Thinking in Numbers: How Maths Illuminates Our Lives (11 page)

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’ children 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.

What is this strange building to which every street in the city addresses itself? Filarete called it the ‘House of Vice and Virtue’. Every floor housed a different class of activity. A brothel, on the ground floor, would entertain the majority of the building’s callers. Alcoholic drinks and games could be had on the floors immediately above. Ascending further, a university and lecture halls offered its few visitors instruction. An observatory topped them all.

The homes to which the citizens would return after their day’s work or play had been just as intricately imagined. Filarete planned his houses according to the resident’s social rank: the artisans’ quarters taking up far less space than the houses of the city’s merchants or gentlemen. In comparison with his artist and painter neighbours the architect’s own house would be twice as spacious.

Filarete’s plans are long, his writing spidery. Spread across its twenty-five volumes his treatise contains an entire city in waiting. But not long after its completion, in 1466, the Duke Sforza died and Filarete’s vision survived only on paper.

Plato’s ideal limits and Filarete’s volumes sought out new dreamers. They passed from mind to mind, and from century to century. It should perhaps come as no surprise that they would go on to inspire the grandest and most ambitious city plans, those drawn up in the United States of America.

King Camp Gillette, who would later invent the safety razor, once dreamed of an immense city he called Metropolis. In 1894, he published a short illustrated book aimed at its promotion. The city, Gillette wrote, would be ‘situated in the vicinity of Niagara Falls, extending east into New York State and west into Ontario’. It would take the shape of a rectangle, sixty miles long and thirty miles wide. Gillette considered its construction, ‘In the light of a machine, or rather a part of the machine of production and distribution; and, as such, the objects to be attained must be known and understood. It must have no unnecessary parts to cause friction or demand unnecessary labour, and yet it must combine within itself all the necessary parts which will contribute to the happiness and comfort of all.’

Sixty million inhabitants would live in circular skyscrapers, six hundred feet in diameter, ‘upon a scale of magnificence such as no civilisation has ever known.’ A beehive-like distribution of apartments throughout the city would lend space enough between the buildings for wide avenues and parks. No citizen, it also ensured, would reside any nearer or farther away from a school, shop, or theatre than any other.

Elevators, a still recent invention in Gillette’s day, had gradually shifted city design from the horizontal toward the vertical. But Metropolis took the idea of a vertical city to a whole other level. Its skyscrapers would be truly colossal, attaining a height of twenty-five storeys. Habitable monoliths, lots and lots of them, shining with glass and progress, smooth and steel-coloured, a monotony of monuments.

The city’s highly regular order appears again in miniature in Gillette’s plan for every home. For Gillette, as for Filarete, the home is a tiny city. The interior would be completely symmetrical, with parallel sitting rooms, bedrooms, and bathrooms on either side. And in each room, the windows would be so arranged as to make it impossible to look out from one apartment into a neighbour’s.

Conscious of the artificiality of his vision (even the hexagonal lawns that surrounded each building would be composed of artificial grass), Gillette proposed distributing thousands of public gardens filled with trees and ‘urns of flowers’ at regular intervals throughout the city. Being constructed with complete regularity did not mean that it would suffer from sameness, he insisted. Looking out from his window, the citizen’s roving eye would encounter ‘a continuous and perfectly finished facade from every point of view, each building and avenue surrounded and bordered by an ever changing beauty in flowers and foliage.’