Thinking in Numbers: How Maths Illuminates Our Lives (10 page)

What did the mathematician make of his nephew’s ‘googolplex’ as the biggest number he could conceive of? His reply is not recorded, but he might have told him about some of the infinitely many numbers that exceed the googolplex’s scope. He might, for example, have mentioned ‘googol factorial’, being the product of multiplying every whole number between 1 and a googol (1 × 2 × 3 x . . . 950,345 x . . . 1,000,000,000,000,008,761 x . . . googol). This number, which computers tell us begins 16294 . . . easily surpasses every other number that we have encountered in these pages.

For a universe of such limited dimensions, these monstrous numbers seem quite useless. Worse, they can appear to us excessive, disproportionate. Every number, after a certain point, feels gratuitous as a joke. Who knows? It is possible they are not intended for our attention. The Flower Adornment sutra speaks of immense aeons,
kalpas
, in which the universe is continuously destroyed and reborn. At the kalpa’s peak, men live for an average of eighty-four thousand years. In other realms, so the
Heart
sutra reports, a single life spans eighty-four thousand
kalpas
– that is to say, eighty-four thousand epochs, each one many zeroes long. For such beings, a googol or its factorial would belong merely to the tangible and the convenient.

Mathematicians aspire to these heavenly realms. Vast numbers that split our senses, enrich their work. But they also produce paradoxes. For instance, which is greater: 10 or 27, when each is multiplied by itself exactly a googolplex number of times? The latter, of course, although even the most powerful calculators – plunging one hundred digits deep – struggle to tell the two apart. This difficulty confounds our expectations: intuitively, we feel that the ordering of a number should remain straightforward, even when the number’s precise value cannot be known. And yet, there exist numbers so large that we cannot easily distinguish them from their double, or triple, or quadruple or any other amount. There exist magnitudes so immense that they escape all our words, and all our numbers.

The most famous paradox concerning big numbers takes us back once more to the ancient Greeks. Tradition attributes it to the philosopher Eubulides. It has been suggested that Eubulides’s inspiration owes something to his fellow sceptic Zeno, who argued that every falling grain of wheat makes a noise proportionate to the noise made by a falling bushel. Eubulides’s formulation does not feature wheat, however. As would Archimedes a century later, Eubulides built his argument on sand.

It goes as follows: first, we agree that one grain of sand does not make a heap. Adding a second grain does not make a heap either. Nor do we produce a heap with the third grain. From this it follows that adding one to any small number generates another number we call ‘small’. But if this is true, a billion is a small number. So, too, is a googolplex.

Understandably wary of this conclusion, the reader might propose that a heap of sand, like a big number, begins at a certain point: say, ten thousand. As a resolution of the paradox, this answer is unsatisfactory. It is unclear why nine thousand nine hundred and ninety nine should be considered small, but not nine thousand nine hundred and ninety nine plus one.

Of course, in at least one sense all numbers are small. Given any number ‘n’, there will be only n-1 numbers less than n, but an infinite group of numbers greater than it.

Addressing the unjustly forgotten mathematician Archytas, the Roman poet Horace, who was regarded as the finest lyric poet of the era of Emperor Augustus, expressed in his verse perhaps the greatest paradox of them all: that of finite men who spend their lives attempting to scale the infinite.

You who measured the sea and the earth and the numberless sands,

you, Archytas, are now confined in a small mound of dirt

near the Matine shore, and what good does it do you that you

attempted the mansions of the skies and that you traversed

the round celestial vault – you with a soul born to die?

Snowman

Outside it is cold, cold. Ten degrees below, give or take. I step out with my coat zipped up to my chin and my feet encased in heavy rubber boots. The glittering street is empty; the wool-grey sky is low. Under my scarf and gloves and thermals I can feel my pulse begin to make a racket. I do not care. I observe my breath. I wait.

A week before, not even a whole week, the roads showed black tyre tracks and the trees’ bare branches stood clean against blue sky. Now, Ottawa is buried in snow. My friends’ house is buried in snow. Chilling winds strafe the town.

The sight of falling flakes makes me shiver; it fills the space in my head that is devoted to wonder. How beautiful they are, I think. How beautiful are all these sticky and shiny fragments. When will they stop? In an hour? A day? A week? A month? There is no telling. Nobody can second-guess the snow.

