Once Upon a Prime, page 4
The basic patterns of stressed and unstressed syllables are called feet. Two common examples, along with the iambs we have just seen, are trochee (), as in “Quoth the Raven ‘Nevermore,’” and dactyl (), as in “The Lost Leader,” by Robert Browning, which begins, “Just for a handful of silver he left us”—actually this is three dactyls and a trochee at the end. How many possible meters are there for a given number of syllables? There are two possibilities for each syllable—stressed or unstressed—so the number of one-syllable feet is two ( or ). To get to two syllables, we can add either a or a to either of these, so the total is four. We can add a or a to each of these four to get eight possible three-syllable meters, and it just keeps doubling—we end up with a sequence 1, 2, 4, 8, 16, and so on, the powers of 2.
But there’s a form of poetry in which something very different happens. I first read about it in Jordan Ellenberg’s excellent paean to geometry, Shape. He recounts how a mathematician friend, Manjul Bhargava, told him about the meters of Sanskrit poetry. As in English poetry, the pattern of syllables is important, but while with English we look at where the stresses lie, in Sanskrit it’s the length that matters. Syllables are either laghu (light) or guru (heavy). Crucially, laghu syllables count as one unit, and guru as two. This means it’s a bit more complicated to work out, for instance, how many four-syllable meters are possible. We can’t just take the number of three-syllable meters and double it. So what do we do? Well, there’s just one one-syllable possibility: laghu. There are two two-syllable options: laghu laghu, or guru. For three syllables, you can check that the three possibilities are laghu laghu laghu, laghu guru, or guru laghu. For four syllables, let’s get a bit clever and divide the problem into two. Either the meter starts with laghu, or it starts with guru. If it starts with laghu, then we can choose from any of the three three-syllable meters to add on to it, to arrive at four syllables. If it starts with guru, then we can choose either of the two two-syllable meters to add on. So the total is 3 + 2 = 5:
laghu laghu laghu laghu
laghu laghu guru
laghu guru laghu
guru laghu laghu
guru guru
What’s more, you can always play this trick. Five-syllable meters are either laghu + (a four-syllable meter) or guru + (a three-syllable meter). So the number of five-syllable meters equals the number of four-syllable meters plus the number of three-syllable meters, which is We can carry on like this. The next number is just the sum of the previous two numbers. So we get a Sanskrit meter sequence like this:
1, 2, 3, 5, 8, 13, 21, …
You may have encountered this sequence before. It’s better known in English-speaking countries as the Fibonacci sequence, popularized in Europe in the thirteenth century by Leonardo of Pisa, whose nickname was Fibonacci. (Sometimes it’s shown as beginning with two 1s, but it’s the same basic principle.) Each term after the first two, as we’ve said, is the sum of the two previous terms. For example, The next term in the sequence after 21 will therefore be The Fibonacci sequence has many interesting properties. One is that the sequence of ratios of consecutive terms converges to the famous golden ratio ≈1.618.
When Fibonacci introduced the sequence in his 1202 book Liber Abaci (“The Book of Calculation”), it was in the context of a rather fatuous puzzle about rabbits. You start with one breeding pair of newborn rabbits. A breeding pair mates after one month, and the female gives birth to a new breeding pair one month after that. Rather unrealistically, the rabbits never die, they keep breeding forever, and we have to ignore minor concerns like rabbit incest. The question is, how many pairs of rabbits are there after one year? We can see that the same rule applies to this sequence. In any given month, the total number of pairs will be the number there was a month ago plus the number of newborn pairs, which (since it takes two months from birth to produce a new pair) is the number of pairs there were two months ago. So each term is the sum of the previous two terms. But this sequence had been known to poetry scholars in India for centuries before Fibonacci. The metrical experts Virahanka (sometime between 600 and 800 CE), Gopala (sometime before 1135 CE), and Hemachandra (around 1150 CE) all knew the sequence and how to produce it, and there’s some evidence that it was known even earlier, in the writings of Pingala (around 300 BCE). Perhaps it’s time to rename the Fibonacci numbers.
