Thinking in Numbers: How Maths Illuminates Our Lives

by Daniel Tammet

  Of course, many of these potential Lolitas would simply not have been viable. And yet among the bewildering, nonsensical or ham-fisted editions, readable alternatives must exist. How many? A hundred? A thousand? A million? More. Many more. Publishers could produce enough of them to give every reader on the planet his or her very own Lolita. In one, the famous opening couplet, ‘Lolita, light of my life, fire of my loins. My sin, my soul’ would appear halfway down page thirty-nine (perhaps replaced with a line that Nabokov placed in his chapter two: ‘My very photogenic mother died in a freak accident (picnic, lightning) . . .’). In another reader’s edition, the couplet shows up at the top of page 117. This Lolita begins instead, ‘I saw her face in the sky, strangely distinct, as if it emitted a faint radiance of its own.’ In a third version, the original couplet serves as the story’s closing lines.

  For all I know, some of these incalculably many editions were actually published, each with their subtle yet striking alterations. Perhaps this would explain why the Atlantic Monthly’s reviewer called the book ‘one of the funniest serious novels I have ever read’, the Los Angeles Times declared it ‘a small masterpiece . . . an almost perfect comic novel’, and the New York Times Book Review announced, ‘technically it is brilliant . . . humor in a major key’, whereas Kingsley Amis read a book leading to ‘dullness, fatuity, and unreality’ and Orville Prescott, writing for the New York Times, found the story ‘dull, dull, dull.’

  Which Lolita did they read?

  It is the writer and reader together who compose their infinite tale. The Argentinian writer Julio Cortázar created a novel in which he made this principle explicit. Rayuela (Hopscotch) was published fifty years ago, not long after Lolita. It contains 155 chapters (over some 550 pages), which can be read in two distinct ways. Either the reader starts at chapter one and continues reading linearly until the end of chapter fifty-six (the chapters and 200 pages that remain being considered ‘expendable’), or else he begins at chapter seventy-three, then turns back to chapter one, continues to chapter two before jumping forward to chapter 116, then back to chapter three, forward to chapter eighty-four, and so on back and forth between the chapters according to a ‘Table of Instructions’ at the front of the book.

  In one of his ‘expendable’ chapters, Cortázar describes the book’s goal.

  It would seem that the usual novel misses its mark because it limits the reader to its own ambit; the better defined it is, the better the novelist is thought to be. An unavoidable detention in the varying degrees of the dramatic, the psychological, the tragic, the satirical, or the political. To attempt on the other hand a text that would not clutch the reader but which would oblige him to become an accomplice as it whispers to him underneath the conventional exposition other more esoteric directions.

  As Cortázar’s accomplice, we follow the novel’s hero – an Argentine bohemian – through the streets of Paris as he contemplates his life and its inexhaustible potential paths. We begin the book either at chapter one, like this: ‘Would I find La Maga?’ Or else at chapter seventy-three, on page 383: ‘Yes, but who will cure us of the dull fire, the colourless fire that at nightfall runs along the Rue de la Huchette . . .’

  Turning the pages, reading different stories. For instance, the reader who begins at chapter one will soon reach this line in the fourth chapter: ‘[She] picked up a leaf from the edge of the sidewalk and spoke to it for a while.’ For the other reader, however, ‘chapter four’ is really the seventh chapter of the story, preceded by the sixth chapter, which is labelled ‘chapter eighty-four’. In this chapter, on page 405, he reads, ‘I keep on thinking of all the leaves I will not see, the gatherer of dry leaves, about so many things that there must be in the air and which these eyes will not see . . . there must be leaves that I will never see.’ These lines enrich the second reader’s understanding of the woman who, several pages later, on page twenty-five, will pick up a leaf from the edge of the sidewalk and speak to it.

