Do You Think You're Clever? (8 page)

(Social and Political Sciences, Cambridge)

‘Surely,’ said the nineteenth-century American poet and naturalist Henry David Thoreau, ‘the apple is the noblest fruit.’ And there is no fruit that has acquired such symbolic status and been so overlaid with meaning as this ball of pulp, seeds and skin. From New York City to Macintosh computers, it stands for everything from true knowledge to all that’s wholesome, all in a tidy little package. A child is the ‘apple of his parents’ eyes’. A good kid gone rotten is ‘a bad apple’. Apples are the comfort for those sick of love in the ‘Song of Solomon’. In fact, apples are pretty much anything you want them to be. And everyone has their own way of describing them.

If you were an artist, you might describe an apple as a roundish fruit typically about the size of a tennis ball (5 to 9 centimetres in diameter). It’s not quite round, though, as a closer look would show, for there are indentations on opposite sides – a shallow one at the bottom and a deeper one at the top where the stalk is attached. Some apples, you’d observe, have a glossy, waxy look that glistens with brilliant surface highlights, while others have a rougher,
mottled matt texture that is richly and densely coloured. They vary in hue from a whitish green to golden yellow, from russet brown to bright crimson – but the colour is very rarely even. On some shiny green apples, the variations are subtle, with just paler spots and stripes visible here and there, with the occasional small dark spots. On others there are marked differences, especially in ripe apples, with some patches turning bright, warm scarlet and others remaining a sharp green.

A mathematician might come at the description from a different angle. This would be tricky because the shape is variable, and it’s a complex shape. It’s only very roughly approximate to a sphere. You could say it’s an oblate spheroid, since it’s slightly flattened at the ends. But that would not allow for the flattening around the sides on many apples, too. And of course, this completely ignores the toroidal indenting at top and bottom. It might be simpler to sum up its shape as apploid. We might even be able to come up with an equation to describe an apploid, by assuming an imaginary, perfectly symmetrical apple. Of course, real apples are never perfectly symmetrical, but our mathematical apploid could be a good model of the average apple.

A botanist might have yet another description: An apple is a pome, a special kind of fruit that grows on
Rosaceae
trees of the Maloid family, which also includes pears, quinces, medlars and rowans. Pomes are distinctive because they are ‘accessory’ fruits, in which the flesh is
not made from the carpels (the fertilised ovary) but grows around them. In an apple, the carpels are the five-pointed star of the core alone, each with its seeds or ‘pips’. The shrivelled remains of sepals, style and stamen often form a little complementary five-pointed star at the base of the apple. The apples we eat are cultivars, varieties developed from
Malus domestica
, the cultivated tree taken from the wild
Malus sieversii
, which grows in the mountains of Central Asia.

An organic chemist might describe an apple as an arrangement of cellulose in cells containing water, sugars such as fructose and glucose, mild acids such as malic acid, Vitamin C, minerals and amino acids.

A mythologist could describe an apple in many ways – as the forbidden fruit, the fruit of the tree of knowledge, for instance. It was said to be an apple that the Biblical Eve coaxed Adam to share with her, so that they both learned they were naked and were expelled from the Garden of Eden. Golden apples were the fruit of the Greek Tree of Life in the Garden of the Hesperides, which Hercules had to pluck as one of his Twelve Labours. They were the gifts of Aphrodite, the goddess of love, too. Indeed, pretty much every culture has its own special apple stories.

A greengrocer might describe an apple as ‘a lovely eater, really sweet and crisp – delicious’ or ‘a great cooker, big and juicy’. A cook might add that it’s very versatile fruit, which can be used in a huge variety of dishes, including apple pie and apple crumble – not to mention as an accompaniment
to pork. A wine merchant could say it’s the basic ingredient of cider or calvados. To a harassed parent, an apple might be a healthy way to keep the kids quiet in between meals. For disappointed medieval audiences, a (rotten) apple might be a critical theatre review. William Tell’s son might describe an apple as the difference between life and death, or ‘dad’s psychopathic moment’. To Isaac Newton, it was a matter of some gravity. And for Salvador Dali it could be a nun’s posterior or a skull’s eye socket …

And finally, listen to the very bad but aptly named romantic poet, Bramley Laxton:

(Education Studies, Cambridge)

The stage today can be pretty much what the performer wants it to be, yet it’s rarely either opinion or entertainment alone. There is quite a difference, for instance, between even the most undiluted piece of ‘opinion’ theatre and someone giving a political speech in parliament or haranguing passers-by at Speaker’s Corner, even though all can be highly theatrical.

A speech seeks to communicate the speaker’s ideas directly to listeners. A stage performance seeks to engage the audience’s imagination, and help them understand the ideas by living through them in their imagination. This is why, generally, stage performances use characters to tell a story or create a picture, rather than talk to the audience directly. When they do appear to address the audience directly, it’s in character or within the imaginary world created for the performance.

It’s this creation of an imaginary world that marks out the stage from the direct address of a public speaker. The stage is the arena for the creation of this world, a defined area in which the vision is created – whether it’s an actual
stage, or simply a subtle imaginary barrier set up by the performers between them and their audience as they wander among them in ‘promenade’ performances. The stage in essence, then, is a playground for the imagination. Stand-up comedians, singers and musicians all perform upon stages, and all to a lesser or greater extent engage the imagination. What makes the stage in the sense of theatre different is that a story is enacted and a picture embodied.

