Once Upon a Prime, page 3
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Rhyme schemes are among the defining characteristics of poetic forms—sonnets, villanelles, alexandrines, and so on. A villanelle, for example, is a nineteen-line poem consisting of five three-line stanzas with the aba rhyme scheme and a final abaa quatrain. There is additional structure: the first and third lines of the opening stanza repeat, alternately, as the final line of successive stanzas and as the last two lines of the quatrain. Probably the most famous villanelle is “Do not go gentle into that good night,” Dylan Thomas’s wonderful anthem to the human spirit. Sonnets, meanwhile, consist of fourteen lines. There are different traditional rhyme schemes in different languages, but Shakespeare and most other English-speaking writers have used three abab quatrains, followed by a rhyming couplet.
Shakespeare was a prolific poet—the 1609 edition of his collected sonnets contains 154 of them. But this is nothing compared to the French author Raymond Queneau’s Cent mille milliards de poèmes, which uses the mathematics of randomness to fit 100 trillion sonnets into a single book. How is this possible? Let me explain. Everyone loves a sonnet, but my editor would kill me if I wanted to include 100 trillion of them in this book, so I decided to prolong my life by giving a smaller example to set the scene. To that end, I’ve deployed my amazing poetry skills to write some limericks for you instead.
Limericks are short, usually humorous poems consisting of five lines with the rhyme scheme aabba, popularized in England in the nineteenth century by the Victorian writer Edward Lear. Here’s a typical example from his bestselling 1861 Book of Nonsense:
There was an Old Lady whose folly,
Induced her to sit on a holly;
Whereon by a thorn,
Her dress being torn,
She quickly became melancholy.
Lear is sometimes called the Father of Limericks, although he didn’t use the term “limerick” himself (it’s first recorded in 1898) or even invent them. However, with his much-loved books he certainly popularized the form, writing an impressive 212 limericks along the way. It’s rather unclear how they ended up being named for an Irish county. One theory is that the name arose from a particularly popular example (not one of Lear’s) that featured the line “Will (or won’t) you come to Limerick?”
With the amazing power of randomness, I hereby scoff at 212 limericks and present a means to writing many more with a minimum of effort and artistic ability. Here are two not very good limericks (shown on the left and on the right, below) that I’ve invented to show you the method:
There once was a woman called Jane
There once was a person from Maine
Who constantly traveled by train
Who never went out in the rain
When going abroad
Damp days left her bored
She couldn’t afford
Oh how she adored
A wonderful journey by plane
A week in the sunshine in Spain
From these two starting points, you can construct many more limericks. You do it by randomly picking lines from the two choices you have at each point. You can, for example, toss a coin to determine each line. If it’s heads, you read the left-hand line; if tails, the line on the right. Brilliantly, there is a website, justflipacoin.com, that allows you to do this even without taking the trouble to find a physical coin. I tried it just now and got heads, tails, tails, heads, tails. So my new limerick reads:
There once was a woman called Jane
Who never went out in the rain
Damp days left her bored
She couldn’t afford
A week in the sunshine in Spain
Since the poem has to “work” whichever option you pick for each line, if you want to try doing something like this, you need to understand the structure of the poem. As I noted already, the limerick has the rhyme structure aabba, so you need three a rhymes in each limerick. That means for two limericks you’ll need six a rhymes. In this toy example, I chose “Jane,” “train,” “Maine,” “rain,” “plane,” and “Spain.” If you wanted a third limerick you could weave in words like “drain,” “pain,” “complain,” “feign,” “rein,” and so on.
Our little poem set of two limericks has two choices for each of the five lines. There are two possible first lines. Each of these can be followed by two possible second lines. This means we have possibilities for the first two lines. Each of these can in turn be followed by two options for line 3, giving possibilities for the first three lines. At each stage, the number of possible poems doubles. With our five lines to choose, we end up with a total of bona fide limericks. But if we wrote just one more limerick, we’d have three choices for each line, meaning a total of
limericks. Here’s a third limerick for your delectation:
There once was a girl from Bahrain
Who viewed snow and hail with disdain
The cold she abhorred
She cheered when she scored
A trip to the African plain
Congratulations, you are now the proud owner of thirty-one more limericks than are contained in the entire oeuvre of Edward Lear. If you can add a fourth limerick to this set, then the total number will leap to which is 1,024, and since I wrote only 243 of these, you are morally entitled to more than 75 percent of the worldwide fame that will surely result from the composition of over a thousand limericks.
