Once upon a prime, p.15
Once Upon a Prime, page 15
Now, this is slightly silly of Bloom, because he doesn’t actually have to do the calculation to know that the 9th power of 9 (or 99), whatever it is (okay, it’s 387,420,489), is definitely less than the 9th power of 10, which is 1,000,000,000. So, the 9th power of the 9th power of 9 is going to be less than which is a 1 with 81 zeros after it. (It’s 196,627,050,475,552,913,618,075,908,526,912,116,283,103,450,944,214,766,927,315,415,537,966,391,196,809, if you want to know.) But let’s be kind and assume that what Bloom meant to say was not the 9th power of the 9th power of 9, but 9 to the 9th power of 9. It’s a curious fact about powers that if you try to do a power of a power, you have to be really careful what you mean by it. What is ? Does it mean 33 which is 27, raised to the power 3? This would be which is 19,683. Or does it mean 3 raised to the power 33 which is 327 or just over 7.5 trillion? With exponents, it really matters where you put your brackets:
In honor of Joyce, mathematicians have named numbers like Joyce numbers. The nth Joyce number is If you thought powers of two grew quickly, these Joyce numbers, being exponents of exponents, grow even faster. The first Joyce number, is 1. The second, is 16. The third is 7.5 trillion, and the fourth is already too long sensibly to write, with 155 digits. If Bloom had been thinking of the ninth Joyce number, then he wasn’t too far off with his estimate of the number of books required to contain it. It’s possible that Joyce had read about this number somewhere because in 1906 the mathematician C. A. Laisant proved that has 369,693,100 digits. Over the thirty-three volumes of a thousand pages that Bloom recalls, this would mean squeezing around eleven thousand digits onto each page—just about possible, with a tiny typeface, no line spacing, large pages, and narrow margins.
This is certainly not the only large number in the Joycean oeuvre, and it is a mathematically sophisticated example of the tradition of those “upper limit” numbers like 99 and 999 that we saw in Chapter 5, because the number is very large but not infinite. It is enormous, but bounded. We’ll leave to the more arcane academic journals the joys of deciphering the mathematics of Finnegans Wake, but I can’t help mentioning, in the context of symbolic numbers, the novel’s famous hundred-letter words, like this one: bababadalgharaghtakamminarronnkonnbronntonnerronntuonnthunntrovarrhounawnskawntoohoohoordenenthurnuk, which I’m sure you can tell is the sound of a thunderclap. Specifically, the one that reverberated around the heavens at the moment of the fall of Adam and Eve. There are ten of these “thunder words,” but in fact they don’t all have exactly one hundred letters. The first nine do, and then the final one has 101 letters, making a grand total of 1,001, another symbolic number with many cultural resonances.
Coming back to “Ithaca,” Stephen departs as he entered, with geometry:
How did they take leave, one of the other, in separation?
Standing perpendicular at the same door and on different sides of its base, the lines of their valedictory arms, meeting at any point and forming any angle less than the sum of two right angles.
This is a conscious mangling of Euclid’s fifth postulate: If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.6 If the men had remained parallel, as they were at the start of the chapter, then these interior angles would add up to 180°, which is two right angles, and the lines would not be able to meet. Or at least they wouldn’t in standard Euclidean geometry. Joyce knew that kinds of geometry had been discovered in which the parallel postulate does not hold, but the setup has given us parallel lines, so the mathematical catechism has resulted in a contradiction—another in-joke from Joyce for the mathematically inclined.
* * *
For the writers in this chapter, mathematics is more than a way to communicate: it is a vital way of understanding the world. Mathematics has meaning, whether you are a village carpenter like Adam Bede or a deckhand like Ishmael. It is a refuge, a solace. Still, there are risks. Melville shows us the tragic outcome of assuming, like Ahab, that statistics give us complete control, and Joyce’s absurdist calculations remind us that just because a number sounds impressive, that doesn’t make it correct. The novels in this chapter have shown life through the prism of mathematics, from the smallest scale to the largest—from late-night rambles in Dublin to the entire sweep of human history. For these novelists, mathematics is the key.
