a collection of prices – just add up the prices and divide by the number of different prices you totalled. This is what statisticians call the arithmetic mean, or simply the ‘average’. Typically, you want to see how things are changing over time – is the cost of the same basket of goods going up or down from month to month? – so you want to compare the average last year with its value this year by dividing the first by the second. If the result is bigger than 1, then prices are going down; if it is less than 1, then they are going up. This is simple enough, but are there hidden problems?
Suppose that a family habitually spends the same amount of money each week on beef and fish and then the price of beef doubles while fish costs the same. If they keep buying the same amount of beef and fish, their total bill for beef and fish will be 1.5 times their old one, an increase of 50%. The average of the price changes will be ½ × (1 + 2) = 1.5. The ½ is just dividing by the number of products (two: fish and beef ); 1 is the factor by which the price of fish changes (it stays the same) and 2 is the factor by which the price of beef changes (it doubles).
This 1.5 inflation factor, an increase of 50 per cent, will be the headline statistic. But for a non-meat-eating family that eats no beef it will be meaningless. If they were only eating fish they will have seen no change in their weekly bill at all. The inflation factor is also an average over all possible family eating choices. It is based on an assumption about human psychology. It assumes that the family will go on eating the same amount of beef as fish, despite the rise in the relative cost of beef compared to fish. In reality, families might behave differently and decide to adjust the amount of fish and beef they buy, so that they still spend the same amount of money on each. This will mean they buy less beef in the future because of its increased cost.
The assumption that families respond to price changes by keeping constant the fraction of their budget that they spend on each commodity suggests that the simple arithmetic mean price index should be replaced by another sort of mean.
The geometric mean of two quantities is the square root of their product.fn1 The geometric mean index of the price change in beef and no change in the price of fish is therefore:
√(New beef cost/old beef cost) × √(New fish cost/old fish cost) = √1 × √2 = 1.41
The interesting thing about these two measures of inflation is that the geometric mean of any number of quantities is never greater than the arithmetic mean21 so governments certainly prefer the geometric mean.fn2 It suggests lower inflation and results in a lower inflationary index for wage increases and social security benefits.
Another benefit of the geometric mean is practical rather than political. To determine the inflation rate you compare the index at different times. If you were using the arithmetic mean then you would need to work out the ratio of the arithmetic means in 2008 and 2007 to find out how much the ‘basket’ of prices inflated in price over the last year. But the arithmetic mean involves the sum of all sorts of different things that can be measured in different units: £ per Kg, £ per litre, etc; some involve price per unit weight, some price per unit volume. It’s a mixture, which is a problem because in order to calculate the arithmetic mean you have to add them together, and this doesn’t make any sense if the units of the different commodities that go into it are different. By contrast, the beautiful feature of using the geometric mean is that you can use any units you like for the commodity prices in 2008 as long as you use the same ones for the same commodities in 2007. When you divide the geometric mean value for 2008 by its value for 2007, to find the inflation factor, all the different units cancel out because they are exactly the same in the denominator and in the numerator. So, it’s a pretty mean index.
fn1 More generally, the geometric mean of n quantities is the nth root of their product.
fn2 In the United States the Labor Bureau changed the calculation of the Consumer Price Index from the arithmetic to the geometric mean in 1999.
86
Omniscience can be a Liability
For sale by owner, Encyclopaedia Britannica, excellent condition. No longer needed. Husband knows everything.
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Imagine what it must be like to know everything. Well, maybe it’s not so easy. Perhaps it’s more manageable to imagine what it would be like to know everything that you wanted to know – or needed to know. Even that sounds like being in a commanding position: you know next week’s winning lottery numbers, which train to take so as to avoid delays, who is going to win the big football game. There are huge advantages to be had, although life could eventually prove miserable without the benefit of the odd pleasant surprise.