  A consequence of reading in this way is disorientation; the leapfrogging reader lacks any sense of having completed the book. He reads the final lines on the final page of the physical book long before he comes to any conclusion of the story. Arriving later at the one hundred and fifty-third chapter (labelled ‘chapter 131’), he proceeds to the following chapter (labelled ‘chapter fifty-eight’), only to discover that he should return again to chapter 131. An interminable loop between the two ‘final’ chapters appears. What is more, assuming he has kept count, the reader notices that the chapters – read in this order – total 154. One of the chapters – ‘chapter fifty-five’ – is absent from the list.

  Hopscotch’s structure demands that readers make their own sense of the story. One might decide to read the chapters consecutively, but in descending order, starting at chapter 155. Another decides to read all the even chapters before the odd: two, four, six, eight . . . one, three, five, seven . . . A third does the same, but in reverse, reading all the odd chapters before the even. A fourth reads only the prime numbered chapters: two, three, five, seven, eleven, thirteen, seventeen, nineteen, twenty-three, twenty-nine, thirty-one . . . finishing at chapter 151 (thirty-six chapters in all). A fifth begins at the first chapter, then reads the third (1 + 2), turning next to the sixth (1 + 2 + 3), followed by the tenth (1 + 2 + 3 + 4) and so on.

  Just when the plucky reader has attained the end of one story, another story beckons him to pick up its pages and start again. The book of ascending chapters becomes a book of descending chapters. The book of odd number chapters becomes a book of even number chapters. Every reading differs; every reading offers something new. It is impossible to dip into the same book twice.

  I am reminded of Nabokov’s view that we can never read a book: we can only reread it. ‘A good reader, a major reader, an active and creative reader,’ says Nabokov, ‘is a re-reader.’ Initial readings, he explains, are always laborious, a ‘process of learning in terms of space and time what the book is about, this stands between us and artistic appreciation.’

  Think of the countless stories of Chekhov, of the innumerable editions of Lolita and Hopscotch, which lie before every reader’s eyes unnoticed, unloved, unread.

  Flaubert, in a letter to his mistress, wrote, ‘How wise one would be if one knew well only five or six books.’ It seems to me that even this figure is an exaggeration. To learn infinitely many things, we would only ever need perfect knowledge of one book.

  Poetry of the Primes

  Arnaut Daniel, whom Dante praised as ‘il miglior fabbro’ (‘the better craftsman’), sang his love poems in the streets of twelfth-century southern France. Of his life not much is known, but I find it tempting to link a brief and rare report about the troubadour with the sestina form of poetry (six stanzas, each containing six lines, plus a concluding half stanza) that he invented.

  A contemporary, Raimon de Durfort, called Arnaut ‘a scholar undone by dice.’ This alleged acquaintance with gambling suggests a possible influence for the sestina’s form. A die, as everyone knows, has six faces. The throw of a pair of dice creates a range of outcomes amounting to thirty-six, which is the total number of lines in the poem’s six stanzas. So far as I can tell, no one has made this connection between the sestina and the die before – maybe because it is a connection little worth making. I leave it for the reader to decide.

  Unusually, the sestina does not run on rhyme, symbolism, alliteration or any other of the poet’s typical devices. Its power is the power of repetition. The same six words, one at the tail of each line, persist and permute across every stanza (in the final half stanza, the words appear two to a line). The order in which the concluding words for each line rotate is fixed according to an intricate pattern.

  First stanza: 1 2 3 4 5 6

  Second stanza: 6 1 5 2 4 3

  Third stanza: 3 6 4 1 2 5

  Fourth stanza: 5 3 2 6 1 4

  Fifth stanza: 4 5 1 3 6 2

  Sixth stanza: 2 4 6 5 3 1

  Conclusion: 2 1, 4 6, 5 3<
br />
  Which is to say, the final word in line six of the first stanza (1 2 3 4 5 6) reappears as the last word of the next stanza’s opening line (6 1 5 2 4 3), and at the close of the second line of stanza three (3 6 4 1 2 5), and so on. It will be clearer if I give an example, from Dante (here employing the six words ‘shadow’, ‘hills’, ‘grass’, ‘green’, ‘stone’ and ‘woman’).

