Once Upon a Prime, page 7
A safer comparison is between the difficulty of La Disparition and the difficulty of Les Revenentes, Perec’s follow-up novel that he joked used up all the e’s missing from La Disparition. This time, to work out the difficulty we have to add up the frequencies in French of all the other vowels together. I’ve checked this and come up with a total frequency of 0.28018; I’ve also done a very rough word count of Les Revenentes and arrived at a total of thirty-six thousand words—that means a difficulty level of 10,086. It’s obvious why Les Revenentes is shorter. If it were the same length as La Disparition, it would be verging on twice as hard to write.1
A more recent lipogrammatic text that has the same self-referential quality as La Disparition is 2001’s Ella Minnow Pea by Mark Dunn—there’s a hint of what is to follow even in the title character’s name, which sounds like the sequence l, m, n, o, p. The book is set on the fictional island of Nollop, whose inhabitants revere Nevin Nollop, putative inventor of the pangram “The quick brown fox jumps over the lazy dog.” (In case you’re unfamiliar, a pangram is a phrase or sentence featuring every letter of the alphabet.) There’s a statue of Nollop on the island, with the pangram inscribed below it. One day, one of the tiles on which the letters of the pangram are written falls off, and the island’s rulers take this as a sort of divine instruction that this letter must be stricken from the alphabet and banned. At this point, it disappears from the text in the book. As more letters fall from the statue, they are banned too. The only way for this process to cease, decides the government, is if it were to turn out that Nollop is not actually a deity—and that can be true only if a shorter pangram is found. At the most desperate moment, when only the letters l, m, n, o, and p remain, the eponymous Ella manages to find a thirty-two-letter pangram (three letters shorter than Nollop’s), the full alphabet can be restored, and they all live happily ever after.
Before moving on from lipograms, I’ll just mention Eunoia, by the Canadian author Christian Bök, which won Canada’s Griffin Poetry Prize in 2002. There are five chapters in the main part of the book that each use only one vowel, with the letter y being omitted throughout. A sample sentence from Chapter A is “A law as harsh as a fatwa bans all paragraphs that lack an A as a standard hallmark.” The title, Eunoia, is the shortest English word that contains all the vowels—it means a state of good health. The shortest such word in French is somewhat better known—oiseau, meaning “bird”—which is the title of the second part of the book. As the final section, “The New Ennui,” states, the text “makes a Sisyphean spectacle of its labour, wilfully crippling its language in order to show that, even under such improbable conditions of duress, language can still express an uncanny, if not sublime, thought.” This is beautifully put, and there’s certainly some lovely imagery in the book. But I have to say that, though we can admire the extraordinary technical accomplishment required to produce such a lipogrammatic tour de force, the ratio of clever technique to emotional punch in the work is a little too high in places. With Eunoia, I think it’s time to draw our discussion of lipograms to a close.
* * *
There is something so French, somehow, about the Oulipo—where else could it possibly have been formed than in a Parisian café? Yet probably the most famous Oulipian was a Cuban-born Italian. His mother gave her son a name intended to remind him of his heritage, only to move back to Italy almost immediately, meaning that Italo Calvino (for it is he) was saddled forever with a name that he described as “belligerently nationalistic.”
Calvino’s best-known work is If on a Winter’s Night a Traveler, one of those rare books written in the second person. It’s about a reader (you) trying to read a book called If on a Winter’s Night a Traveler. You buy it, but it has the same sixteen pages repeated over and over again. When you return it, it turns out that actually these pages are copies of a different book called Outside the Town of Malbork. But something goes wrong when you try to find that book too, leaving you with the tantalizing start of a third book. Sections recounting your attempts to get hold of these books alternate with the beginning chapter of each. It’s clever and funny and includes a list of book categories that is instantly recognizable to the inveterate book buyer (including Books You Could Put Aside Maybe to Read This Summer, Books You Want to Own So They’ll Be Handy Just in Case, and Books That Everybody’s Read So It’s as If You Had Read Them Too).
Perhaps you have already put aside If on a Winter’s Night a Traveler, maybe to read this summer, so let me persuade you also to look at Calvino’s beautiful, melancholy book Invisible Cities. To use another of his categories, it is definitely a Book You Need to Go with Other Books on Your Shelves: Invisible Cities nods both to the travels of Marco Polo and to Thomas More’s Utopia, with a hint of One Thousand and One Arabian Nights. The book contains fantastical descriptions of fifty-five cities that are supposed to have been in Kublai Khan’s empire, ranging in length from a paragraph or two to a couple of pages. Argia, the underground city, for example, gets just fourteen lines—nothing can be seen from above ground, and it’s hard to know whether the city is there at all. “The place is deserted,” Calvino writes. “At night, putting your ear to the ground, you can sometimes hear a door slam.” Behind all of these cities is the one city that Marco Polo never speaks about, but of which every other city is just a reflection: his home. “Every time I describe a city I am saying something about Venice,” he tells the Khan.
Invisible Cities is divided into nine chapters, but the way the accounts of the cities are divided up and numbered is rather curious. Each city falls into a particular one of eleven categories (such as “Cities & the Dead,” or “Continuous Cities”), with five cities of each type. Chapter 2, for example, is shown like this in the table of contents:
2.
