Thinking in Numbers: How Maths Illuminates Our Lives (16 page)

The people are all different. They have various motives for being here, and various goals. A teenager finds in the hall a hideout from his Sunday boredom; a manual labourer, having donated the equivalent of a fag packet in salary, sticks around to get his money’s worth; an American tourist in shorts and a Mickey Mouse hat cannot wait to recount the spectacle to her family.

An hour passes, and then another.

‘Zero, five, seven, seven, seven, seven, five, six, zero, six, eight, eight, eight, seven, six . . .’

I head further and further inside the number, exhaling effort, rhythm and precision with every breath. The decimals exhibit a kind of deep order. Fives never outstrip sixes for long, nor do the eights and nines lord it over the ones and the twos. No digit predominates except for brief and intermittent instants. Every digit, in the end, has more or less equal representation. Every digit contributes equally to the whole.

Halfway through the recitation, more than ten thousand decimals in, I stop to stretch. I push back the chair, stand and shake out my limbs. The mathematicians put down their sharp pencils and wait. I bring a bottle to my lips and swallow the plastic bottle-tasting water. I eat a banana. I fold my legs, resume my position at the desk, and continue.

The silence in the hall is total. It reigns like a tsar. When a young woman’s mobile telephone suddenly starts up, she finds herself promptly ejected.

Despite such rare commotions, a sly complicity establishes itself between the public and me. This complicity marks a vital shift. At the beginning, the men and women beamed confidence, listening expectantly and taking pleasure in the digits’ familiar sounds as those shoe sizes, historical dates and car registration plates reached their ears. But, slowly, imperceptibly, something changed. Consternation grew. They could not follow the rhythm of my voice, they realised, without making continuous minor adjustments. Sometimes, for example, I recited the digits fast, and sometimes I recited them slow. Occasionally, I recited in short bursts interleaved with pauses; at other moments, I recited the digits in a long, unbroken phrase. Sometimes the digits sounded thinly, accented by some inner agitation in my voice; instants later, they would soften to a clear and undulating beat.

Consternation now turns increasingly to curiosity. More and more, I feel the timing of their breathing coincide with mine. I sense their raw intrigue at the sound, and sweep, of every digit as it passes and makes way for the next. When the digits darken in my mouth – heavy eights and nines packed together – the tense distant faces grow tenser still. When a sudden three emerges from a series of zeroes and sevens, I hear something like a faint collective pant. Silent nods greet my accelerations; warm smiles welcome my slowdowns.

Between the moments when I stop reciting to sip water or take a bite, and continue reciting, I hardly know where to look. My solitude is absolute; I do not want to return the people’s stares. I look down at the bones and veins in my hands, and at the scuffs in the wooden desk on which they rest. I notice the glimmers of shiny metal that dapple the display cases. On a cheek, here and there, I cannot help noticing, tears.

Perhaps the experience has taken the audience by surprise. No one has told them that they will find the number tangible, moving. Yet they succumb gladly to its flow.

I am not the first person to recite the number pi in public. I know there are a few ‘number artists’ – men who recount numbers as actors recall their scripts. Japan is the centre of this tiny community. In Japanese, spoken digits can sound like whole sentences; pronounced a certain way, the opening digits of pi, 3.14159265, mean ‘An obstetrician goes to a foreign country’. The digits 4649 (which occur in pi after 1,158 decimal places) sound just like ‘nice to meet you’, while a Japanese speaker pronouncing the digits 3923 (which occur in pi after 14,194 decimal places) simultaneously says, ‘Thank you, brother.’

Of course, such verbal constructions always suffer from arbitrariness. The short, stiff phrases stand apart, with only the speaker’s ingenuity to splice them together. Japanese spectators, I have heard, watch these men perform as they might watch a tightrope walker; listening only in case of a blunder, as others watch only in case of a fall.

The relationship these artists have with numbers is complicated. Many years of repetitive learning hone their technique; but they also produce a nagging feeling of duplicity: repeated numbers (and words) often finish by losing all their sense. It is not uncommon, after each public display, for the performer to impose a months-long fast of every digit. Benumbed by numbers, even a price tag, a barcode, an address sickens him.