We can now see just how Raymond Queneau managed to construct his 100 trillion poems. It’s exactly the same principle, just on a bigger scale. The poems are sonnets, so they have 14 lines. Queneau chose the rhyme scheme abab abab ccd eed. (Translations into English have tended to use the Shakespearean abab cdcd efef gg.) Cent mille milliards de poèmes consists of ten sonnets, printed on ten consecutive sheets. All the first lines rhyme with each other, all the second lines rhyme with each other, and so on. In effect, the ten sonnets line up to create a three-dimensional poem. This means, for instance, that of the 140 total lines, 40 of them, 4 in each poem, must end with rhyme a. Sonnets can then be made by choosing any of 10 possible lines at each point. So I might choose line 1 from poem 3, line 2 from poem 1, line 3 from poem 4, and so on. If I continued selecting the poem numbers by following the digits of nobody could stop me from then saying I have produced a em (sorry).
How many poems are contained in this little book, then? Well, the number of possible first lines is 10. Each can be followed by any one of ten second lines, giving possibilities for the first two lines. With fourteen lines altogether, the total number of possibilities is ten multiplied by itself fourteen times, or 100,000,000,000,000. In other words, 100 trillion. Is this the longest book ever written? If you read a different sonnet every minute without stopping, it would take 190,128,527 years to read them all. (Raymond Queneau did this calculation too, but he arrived at an answer of 190,258,751 years, which made me doubt my arithmetic skills. But a quick check shows that his is the answer you get if you read one sonnet per minute but forget about leap years. Perhaps Queneau was very generously allowing his readers to take a day of rest on February 29.) A philosopher might ask: Did Queneau write all these poems? In what sense do they exist at all? I don’t know, but Queneau was a member of a group of writers and poets experimenting with what they termed “potential literature.” This group was known as the Oulipo—I’ll be showing you more of their work and ideas later. But a book of 100 trillion poems is certainly an excellent example of potential literature.
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The mathematics of poetry does not stop with rhyme schemes; wherever there is structure, there is mathematics, and rhyme schemes are just one way to impose structure. If we abandon rhyme, then something else needs to take its place. One possibility that dates back to medieval times is the sestina, and I want to talk about this form in particular because its elegant structure works thanks only to some curious mathematics involving the number six.
A sestina consists of six stanzas, each of six lines. The last words of each line in the first stanza reappear as the last words of the lines in subsequent stanzas, in a different (but specific) order. Then the whole thing is usually finished with a three-line “envoy” that features all six end-words somewhere in it.
I’d like to give you a complete example, if I may, so that you can see what is going on. There’s a lot of choice, because even though this form was first used more than eight hundred years ago, it is still in use and has enjoyed periods of great popularity. The 1950s were even described as the “age of the Sestina” by James Breslin (at the time, a professor of English at UC Berkeley). There are sestinas by poets from Dante to Kipling, Elizabeth Bishop to Ezra Pound, through to contemporary works by the American poet David Ferry (“The Guest Ellen at the Supper for Street People”) and by the English “thingwright”—this is the marvelous way she describes herself on her website—Kona Macphee (the desperately sad 2002 poem “IVF”). The example I’ve chosen is a poem by Charlotte Perkins Gilman, who is best known nowadays for her 1892 short story, “The Yellow Wallpaper.”
To the Indifferent Women
A Sestina
by Charlotte Perkins Gilman
You who are happy in a thousand homes,
Or overworked therein, to a dumb peace;
Whose souls are wholly centered in the life
Of that small group you personally love—
Who told you that you need not know or care
About the sin and sorrow of the world?
Do you believe the sorrow of the world
Does not concern you in your little homes?
That you are licensed to avoid the care
And toil for human progress, human peace,
And the enlargement of our power of love
Until it covers every field of life?
The one first duty of all human life
Is to promote the progress of the world
In righteousness, in wisdom, truth and love;
And you ignore it, hidden in your homes,
Content to keep them in uncertain peace,
Content to leave all else without your care.
Yet you are mothers! And a mother’s care
Is the first step towards friendly human life,
Life where all nations in untroubled peace
Unite to raise the standard of the world
And make the happiness we seek in homes
Spread everywhere in strong and fruitful love.
You are content to keep that mighty love
In its first steps forever; the crude care
Of animals for mate and young and homes,
Instead of pouring it abroad in life,
Its mighty current feeding all the world
Till every human child shall grow in peace.
You cannot keep your small domestic peace,
Your little pool of undeveloped love,
While the neglected, starved, unmothered world
Struggles and fights for lack of mother’s care,
And its tempestuous, bitter, broken life
Beats in upon you in your selfish homes.
We all may have our homes in joy and peace
When woman’s life, in its rich power of love
Is joined with man’s to care for all the world.
