Tag Archives: number

Uncategorized

How Many Novels Can There Be?

I like reading. I like writing. When you’ve been writing for a while, you start to get really obsessed with word counts. Anybody you talk to about publishing something you’ve written will want to know your word count. For short fiction, you sometimes get paid by the word. And the number of words in the thing you’ve written determines whether it counts as a short story, a novella, a novel, as War and Peace, or as an encyclopedia.

Every year, I participate in National Novel-Writing Month. Unless, you know, I don’t feel like it. But I’ve participated more years than not, and I’ve produced a surprising number of novels. Every single one of them terrible, but that’s not NaNoWriMo’s fault. The goal in NaNoWriMo is to write a novel of at least 50,000 words in 30 days. And I got to thinking: how many novels that length are there?

Well, in the English language, there are somewhere between 100,000 and 1,000,000 words. But you’ll be able to understand 95% of everything written in English by knowing only the 3,000 most common ones. After all, even though it’s a valid word, people generally don’t go around calling each other antipodean anymore.

The question is: “How many 50,000-word novels are possible, using mostly the 3,000 most common words?” The naive answer is to allow each word to be any of those 3,000, which means the number of possible novels is 3,000^(50,000). That’s 1.155 x 10^173,856. You’ll be happy to know that this number is so large that, when I tried to copy and paste the full thing into this article, it crashed my browser.

Of course, this will include novels that consist entirely of the sentence “Anus anus anus anus anus!” over and over again, which is so avant-garde it makes me want to go pee on Samuel Beckett. The list will also contain more coherent, although still somewhat dubious works, like Stuart Ashen’s peerless desk reference, Fifty-Thousand Shades of Grey. But Fifty-Thousand Shades of Grey is actually constructed of coherent sentences. (Well, one coherent sentence, at least…) Most of the novels in this ridiculously long list will be more along the lines of “Him could carpet but also because you die but but the but the but the butt.”

We’re working from a flawed assumption: that a text is just a bunch of words stuck together. But unless you’re James Joyce (or, to a lesser extent, Stephanie Meyer), that’s not how it works. A novel is a bunch of words stuck together in a particular way. Although “that that” is grammatically valid (even though it looks weird on the page), “the the” isn’t, and “centipede cheese carpet muffin” is the kind of thing I say when I haven’t been getting enough sleep.

We’ve been working from the assumption that any word is equally likely to follow any other word. That is, that all word-pairs are equally likely. They’re not. “Our way” is a lot more common than “our anus,” for instance. Naively, the probability of any two-word combination is (1 / 3,000)^2, or 1 in 9,000,000. To put it another way, there are 9,000,000 two-word pairs, 25,000 of which would make up our nonsensical novel. It’d be much closer to reality to assume that, on average, there are only 50 words that make sense after a given word (the number will be much higher (in the thousands, I’d imagine), for words like “the”, and lower for words like “hoist.”) So, in reality, there are only 150,000 two-word combinations that make sense.

We could extend this to three-word combinations, but there are two problems with that: 50,000 isn’t evenly divisible by three, and that repeating decimal will drive me crazy. More importantly, the longer your word-block, the more words become possible at the end, until you’re getting close to 3,000 possibilities again. For example: “The” could be followed by any noun in our 3,000-word list. “The man” must be followed by a verb, the start of an adjective phrase (example: “The man I met last summer“), or something like that. “The man talked” will likely be followed by a word like “to” or “about.” But there’s an enormous range of things that the man could be talking to or about, so pretty much any noun or participle is fair game, bringing the number of possibilities back up into the thousands again.

So how many novels can there be? Well, the upper bound is probably (as we’ve seen), (3,000 * 50)^25,000, which is 1.912 x 10^129,402. That’s still a number so large there’s no name for it, but it’s smaller than our first number by almost fifty thousand orders of magnitude, which is something.

But let’s take it one step further. To simplify the math, I’m going to skip right to four-word combinations. And let’s say that any two-word combination forms the start of a phrase, and that the third word in the phrase can only be one of 10 words, on average. And, to take into account the fact that the number of choices start rising again with a long enough phrase, let’s say the fourth word can be any one of 500 words. The number of possible 50,000-word novels is now (3,000 * 50 * 10 * 1,000)^(12,500), or 1.382 x 10^114,701. So we’ve chopped off another ten thousand orders of magnitude. Still, that’s a big number. And, although I don’t have the math or linguistics background to prove it, I’m guessing that’s pretty close to the number of actual, sensible novels you could construct with 50,000 words: it takes into account the rough structure of the English language. This is related to the idea of a Markov Chain, which is a mathematically-formal way of saying “where you’re likely to go next depends on where you’re at now.”

