Thinking in Numbers: How Maths Illuminates Our Lives (20 page)

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

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 (14285
7
) reappears in the fourth position of the second answer (285
7
12), and in the fifth position in the third (4285
7
1), 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 blank 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.

One possible (if partial) explanation for this structure is the marked Japanese preference for odd numbers. In the annual
Shichigosan
(Seven-Five-Three) festival, three-year-old children of both sexes, five-year-old boys and seven-year-old girls visit shrines to celebrate their growth. Cheer groups at a sports match clap in three-three-seven beats. Even numbers, meanwhile, are virtual bogeymen. The number two represents parting and separation, while four is associated with death. In several phrases the number six translates roughly as ‘good-for-nothing’.