1One, Two, Buckle My Shoe
The Patterns of Poetry
The connections between mathematics and poetry are profound. But they begin with something very simple: the reassuring rhythm of counting. The pattern of the numbers 1, 2, 3, 4, 5 appeals to young children as much as the rhymes we sing with them (“Once I caught a fish alive”). When we move on from nursery rhymes, we satisfy our yearning for structure in the rhyme schemes and meter of more sophisticated forms of poetry, from the rhythmic pulse of iambic pentameter to the complex structure of poetic forms like the sestina and the villanelle. The mathematics behind these and other forms of poetic constraint is deep and fascinating. I’ll share it with you in this chapter.
Think of the nursery rhymes of your childhood. I bet you can still remember the words. That’s the power of pattern—our mathematical brains delight in it. The subliminal counting of rhythm and rhyme feels so natural that it helps us remember, hence the oral tradition of poems telling the deeds of great heroes. Many traditional rhymes involve counting up cumulatively, adding a new line with each verse and counting back down to one every time. There’s an old English folk song, “Green Grow the Rushes, O,” which builds up to twelve—the last line of every verse is the melancholy “One is one and all alone and ever more shall be so.” Meanwhile, the Hebrew Echad Mi Yodea (“Who Knows One”) rhyme, traditionally sung on Passover, uses rhythm and counting to teach children important aspects of the Jewish faith. It ends with “four are the matriarchs, three are the patriarchs, two are the tablets of the covenant, One is our God, in heaven and on earth.”
There are many mathematical mnemonics that we may have learned at school for remembering things like the first few digits of “How I wish I could calculate pi”: that’s not me expressing a desire to calculate it’s the mnemonic. The number of letters in each word tells you the next number in the decimal, which begins 3.141592. If you need more digits, a longer mnemonic is “How I need a drink, alcoholic in nature, after the heavy lectures involving quantum mechanics!” That one has been around for at least a century and is credited to the English physicist James Jeans. In fact, it’s now a niche hobby to compose verse in “pilish,” in which the word lengths are defined by the digits of 1 My favorite example of this is “Near a Raven,” a pilish version of Edgar Allan Poe’s “The Raven,” by Michael Keith:
Poe, E.
Near a Raven
Midnights so dreary, tired and weary.
Silently pondering volumes extolling all by-now obsolete lore.
During my rather long nap—the weirdest tap!
An ominous vibrating sound disturbing my chamber’s antedoor.
“This,” I whispered quietly, “I ignore.”
There’s no need to learn this poem in its entirety, though—it’s been estimated that a mere forty digits of are enough to calculate the circumference of the entire known universe accurate to less than the size of a hydrogen atom. So the first verse alone is more than enough for all practical purposes.
The pilish “Raven” is based on a mathematical constant, but its contents aren’t mathematical. There is, however, at least one well-known poem that poses a mathematical puzzle. You may know it:
As I was going to St. Ives,
I met a man with seven wives.
Each wife had seven sacks,
Each sack had seven cats,
Each cat had seven kits.
Kits, cats, sacks, and wives,
How many were going to St. Ives?
I remember trying to multiply all those sevens as a kid—only to realize I’d fallen for the oldest misdirection trick in the book.
Much more sophisticated mathematical problems have been expressed in verse, though. As I mentioned in the introduction, it was the standard format for mathematics in the Sanskrit tradition. The twelfth-century Indian mathematician and poet Bhaskara wrote all his mathematical works in verse. Here is one of the poems in a book he dedicated to his daughter Lilavati:
Out of a swarm of bees, one fifth part settled on a blossom of Kadamba,
and one third on a flower of Silindhri;
three times the difference of those numbers flew to the bloom of a Kutaja.
One bee, which remained, hovered and flew about in the air,
allured at the same moment by the pleasing fragrance of jasmine and pandanus.
Tell me, charming woman, the number of bees.
What a lovely way to write about algebra!
We don’t tend to write our mathematics in verse nowadays, more’s the pity, but the aesthetic link with poetry remains: the goal of both is beauty, a beauty that makes a virtue of economy of expression. Poets and mathematicians alike have praised each other’s specialisms. “Euclid alone has looked on Beauty bare,” wrote the American poet Edna St. Vincent Millay in a 1922 sonnet paying homage to Euclid’s geometry. For the Irish mathematician William Rowan Hamilton, both mathematics and poetry can “lift the mind above the dull stir of Earth.” Einstein is reported to have said that mathematics is the poetry of logical thought. A mathematical proof, for example, if it’s any good, has a lot in common with a poem. In both cases, each word matters, there are no superfluous words, and the goal is to express an entire idea in a self-contained, usually fairly short, and fairly structured way.