  I have come, alas, to the great circle of shadow,

  to the short day and to the whitening hills,

  when the colour is all lost from the grass,

  though my desire will not lose its green,

  so rooted is it in this hardest stone,

  that speaks and feels as though it were a woman.

  And likewise this heaven-born woman

  stays frozen, like the snow in shadow,

  and is unmoved, or moved like a stone,

  by the sweet season that warms all the hills,

  and makes them alter from pure white to green,

  so as to clothe them with the flowers and grass.

  When her head wears a crown of grass

  she draws the mind from any other woman,

  because she blends her gold hair with the green

  so well that Amor lingers in their shadow,

  he who fastens me in these low hills,

  more certainly than lime fastens stone.

  Her beauty has more virtue than rare stone.

  The wound she gives cannot be healed with grass,

  since I have travelled, through the plains and hills,

  to find my release from such a woman,

  yet from her light had never a shadow

  thrown on me, by hill, wall, or leaves’ green.

  I have seen her walk all dressed in green,

  so formed she would have sparked love in a stone,

  that love I bear for her very shadow,

  so that I wished her, in those fields of grass,

  as much in love as ever yet was woman,

  closed around by all the highest hills.

  The rivers will flow upwards to the hills

  before this wood, that is so soft and green,

  takes fire, as might ever lovely woman,

  for me, who would choose to sleep on stone,

  all my life, and go eating grass,

  only to gaze at where her clothes cast shadow.

  Whenever the hills cast blackest shadow,

  with her sweet green, the lovely woman

  hides it, as a man hides stone in grass.

  An air of expectancy permeates the text: since the reader knows what is coming, the poem must rise to the challenge of delivering surprise. The sestina plays with meaning, conferring new aspects, in changing contexts, on the same word. A tension between the law of the numerical pattern and the liberty of the author is ever present, ever palpable.

  Artists and mathematicians alike have been drawn to the sestina’s numerous properties. In their wonderful book, Discovering Patterns in Mathematics and Poetry, the mathematician Marcia Birkin and poet Anne C. Coon compare the rotation of words in a sestina to the shifting digits in a cyclic number.

  Cyclic numbers are related to primes. Division using certain prime numbers (such as 7, 17, 19 and 23) produces decimal sequences (the cyclic numbers) that repeat forever. For example, dividing one by seven (1/7) gives the decimal expansion 0.142857142857142857 . . . where the six digits 142857 – the smallest cyclic number – continue round and round in a never-ending ring dance.

  When we now multiply 142,857 by each of the numbers below 7, we see that the answers are permutations of the same six digits.

  1 × 142857 = 142857

  2 × 142857 = 285714

  3 × 142857 = 428571

  4 × 142857 = 571428

  5 × 142857 = 714285

  6 × 142857 = 857142

  In this instance, the digit 7 at the end of the first answer (142857) reappears in the fourth position of the second answer (285712), and in the fifth position in the third (428571), and so on. Each digit rotates through every answer, changing place at every turn, like the end words in the stanzas of a sestina.

  Hazard has no place in the sestina. Its end words in every stanza fall at once into line, the position of each determined before the poem begins. Algebraically, we can describe the sestina’s structure (from the second stanza onwards) like this.

  {n,1, n-1, 2, n-2, 3} where n refers to the number of stanzas (six).

  So, following the first stanza (1 2 3 4 5 6), it is the sixth (or, for the purpose of the formulation, the n-th) terminating word that now ends the second stanza’s first line:

  6 . . . . . . . . . . . . . . . . . .

  Followed by the first word at the end of the second stanza’s line two:

  6 1 . . . . . . . . . . . . . . .

  Then the n-th (sixth) minus one, i.e. the fifth word at the end of line three:

  6 1 5 . . . . . . . . .