7
Travels in Fabulous Realms
The Math of Myth
In Jonathan Swift’s 1726 novel, Gulliver’s Travels, the intrepid traveler Lemuel Gulliver visits the miniature land of Lilliput. He gives lots of detail about the precise dimensions of the people there, and he describes how the king of Lilliput arranges for Gulliver to be fed:
His Majesty’s mathematicians, having taken the height of my body by the help of a quadrant, and finding it to exceed theirs in the proportion of twelve to one, they concluded from the similarity of their bodies, that mine must contain at least 1724 of theirs, and consequently would require as much food as was necessary to support that number of Lilliputians.
While we do not judge satirical novels by the plausibility of their science, this is still an irresistible challenge. Where does this 1,724 come from, and is it correct? Spoiler alert: no, it’s not, and if Mr. Gulliver is going to bring my Lilliputian colleagues’ academic integrity into question with a howler like this, it’s my duty as a mathematician to defend them. Previously in this part of the book, I’ve shown you how mathematics makes itself visible in fiction in several ways, from symbolic pattern numbers to lovely mathematical metaphors. In this chapter, we’re going to explore another way that mathematics can be deployed: the narrative technique that I call performative arithmetic. As in the calculation above, it’s often used when the narrator is recounting something that may appear unbelievable. A dash of solid fact, in the form of a mathematical calculation, gives plausibility to proceedings.
This is exactly what’s going on when Gulliver visits the floating island of Laputa later in his travels. He gives us another calculation. The island is, he says, “exactly circular, its diameter 7,837 yards, or about four miles and a half, and consequently contains ten thousand acres.” We readers can check this calculation for ourselves. An acre is 4,840 square yards, so ten thousand acres is a good approximation for the amount of land such a circle can contain—to the nearest whole number, it is 9,967 acres. The sleight of hand here is the elision between the verifiability of the arithmetic and the verifiability of the narrative. The mathematics is (roughly) correct, but that does absolutely nothing to establish that such a circular island exists. The spurious precision of 7,837 yards is probably designed to increase the illusion that this is a report of reality. It actually makes the calculation less accurate because the much rounder number of 7,850 would have given an area of 10,000 acres almost exactly—less than half an acre short.
In this chapter, I’ll give you the tools to turn the tables on some literary logic and ask: Does this really stack up? We can check the working of the Lilliputian mathematicians and can laugh with Voltaire at the self-aggrandizing antics of the humans who turn out to be the tiniest of creatures in comparison to the giant visitor Micromégas, from his planet near Sirius. Are these fantastical lands possible, and what would life be like for their inhabitants? I’ll show you the math that proves just how magical these creatures must be.
As Peter Pan tells Wendy, “You see children know such a lot now, they soon don’t believe in fairies, and every time a child says, ‘I don’t believe in fairies,’ there is a fairy somewhere that falls down dead.” Not wanting to have any fairy slaughter on my conscience, I must emphasize that if anything I say makes it seem as if flying horses or giants or tiny people can’t exist, all I mean is that if you do encounter one, something beyond our normal laws is occurring. As we’ll see, creatures like the giant spiders that live in the Forbidden Forest at Hogwarts must be highly magical beings to defy all the mathematics that might otherwise “prove” they can’t exist. Which is absolutely fine by me (as long as I don’t have to be in a room with one).