Let me show you how a sestina is constructed. To move from one stanza to the next, you move around the end-words in precisely the same way each time, a sort of ordered disorder created by working in reverse from the last end-word backward, and interleaving them with the first end-words in the right order, until we’ve used them all up. We can see this in Charlotte Perkins Gilman’s sestina. The end-words in the first verse are homes/peace/life/love/care/world. Reversing the last words gives world/care/love…, and we interleave these with homes/peace/life…, so as to obtain
world care love
homes peace life
That is, world/homes/care/peace/love/life. And these are, as you can see, exactly the end-words in the second stanza. This specific shuffling gives a nice continuity between the stanzas, because the end of the last line in one stanza is the end of the first line of the next. The structure continues, though, because we repeat this same reverse interleaving on the end-words of the second stanza to obtain the ordering of the end-words in the third stanza. If you try this, you’ll find that it turns world/homes/care/peace/love/life into life/world/love/homes/peace/care. And we repeat this process to obtain the orderings for the fourth, fifth, and sixth stanzas. There is a beautiful bit of unseen structure here, too, in that if we were to continue to a seventh stanza, our interleaving process applied to the sixth stanza’s ordering of peace/love/world/care/life/homes would result in the end-words homes/peace/life/love/care/world. If this looks familiar, it should—it’s the same ordering as we started with. The six stanzas therefore give us, even though we don’t consciously recognize it, a complete circle of six iterations, which if continued would bring us exactly back to our starting point. I think we do experience and appreciate this mathematical structure subconsciously, even though we may not detect it consciously. The shuffling also has pleasing internal symmetries—every end-word appears at the end of every different possible line, from first to last, in precisely one stanza. It’s a compelling design.
Unusually for so ancient a form, we have a plausible candidate for who invented it—the twelfth-century poet Arnaut Daniel. It was viewed as a very refined form of poetry that only the expert troubadour could master. I don’t know how Daniel came up with the idea—it’s a really simple permutation, very easy to remember, and you might think, once you hit on the process to follow, that given that the number of stanzas and the number of lines in each stanza are equal, both six, then you’ll naturally come back to where you started with after six shuffles. But let’s see what happens when we try to create a “quartina” with the same process. We start with a four-line stanza. Let’s suppose our end-words are north/east/south/west. Remember the rule—we work in reverse order from the end, interleaving with words from the start. So we get west/north/south/east for our second stanza. We repeat the process to get east/west/south/north for the third stanza, then again to get north/east/south/west for the fourth stanza. Oh, no! We have regained our original order in the fourth stanza! So this process would not give us four different stanzas. Even worse, you can see that the end-word “south” gets stuck—it’s the end-word of the third line in every stanza.
If you try to create a sestina-like poem with numbers other than six, you’ll find that sometimes it works and sometimes it doesn’t. In the 1960s, people started to try to figure out which values of n work. These “generalized sestinas” were named queninas by the Oulipo, in honor of Raymond Queneau. It turns out to be a really tricky problem. It works, for instance, for 3, 5, 6, 9, and 11, but not for 4, 7, 8, and 10. Amazingly, it is still an unsolved problem whether there are infinitely many values of n for which a quenina is possible, although a 2008 paper by the mathematician Jean-Guillaume Dumas described exactly the properties that such n would have to have. There is a particularly nice kind of number that will always have a quenina, a prime number called a Sophie Germain prime. It was named after a remarkable mathematician who did brilliantly innovative work in several areas of mathematics despite having to register at university under a false name and get other students to send her the course notes, due to the dreadful failing of being a woman—this was eighteenth-century Paris, after all. A prime number is called a Germain prime if, when you double it and add 1, the answer is again prime. The number 3, for example, is a Germain prime because is again prime, but 7 is not a Germain prime because is not prime. I can’t prove it for you, but it turns out that a quenina is possible for every Sophie Germain prime, which I love. Indeed, I know of at least one published “tritina” (three stanzas of three verses; the envoy is one line that includes all three of the end-words), by the English poet Kirsten Irving.
Talula-Does-the-Hula-from-Hawaii
by Kirsten Irving
Where do stupid names end up, these shorn tags
tied on toes by parents with the abandon
and foresight of tyrants annoying their court?
Today the three of you, now strangers, leave court
in opposite directions, untying cloakroom tags
from belongings, as you abandon
what passed for a name. That punchline abandoned
to the playground’s corrupt court
and the toilet wall’s smeared tags.
Tags abandoned, a girl who’s not Talula courts the world.
Rhyme schemes and queninas impose structure on the ending of lines, and they already give us some fascinating mathematics to play with. But there’s even more to explore when we consider the patterns within lines of poetry, and that’s what we’ll turn to next.
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In addition to the rhyme scheme, poetic forms often have a specific rhythm in their lines, which we call meter. Shakespeare’s plays are full of iambic pentameters, for instance. The “penta” bit is from the Greek word for five, and an “iamb” is a two-syllable phrase of which the second syllable is stressed. Thus an iambic pentameter has ten syllables, with the second one in each pair being stressed. I’ve underlined the stressed syllables in the following example, from the balcony scene in Romeo and Juliet.
But soft, what light through yonder window breaks?
It is the East, and Juliet is the sun.
This “di-dum di-dum di-dum di-dum di-dum” can be represented visually using dots and dashes, just like Morse code. An iamb is and an iambic pentameter looks like this:
