‘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 ot
hers, resembling butterfly wings, that no one had identified before.
My friends take me on a trek through the nearby forest. The flakes are falling intermittently now; above our heads, patches of the sky show blue. Sunlight glistens on the hillocks of snow. We tread slowly, rhythmically, across the deep and shifting surfaces, which squirm and squeak under our boots.
Whenever snow falls, people look at things and suddenly see them. Lamp posts and doorsteps and tree stumps and telephone lines take on a whole new aspect. We notice what they are, and not simply what they represent. Their curves, angles, repetitions, command our attention. Visitors to the forest stop and stare at the geometry of branches, of fences, of trisecting paths. They shake their heads in silent admiration.
A voice somewhere says the river Hull has frozen over. I disguise my excitement as a question. ‘Shall we go?’ I ask my friends. For where there is ice, there will inevitably dance ice skaters, and where there are ice skaters, there will be laughter and light-heartedness, and stalls selling hot pastries and spiced wine. We go.
The frozen river brims with action: parkas pirouette, wet dogs give chase and customers line up in queues. The air smells of cinnamon. Everywhere, the snow is on people’s lips: it serves as the icebreaker for every conversation. Nobody stands still as they are talking: they shift their weight from leg to leg, and stamp their feet, wiggle their noses and exaggerate their blinks.
The flakes fall heavier now. They whirl and rustle in the wind. Everyone seems in thrall to the tumbling snowflakes. Human noises evaporate; nobody moves. Nothing is indifferent to its touch. New worlds appear and disappear, leaving their prints upon our imagination. Snow comes to earth and forms snow lamp posts, snow trees, snow cars, snowmen.
What would it be like, a world without snow? I cannot imagine such a place. It would be like a world devoid of numbers. Every snowflake, unique as every number, tells us something about complexity. Perhaps that is why we will never tire of its wonder.
Invisible Cities
‘We wish to see ourselves translated into stone and plants, we want to take walks in ourselves when we stroll around these buildings and gardens.’ This, says Nietzsche, is the purpose of the city: to create space and structure in which a person might think. Ostentatious church buildings, he complained, inhibited free thought. He argued for the ideal of a ‘wide’ and ‘expansive’ city, expansive in every sense of the word.
I remember these words every time I fly to New York City, a place where tall buildings aspire to the sky. Long shadows, the shape of skyscrapers, alight on yellow taxis and hotdog stands. The city’s buildings are home to eight million human beings. Among them number some of the most creative people in the world. The people come here from every land, and from every language, for what reason? Quite possibly, they come here in order to think.
But New Yorkers, like the rest of us, do not pay much attention to their surroundings, how the city incites and informs their thoughts. There are exceptions, of course, and I am not only talking about those who are newly arrived. I am talking about mathematicians, who are tourists in every place. What with its towering buildings, and its grid’s rectilinear streets, and the intersections named after numerals (Ninety-Third and Fifth Avenue), NYC was made for mathematicians.
Planning a city, or dreaming about one, invites us to think by numbers, to borrow some of the mathematicians’ delight. Architects of cities and of individual buildings divide and categorise the air. In this portion: morning traffic, in that section: jogging in the park. Up on this level: office computers, beneath it: a parking lot. The designers translate numbers into symmetry, into shape and order and liveable form. Cities are the embodiments of numerical patterns that contain and direct our lives. But all cities are invisible to start with.
Before New York the city, there was New York the idea: a mere glint in the European settlers’ eye. They christened the woodlands and rivers and the trails of native clans that they discovered, Nieuw Amsterdam. Many years later, following the War of Independence, the nascent settlement would serve briefly as the fledgling Union’s capital. Intangible visions could now be rendered concrete.
A commission, formed in 1811, found in favour of a plan for the mass building of ‘straight-sided and right-angled houses’. Avenues were laid, precisely one hundred feet wide, and numbered (beginning with the easternmost) from one to twelve. Running at right angles to the avenues were passages turned into symmetrical streets (sixty feet in width), each allotted a consecutive number between one and one hundred and fifty-five. Street names acted like compass points, shepherding strangers and the easily-lost to their destination. The rigid geometry imposed order, efficient commerce and cleanliness, but it also obliterated many of Manhattan Island’s natural spaces. In the words of one commissioner, the grid system became ‘the day-dawn of our empire’.
New York City is an exception, though. Not all cities find their territory; many remain forever orphans, existing only in their inventor’s dreams. I would like to sketch a brief history of these invisible cities.