For your amusement, I’m going to back up this post, and try to copy and paste (3,000 * 50 * 10 * 1,000)^(12,500) just below. If you see a horrific salad of numbers, you’ll know it worked. If you see an apology, you’ll know it crashed my browser again. Wish me luck!

…

Sorry. It didn’t work. Browser crashed again. But that’s probably good news for you, the reader, since, when I pasted the number of possible sensible novels into my word processor, it produced a document 32 pages long consisting of nothing but digits in 12-point Helvetica. I think that’d make most people’s eyes bleed. Or explode. Or sprout wings and fly away.

The moral of this story is: don’t worry about machines taking over the writing of novels. If a computer could output one word of its current novel every Planck time (which is generally agreed to be close to the shortest time interval that makes sense in our physics), the time it would take would be larger than the current age of the universe. And that’s an understatement. It would actually be so much larger than the current age of the universe, that if I were to express it as a multiple (in the same way I say 10^24 is a trillion trillion times larger than 1), then I’d have to write out the word “trillion” 9,558 times just to express it. If I allow the convention that 1 googol googol is (10^100) * (10^100), or 10^200 times bigger than 1, then I’d need to write “googol” over 1,100 times. There is simply no good way to express the size of this number. It’s 10^110,000 times larger than the age of the universe in Planck times, the diameter of the observable universe in Planck lengths, and the number of particles in the universe.

Boy oh boy. I started out talking about novels, and now I’m getting into numbers that trip the circuit breakers in my brain. Math can be scary sometimes. And you wanna know the scariest thing? There are numbers, like Graham’s number and the outputs of the Ackerman function for inputs larger than (6,6), that make the number of possible novels look exactly like zero by comparison, for any practical definition.

…I need to go lie down now. Although I’m probably going to come back later and talk about really enormous numbers, because part of my brain seems to want me to have a stroke.

Standard
Uncategorized

Give me a Trillion of Everything!

I think one trillion is my favorite number. The word “trillion” just has a nice sound to it. It could be the name for a classy British stripper. But it’s also a good number for bridging the gap between “numbers human beings can kinda get their heads around” and “numbers that make human beings’ eyes glaze over.” One trillion certainly isn’t as intuitive as three, or one thousand, but it’s more intuitive than something like 10^80. So today (in a slight nod to an earlier post), I want to see what it’s like to have a trillion of various things just lying around.

Gold atoms. Who wouldn’t want a trillion gold atoms? If you just stack them in a cube, that cube will have edges 10,000 atoms long. That works out to a cube 2.72 microns on an edge. Okay, so a trillion atoms of gold isn’t as impressive as I thought. I don’t think it’ll ever catch on as a currency, seeing as if you handled it too roughly, it’d slip between the cells of your finger and disappear into your bloodstream.

But actually, gold atoms don’t stack in a neat cubic lattice. They, like many other metal atoms, arrange themselves in a face-centered cubic lattice, which is also the most efficient way to stack oranges or cannonballs (hands up everybody who can’t help but think of Kurt Vonnegut). So a real trillion-atom gold bar (still measuring 10,000 atoms along every edge) would actually be 2.72 microns long, 2.36 microns wide, and 2.22 microns thick. My entire investment could be eaten by a particularly zealous amoeba.

Bacteria. E. coli is a useful bacterium. For one thing, it’s easy to grow in the lab, which lets scientists observe bacteria without the hassle of giving them weird shit like quaternary ammonium salts or whatever those other stubborn bacteria like. For another, their dimensions are conveniently close to nice, round numbers: the average E. coli bacillus is about 1 micron in diameter and 2 microns long. So what if you had a trillion of them? Well, let’s stack them in a 10,000 x 10,000 x 10,000 cube again. This bacterial block would be one centimeter wide, one centimeter high, and two centimeters long. It would be a slimy, milky mass the color of bad mozarella. You could eat it on a cracker. (Sorry.)

Human cells. I don’t like human cells, because they’re all lumpy and irregular and hard to measure. Lucky for me, there’s a kind of tissue called simple cuboidal epithelium, found in the ovaries, the thyroid glands, and, as pictured below, the little pee-secreting tubules of the kidney:

(Source.)

In addition to being nicely-shaped, keeping our bodies from filling with urine (that’s how kidneys work, right?), and having a cool name (“Cuboidal” is a great word. Even better than “furan.”), they’re also pretty regular in size: about 10 microns on an edge. And being cubes, I don’t have to bother with all that hexagonal-close-packing math. I can just multiply 10 microns by 10,000. One trillion human cells would form a gelatinous cube (no, not that kind) ten centimeters on an edge. It would be a horrible meaty cube weighing around a kilogram. You could eat the whole thing if you were really hungry (it’d weigh the same as a 35-oz porterhouse). I don’t recommend that, though: I tried some cooked kidney once, and I couldn’t quite get over the fact that, with every bite, I got the stench of one of those public urinals that hardly ever gets flushed.