..…
Cities & Memory • 5
Cities & Desire • 4
Cities & Signs • 3
Thin Cities • 2
Trading Cities • 1
..…
The dotted lines represent unnamed sections in each chapter that contain conversations between Marco Polo and Kublai Khan. Chapters 3 through 8 also have five cities, numbered 5, 4, 3, 2, 1. But Chapter 1 and Chapter 9 have ten cities each, numbered seemingly (but in fact not) at random. Chapter 1 does not contain any 5s and Chapter 9 has no 1s. What is going on? Why this descending order 5, 4, 3, 2, 1? Why not just have eleven chapters with five cities in each, or five chapters with eleven? Why have fifty-five cities in the first place? Let’s begin with that last question.
One of the inspirations for Invisible Cities was Utopia, by Thomas More. Thomas More was a Tudor writer and statesman, eventually becoming lord chancellor of England under Henry VIII. Unfortunately, he opposed Henry’s decision to separate England from the Catholic Church and was executed for treason because of it. His 1516 book, Utopia, is the account of an imagined perfect country. (“Utopia,” a word Thomas More coined, is derived from the Greek for either “no place” or “good place,” depending how you convert that “U” into Greek.) Only one city, Amaurot, is described in detail because we are told that all the others are similar. So Calvino is filling in the gaps in More’s work by telling us about all fifty-five cities.
Hold the front page, though. When I looked at English translations of Utopia (it was written in Latin), they all said that there are 54 cities. This is rather curious. I don’t know if Calvino owned an Italian translation in which there are erroneously 55 cities. Or are we to understand that there are 54 cities in addition to Amaurot? One edition has a footnote that the 54 cities of Utopia “parallel the fifty-three counties that made up England and Wales in More’s time, plus one for London.” My Latin’s not up to much, but the original “quatuor et quinquaginta” does look awfully like 54, even to me. I don’t want to start an international incident here, so to keep the peace let me suggest that if Utopia’s 54 comes from 53 + 1, perhaps Invisible Cities is 54 + 1, in tribute.
Now that we have 55 cities (somehow), how shall we arrange them into chapters? Well, we have eleven kinds of city, with five of each kind. So we could have a structure like a rectangle, with each row representing a chapter, and each column a type of city, like this. Here, the numbers 1, 2, 3, 4, and 5 are the five cities of each type. The first column is “Cities & Memory,” the second is “Cities & Desire,” and so on to the eleventh column, which is “Hidden Cities.”
Chapter 1 here has City 1 of each type; Chapter 2 has City 2; and so on. Let’s not mince words: This structure is boring. Cycling through the same eleven elements in the same order each time does not give a feeling of progression and does not allow for different chapters to have different flavors.
A clue to the structure chosen by Calvino is given in the text. “My Empire,” says Kublai Khan, “is made of the stuff of crystals, its molecules arranged in a perfect pattern. Amid the surge of the elements, a splendid hard diamond takes shape.” What Calvino does is shift each of the columns successively downward like this:
In order to avoid chapters with just one or two cities in them, and to create a pleasing symmetry, Calvino gives Chapter 1 and Chapter 9 the first and last four rows of this structure. The intervening chapters now all have a 5, 4, 3, 2, 1 pattern, in which we see the fifth example of one type, then the fourth of the next, and so on. Each chapter, we visit a given type for the final time and introduce a new type. This mix of old and new, familiar and unfamiliar, gives a subtle momentum to the book’s framework.
We can also notice that if we take Chapter 1 and Chapter 9 together, they fit together to form a microcosm of the whole, with exactly four copies of the “5 4 3 2 1” motif. And there we are: “A splendid hard diamond takes shape.” It is a very elegant design. Indeed, Calvino himself said that Invisible Cities was one of the works with which he was most pleased, because in it he managed to say the “maximum number of things in the smallest number of words.”
There’s one more Easter egg in the book’s structure for the mathematical reader. In Chapter 8 of Invisible Cities, Kublai Khan meditates on the game of chess (of course it’s in Chapter 8, because chess is played on an 8 × 8 square): “If each city is like a game of chess, the day when I have learned the rules, I shall finally possess my empire, even if I shall never succeed in knowing all the cities it contains.” We have seen the patterns drawn in the book’s structure, and look—55 cities plus 9 chapters equals the 64 squares on a chessboard. Coincidence? Not a chance.
* * *
Like any successful franchise, the Oulipo has several spin-offs. “Oulipo,” remember, stands for “Ouvroir de littérature potentielle,” or “workshop for potential literature.” Any creative endeavor can have a “workshop for potential X,” or “Ou-X-po.” There’s the Oubapo (bandes dessinées—comic strips), the Oupeinpo (peinture—painting), and even the Oulipopo, the Ouvroir de littérature policière potentielle: the workshop for potential detective fiction. There are many potential potential workshops. What’s really needed, naturally, is an Ou-ou-X-po-po, and that would have to be followed by an Ou-ou-ou-X-po-po-po, and so on … but I digress.