Plato, in his Laws, gives a recipe for the ideal city. Like any recipe maker, he puts great store by the precision with which he delineates its ingredients. At various points in the text, he insists rather heavily on a particular number. No margin exists for approximation in the Platonic design. Neither is there any room for discussion, since for Plato, the self-evident quality of his city is ‘as plain as the fact that Crete is an island’.
By his laws, Plato really intended limits. Without a city, he argues, man would dwell in a ‘fearful, illimitable desert’. Such a man would know nothing of art or science; worse, he would hardly know himself.
But too large a city would be no help either. A city’s limits should thus be carefully demarcated, set neither too big nor too small, so that its citizens might, with time and sufficient effort, be capable of putting a name to every face. This, in Plato’s judgment, would prevent the blight of war, which had struck down so many great cities of the past. He quotes with approval the poet Hesiod’s praise of moderation: ‘the half is often greater than the whole’.
Starting from the principle that ‘numbers in their divisions and complexities are useful for everything,’ Plato proposes limiting his ideal city to exactly 5,040 landholding families. Why 5,040? It is what mathematicians call a ‘highly composite number’, meaning that it can be divided in multiple ways. In fact, no fewer than sixty numbers divide into it, including every number from one to ten.
Twelve can also divide evenly into 5,040. Plato divides the total number of households into twelve tribes, each therefore consisting of 420 families. While interdependent, each tribe is conceived as being fixed and self-sufficient, like the months in a solar year.
Using highly composite numbers facilitates the subdivision of land and property among the citizens. Each family, in each tribe, receives an equal lot of land, beginning from the city centre and radiating out to the countryside. In this way the city fairly distributes the fertility of its soil: half of each lot shall contain the city’s richest ground, while the other half shall consist of the rockiest.
For modern statisticians, Plato’s ideal number of 5,040 families intrigues. They calculate that such a population would require 164 (or 165) births per year to sustain itself. Following the ancient Greek logic that treated men as the head of every household, they also calculate the city’s annual number of potential fathers to be 1,193. Plato believed that one marriage in seven would be fruitful per year, suggesting an anticipated annual birth rate of 170 – which corresponds almost exactly to our statisticians’ calculations.
How does Plato count on keeping his ideal city’s number of households in check? He proposed that each inheritance should pass into the hands of a single ‘best loved’ male heir. Any remaining sons would be distributed among the childless citizens, while the daughters should be married off.
Large families had no place in Plato’s city. Fecundity would be illegal: any couple producing ‘too many’ ch
ildren was to be rebuked. The city’s precise limit of 5,040 households was inviolable: all surplus members were to be sent packing.
Plato imagined that his limits would ensure equality and security for every citizen. In his bucolic vision, men and women would ‘feed on barley and wheat, baking the wheat and kneading the flour, making noble puddings and loaves; these they will serve up on a mat of reeds or clean leaves; themselves reclining the while upon beds of yew and myrtle boughs. And they and their children will feast, drinking of the wine which they have made, wearing garlands on their heads, and having the praises of the gods on their lips, living in sweet society, and having a care that their families do not exceed their means; for they will have an eye to poverty or war.’
Perhaps. But it is also conceivable that Plato’s city would have encouraged just the kind of petty-mindedness among its citizens that exact calculation often fosters.
During the Renaissance, when Plato and his ideas were rediscovered by humanist scholars, we find an Italian architect similarly moved to envision his own perfect city. His name was Antonio di Pietro Averlino, though he is better known today by the Greek name Filarete (meaning ‘lover of virtue’). Unlike Plato, Filarete was an architect, albeit one with a rich and complicated past: he had once been arrested and barred from working in Rome for allegedly stealing the head of Saint John the Baptist.
Filarete described his city Sforzinda (the name a flattery aimed at his patron Francesco Sforza of Milan) at some length in his Trattato di architettura. Its thick symmetrical outer walls formed an eight-pointed star. Though attractive, the unusual choice of shape was also intended to be defensive: invaders mounting its angles would find themselves exposed on multiple sides.
Like spokes in a vast wheel, eight straight roads led from the walls to the city centre. The roads were studded with small piazzas, surrounded by shops and markets. A visitor treading his path downtown from the city gates would pass pyramids of apples, and stacks of loaves, and multi-coloured garments spilling over tables. Merchants, their eyes enlarged by expectation, shouting: ‘Signore, signore!’ At last the city centre appears. Three vast interconnecting piazzas greet him. Here the market noises fade before the imposing ducal palace to his left, and a massive cathedral on his right. In between the palace and the cathedral, on the main piazza, stands another lofty building, ten storeys high.
Thinking in Numbers: How Maths Illuminates Our Lives Page 8