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Mathematics and poetry are two of our most ancient forms of creative expression, and their connections reach back to the very beginnings of writing itself. The earliest known works by a named author in the whole of human history were created by a remarkable woman named Enheduanna, who lived over four thousand years ago in the Mesopotamian city of Ur. She wrote perhaps the very first collection of poems—a cycle of forty-two “Temple Hymns.” But as high priestess of the moon god Nanna, she would have needed knowledge of astronomy and mathematics as well. These come together in her poetry, both in her use of numbers, particularly the number seven, and in mention of calculation and geometry. The final Temple Hymn speaks of the mathematical activities of the “true woman of unsurpassed wisdom”:
She measures the heavens above
and stretches the measuring cord on the earth.6
From these earliest beginnings, the love affair between poetry and mathematics has flourished. Mathematics has been there in the deep currents of verse, underpinning its rhymes and hidden in its structures. As the great nineteenth-century mathematician Karl Weierstrass wrote, “A mathematician who is not somewhat of a poet, will never be a perfect mathematician.” And poetry? It’s simply the continuation of mathematics by other means.
2
The Geometry of Narrative
How Mathematics Can Structure a Story
At a public lecture in 2004, Kurt Vonnegut gave illustrations of the “graphs” of some possible stories.1 The first of these was “Man in a Hole”:
In Vonnegut’s graphs, the vertical axis measures good fortune and the horizontal axis measures time passing—a rising curve means improving fortunes, a falling curve means things are getting worse. In “Man in a Hole,” for instance, we start with someone going along happily when suddenly disaster strikes, but everything works out wonderfully in the end. A novel in this category might be David Copperfield—or, to give it its full, glorious title, The Personal History, Adventures, Experience and Observation of David Copperfield the Younger of Blunderstone Rookery (Which He Never Meant to Publish on Any Account). The young David has a very happy childhood until he is seven, when his mother first marries beastly Mr. Murdstone, then soon afterward dies, leaving poor David orphaned. But after many reverses and trials, David eventually finds happiness. Vonnegut gave three other graphs, which I’ve sketched below:
“Boy meets girl,” of course, is a feature of most romantic novels. Boy meets girl, boy loses girl, boy gets girl in the end. Happiness all around. To pick a random example, take the story line of Jane Bennet and Mr. Bingley in Jane Austen’s Pride and Prejudice. Jane and Bingley are fairly content already at the start of the novel. Then they meet and fall in love, and life looks even better. But they are separated by the machinations of proud Mr. Darcy and snobbish Miss Bingley. Misery ensues. In the end Darcy realizes the error of his ways and confesses all to Bingley, who at once returns to get his girl. And they all live happily ever after.
In “Cinderella,” by contrast, the starting point is unhappiness. Poor Cinders sleeps in the ashes of the fire (hence her name) and works all day for her horrid stepsisters. But then things start looking up. Off she goes to the ball, where she meets Prince Charming, but then—disaster! Midnight strikes and all appears lost. Fortunately, her feet are so freakishly shaped that she’s the only girl in the kingdom who can fit into the glass slipper left behind when she fled. She marries the prince, and her happiness becomes infinite.
The last of Vonnegut’s graphs is “Metamorphosis,” which refers to the darkly comic story by Franz Kafka. You will remember that this is the tale of Gregor Samsa, unhappy and alienated in his job as a traveling salesman. One morning he wakes up and finds that during the night he has turned into a gigantic “vermin” (usually assumed to be a cockroach). There follows a degrading and painful descent into illness and death. Good old Kafka.