Eugh. Let’s move on to something less horrible.

(Left picture source. Right picture source.)

I might have been lying about that. These are Dust mites, which I never realized bear a very suspicious resemblance to the headcrabs from Half-Life 2. I honestly think the one on the right is cute, but then again, I’ve probably got so many holes in my brain a squirrel could use it to store nuts for the winter. I had to include both of these pictures because the one on the right is a classic well-proportioned dust mite (lovely eyeless arachnids that they are). But the one on the left is posing with a motor and gearbox whose gears are the size of blood cells. How could I not include that?

But back to the matter at hand: I want a trillion dust mites. (Leave them at the corner of 7th and Walthrop and we’ll release your mother unharmed.) An average dust mite measures around 300 microns wide, 300 microns tall, and 400 microns long. My trillion-mite brick measures 2 meters by 2 meters by 4 meters. In other words, I’ve now got a heap of creepy-crawlies big enough to fill a medium-sized closet. I’d be concerned about them giving me allergies, but imagining a cube of mites larger than me has made me pee my pants, which is a more immediate concern.

Peas. (I was tempted to use a horrible pun as a segue, but the thought made me throw up in my mouth.) I really like peas. My parents never had to tell me to finish my peas. I ate the little buggers right up. Plus, they’re another conveniently uniform object we can use to demonstrate just how massive a trillion is. According to this USDA pea-grading document (the existence of which pleases me), a green pea averages 8 millimeters in diameter. Since peas are irregular and don’t stack as neatly as gold atoms, I’m going to come up with the volume of my trillion-pea cube a different way. We’ll assume they’re roughly hexagonally-close-packed, which means that only 74% of the volume will be occupied by peas. The rest will be air. We end up with about 362,300 cubic meters of peas. They would form a cube big enough to span the width of a football field (either kind), and most of the length: 71 meters on an edge. The peas would weigh 180,000 metric tons, apparently half as much as the Empire State Building. Here’s the kind of vehicle you’d need to transport that much stuff:

(Source.)

I’m now imagining an alternate reality in which the Persian Gulf is a global center of pea production. In which the economic fate of the world is determined by the price of a barrel of green peas. In which pea tankers with armed guards criss-cross the Pacific. It wouldn’t be any weirder than this reality.

 People. There are not a trillion human beings on Earth. According to most of the estimates I’ve ever read, the Earth can’t support a trillion people. But that doesn’t mean we couldn’t make a trillion clones and plop them down. Let’s be all sociological and assume that every human being belongs to a family of five, and that one hectare (10,000 square meters) is enough space for their house and enough crops to keep them alive. (One hectare is about the area of the grassy center of a standard athletic track, or the area of a rugby field, or the area of a square that could enclose the base of the Statue of Liberty.) So we have 200 billion families, each requiring 10,000 square meters.

People can live in the ocean, right? And you can grow a pretty good selection of crops in Antarctica, right? Because it turns out that a trillion humans requiring a fifth of a hectare each would require thirteen times the Earth’s land area (arable or not). They’d require ten times the Earth’s surface area. Probably not going to happen, although I really, really like the image of farmers in wetsuits paddling around their kelp fields with rakes.

But if it was just a trillion-person crowd (doomed to starvation), what would that look like? We’ll assume they’re all pretty friendly, and so they only require a square one and a half meters on a side. Whether they assembled in a square, a circle, or some other roughly-symmetric shape, the crowd would be about 1500 kilometers across. That’s a quarter of the size of the United States, China, or the Roman Empire at its peak (thanks, Wolfram Alpha).

Now, a human being breathes something like 11,000 liters of air in a day. That means our crowd is going to be breathing the volume of Lake Superior daily. An average resting adult requires something like 210 milliliters of oxygen per minute. Taken together, they’d require sixteen Amazon Rivers’ worth of oxygen flow. Much to my surprise, they probably wouldn’t suffocate. If we assume each person has access to the air in their square from sea level to the “death zone” (an altitude of about 8,000 meters, where the oxygen concentration is low enough to become acutely lethal), they’d all be able to breathe for about 20 years. Plenty of time for photosynthesis to replace the oxygen. And plenty of time for that crowd to become very, very smelly. And hungry. And violent. We probably shouldn’t have made a trillion clones. I blame you for giving me the idea.

And now, we’ve hit another limit. We’re already talking about continent-scale collections of objects, and like I said, human beings aren’t all that good at understanding things on really large scales. So I’ll stop at our trillion-person Woodstock before I start imagining the stink of two trillion unwashed feet.

Standard