There is a strange paradox about knowing everything that shows that you can find yourself worse off than if you didn’t know everything. Suppose you are watching a dare-devil game of ‘chicken’ in which two stunt pilots fly aircraft towards each other at high speed (it’s like aerial jousting without the horses and lances). The loser is the flyer who first swerves off to the side. What should a pilot’s tactic be in such a game? If he never swerves, then he w
ill end up dead if the other flyer has the same tactic – no one wins then. If he always swerves he will never win – only draw when the other pilot swerves as well. Clearly, always swerving is the only sure strategy that minimises loss. Some mixed strategy of sometimes swerving and sometimes not swerving will produce some wins, but ultimately will result in death unless the other pilot always swerves when the other does not. If the other pilot thinks the same way (he is thinking this through independently and doesn’t know what his rival has decided), he should draw the same conclusions.
Now play this game with an omniscient opponent. He knows what your strategy is going to be. So you should choose never to swerve. He will know that your strategy is never to swerve and will therefore choose to swerve every time. The omniscient pilot will never win!
This story presumably has an application to the world of espionage. If you are listening-in on all your enemies’ communications and they know that you are listening, then your omniscience may put you at a disadvantage.
87
Why People aren’t Cleverer
God help in my search for truth, and protect me from those who believe they have found it.
Old English Prayer
When astronomers speculate about the nature of advanced extraterrestrials or biologists contemplate a future where humans have evolved to be smarter than they are today, it is always assumed that increased intelligence has to be a good thing. The evolutionary process increases the likelihood of passing on traits that increase the chance of survival and having offspring. It is hard for us to imagine how it might become a liability to be more intelligent on the average as a species than we are.
If you have ever had experience of trying to manage a community of cleverer-than-average individuals, then you could easily be led to think otherwise. A good example might be the challenge faced by the chairperson of a university department or the editor of a book to which many authors are contributing. In such situations you soon realise that this type of high intelligence tends to accompany a tendency to be individualistic, to think independently and to disagree with others who think differently. Perhaps the ability to get on with others, to work with others rather than against them, was more important during the early evolution of intelligence. If intelligence now evolves rapidly to superhuman levels, perhaps the effects would be socially disastrous? Then again, low levels of average intelligence are clearly disadvantageous when it comes to dealing with potentially foreseeable hazards. There might well be an optimal level of intelligence for life in a given environment in order to maximise the chances of long-term survival.
88
The Man from Underground
Art has to move you and design does not, unless it’s a good design for a bus.
David Hockney
Once I saw two tourists trying to find their way around central London streets using an Underground train map. While this is marginally better than using a Monopoly board, it is not going to be very helpful. The map of the London Underground is a wonderful piece of functional and artistic design, but it has one striking property: it does not place stations at geographically accurate positions. It is a topological map: it shows the links between stations accurately, but distorts their actual positions for aesthetic and practical regions.
When Harry Beck first proposed this type of map to the management of the London Underground Railway, he was a young draughtsman with a background in electronics. The Underground Railway was formed in 1906, but by the 1920s it was failing commercially, not least because of the apparent length and complexity of travelling from its outer reaches into central London, especially if changes of line were necessary. A geographically accurate map looked a mess because of the higgledy-piggledy nature of inner London’s streets that had grown up over hundreds of years without any central planning. It was not New York, nor even Paris, with a simple overall street plan. People were put off.
Beck’s elegant 1931 map – although initially turned down by the railway’s publicity department and the Underground’s managing director, Frank Pick – solved many problems at one go. Unlike any previous transport map, and reminiscent of an electronic circuit board, it used only vertical, horizontal and 45-degree lines. It also eventually drew in a symbolic River Thames, introduced a neat way of representing the exchange stations and distorted the geography of outer London to make remote places like Rickmansworth, Morden, Uxbridge and Cockfosters seem close to the heart of London. Beck continued to refine and extend this map over the next 40 years, accommodating new lines and extensions of old ones, always striving for simplicity and clarity. It was always referred to by him as the London Underground Diagram, or simply ‘The Diagram’, to avoid any confusion with traditional maps.
Beck’s classic piece of design was the first topological map. It can be changed by stretching it and distorting it in any way that doesn’t break the connections between stations. Imagine it drawn on a rubber sheet, which you then stretch and twist in any way you like without cutting or tearing it. Its impact was sociological as well as cartographical. It redefined how people regarded London. It drew in the outlying places on the map and made their residents feel that they were close to central London. It even defined the house price contours. For most of us, it is the picture of how London ‘is’.