  Next the second word closes line four:

  6 1 5 2 . . . . . .

  Now the n-th (sixth) minus two, i.e. the fourth word ends line five:

  6 1 5 2 4 . . .

  Finally the third word concludes line six:

  6 1 5 2 4 3

  The same shifting pattern applies to all the stanzas that follow, so that the third stanza’s first line closes with the n-th (i.e., sixth) terminating word from the second stanza (this time, 3), then the second line ends on the first word (now, 6), then the third line’s last word is the sixth minus one – the fifth – word (4), and so on.

  How our mediaeval troubadour concocted this clever pattern is unknown. His deep familiarity with the rhythms of words and music likely helped. In one of his few surviving songs he says:

  Sweet tweets and cries

  and songs and melodies and trills

  I hear, from the birds that pray in their own language,

  each to its mate, just as we do

  with the friends we are in love with:

  and then I, who love the worthiest,

  must, above all others, write a song contrived so

  as to have no false sound or wrong rhyme.

  Of course, that Arnaut plumped for six stanzas, and not five or seven, probably owes as much to chance as the outcome of the toss of a die. In fact, a small number of poets have tried their hands at tritinas (containing three stanzas) and quintinas (containing five), with some success. Raymond Queneau, a French poet with a mathematician’s itch for generalisation, eager to understand how the pattern works, explored the limits of the form. In the 1960s, he worked out that only certain numbers of stanzas could permute like the words in a sestina. A four-stanza poem, for example, produced jarring alignments of the same word.

  {n, 1, n-1, 2}

  First stanza: 1 2 3 4

  Second stanza: 4 1 3 2

  Third stanza: 2 4 3 1

  Fourth stanza: 1 2 3 4

  The same went for a poem of seven stanzas.

  {n, 1, n-1, 2, n-2, 3, n-3}

  First stanza: 1 2 3 4 5 6 7

  Second stanza: 7 1 6 2 5 3 4

  Third stanza: 4 7 3 1 5 6 2

  Fourth stanza: 2 4 6 7 5 3 1

  Etc . . .

  After much trial and error, Queneau determined that only thirty-one of the numbers smaller than 100 produced the sestina’s patterns. His observation led mathematicians to discover a surprising relationship between the sestina and the primes. Namely, poems containing three or five stanzas behave like the six stanzas in a sestina, because 3 (or 5, or 6) x 2, + 1, always equals a prime number. For the same reason, sestina-like poems of eleven, thirty-six, or ninety-eight stanzas are all possible, but not those containing ten, forty-five, or one hundred.

  Sestinas are not the only form of poetry to be shaped by primes. Brief and glancing, haiku poems also derive their strength from these numbers.

  The Japanese have long been disposed to brevity. Enquiries for the name of ‘Japan’s Shakespeare’ or ‘Japan’s Stendhal’ will be, in the best of cases, greeted with bl
ank stares. Oriental epics fell into complete neglect at about the same time that the Viking Snorri Sturluson was putting the finishing touches to his saga. Courtiers of the Heian period (dating from the eighth to the twelfth centuries, a period which the Japanese consider a high point in their history) worked up the most extended pieces by concatenating dozens of short verses, by many hands. Dignitaries alone, however, had the right to commence these chain poems by inventing their opening three lines (called the ‘hokku’). Among images of romantic love and soul searching, these opening lines always contained a reference to the seasons, and an exclamation such as ‘ya’ (‘!’) or ‘kana’ (how . . . ! what . . . !). However, even this carefully wrought convoy of miniature verses became too cumbersome in the end for Japanese tastes, so that generations of mouths gradually eroded them to the triplet of lines that we know today as ‘haiku’.

  Like the sestina, the haiku uses no rhyme. Its three lines contain five, seven and five onji (syllables), seventeen in all. Three, five and seven are the first odd prime numbers. Seventeen, too, is prime.