Geekspeak: Why Life + Mathematics = Happiness

When you fly to Rome for a weekend break, the plane will burn aviation fuel at about the same rate as when you drive your car: 30–40 miles per gallon (10–14 km per litre). Of course, that’s just for you and your baggage; the plane actually does about 0.2 mpg (less than 0.1 km/litre), but fortunately it holds more people than your car, so the consumption per person is much less.

Flying the round trip of 2,000 miles (3,000 km) to Rome and back will use 50 gallons (240 litres) of fuel on your personal account, and each litre of aviation spirit burnt releases about 2.5 kg of CO

. Overall, your return trip will put approximately 600 kg into the atmosphere.

Will you forgo your new house for a tent so that you can have 37,000/600 = 61 weekend breaks in Rome and still hold your carbon-neutral head up high?

SPEAK GEEK

AN AESTHETICALLY PLEASING WINDOW HAS A HEIGHT THAT IS 1.618 TIMES GREATER THAN ITS WIDTH.

Some houses look right and some look wrong. It often depends on the ratio of the length and width of the building’s facade, and of the windows and doors and their positioning in the walls.

To many people, windows and other rectangular shapes look ‘right’ only when their width-to-height ratio is close to 1.618. This number is based on a mathematical rule called the Golden Section that has been used by everyone from the Ancient Greeks to Norman Foster. Using the Golden Section ensures an eye-pleasing harmony in the shape of the window.

Check your house-harmony by measuring the height and width of its window frames; divide the numbers, and give the result to your estate agent.

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WELL CONNECTED

Do you nkow the Queen?

When I was about seven or eight, I went to play with my elder brother on a railway track. My brother, deliberately winding me up, announced that he was going to derail the next train by placing an old penny on the rail. I believed him, and silently prayed for the imminent disaster to somehow be averted. Penny in place, we hid at the bottom of the embankment. The sound of the steam locomotive grew louder as it approached. There was a terrifying sound of steel on steel, of the snorting smoke box, and the train arrived… and passed by, oblivious to our presence. There were no newspaper headlines. The only evidence of our misdemeanour was the penny, now a flattened piece of copper about two inches by one inch, with a slight curl, lying quietly on the ballast between the rails. To this day I still feel a tingle of relief.

I tell this story because we, like other children of our generation, were pretty free of adult supervision. There were paedophiles, transformer yards and rail tracks waiting to catch the unwary child, but the sense of anxiety about strangers and the world’s danger barely registered.

One cause of our present-day unease is maybe down to not knowing who is who in our community. In a village of three hundred people a hundred years ago, you would probably know which men had a predatory eye for children. What’s more, you would know their name, and their parents and their brothers and sisters. Nowadays, you might not even know who lives two doors away in your street.

If we look at the number of children born to each couple and do some simple arithmetic, we can see why this has happened, and consequently why we are now faced with the prospect of identity cards, biometric devices and databases with information about our DNA.

The story goes like this.

How often has someone you know done some name-dropping by telling you about a friend of a friend who has met someone famous, or an eighth cousin twice removed who’s starred alongside Tom Cruise? Maybe you know someone whose friend has a relative who works at Buckingham Palace and has spoken to the Queen.

I come from a middle-class family, but went to a secondary modern school. I have no obvious links to the aristocracy. But now I come to think of it, I do know a woman who was married to a man whose father was General Montgomery’s adjutant during the Second World War, and who as a result was invited to tea with the Queen. That’s just four links between me and the throne.

Try to forge those links for yourself, and you’ll almost certainly succeed in finding a connection. In fact there’s a good chance of making a chain of personal connections between you and anyone else on Earth using no more than six links in the chain. That means that a friend of a friend of a friend of a friend of a friend knows anyone you care to name – George Bush, Naomi Campbell, Osama bin Laden…

Osama bin Laden probably doesn’t have mates that he meets in the pub, but he certainly has lots of other personal relationships that eventually lead back to you.

The fact that anyone on the planet is at most only six links away in a chain of relationships seems unlikely until you think about how many people you know on first-name terms. It’s simple to estimate the total by dividing them into a few broad groups that can be enumerated. For example, family, friends, casual acquaintances and work colleagues probably cover most of the people you know by their first-name.

Start with an average family in which parents have about two children. We’ll assume that grandparents, uncles, aunts and cousins are included in the extended family. After all, you’ll almost certainly be on first-name terms with all those people. The list will look something like this:

So, if you are a child or young adult, there are probably around 16 members in your extended family. Now imagine going back two or three generations in Britain. A century ago, most couples had four or five children who survived into adulthood. It makes the table look dramatically different:

Evidently, in societies with large families it’s the cousins who multiply almost exponentially with family size, and it is they who most likely provide the connectivity that holds communities together.