In the number artist’s brain, pi can be reduced to a series of phrases. In my mind, it is I, not the number, who grows small. Before the mystery of pi, I diminish myself as much as possible. Emptying myself, I perceive every digit up close. I do not wish to fragment the number; I am not interested in breaking it up. I am interested in the dialogue between its digits; in the unity and continuity that underlie them all.

A bell cannot tell time, but it can be moved in just such a way as to say twelve o’clock – similarly, a man cannot calculate infinite numbers, but he can be moved in just such a way as to say pi.

‘Three, one, two, one, two, three, two, two, three, three, one . . .’

Reciting, I try to summon up a true picture of what I see and feel. I want to convey the shapes, and colours, and emotions that I experience, to everyone in the hall. I share my solitude with those who watch and listen to me. There is intimacy in my words.

A third hour comes and goes; the recitation enters its fourth hour.

More than sixteen thousand decimal places have escaped my lips. Their swelling company presses me on. But exhaustion also grows within my body, and all of a sudden my mind goes blank. I feel the blood falling out of my head. Up until only a few moments ago the digits had accompanied me; now they make themselves scarce.

In my mind’s eye, ten identical-looking paths stretch out before me; each path leading on to ten more. One hundred, a thousand, ten thousand, one hundred thousand, a million paths beckon me out of the impasse. They stream in every possible direction. Which way to go? I have no idea.

But I do not panic. What good did panicking ever do anyone? I shut my eyes tight, and coaxingly rub the skin around my temples. I take a deep breath.

Green-tinted blackness pervades my mind. I feel disorientated, lost. A filmy white surfaces over the black, only to be recovered by a rolling grey-purple. The colours bulge and vibrate but resemble nothing.

How long did these maddening misty colours last? Seconds, but they each seemed agonisingly longer.

The seconds pass indifferently; I have no choice but to endure them. If I lose my cool, all is finished. If I call out, the clock comes to a halt. If I do not give the next digit, in the next few moments, my time will be up.

No wonder this next digit, when finally I release it, tastes even sweeter than the rest. This digit requires all my force and all my faith to extract it. The mist in my head lifts, and I open my eyes. I can see again.

The digits flow fleet and sure, and I regain my composure. I wonder if anyone in the hall noticed a thing.

‘Nine, nine, nine, nine, two, one, two, eight, five, nine, nine, nine, nine, nine, three, nine, nine . . .’

Quickly, quickly, I must keep going. I must not let up. I cannot linger, not even before the most outstanding glimpses of the number’s beauty; the joy I feel is subordinate to the need to reach my goal and recite the final digit in my mind. I must not disappoint all who are standing here, watching me and listening to me, waiting for me to bring the recitation to its fitting conclusion. All the preceding thousands have no value in themselves: only once I have wrapped everything up can they successfully count.

Five hours have now elapsed. My speech begins to slur; I have got drunk on exhaustion. The end, however, is in sight. The end generates fear: am I up to it? What if I fall short? Tension stirs me for this culminating burst.

And then, minutes later, I say, ‘Six, seven, six, five, seven, four, eight, six, nine, five, three, five, eight, seven,’ and it is over. There is nothing more to say. I have finished recounting my solitude. It is enough.

Palms come together; hands clap. Someone lets out a cheer. ‘A new record,’ someone else says: 22,514 decimal places. ‘Congratulations.’

I take a bow.

For five hours and nine minutes, eternity visited a museum in Oxford.

Einstein’s Equations

Speaking about his father, Hans Albert Einstein once said, ‘He had a character more like that of an artist than of a scientist as we usually think of them. For instance, the highest praise for a good theory or a good piece of work was not that it was correct nor that it was exact but that it was beautiful.’ Numerous other acquaintances also remarked on Einstein’s belief in the primacy of the aesthetic, including the physicist Hermann Bondi, who once showed him some of his work in unified field theory. ‘Oh, how ugly,’ Einstein replied.