We might place works like The Metamorphosis at the pessimistic end of the fine tradition of absurdism in literature, a style of writing amusingly described by author Patricia Lockwood as “novels where a man turns into a teaspoonful of blackberry jam at a country house.”2 For a truly absurd story graph, there’s no better place to turn than the brilliant, anarchic work of genius that is Tristram Shandy. Laurence Sterne’s novel appeared originally in nine volumes, published over eight years from 1759 to 1767. The narrator is Tristram Shandy, a gentleman who has decided to write his autobiography but is continually thwarted in this aim by the intrusions of other characters into the story. Tristram gets sidetracked by so many digressions and diversions that he doesn’t even manage to be born until Volume III. It’s a joyously chaotic read. Toward the end of Volume VI, Tristram Shandy draws a diagram of his narrative “lines” so far:
“These were the four lines I moved in,” he writes, “through my first, second, third, and fourth volumes. In the fifth volume I have been very good,——the precise line I have described in it being this:”
He claims that this is an improvement: “except at the curve, marked A, where I took a trip to Navarre,—and the indented curve B, which is the short airing when I was there with the Lady Baussiere and her page,—I have not taken the least frisk of a digression, till John de la Casse’s devils led me the round you see marked D.—for as for c c c c c they are nothing but parentheses.” “If I mend at this rate,” he says, “it is not impossible but I may arrive hereafter at the excellency of going on even thus:
* * *
which is a line drawn as straight as I could draw it.… The best line! say cabbage planters—is the shortest line, says Archimedes, which can be drawn from one given point to another.” You will be pleased to hear that this optimistic prediction proves entirely false, and the last volumes of the novel romp around as gleefully as the first.
Vonnegut’s graphs and Shandy’s crazy narrative “lines” are amusing, but are there more sophisticated, genuinely mathematical takes on narrative and plot? This chapter takes its title from Hilbert Schenck’s story “The Geometry of Narrative” (1983), in which a student suggests that simple plot “lines” are just the start. He finds a way to link Shakespeare’s Hamlet to a four-dimensional “hypercube” by arguing that we should think of instances of a story within a story as adding a dimension. That is, instead of time being the fourth dimension, Schenck’s protagonist, Frank Pilson, suggests we use what he calls narrative distance:
Here are two separate three-dimensional realities: the play, Hamlet itself, with old Claudius popping his mental cork when he sees the Hamlet-buggered script acted out, and the shorter, smaller, on-stage murder-of-Gonzago play. But the little play is at a greater distance from Hamlet, both from the real audience and from the Court of Denmark watching it on stage, since it is presented as a created artefact within the “true” or “real” drama. So not only is this part of Hamlet modelled by a four-dimensional geometrical object, but the staging assumes the exact projected form of the hypercube, with one small stage located in the middle of the other, larger one.
The rest of the story very cleverly sees the narrative camera repeatedly zooming in, so to speak, so that the frame of reference is constantly shifting. The part of the story you encounter first can change your understanding of the plot—is the story a first-person account by Pilson, telling us about his literature seminar and quoting excerpts from a story, or are we actually reading a story about an author who happens to be working on his novel about a student called Pilson? Our understanding of these different levels of narrative may prompt us to revisit the text and read it again, but from a different viewpoint or in a different order.
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Shakespeare was not thinking of hypercubes when he wrote Hamlet, but many authors have consciously chosen to impose mathematical constraints on their narratives. As the author Amor Towles said in a 2021 interview,3 “Structure can be very valuable in artistic creation. Much as the rules of the sonnet are valuable to the poet, adopting the rules and trying to invent, within those rules, something that’s new and different, the structure of a novel can do the same kind of thing.” You may be thinking, Why would a writer bother with some fancy structure? Why not just write a good story? This, I would argue, is a false dichotomy. All writing has structure from the get-go. Language itself is built of component parts, each of which has patterns. Letters make up words, words form sentences, sentences form paragraphs, and so on. This is already a structure, analogous to the hierarchy of point, line, plane in geometry. At each stage, further structures can be imposed. Paragraphs, for instance, can be joined together to form chapters. The decision is not whether to structure your work; rather it’s what structure to choose. Within each of these levels, writers may choose to add additional structural constraints. This added structure works best when it feels most natural, when it fits with the narrative themes or the design of the plot.
