Thinking in Numbers: How Maths Illuminates Our Lives Page 12
A painter exhibits his artwork. What was I to do? After three months of preparation, I took the number to a museum, the sprawling digits tucked inside my head. My aim: to set a European Record for the recitation of pi to the greatest number of digits.
March is the month of spring showers, and school holidays, and spick and span windows. It is also the month when people the world over celebrate ‘Pi Day’, on 14 March. So on that day, in 2004, I travelled north from London to the city of Oxford. Members of staff at the university’s Museum for the History of Science were waiting for me. Journalists too. An article in The Times, complete with my photo, announced the upcoming recitation.
The museum lies in the city centre, in the world’s most ancient surviving purpose-built museum building, the Old Ashmolean. Iconic stone heads, wearing stone beards, peer down at visitors as they pass through the gates. The walls are thick, the colour of sand. Approaching the building, a snap of photographers appear as if from nowhere, holding cameras, like masks, up to their faces. The piercing flashes momentarily petrify my expression. I stop and raise my features into a smile. A minute later they are gone.
The record attempt’s organisers have occupied the museum building. Television camera wires snake the length of the floor. Posters requesting donations (the event is raising money for an epilepsy charity, at my request, since I suffered from seizures as a young child) dress the walls. A table and chair, I see on entering, have already been set out for me on one side of the hall. Before it, a longer table awaits the mathematicians who will verify my accuracy. But there is still an hour before the recitation is due to start, and I find only a trio of men talking together. One has a full head of wiry hair, one has a multi-coloured tie, and one has neither hair nor tie. The third steps briskly forward and introduces himself as the main organiser. I shake hands with the museum’s curator and his assistant. Their faces show mild puzzlement, curiosity and nerves. Shortly afterwards, reporters arrive to hold the microphones and man the television cameras. They film the display cases containing astrolabes, compasses and mathematical manuscripts.
Someone asks about the blackboard that hangs high on the wall opposite us. Albert Einstein used it during a lecture, the curator explains, on 16 May 1931. What about the chalky equations? They show the physicist’s calculations for the age of the universe, replies the curator. According to Einstein, the universe is about ten, or perhaps one hundred, thousand million years old.
Footfalls increase on the museum’s stone steps as the hour approaches. The mathematicians duly arrive, seven strong, and take their seats. Men, women and children keep coming; it is soon standing room only. The air in the hall grows thick with hushed talk.
At last, the organiser calls everyone to silence. All eyes are on me; nobody moves. I sip a mouthful of water and hear my voice begin. ‘Three point one four one five nine two six five three five eight nine seven nine three two three eight four . . .’
My audience are only the second or third generation able to hear the number pi beyond the first few tens or hundreds of decimal places. For millennia it existed only in a breathful of digits. Archimedes knew pi to only three correct places; Newton, almost twenty centuries later, managed sixteen. Only in 1949 did computer scientists discover pi’s thousandth digit (following the decimal point): nine.
It takes about ten minutes, at a rate of one or two digits per second, for me to reach this nine. I do not know how long exactly; an electronic clock records the seconds, minutes, hours of the recitation for the public to watch, but I cannot see it from my chair. I stop reciting to sip water and catch my breath. The pause feels palpable. Dolorous even. I feel completely, oppressively alone.
The rules for the recitation are strict. I cannot step away from the desk, except to use the bathroom, and then always accompanied by a member of the museum’s staff. No one may talk to me, not even to cheer me on. I can stop reciting momentarily to eat fruit or a piece of chocolate, or drink, but only at pre-agreed intervals a thousand digits apart. Cameras record my every sound and gesture.
‘Three eight zero nine five two five seven two zero one zero six five four . . .’
An occasional cough or sneeze from the audience punctuates the flow of digits. I do not mind. I meditate on the colours and shapes and textures of my inner landscape. Calmness gains on me; my anxiety falls away.
Most of the spectators know nothing of Archimedes’s polygons, have no idea that the ten digits they have just heard will eventually repeat an infinite number of times, have never thought of themselves as being in any way susceptible to maths. But they listen attentively. The concentration in my voice seems to communicate itself to them. Faces, young and old, round and oval, all wear delicate frowns. Listening to the digits, they hear their dress sizes, their birthdays, their computer pass codes. They hear excerpts – both shorter and longer – from a friend’s, or parent’s, or lover’s telephone number. Some lean forward in expectation. Patterns coalesce, and as quickly disperse, in their minds.
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 dap
ple 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)
&nbs
p; 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.