It is a mostly thankless task to try to assign to mathematicians some universal trait. Einstein’s famous predilection for beauty offers one rare exception. Mathematicians can be tall or short, worldly or remote, bookworms or book-burners, multilingual or monosyllabic, tone-deaf or musically gifted, devout or irreligious, hermit or activist, but virtually all would agree with the Hungarian mathematician Paul Erdos when he said, ‘I know numbers are beautiful. If they are not beautiful, nothing is.’

Einstein was a physicist, yet his equations inspired the interest and admiration of many mathematicians. His theory of relativity drew their praise for combining great elegance with economy. In a handful of succinct formulas, every symbol and every number obtaining its perfect weight, Newtonian time and space were recast.

Books on popular mathematics abound with discursive explanations of technical proofs to illustrate their beauty. I cannot help but wonder if this might not be a mistake. I suspect that, more usually, what we laymen really admire in the work of a Euclid or an Einstein is its ingenuity, rather than its beauty. We are impressed, and yet unmoved by them.

The barrier to an appreciation of mathematical beauty is not insurmountable, however. I would like to suggest a more indirect approach. At a remove from the technical acumen of a theorist, my suggestion is more intuitive. The beauty adored by mathematicians can be pursued through the everyday: through games, and music, and magic.

Take the game of cricket, which was the frequent inspiration for G.H. Hardy, a major number theorist and the author of
A Mathematician’s Apology
, who scoured the newspaper for cricket scores over breakfast every morning. In the afternoon, after some hours at his desk, he would furl his theorems and transport them in his pockets (in case of rain) to see a local match. Among his papers he sketched the following cricket ‘dream team’:

Hobbs

Archimedes

Shakespeare

M Angelo

Napoleon (Capt)

H Ford

Plato

Beethoven

Johnson (Jack)

Christ (J)

Cleopatra

Cricket matches presented Hardy the spectator with the same ‘useless beauty’ that he so cherished in his theorems. By useless, he meant only that neither had any goal beyond the pursuit for its own sake. He would also frequently stand at the stumps himself, surrounded by the other team’s fielders, watching the red ball expand as it flew towards his bat. Both experiences seemed to stimulate his mathematical antennae for order, pattern and proportion.

At its best, a well-executed, smooth-flowing cricket match can replicate the sense of harmony that we most often associate with music. The tension mounts and falls tidally, like the notes in a song. Time elapses differently on a cricket ground or in a concert hall. A five-day match is adept at slackening and pulling tight the outline of its hours, while every musical composition bears its own time within the structure of its notes. The unique tempo is also a part of the experience of mathematical beauty.

Gottfried Leibniz wrote that music’s pleasure consisted of ‘unconscious counting’ or an ‘arithmetical exercise of which we are unaware.’ The great philosopher-mathematician meant, I suppose, that the numerical ratios that underlie all music are grasped intuitively by our minds. Every instant the listener mentally resolves the relation between the various notes – the fourths and fifths and octaves – as though they were objects all laid out before him side by side in some gigantic illustration. This ‘grasping’ of the music – however fleeting and transient – is something we can all experience as beautiful.

On the relationship between musical and mathematical beauty, we can learn more from writings about the ancient Greek philosopher-mathematician-mystic, Pythagoras. It is said that he possessed a musician’s ear. From boyhood he showed a flair for the lyre. Perhaps he first heard its seven strings played by a travelling
citharede
, a female performer dressed in long curls and bright colours; the
citharedes
were the divas of their day.

Pythagoras discovered that the most harmonious notes result from the ratios of whole numbers. A vibrating string exactly halved or doubled, for example, produces an octave (ratio 1:2 or 2:1). If precisely one-third of the string is held down, or when the string is tripled in length, a perfect fifth (an octave higher) results. A perfect fourth can be obtained by holding down one quarter of the string, or stretching it out four times longer. The whole harmonic scale was constructed in this way. Pythagoras observed that all of music depended on the first four numbers and their interrelations. Ten he worshipped as the most perfect number, reflecting the unity of all things, it being the sum of one and two and three and four.