All stage performances must entertain in the broadest sense. ‘Entertain’ is a word that came originally from the French
entretenir
, meaning ‘hold together’ – and that’s what theatre seeks to achieve: binding performers and audience together in a communal act of imagination. But of course, by ‘just’ entertainment, people generally mean something rather less – something that merely makes the audience laugh or smile but rarely makes them think. By contrast, then, ‘opinion’ theatre would make them think, but maybe not laugh or smile. The juxtaposition, though, is false. Even the crudest jokers are expressing an opinion – and their audience have to think, however shallowly and briefly, in responding. And if the audience doesn’t fall asleep, they are entertained in the sense of engagement by even the most undiluted opinion piece.

What really matters is the quality of thought and the richness and worth of the ideas involved, and this may be where modern theatre often loses its way, setting up false opposites between the brash, flashy world of ‘sheer entertainment’ in West End musicals and the intensely
serious world of ‘challenging’, ‘relevant’ drama. The irony, then, is that often both end up simply poor – being neither great entertainment nor particularly interesting opinion. (Opinion without knowledge or wisdom is facile and attention-seeking rather than instructive and engaging – and ultimately no more profound or valuable than ‘sheer entertainment’, besides being much less enjoyable.)

Back in the earliest days of theatre in Ancient Greece, Aristotle in his
Poetics
highlighted the difference between tragedy and comedy, and until recently these were seen as the two kinds of theatre, with countless theatre companies carrying as their trademark the opposite weeping mask of tragedy and the smiling mask of comedy. The essence of comedy was a happy ending, and the audience was intended to enjoy the triumph of a lowly, often mischievous character or outcast. The essence of tragedy was the downfall of a good or noble person through some fatal error, and the audience was intended to feel pity and fear that led them to what Aristotle called ‘catharsis’ – a purging of these negative emotions. Yet most of the best tragedies are leavened with laughter, because humour helps engage an audience; and many of the best comedies have a darker or sadder side that likewise helps engage the audience’s sympathies.

Aristotle traced the origins of both tragedy and comedy to religious rituals, and if the overtly religious element has long since vanished, there remains an element of ritual. Without really thinking about it, as an audience we will
ingly fall hushed at the beginning of the performance and sit rapt in silence in the dark – a condition that has distant echoes of the ritual trances of primitive religions in which people voyaged in their minds into visions and other worlds.

The Ancient Greeks called dramatists
didaskalos
, which is typically translated as teachers, but a better word might be guides. Dramatists are guides through these visions, and they may be opinionated or entertaining, reflective or passionate, thoughtful or impetuous – in fact, anything they may wish to be. Above all, though, they must, like all the best guides, know their way …

(Mathematics, Oxford)

This appears be intuitively simple. A first guess might be that the tank should be located somewhere on the line between the four towns, at the mid-point between one of
the two central towns. Then, if the towns are 1 km apart, this would give two 1-km round trips and two 3-km round trips, a total of 8 km. However, intuition is not always right, and proving that answers like these are correct has led mathematicians into some of the most complex and important of all mathematical problems. Known as optimisation, these are about mathematically finding the best solution to a problem, whether it’s the shortest route or the optimum structure.

The shortest distance problem has always fascinated mathematicians, not only because it’s an interesting geometrical conundrum, but also because it has clear practical applications. The point we are looking for to locate our oil baron’s tank is known as the geometric median, which is the point which minimises the distance to a set of sample points.

Back in the seventeenth century, the great French mathematician Pierre de Fermat posed a similar problem to Evangelista Torricelli, the Italian inventor of the barometer. Fermat’s problem was to find the shortest distance from a single point to three sample points. Torricelli ingeniously solved the problem, and the point is now known as the Fermat or the Torricelli point. (German economist and pioneer of globalisation theory Alfred Weber also gets in on the act because of his 1909 thesis on the location of industries, so this question is sometimes called the Fermat-Weber problem.)

Fermat’s work gave a solution for three points; the problem of finding the geometric median for four points was actually solved by the brilliant Italian priest and mathematician Giovanni Fagnano dei Toschi around 1750. Fagnano realised that when the fixed points form a convex quadrangle, the geometric median is the intersection point of both diagonals; otherwise it’s the fixed point in the triangle formed by the three other fixed points.

Mathematicians now know that there is only one geometric median when the sample points are not in a line. In the special case, when the points are in a line or ‘colinear’ as in the oil baron’s problems, the geometric median is the median. The median is the number that divides the population in half. In the case of the oil baron, it is indeed as we guessed – the midpoint between the two middle towns.

Interestingly, the idea of optimising routes is not simply a geometrical problem. It leads into network theory, a branch of mathematics that has attracted extraordinary interest in recent years. The more people study it, the more relevant it seems as an organising principle to explain how the world really works. Networks crop up everywhere. People network socially. The internet is a network. Transport links form networks. Ecosystems form networks. Computers depend on them. What is really exciting people is how lessons learned in one discipline, such as biology, are feeding into others, such as economics.

The recent banking crisis, for instance, has provoked a number of economic theorists such as Domenico Delli
Gatti and Joseph Stieglitz to try to tease out the problems using network theory, showing how a few banks in a network emerge as connecting hubs with more than their share of links. The problem with this is that when such hubs collapse, they can take down the whole network with them. So the network economists are looking to ecological food webs for remedies, because these have evolved ways of coping with crises better. Species with many links, for instance, are often connected to species with few. An insect may pollinate a wide variety of plants, but each plant may be pollinated by only one or a few insects. That way a catastrophe for even the multi-connected insect affects only a limited part of the network.

As director of network research at the University of Notre Dame, Albert-László Barabási writes: ‘The diversity of networks in business and the economy is mind-boggling. There are policy networks, ownership networks, collaboration networks, organizational networks, network marketing – you name it. It would be impossible to integrate these diverse interactions into a single all-encompassing web. Yet no matter what organizational level we look at, the same robust and universal laws that govern nature’s webs seem to greet us.’