The neighbours have not seen its like in a generation, they tell me. Shovels in hand, they dig paths from their garage doors out to the road. The older men affect expressions both of nonchalance and annoyance, but their expressions soon come undone. Faint smiles form at the corners of their wind-chapped mouths.

Granted, it is exhausting to trudge the snowy streets to the shops. Every leg muscle slips and tightens; every step forward seems to take an age. When I return, my friends ask me to help them clear the roof. I wobble up a leaning ladder and lend a hand. A strangely cheerful sense of futility lightens our labour: in the morning, we know, the roof will shine bright white again.

Hot under my onion layers of clothing, I carry a shirtful of perspiration back into the house. Wet socks unpeel like plasters from my feet; the warm air smarts my skin. I wash and change my clothes.

Later, round a table, in the dusk of a candlelit supper, my friends and I exchange favourite recollections of winters past. We talk sleds, and toboggans and fierce snowball fights. I recall a childhood memory, a memory from London: the first time I heard the sound of falling snow.

‘What did it sound like?’ the evening’s host asks me.

‘It sounded like someone slowly rubbing his hands together.’

Frowns describe my friends’ concentration. Yes, they say, laughing. Yes, we can hear what you mean.

One man laughs louder than the others. Above his mouth he sports a grey moustache. I do not catch his name; he is not a regular guest. I gather he is some kind of scientist, of indeterminate discipline.

‘Do you know why we see snow as white?’ the scientist asks. We shake our heads.

‘It is all to do with how the sides of the snowflakes reflect light.’ All the colours in the spectrum, he explains to us, scatter out from the snow in roughly equal proportions. This equal distribution of colours, we perceive as whiteness.

Now our host’s wife has a question. The ladle with which she has been serving bowls of hot soup idles in the pan. ‘Could the colours never come out in a different proportion?’ she asks.

‘Sometimes, if the snow is very deep,’ he answers. In which case, the light that comes back to us can appear tinged with blue. ‘And sometimes a snowflake’s structure will resemble that of a diamond,’ he continues. Light entering these flakes becomes so mangled as to dispense a rainbow of multicoloured sparkles.

‘Is it true that no two snowflakes are alike?’ This question comes from the host’s teenage daughter.

It is true. Imagine, he says, the complexity of a snowflake (and enthusiasm italicises his word ‘complexity’). Every snowflake has a basic six-sided structure, but its spiralling descent through the air sculpts each hexagon in a unique way: the minutest variations in air temperature, or moisture can – and do – make all the difference.

Like mathematicians who categorise every whole number into prime numbers or Fibonacci numbers or triangle numbers or square numbers (and so on) according to its properties, so researchers subdivide snowflakes into various groupings according to type. They classify the snowflakes by size, and shape and symmetry. The main ways in which each vaporous hexagon forms and changes, it turns out, amount to several dozen or several score (the precise total depending on the classification scheme).

For example, some snowflakes are flat and have broad arms, resembling stars, so that meteorologists speak of ‘stellar plates’, while those with deep ridges are called ‘sectored plates’. Branchy flakes, like the ones seen in Christmas decorations, go by the term ‘stellar dendrites’ (dendrite coming from the Greek word for a tree). When these tree-like flakes grow so many side branches that they finish by resembling ferns, they fall under the classification of ‘fernlike stellar dendrites’.

Sometimes, snowflakes grow not thin but long, not flat but slender. They fall as columns of ice, the kinds that look like individual strands of a grandmother’s white hair (these flakes are called ‘needles’). Some, like conjoined twins, show twelve sides instead of the usual six, while others – viewed up close – resemble bullets (the precise terms for them are ‘isolated bullet’, ‘capped bullet’, and ‘bullet rosette’). Other possible shapes include the ‘cup’, the ‘sheath’, and those resembling arrowheads (technically, ‘arrowhead twins’).

We listen wordlessly to the scientist’s explanations. Our rapt attention flatters him. His white hands, as he speaks, draw the shape of every snowflake in the air.

Complexity. But from out of it, patterns, forms, identities that every culture can perceive and understand. I have read, for instance, that the ancient Chinese called snowflakes blossoms and that the Scythians compared them to feathers. In the Psalms (147:16), snow is a ‘white fleece’ while in parts of Africa it is likened to cotton. The Romans called snow
nix
, a synonym – the seventeenth-century mathematician and astronomer Johannes Kepler would later point out – of his Low German word for ‘nothing’.