Beck’s original idea makes good sense. When you are below ground on the Underground you don’t need to know where you are, as you do if you are travelling on foot or by bus. All that matters is where you get on and off and how you change onto other lines. Pushing far away places in towards the centre doesn’t only help Londoners feel more connected, it helps create a neat beautifully balanced diagram that fits nicely on a small fold-out sheet that will go into your jacket pocket.
89
There are No Uninteresting Numbers
Everything is beautiful in its own way.
Ray Stevens
There is no end to the list of numbers. The small ones, like 1, 2 and 3, are in constant use to describe the small numbers of life: the number of children, cars, items on a shopping list. The fact that there are so many words for groups of small quantities that are specific to their identities – for instance, double, twin, brace, pair, duo, couple, duet, twosome – suggests that their origins predate our decimal counting system. Each of these small numbers seems to be interesting in some way. The number 1 is the smallest of all, 2 is the first even number, 3 is the sum of the previous two, 4 is the first that is not prime and so can be divided by a number other than itself, 5 is the sum of a square (22) plus 1. And so we might go on. Gradually, you begin to wonder whether there are any completely uninteresting numbers at all, sitting there unnoticed like wallflowers at the numbers ball.
Can you prove it? Well, yes you can if you approach the question in the manner of many mathematical arguments. You start by assuming that the opposite is true and then use that assumption to deduce something that contradicts it. This means that your first assumption must have been false. It is an ultimate version of the gambit in chess wherein a player offers a piece to an opponent in the knowledge that, if it is taken, it opens the way for a much bigger gain in the future. It is the ultimate version of this manoeuvre because it is the whole game, rather than simply a single piece, that is being offered by this logical gambit.
Let us assume that there are uninteresting positive whole numbers and collect them together. If such a collection exists, then it will have a smallest member. But that smallest member is by definition interesting: it is the smallest uninteresting number. This contradicts our initial assumption that it is an uninteresting number. So our first assumption, that there are uninteresting numbers, was false. All numbers must be ‘interesting’.
Just to prove it, here is a story, well known among mathematicians, that is told about the English mathematician Godfrey Hardy when he went to visit his friend, the remarkable Indian mathematician Srinivasa Ramanujan, in a London hospital: ‘In the taxi from London, Hardy noticed its number, 1729. He must have thought about i
t a little because he entered the room where Ramanujan lay in bed and, with scarcely a hello, blurted out his disappointment with it. It was, he declared, “rather a dull number”, adding that he hoped that wasn’t a bad omen. “No, Hardy,” said Ramanujan, “it is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.”fn1 Such numbers are now known as ‘taxicab’ numbers in memory of this incident.
fn1 Because 1729 = 13 + 123 = 93 + 103. The cubes have to be positive in these examples. If you allow negative numbers to be cubed, then the smallest such number is 91 = 63 + (-5)3 = 43 + 33.
90
Incognito
The number you have reached does not exist.
Recorded message for unobtainable Italian phone numbersfn1
In the sixteenth and seventeenth centuries it was not uncommon for the leading mathematicians of the day to publish their discoveries in code. This seems very strange to modern scientists, who clamour for recognition and priority in being the first to discover something, but there was some method to the seeming madness of those early mathematicians. They wanted to have their proverbial cake and eat it. Publishing a discovery that makes use of a new mathematical ‘trick’ establishes you as its discoverer, but it also reveals the trick so that others can use it and beat you to other, possibly greater, discoveries. You have a choice: hold off announcing the first discovery until you have given yourself time to investigate other possibilities more closely – and run the risk that someone else will discover and publish your first result – or you could publish an encoded version of your discovery. Assuming no one breaks the code, your new trick is safe from exploitation by others, and if anyone else comes along and announces they have discovered what you have already found, you can just apply the decrypt to show that you made the discovery long before. Very devious – and I hasten to add that such behaviour does not go on today in science and mathematics, and probably wouldn’t be tolerated if it was attempted. However, it does happen in the world of literature. Books like the political novel Primary Colours, about the Presidential election campaign of Bill Clinton, written by an initially unidentified journalist using a pseudonym, look a little like an attempt to have it both ways.