Let’s start at the highest level usually used in novels: the chapter. Eleanor Catton’s The Luminaries, which was published in 2013, is an astonishing achievement. Catton was the youngest finalist ever for the Booker Prize. She then became, at twenty-eight, its youngest winner ever. The judges described the book as a “dazzling,” “luminous” work, “vast without being sprawling.” And it is indeed vast. At 832 pages, it was the longest book ever to win the prize. The events of the novel are centered on the gold rush town of Hokitika, New Zealand, in the mid-1860s. The first chapter, mathematically titled “A Sphere Within a Sphere,” opens with the prospector Walter Moody arriving in Hokitika on January 27, 1866, and walking in on a meeting of twelve local men who have gathered to discuss a series of recent crimes. He becomes entangled in a web of murder, strange disappearances, attempted suicide, opium dealing, and the discovery of £4,096 worth of stolen gold.
There are twelve chapters, or parts, each taking place over the course of a single day in 1865 or 1866 (the novel’s first chapter begins, chronologically, at the midpoint of events). The twelve men whom we meet at the start of the novel are each associated with a specific sign of the zodiac. Their actions and behavior in each of the twelve chapters are determined in part by that sign’s astronomical configuration on the date of the chapter. Catton did careful research into the positions of the stars and planets in the night sky of Hokitika on those precise dates. By the way, I don’t think that this is because she is necessarily a believer in astrology. She says of Walter Moody that he was “not superstitious, though he derived great enjoyment from the superstitions of others.” The astrological and astronomical information is both a way to give structure and a way to inform the broader meditation in the book about the interplay between fate, circumstance, and free will.
In The Luminaries, each chapter is divided into a specific number of sections, and in every case, the number of sections, added to the number of the chapter, is the same: thirteen. Thus the first chapter has twelve sections, the second chapter has eleven sections, and by the time we get to the twelfth and final chapter, it has just one section. This kind of pattern, in which we see the same increase or decrease each time, as in the sequence 12, 11, 10, 9,…, is known in mathematics as an arithmetic progression. Hidden in the thirteenness of chapter number plus number of sections is a really simple trick to add up the total number of sections in the book. It would be annoying to have to calculate the sum by laboriously adding the numbers one by one. But if you go over all twelve chapters, in each case we know that the chapter number plus the number of sections equals 13. So the total of these thirteens, over the twelve chapters, is This picture shows, on the left, the chapter numbers, and on the right, the sections, adding up to 13 every time.
But this total is double what we need, because it’s also got the of the chapter numbers in there. All we have to do is halve it: the total number of parts is (12 × 13) = 78.
This trick is one of my first mathematical memories—my mother taught it to me when I was a kid, and I thought it was pretty amazing. She recounted the (possibly apocryphal) story of how the great mathematician Carl Friedrich Gauss, while still in elementary school, ruined a teacher’s attempt to get a bit of peace and quiet one afternoon, when the teacher set Gauss’s class the task of adding up all the numbers from 1 to 100. The young Carl apparently invented on the spot this little trick I’ve just explained. If our book had 100 chapters with the same pattern, then the sum is (100 × 101) = 5,050. Cool! I feel a bit sorry for the poor teacher, though—all he wanted was half an hour of quiet.
The most interesting and impressive aspect of the mathematical structure of The Luminaries is that each chapter is half the length of the last. That constraint has significant implications for the length of the novel. We can represent the length of the first chapter with a rectangle (we might measure the length in words, characters, lines, pages—whatever you prefer; it doesn’t make much difference). Here it is:
Now, the next chapter is half as long, so we can slot it in with a half-sized rectangle on the right. Chapter 3 is half the length of Chapter 2, and Chapter 4 is half the length of Chapter 3. I’ve shown the first few chapters in the picture:
We can keep on going, slotting ever smaller and smaller rectangles into this picture and never escaping from the outer square boundary. I’ve added Chapters 5, 6, 7, and 8 to the left-hand diagram and Chapters 9 to 12 to the right-hand diagram just to show you.