Kepler was the first scientist to describe snow. Not as flowers or fleece or feathers, snowflakes were at last perceived as being the product of complexity. The reason behind the snowflake’s regular hexagonal shape was ‘not to be looked for in the material, for vapour is formless’. Instead, Kepler suggested some dynamic organising process, by which frozen water ‘globules’ packed themselves together methodically in the most efficient way. ‘Here he was indebted to the English mathematician Thomas Harriot,’ reports the science writer Philip Ball, ‘who acted as a navigator for Walter Raleigh’s voyages to the New World in 1584-5.’ Harriot had advised Raleigh concerning the ‘most efficient way to stack cannonballs on the ship’s deck,’ prompting the mathematician ‘to theorize about the close packing of spheres’. Kepler’s conjecture that hexagonal packing ‘will be the tightest possible, so that in no other arrangement could more pellets be stuffed into the same container’ would only be proven in 1998.

That night, the snow reached even into my dreams. My warm bed offered no protection from my childhood memories of the cold. I dreamed of a distant winter in my parents’ garden: the powdery snow, freshly fallen, was like sugar to my younger brothers and sisters, who hastened out to it with shrieks of delight. I hesitated to join them, preferred to watch them playing from the safety of my bedroom window. But later, after they had all wound up their games and headed back in, I ventured out alone and started to pack the snow together. Like the Inuit (who call it
igluksaq
, ‘house-building material’), I wanted to surround myself with it, build myself a shelter. The crunching snow gradually encircled me, accumulating on all sides, the walls rising ever higher until at last they covered me completely. My boyish face and hands smeared with snow, I crouched deep inside feeling sad and feeling safe.

‘
On t’attend!
’ my friends call up in the morning to my room. ‘We are ready and waiting!’ I am the English slowcoach, unaccustomed to this freezing climate, to the lethargy it imposes on the body, and the dogged, unshakeable feeling of being underwater. What little snow I have experienced all these years, I realise now, has been but a pale imitation of the snow of my childhood. London’s wet slush, quick to blacken, has muddied the memory. Yet here the Canadian snow is an irresistible, incandescent white – its glinting surfaces give me back my young days, and alongside them, a melancholic reminder of age.

After my sweater, I pull on a kind of thermal waistcoat, then a coat, knee-long. My neck is wrapped in a scarf; my ears vanish behind furry muffs. Mitten fingers tie bootlaces into knots.

Fortunately, the Canadians have no fear of winter. The snow is well superintended here. Panic, of the kind that grips Londoners or Parisians, is unfamiliar to them; stockpiling milk, bread and tinned food is unheard of. Traffic jams, cancelled meetings, energy blackouts are rare. The faces that greet me downstairs are all kempt and smiling. They know that the roads will have been salted, that their letters and parcels will arrive on time, that the shops and schools will be open as usual for business.

In the schools of Ottawa, children extract snowflakes from white sheets of paper. They fold the crisp sheet to an oblong, and the oblong to a square, and the square to a right-angled triangle. With scissors, they snip the triangle on all sides; every pupil folds and snips the paper in their own way. When they unravel the paper different snowflakes appear, as many as there are children in the class. But every one has something in common: they are all symmetrical.

The paper snowflakes in the classroom resemble only partly those that fall outside the window. Shorn of nature’s imperfections, the children’s unfolded flakes represent an ideal. They are the pictures that we see whenever we close our eyes and think of a snowflake: equidistant arms identically pliable on six sides. We think of them as we think of stars, honeycombs and flowers. We imagine snowflakes with the purity of a mathematician’s mind.

At the University of Wisconsin the mathematician David Griffeath has improved on the children’s game by modelling snowflakes not with paper, but with a computer. In 2008, Griffeath and his colleague Janko Gravner, both specialists in ‘complex interacting systems with random dynamics’, produced an algorithm that mimics the many physical principles that underlie how snowflakes form. The project proved slow and painstaking. It can take up to a day for the algorithm to perform the hundreds of thousands of calculations necessary for a single flake. Parameters were set and reset to make the simulations as lifelike as possible. But the end results were extraordinary. On the mathematicians’ computer screen shimmered a galaxy of three-dimensional snowflakes – elaborate, finely-ridged stellar dendrites and twelve-branched stars, needles, prisms, every known configuration, and others, resembling butterfly wings, that no one had identified before.