Introduction



“Call me Ishmael.” This has to be one of the most famous opening sentences in literature. I’m embarrassed to say that I didn’t get past it for a long time—Moby-Dick was in the guilt-inducing category of “books you should have read,” which obviously made me rebel against doing so, as I feared it would be the worst of all things: worthy. Thank goodness I decided one day to finally take the plunge, because it’s probably not an exaggeration to say it changed my life. It set me thinking about the links between mathematics and literature, which led ultimately to this book.

It all started when I heard a mathematician mention that Moby-Dick contains a reference to cycloids. The cycloid is a beautiful mathematical curve—the mathematician Blaise Pascal found it so distractingly fascinating that he recounted thinking about it to relieve the pain of toothache. But applications to whaling are not usually listed on its résumé. Intrigued, I decided it was high time to finally read this Great American Novel. To my surprise and delight, I found that right from the start, Moby-Dick abounded with mathematical metaphors. The more Melville I read, the more mathematics I discovered. And it wasn’t just Melville. Leo Tolstoy writes about calculus, James Joyce about geometry. Mathematicians appear in work by authors as disparate as Arthur Conan Doyle and Chimamanda Ngozi Adichie. And how about the fractal structure that underlies Michael Crichton’s Jurassic Park or the algebraic principles governing various forms of poetry? Mathematical references in literary works go back at least as far as Aristophanes’s The Birds, first performed in 414 BCE.

There have been occasional academic studies on mathematical aspects of specific genres or authors. But even in the case of Melville, whose affinity for mathematics is (for me) so obvious in his work, I could find just a handful of scholarly articles. The more holistic connections between mathematics and literature have not received the attention they deserve. My goal in this book is to convince you not only that mathematics and literature are inextricably, and fundamentally, linked, but that understanding these links can enhance your enjoyment of both.

Mathematics is often viewed as being quite separate from literature and other creative arts. But the perceived boundary between them is a very recent idea. For most of history, mathematics was part of every educated person’s cultural awareness. More than two thousand years ago, Plato’s Republic put forward the ideal curriculum of arts to be studied, which medieval authors split into the trivium (grammar, rhetoric, logic) and the quadrivium (arithmetic, music, geometry, astronomy). Together, these are the essential liberal arts. There is no artificial dichotomy here between “mathematics” and “art.”

The eleventh-century Persian scholar Omar Khayyám, to whom the poetry collection known as the Rubaiyat is attributed (modern scholars believe it to be the work of several authors), was also a mathematician, creating beautiful geometrical solutions to mathematical problems whose full algebraic solutions would not be found for another four hundred years. In the fourteenth century, Chaucer wrote both The Canterbury Tales and a treatise on the astrolabe. There are innumerable such examples, not least that of Lewis Carroll, who, of course, was a mathematician first and author second.

But there is a deeper reason why we find mathematics at the heart of literature. The universe is full of underlying structure, pattern, and regularity, and mathematics is the best tool we have for understanding it—that’s why mathematics is often called the language of the universe, and why it is so vital to science. Since we humans are part of the universe, it is only natural that our forms of creative expression, literature among them, will also manifest an inclination for pattern and structure. Mathematics, then, is the key to an entirely different perspective on literature. As a mathematician, I can help you see it too.

I’ve always loved patterns—whether that’s patterns of words, numbers, or shapes. I’ve loved patterns since before I knew that what I was doing was mathematics. Gradually it became clear that I was going to be a mathematician, but that came with consequences. In the British education system in recent decades, mathematics has come to be treated exclusively as a science subject, far removed from the humanities. If you want to study mathematics after the age of sixteen, you probably have to pick the “science” stream. At the end of my very last English class at school, in 1991, the teacher gave me a lovely handwritten note with a long list of books she thought I might like, saying, “Sorry to lose you to the lab.” I was sorry to be considered lost, too. But I wasn’t lost—and if you’ve ever had to “choose” one subject over another, you aren’t lost either. I love language; I love the way words fit together; I love the way that fiction, like mathematics, can create, play with, and test the limits of imaginary worlds. I went off to Oxford to study mathematics, very happy to be living one street away from the pub where my childhood literary heroes C. S. Lewis and J.R.R. Tolkien had met each week to discuss their work.

After completing a master’s degree and Ph.D. in Manchester, in the north of England, I moved to London for a job at Birkbeck, one of the colleges of the University of London, in 2004, and became a full professor there in 2013. During all this time, although my “day job,” so to speak, has been teaching and research, mainly in the area of abstract algebra known as group theory, I became increasingly interested in the history of mathematics, particularly in how mathematics is part of our broader cultural experience. I’ve always felt that what I do as a mathematician fits in with other creative arts, like literature or music. Good mathematics, like good writing, involves an inherent appreciation of structure, rhythm, and pattern. That feeling we get when we read a great novel or a perfect sonnet—that here is a beautiful thing, with all the component parts fitting together perfectly in a harmonious whole—is the same feeling a mathematician experiences when reading a beautiful proof. The mathematician G. H. Hardy wrote that “a mathematician, like a painter or poet, is a maker of patterns.… The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.”