The reduction in the number of children in each family has had a profound and mathematically inevitable effect on community. In a small town of 10,000 inhabitants at the end of the nineteenth century, only a few hundred extended families would account for the entire population. You, or someone in your extended family, would be related to every other person in town.

Nowadays, with just 16 members in an extended family, you would have to live in a village of fewer than 200 geographically and socially immobile inhabitants to get the same level of coverage.

The tangible result of smaller families is low social connectivity and loss of community. Identity cards with biometric information and links to databases with our health records, past addresses and benefit history may be the inevitable synthetic replacement for first-hand knowledge about our neighbours.

A few years back, the Department of Trade and Industry organised a series of conferences on fraud prevention and security. They centred on the use of biometrics – technologies that use computer recognition of the iris, fingerprints or speech to uniquely identify an individual. One of those techniques is likely to be incorporated into British identity cards if they are ever introduced.

Grey suit after grey suit delivered the same technological gospel. Then, one speaker stood up and delivered a totally unexpected scathing attack on the whole idea. His argument: we have worked hard for our anonymity. It should be cherished as a prize of our development into a society based on individuals living without family. Who wants to go back to a village society where your every move is known to all?

Anyway, let’s get back to analysing your connection to Osama bin Laden or the Queen. We’ve established that a typical extended family has 16 members, or 15 other people you know very well. How about work associates?

Assuming that they are people you know by first name, the number will depend strongly on the kind of organisation you work in. Self-employed people might know just half a dozen customers by first name, whereas someone working in a large insurance office might know 40 or 50. We could take a typical number as 25.

And then there are your friends: most friends are known to you by their first name. You put in a figure for yourself. I think I have about 20 friends who are not anything to do with my work.

So far, the total number of people with whom you have a sufficiently close relationship to call them by their first name stands at about 60. That figure doesn’t include all the casual acquaintances whose name you might know: the man at the post office, the window cleaner, the man you meet walking his dog. Let’s say that makes another 20.

The total number of people with whom you are on first-name terms now stands at 80. Each of those people in your social circle is at the centre of their own personal social circle of about 80 people, and each of those people is at the centre of their circle. To simplify things, I’ll use ‘pal’ to denote anyone who is a friend, a relative, a work colleague or an acquaintance.

As a start, we can say that the number of pals of pals we have is about 80 × 80 = 6,400. But in real life some of the pals’ pals will be members of your own family, work colleagues and circle of friends, and the figure of 6,400 has counted those people twice over.

A simple way to take that overlap into account is to introduce a factor that expresses the percentage of each successive social circle that is common to the previous group. A factor of 50% seems fair. It means that, of the 80 people in the social circle of one of your pals, 40 will also be in your own circle.

So now, very conservatively, each link in the chain of social circles multiplies the number of people connected to you by a factor of just 40, but successive multiplication builds up the numbers very quickly. Try it out for six links in the chain:

The table shows there are a total of 4.1 billion people accessible through a chain of just six links. The Earth’s population is about 6.6 billion, so it seems that a pal of a pal of a pal of a pal of a pal will connect you to three-quarters of humanity.

Does it really work for anyone? Finding a link to Osama might be tricky, although certainly of interest to MI6. Bin Laden is a Saudi citizen, so a good bet would be to look for a link between yourself and someone in Saudi Arabia. Once you have found that link into the country, there are likely to be lots of links between your Anglo-Saudi bridge person and someone in bin Laden’s family.

The Osama example shows how the small number of links could be exploited by anyone using information about phone calls or emails between individuals. The first step is to construct a database of all known names. Listed under each name are the numbers for all incoming and outgoing phone calls or emails for that individual. That’s difficult, but technologically possible. Those phone numbers and email addresses are like the social circle of the named individual. A computer searches for chains with the same links.

We’ve already shown by simple arithmetic that there are probably no more than six links in the chain and that the people in the chain form a community of some kind. It may be benign or malign, but significantly it can be detected automatically by a computer.

It only takes one suspicious individual in the chain to cast suspicion on all the other people in the chain – the Queen, perhaps?

SPEAK GEEK

COUNTING FRIENDS OF FRIENDS, THERE ARE OVER A THOUSAND PEOPLE IN YOUR SOCIAL CIRCLE.

A typical person has 30 to 50 people they consider to be close friends or acquaintances. We could take an average of 40 for the purpose of calculation. If each of these friends has a further 40 friends, excluding you and your shared acquaintances, there will be approximately 40 × 40 people in your social circle. That’s 1,600 people. If you count friends of friends of friends, the number rises to 64,000.

You’d spend a lot of time writing Christmas cards.

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FATAL ATTRACTION

How much are you physically attracted to your partner?