If you like your Oulipo with a side order of murder, look no further than Claude Berge’s Qui a tué le Duc de Densmore? (Who Killed the Duke of Densmore?). Berge was a respected French mathematician who made significant contributions to graph theory, and also a long-standing member of the Oulipo. He loved both mathematics and literature (a man after my own heart) and found it hard to decide which should be his career focus: “I wasn’t quite sure that I wanted to do mathematics. There was often a greater urge to study literature.” Berge’s story about the murder of the Duke of Densmore not only uses a mathematical idea, it also uses a mathematical consequence of that idea. In this way it respects the second of the precepts put forward by Jacques Roubaud.
The first of these, if you recall, says that a text using a particular constraint must mention that constraint in some way. The second says that if a mathematical idea is used, then some consequence of that idea should also be incorporated. Berge’s story involves a famous detective trying to solve an old case—the Duke of Densmore had been murdered years ago, but the culprit is still at large. The pool of suspects is narrowed down to a group of seven lady friends (to put it coyly) of the duke. Each of them visited the duke’s house in the time leading up to the murder. Over the intervening years, they all claim to have forgotten the exact dates of their visits. But they do remember who else was there at the time. If two people met, then their visits must have coincided, if only briefly. What our detective ends up with, then, is a collection of intervals of time, and all he knows about them is which ones overlap.
This doesn’t seem a lot to go on. But there’s a clever way to visualize the connections in a situation like this—it’s called an interval graph. Graphs in this sense of the word are something like a map of a subway system—you have various points (subway stations, or time intervals), and you join together the ones that are connected (adjacent stops on a subway line, or time intervals that overlap). For an example, let’s take a literary family—the March girls from Little Women. Suppose that Meg, Jo, Beth, and Amy all visit their grumpy aunt. Meg says she saw Jo and Amy there, Jo says she saw Meg and Beth, Beth reports meeting Jo and Amy, and Amy sees Beth and Meg. All that information can be captured efficiently in this graph, which, if everyone is telling the truth, is an example of an interval graph:
Here’s the thing, though. This graph has a cycle Meg—Jo—Beth—Amy. But there is a theorem in graph theory that says that every interval graph is “chordal.” What this means is that somewhere in every cycle there has to be a chord—an intermediate line joining two of its points. If a graph doesn’t have that property, then it can’t be a true interval graph. Here, it means there must be a line either joining Meg to Beth, or one joining Amy to Jo. The inescapable conclusion, though it pains me to say so, is that at least one of the March girls isn’t telling the truth. Marmee will be so disappointed. With this example, we can’t prove who is lying. (My money’s on Amy.) But in the Densmore story, there are more suspects, and the graph has the property that there is exactly one person who, if we removed them from the graph, leaves the rest of the graph as a real interval graph. And what better reason to lie than that you murdered the duke? The detective knows the interval graph theorem and catches the killer.
* * *
As we have seen throughout this chapter, the Oulipian approach is at once playful and earnest—one of my favorite combinations. Life, as they say, is too important to be taken seriously. To close our guided tour of all things Oulipo, I’ve been inspired to invent my own piece of potential literature. I don’t think it has been done before, but if it has, then I congratulate my predecessors on their excellent bit of anticipatory plagiarism.
In 1976, Raymond Queneau published a short article titled “Foundations of Literature (after David Hilbert).” David Hilbert was an important nineteenth- and twentieth-century mathematician who did a lot to put mathematics, and especially geometry, on a firm, rigorous footing. In geometry, people had spent the best part of two thousand years trying to sort out once and for all what the hell was going on with Euclid’s parallel postulate, the axiom that says if you start with a line and take any point not on that line, there is exactly one line through that point that is parallel to the line you started with. Nobody could prove this, which is why it had to be taken as an axiom. But it’s less obvious than the other axioms. What people realized in the nineteenth century is that in fact there are versions of geometry—so-called non-Euclidean geometries—in which the parallel postulate actually doesn’t hold, meaning that some of the properties of Euclidean geometry may no longer follow. For example, take planet Earth. Draw a triangle by heading from the North Pole down to the equator, then traveling a quarter of the way along the equator, and then going back to the North Pole. This triangle has three right angles! Have we just destroyed geometry? No. What’s happened is that we have discovered that the geometry of curved surfaces is different from that of flat surfaces. Here’s another example—perspective drawing. In perspective drawing, parallel lines meet at a “vanishing point.” This is a bit of a downer if your definition of “parallel” is “never meeting.”
What Hilbert did, brilliantly, was to set up some rules, or axioms, of geometry that would be general enough to cover all these different examples, and many others, while keeping the things they all have in common. Here are two of Hilbert’s axioms:
1. Given two distinct points, there is always a line containing those points.
2. Given a line, any two points on the line uniquely determine that line.
Together, these rules say that any two points will lie on one, and only one, line. These axioms are true in standard geometry, but they are also true for the curved “lines” on a sphere and for the lines in a perspective drawing. In fact, there are lots of situations in which there is a useful concept of “lines” and “points.” The important insight is that as long as the axioms are true for our particular setup, however weird and wacky, then all the consequences of those axioms will also be true. So we can prove theorems that are true in a bunch of different scenarios, with no extra effort.