Becoming Gresham Professor of Geometry in 2020 gave me the chance to bring together my decades of thinking about mathematics and its place in history and culture. This professorship is one of the few Tudor jobs still around—it was created in 1597 in the will of the Elizabethan courtier and financier Sir Thomas Gresham, and I’m the thirty-third person, and the first woman, to do it. I get to give public lectures on any mathematical subject of my choosing, though fortunately it’s been well over a century since professors were required to deliver each lecture twice: once in English and once in Latin.

So with being a professor of mathematics at Birkbeck and concurrently a professor of geometry at Gresham, as well as raising two wonderful daughters, I know what you are thinking: Sarah, what do you do with all your spare time? The answer is that, as I’ve always done, I read. Constantly and widely. The best thing about e-readers is that there are no pages to turn, which meant I could read even with a sleeping baby in my arms. That’s how I finally found the time to read War and Peace, which was full of mathematical surprises.

Each year my good friend Rachel and I set ourselves the challenge of reading the Booker Prize shortlist before the winner is announced. This gives us about six weeks to read six books. In 2013, one of the shortlisted books (and in fact the eventual winner) was Eleanor Catton’s The Luminaries. Catton made use of several structural constraints in the novel, including a mathematical sequence known as a geometric progression. There are hidden clues and rewards for the reader aware of the mathematics behind the scenes—a haul of stolen gold worth precisely £4,096 is not a coincidence, for example—and understanding the geometric progression unfolding throughout gives you another dimension of enjoyment. This is just one of many literary uses of mathematical structures that I’ll show you in this book.

It’s worth pointing out as well that the links between mathematics and literature do not run in just one direction. Mathematics itself has a rich heritage of linguistic creativity. Going back to early India, Sanskrit mathematics followed an oral tradition. Mathematical algorithms were encoded in poetry so that they could be passed on by word of mouth. We think of mathematical concepts as relating to precise, fixed words: square, circle. But in the Sanskrit tradition, your words must fit into the meter of your poem. Number words, for example, can be replaced with words for relevant objects. The number 1 can be represented by anything that is unique, like the moon or the earth, while “hand” can mean 2, because we have two hands—but so can “black and white,” because they form a pair. An expression like “three voids teeth” doesn’t mean a visit to the dentist, but that three zeros should follow the number of teeth we have: a poetic way to say 32,000. The huge array of different words and meanings lends a compelling richness to the mathematics.

Mathematical language continues to be figurative—when we need new words for things, we reach for metaphors. Once these words have been established for long enough, we tend to forget that they have other layers of meaning. But sometimes circumstances can intervene to remind us. As a master’s student, I spent a semester studying at the University of Bordeaux, in southwest France, and reading mathematics in French lent a slight air of surreality to the mathematics because of the use of words and metaphors I didn’t yet know in a mathematical context. Those few months of study opened my eyes forever to the creative metaphorical language underpinning much of mathematics. Learning a subject called algebraic geometry in French, I got a distinctly agricultural vibe from the word gerbe, which until then I had previously been aware of only in the phrase gerbe de blé (wheat sheaf). Sometimes you can overtranslate—for a while I thought there was a result called the walrus theorem, because the French word morse translates to “walrus,” when in fact it was named after its discoverer, the respected mathematician (and non-walrus) Marston Morse.

Just as mathematics makes use of literary metaphors, literature abounds with ideas that a mathematically attuned eye can detect and explore. This adds an extra dimension to our appreciation of a work of fiction. Melville’s cycloid, for example, is a curious curve with many wonderful properties, but unlike curves such as the parabola and ellipse, you probably haven’t heard of it unless you are a mathematician. That’s a real shame, because the properties of this curve are so beautiful that it was nicknamed “the Helen of geometry.” Making a cycloid is quite easy. Imagine a wheel rolling along a flat road. Now mark a point on the rim somehow, say with a blob of paint. That blob will trace out a path in space as the wheel rolls, and this path is called a cycloid. This is a fairly natural idea, but we don’t have evidence of its being studied until the sixteenth century, and things didn’t really heat up until the seventeenth and eighteenth centuries, when it seemed that everyone who was interested in mathematics had something to say. It was Galileo, for example, who came up with the name “cycloid”: he wrote that he had worked on cycloids for fifty years.