I’m going to show you a proof now, because it’s a beautiful thing and it is pure poetry. It’s the proof, attributed to Euclid (though we don’t really know who came up with it), that there are infinitely many prime numbers. Remember, primes are the numbers, like 2, 3, 5, 7, and so on, that can’t be divided up into smaller whole number parts. The number 4, for instance, isn’t prime because you can break it up as And 6 is Every one of the counting numbers after 1 is either a prime number or can be broken up (the technical term is “factorized”) into prime numbers, and even more brilliantly, this can be done in really only one way, as long as you are happy to say that is basically the same thing as By the way, the number 1 feels as if it ought to be prime because it can’t be divided up, but we exclude 1 from the list because otherwise you’d have to say that and there would be infinitely many ways to factorize every number—yuck! We get around this by defining a prime number as a number greater than 1 whose only factors are 1 and itself.
Understanding the prime numbers is as important to math as understanding the chemical elements in science, because just as every chemical substance is made up of a precise combination of elements (every molecule of water, or H2O, has exactly two hydrogen atoms and one oxygen atom, for instance), every whole number has a particular prime decomposition. One of the most exciting discoveries of early mathematics was that, unlike chemical elements, the prime numbers go on forever. Actually, at the time, the contrast would have been even more stark, because for the ancient Greeks there were just four elements—earth, air, fire, and water—that were believed to make up all things.
Here’s a proof that there are infinitely many prime numbers:
What if we had a list of all primes, a finite list?
It would start with 2, then 3, then 5.
We could multiply all the primes together, and add 1 to make a new number.
The number is 2 times something plus 1, so 2 can’t divide it.
The number is 3 times something plus 1, so 3 can’t divide it.
The number is 5 times something plus 1, so 5 can’t divide it.
None of the primes on our list can divide it.
Either our number is prime, or a new prime not on our list divides it.
Either way, the list isn’t complete. It can’t be done.
There can’t be a finite number of primes.
QED
It’s a poem, I tell you!
The resonances between poetry and mathematics were expressed well by the American poet Ezra Pound in The Spirit of Romance (1910): “Poetry is a sort of inspired mathematics, which gives us equations, not for abstract figures, triangles, spheres and the like, but equations for the human emotions.” Pound made another analogy between mathematics and poetry—the way that both can be open to many layers of interpretation.2 I would say that mathematicians have a very similar understanding of what makes the greatest mathematics: concepts that hold within them many possible interpretations—structures that can be found in different settings and so have a universality to them. The key thing here is that the elegant brevity of a mathematical expression, just like a poem, can encompass multiple layers of meaning, and the more layers and interpretations it can contain, the greater the artistry. Mathematics, like Walt Whitman, contains multitudes, both literally and allegorically. The only difference is that we hope it does not contradict itself!
* * *
It’s quite hard to give a definition of what poetry is. Sometimes it rhymes, there are almost always line breaks, there’s usually a rhythm, a meter, and so on. What we can broadly say is that poems have some sort of constraint, whether that’s a meter (iambic pentameter, for example), or a rhyme scheme, or a given number of lines in each stanza. Even completely free verse will probably have line breaks, stanzas, and rhythm. One occasionally hears expressed that understanding how something is put together takes away the mystery and therefore spoils it. We don’t want to know how the magician does his tricks—we want to believe in magic. The difference is that poetry is more than artifice. How can understanding something do anything other than add to your appreciation of it? That’s how I feel about the underlying mathematics of structure and pattern.
Submitting yourself voluntarily to a particular constraint spurs creativity. The discipline required means you have to be inventive, creative, and thoughtful. In haiku, with their seventeen syllables, no syllable can be wasted. On a rather less exalted level, the humorous limerick form has to get from setup to payoff in just five lines. The Irish poet Paul Muldoon made the brilliant comment that poetic form “is a straightjacket in the sense that straightjackets were a straightjacket for Houdini.” This may set the record for most uses of the word “straightjacket” in a sentence, but the sentiment is exactly right—the constraint itself is part of the genius of the work.