* * *
I want to talk about giants first because my feeling is that, over the course of history, they have been taken more seriously than other fantastical beings as creatures that could potentially exist. There are several giants in the Bible, for example. In children’s literature we meet Roald Dahl’s beloved BFG, the half-giant Hagrid from the Harry Potter series, and many others. Giants have been popular in satirical novels, too. The French author François Rabelais (who gives us the adjective Rabelaisian, meaning “bawdy and crude”) is most famous for his Life of Gargantua and Pantagruel, a five-volume work concerning two giants and their exploits. To give you just a hint of the cheerful tone of voice in which it’s written, the full English title of the volume in which we first meet Pantagruel is The Horrible and Terrifying Deeds and Words of the Very Renowned Pantagruel King of the Dipsodes, Son of the Great Giant Gargantua. The exaggerated size of a giant emphasizes our own inescapable physicality, so it’s a way to poke fun at our occasional coyness about ourselves. Rabelais enjoys being ridiculous. Gargantua (from whom we get the word “gargantuan”) is born by climbing out of his mother Gargamella’s ear, and it only gets sillier from there. The books are full of numbers and calculations about things like how much fabric is needed for Gargantua’s codpiece (sixteen and a quarter ells, or about twenty yards, since you ask), but they are thrown about with gleeful abandon, much as we might now joke that something costs a million bajillion dollars.
None of the numbers given to describe the size of Gargantua make any attempt at consistency—it’s all just exuberant silliness. We hear that the baby Gargantua’s milk was supplied from a herd of “seventeen thousand nine hundred and thirteen cows of the towns of Pautille and Brehemond.” For his shoes “were taken up four hundred and six ells of blue crimson-velvet, and were very neatly cut by parallel lines, joined in uniform cylinders.” He combs his hair with a nine-hundred-foot comb whose teeth are entire elephant tusks. When Gargantua visits Paris and relieves himself in the street, he accidentally drowns “two hundred and sixty thousand four hundred and eighteen, besides the women and little children.” In more numerical bawdiness, when Gargantua’s wife dies, he thinks fondly of a certain “little” part of her anatomy “yet it had in circumference full six acres, three rods, five poles, four yards, two foot, one inch and a half of good woodland measure.” This is all good fun, but because Rabelais doesn’t tell us how big the giants are, it’s pointless even to ask whether they could exist in real life, because we don’t have enough information to make a reasonable assessment.
Let’s visit Brobdingnag, then, because there we have very precise information. Brobdingnag, the land that Lemuel Gulliver visits after Lilliput, is a kind of inverse to that country, because everything in Brobdingnag is twelve times as big in every dimension as in our world. This is rather convenient in that it means anything that would normally be an inch long (a wasp, for example) is now a foot long. So it’s not just the people that are giants, but the plants and animals too, and even the weather. On one occasion, Gulliver is unlucky enough to be caught outside in a hailstorm: “I was immediately by the force of it, struck to the ground: and when I was down, the hailstones gave me such cruel bangs all over the body, as if I had been pelted with tennis-balls.… Neither is this at all to be wondered at, because nature, in that country, observing the same proportion through all her operations, a hailstone is near eighteen hundred times as large as one in Europe.”
Where does this “near eighteen hundred” come from? Well, we know that every dimension is multiplied by 12. So the hailstones are twelve times as long, twelve times as wide, and twelve times as high as ours. This means their volume is not twelve times as much, but times as much, or, “near eighteen hundred” (though really it’s nearer seventeen hundred). This is the start of the problem with giants. If you scale something up in every dimension by the same factor—here it’s 12, but let’s say it is some fixed k—then the volume would change by a factor of which we write in mathematical notation as k3 because there are three k’s multiplied together. In other words, the volume changes with the cube of the scaling factor. Meanwhile, any area connected with the object will only change with the square of the scaling factor. To see what I mean, have a look at the diagram below. I’ve shown what happens to a box if we enlarge it in each dimension by a factor of 2. Imaginatively enough, it has width w, depth d, and height h.
That means the box has volume v, where Now, if we enlarge by a factor of 2, the new, bigger box has width 2w, depth 2d, and height 2h, That gives it a volume of So yes, this agrees with our reasoning, because 8 = 23. On the other hand, the area A of the base of the original box is w × d but the doubled box has base area 2w × 2d = 4A and 22 = 4.