So the fact that the cycloid gets a mention not just in Moby-Dick but in two great works of eighteenth-century literature, Gulliver’s Travels and Tristram Shandy, again shows us mathematics in its rightful place—not “other” but part of intellectual life. When Gulliver visits the land of Laputa, he finds the inhabitants obsessed with mathematics. Dining with the king, he reports that “servants cut our bread into cones, cylinders, parallelograms, and several other mathematical figures.” There is a shoulder of mutton “cut into an equilateral triangle” and “a pudding into a cycloid.” Meanwhile, over at Shandy Hall, Tristram’s uncle Toby is having terrible trouble trying to construct a model bridge. After consulting various learned sources (there’s even a reference to a real-life mathematical paper in the extremely clever-sounding journal Acta Eruditorum), he decides, rather rashly, that a cycloid-shaped bridge is the way forward. But it doesn’t go well: “My uncle Toby understood the nature of a parabola as well as any man in England—but was not quite such a master of the cycloid;—he talked however about it every day—the bridge went not forwards.”

Part of the enjoyment of reading Tristram Shandy and other great books is the dazzling richness and breadth of their allusions—literary, cultural, and, yes, mathematical. If you’re reading classic literature, then it makes sense to be at least a little familiar with works of Shakespeare because of their profound literary and cultural influence. Is there a mathematical equivalent to the works of Shakespeare, references to which abound in classic literature? A strong contender would be the books of Euclid, known collectively as The Elements of Geometry, or just Euclid’s Elements. They are probably the most influential mathematics books of all time.

There’s an anecdote about how the philosopher Thomas Hobbes got hooked on geometry, told by his biographer John Aubrey:

Being in a gentleman’s library Euclid’s Elements lay open, and ’twas the forty-seventh proposition in the first book. He read the proposition. “By G—d,” said he, “this is impossible!” So he reads the demonstration of it, which referred him back to such a proof; which referred him back to another, which he also read.… At last he was demonstratively convinced of that truth. This made him in love with geometry.

This is a nice story, and it tells us a lot about how mathematics was viewed. Euclid’s Elements lay open, notice, because Hobbes was in “a gentleman’s library,” not “a mathematician’s study.” This stuff was considered part of the well-rounded education of an informed person. More than this, Aubrey assumes that we, the readers, are familiar with Euclid. He refers to Book I, Proposition 47, as if we will know it. We do know it, in fact, because it is Pythagoras’s theorem.

The beautiful certainties encapsulated in Euclidean geometry—axioms and definitions leading inexorably to theorems and proofs—have both inspired and consoled literary figures, from George Eliot and James Joyce, both in their different ways lovers of mathematics, whom we’ll meet in Chapter 6, to poets like William Wordsworth and Edna St. Vincent Millay. In his “Prelude,” Wordsworth speaks of geometry bringing “a pleasure quiet and profound” that can “beguile [your] sorrow”:

Mighty is the charm

Of those abstractions to a mind beset

With images, and haunted by herself,

And specially delightful unto me

Was that clear synthesis built up aloft

So gracefully;…

… an independent world

Created out of pure intelligence.

Everybody knew about the perfection of Euclid, so in the nineteenth century the tremendously exciting discovery of geometries beyond the Euclidean world—things like so-called non-Euclidean geometries, where parallel lines can sometimes meet—instantly caught the public’s imagination. I’ll show you how these ideas have been interpreted in literature by everyone from Oscar Wilde to Kurt Vonnegut. By seeing mathematics and literature as complementary parts of the same quest to understand human life and our place in the universe, we immeasurably enrich both fields.

In Part I of this book, we will explore the fundamental structures of literary texts, from plot in novels to rhyme schemes in verse. I’ll show you the underlying mathematics of poetry. I’ll give you the lowdown on writing that, like The Luminaries, deliberately uses constraints, such as the mathematically inspired work of the French literary group the Oulipo, whose members included Georges Perec and Italo Calvino. In the house of literature, these are the foundations, the load-bearing beams. It’s here that we’ll find mathematical ideas hidden in plain sight.

What comes next is the decoration, the wallpaper, the carpets. Many authors have reached for mathematical metaphors in their writing, and the symbolism of numbers is rich and ancient. These turns of phrase, metaphors, and allusions will be our focus in Part II of the book.

But who lives in our house? What is our writing about? In Part III, I’ll show you how mathematics can become part of the story—with novels featuring overtly mathematical themes and sometimes even mathematicians as characters. We will look at mathematical ideas that have caught the public’s imagination, from fractals to the fourth dimension, and how they have been explored in fiction. We’ll look too at how stereotypes of mathematicians, and the idea of mathematics itself, have been used in fiction.

If you don’t yet love mathematics, I want this book to show you the beauty and wonder of it, how it is a natural part of our creative lives, and why it deserves its place with literature in the pantheon of the arts. I want it to give you an extra perspective on the writing and writers you know, introduce you to writing you don’t, and give you a new way of experiencing the written word. If you happen to be a mathematician, then you already have poetry in your soul, but we’ll look at how that is manifested in places you may never have realized, as part of an enduring conversation between literature and mathematics. I warn you: you’re going to need a bigger bookcase.



1 One, 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.


Copyright © 2023 by Sarah Hart