Constraints in poetry come in many flavors. In the Western tradition, particular rhyme schemes have been favored, and a handful of rhythms have been adopted—those iambs and trochees of classical verse. There is counting, pattern, and therefore mathematics behind both types of constraint. But in other traditions, different pattern-creating devices are used that involve more explicit use of numbers. That’s where we’ll begin our discussion of the mathematics of poetic constraints.
Let me tell you a story that begins in the imperial court of eleventh-century Japan. Murasaki Shikibu, a noblewoman at the court and lady-in-waiting to Empress Shoshi, wrote what is thought to be one of the very earliest novels, The Tale of Genji. An epic novel of courtly love and heroism, it is a Japanese classic, still read a millennium after it was written. One of the novel’s distinctive features is characters’ use of poetry in conversation, quoting or modifying well-known verses or saying the first parts of them (just as we might do when we say, for instance, “A stitch in time” rather than “A stitch in time saves nine”). Many of the poems in The Tale of Genji are in what is called the tanka form. This is one example of a more general style of classic Japanese poetry called waka. Like the more modern haiku, such poems feature lines of 5 and 7 syllables, but where haiku has a 5–7–5 pattern with 17 syllables in total, tanka has 5–7–5–7–7, for a total of 31 syllables. (In fact, what is counted are not exactly “syllables” but “sounds,” a subtle but important distinction, which I beg experts in Japanese poetry to forgive me for not making in more detail.)3
For a mathematician, the connection with prime numbers is inescapable. Look at the haiku: 3 lines, lengths 5 and 7 syllables, and a total of 17 syllables. The numbers 3, 5, 7, and 17 are all prime numbers. With the tanka, there are 2 lines of 5 syllables and 3 lines of 7 syllables—and again, 2, 3, 5, 7, and 31 are all prime. Is this significant? I have read that the 5–7 pairing arose from an earlier “natural” 12-syllable entity, which is broken into two parts with a slight pause. Making the break at 5–7 certainly seems to me to be more exciting and dynamic than the dully exact 6–6 split or the too unbalanced 4–8, so perhaps that’s how it came about. Since primes can’t be divided further, the 5–7 break perhaps helps to categorize the lines as separate indivisible entities, whereas 4, 6, and 8 all have “fault lines” that would arguably weaken the structure.
Centuries after The Tale of Genji was written, a game became fashionable in the parlors of sixteenth-century Japanese aristocrats: Genji-ko. The hostess would secretly choose five incense sticks from a selection of different scents; some of the five scents might be the same. She would then burn them one after the other, and the guests would try to guess which scents were the same and which were different. So you might think that all the scents are different. Or perhaps the first and third scents are the same, and all the others are different. The various possibilities would be represented by little diagrams like this:
The far left diagram represents all scents being different; the next has just the first and third matching; in the next the first, third, and fifth match, as do the second and fourth; the far right diagram has the second, third, and fourth matching, as well as the first and fifth. To help people describe what their guess was, each of the different possibilities was named after a chapter from The Tale of Genji—it turns out there are fifty-two possibilities, from “all different” to “all the same” and everything in between.4 Several editions of The Tale of Genji even featured these patterns next to the corresponding chapter headings. The patterns themselves took on a life of their own—they were used as heraldic crests and in kimono designs.
Meanwhile, thousands of miles away in Tudor England, George Puttenham included diagrams like this in his 1589 book The Arte of English Poesie:
They look just like sideways versions of Genji-ko pictures! In particular, compare
What on earth is going on? Well, Puttenham is describing possible rhyme schemes in a five-line stanza, giving diagrams to aid the reader’s comprehension (or as he put it, “I set you downe an occular example: because ye may the better conceive it”).
The rhyme scheme of a poem, or of a stanza within a poem, is simply the pattern of rhymes in the last words of the lines. The earliest poems we encounter are songs and nursery rhymes with simple rhyme schemes:
Mary had a little lamb
Its fleece was white as snow
And everywhere that Mary went
The lamb was sure to go.
This is a four-line poem—a “quatrain”—with the rhyme scheme abcb, which means that the second and fourth lines rhyme with each other, but not with the remaining lines. By contrast, here’s a quatrain from John Donne’s poem “The Sun Rising”:
Copyright © 2023 by Sarah Hart