I said that this squaring factor was true of any area connected with the object. What I meant by that is that it’s not just the base area that increases by the square of the scaling factor, but, for example, the area of any cross section through the box, and also its surface area, that has this property. We don’t need to work out the exact formula for the surface area to know this (but if you want to, it’s ); it’s enough just to realize that the calculation involves adding together a bunch of areas that involve two of the measurements multiplied together, and so doubling each of the measurements will multiply the total by 4. In the more general case, the box enlarged by a factor of k will have volume k3V and area K2A This fact is known as the square-cube law.
Here’s where things get nasty for giants. When humans are moving around, the weight of their bodies must be supported by their skeletal structure. Studies show that the human femur (thigh bone) will break under about ten times the pressure it normally has to carry. You might remember from high school science that pressure is the force per unit area. That is, pressure = . The area here is the cross-sectional area of the femur. The force exerted comes from our mass being pulled down by gravity, and our mass is roughly proportional to our volume. All this means that the pressure on our femurs is proportional to . Now, if we scale up the human body by a factor of k, the square-cube law tells us that the volume increases by a factor of k3 but the area increases by a factor of only k2 Putting this all together shows that the pressure on the scaled-up human’s bones will be proportional not to the original , but to , which, if we cancel a couple of k’s, is k × . In other words, the pressure on the scaled-up human’s bones is k times the pressure on our bones. The Brobdingnagians are twelve times the size of Gulliver. That means the pressure on their bones, just standing still, is twelve times the pressure on his bones. But bone can take only ten times normal pressure before breaking; the Brobdingnagians’ bones would break as soon as they tried to move.
So the Brobdingnagians cannot really exist. The same is unfortunately true for the BFG, and for Giant Pope and Giant Pagan of John Bunyan’s Pilgrim’s Progress, who are an estimated sixty feet tall, ten times the height of the hero, Christian (unless divine intervention is involved—all bets are off with an omnipotent deity). King Kong, if he could exist at all (the films are far from consistent about the size he is supposed to be), would be incredibly weak—he would barely be able to support his own weight, never mind jumping around on skyscrapers and punching airplanes out of the sky. Fay Wray could probably beat him in a fight!
There is some hope, though, for slightly smaller giants. In the Harry Potter books, Rubeus Hagrid, Keeper of the Keys at Hogwarts School of Witchcraft and Wizardry, is a half-giant. He’s described as being twice the normal height but, crucially, three times the normal width. Assuming that he’s also three times the normal depth, that would mean the cross-sectional area of his bones is 9 (or 3 squared) times ours, but his mass is only 18 times ours, not 27 times. That would mean the pressure on his bones is twice what it is on ours. He could definitely still walk around, and maybe even run, but he would likely be prone to broken bones, and he should certainly not take up skipping. The same goes for King Og of Bashan, a biblical giant whom Moses meets in the book of Deuteronomy. His exact dimensions aren’t given, but we are told that his bed was 13 feet long, so he was perhaps scaled up by a factor of two, thereby doubling the pressure on his bones. Again, survivable, but he wouldn’t be a mighty warrior.1
Before we move on, I want to tell you about Micromégas. It’s a short satirical novel by Voltaire, which I am both grateful and annoyed to have heard about accidentally. I’m one of those people whose brain won’t shut up even when I want it to, and so I often listen to an audiobook or the radio to drift off to sleep. Of course, it can’t be anything too exciting. The audiobook company Audible, realizing that lots of people do this, released a series of “Bedtime Stories for Adults” that were deliberately not very exciting. I am appalled to have to tell you that the second of these was A Short Account of the History of Mathematics, by W. W. Rouse Ball. How dare they! In with this, and a book on quilt collecting, was Micromégas, by Voltaire, which I’d never heard of. Imagine my delight when it turned out to be the story of a giant named Micromégas, from a planet orbiting Sirius, who visits Earth. What’s more, Voltaire describes his exact size and talks about the calculations that mathematicians can make to determine the size of his home planet. So, Monsieur Voltaire, you brought mathematicians into this. Let’s see how your calculations fare.
