Irrational Numbers

On the pleasure of procrastinating with absurd back-of-the-envelope calculations.

Photo illustration of a church, bishop's hat, grapefruit, and the Grand Canyon.
Photo illustration by Slate. Photos by Daniel Heighton/iStock/Getty Images Plus, Charles on Unsplash, and Sonaal Bangera on Unsplash.

Rabbit Holes is a recurring series in which writers pay homage to the diversity and ingenuity of the ways we procrastinate now. To pitch your personal rabbit hole, email

What the heck is all this? All this stuff, moving about, fizzing and whirling. Now still, now hurtling past your head. It’s a planet or a pinball or some invisible atom. This, my friends, is the physical world. Welcome. It’s glorious and bewildering. And best of all, we can sort of make sense of it! The downside for me, a scientist, is that the possibility of that sense-making is horribly distracting.

There’s a rich seam in the salt mines of Science, a glittering vein of purest procrastination, and you find it like this. Divide up the world into questions that probably have answers, and those that probably don’t. Throw away that second half—they have no place here—then shut up and calculate! You need something that you can work out without too much hassle. You must be very clear about your assumptions. Very basic math is your trusty steed, your curiosity is the open road. These calculations are rough, but correct. Imprecise, but true. More often than not, it’s far more satisfying than doing “real” work.

To get a good answer, you need to ask a good question. This isn’t difficult—try a “how many” or a “how big.” The best questions unfold as you go, revealing layer upon layer of complexity as you probe the assumptions, only to find you’ve spent the whole afternoon scribbling wildly on the backs of envelopes.

Let’s do one now. Everyone knows that the world is made up of atoms, but it’s really hard to know how big they are. Things are complicated slightly by the fact that different atoms are different sizes. But a value of 100 picometers is a good rough estimate for the size of an atom.
Great. What on earth does that mean? On a clear day, you can see about 12 miles. Obviously, again, this will depend on a whole bunch of things, but 12 miles is about right. Now, imagine you’re holding a pin. Look out to the horizon, the landscape stretched before you. The relative size pinhead in the landscape is the same as atoms on the head of the pin. Atoms are tiny, but not unimaginably small! These are the sort of revelations I crave.

But how about something a little more complex? Things got especially intense one rainy night in Glasgow. Me and my friend Sam were dining at Jist Misto, a fine fusion restaurant. While enjoying the strangely named Grand Canyon burger, Sam said he’d always wondered how many grapefruit would fit into York cathedral, the famous Minster. Naturally, I pounced on the opportunity. This is exactly the kind of question that has an answer, and basic math will get us there. So away we went.

First, the volume of a grapefruit is fairly easy to get: just under a liter, or I-have-no-idea gallons please consider adopting the metric system like the rest of humanity (OK, OK, it’s ¼). York Minster’s volume is harder to estimate. You can do a simple length times width times height, but this doesn’t take into account those nooks and crannies. So I sent a tweet to the Minster, and the Minster didn’t know. I guess that’s fine, you probably don’t know your own volume either. So, disregarding the crannies, we get just over 300,000 m3—and a grand total of 350 million grapefruit per Minster!

But why stop there? A natural follow-up, given those burgers, was how many Minsters can you fit inside the Grand Canyon. Luckily enough, the National Parks Service’s website suggests “The volume of the Grand Canyon is estimated to be 5.45 trillion cubic yards.” Converting, as we must, to metric and dividing by the volume of the Minster gives—drum roll please—a mere 13 million! The canny reader will note I’ve wildly glossed over some practicalities here. For one, there’s the idea of packing fraction: If you pack balls into a volume, they can’t fill all the space. But if you think this is a problem for packing grapefruit, you’ve never tried to pack Minsters. So either way, the grapefruit win.

Or do they? What about the practicalities? How do we actually get grapefruit into the Minster? Picture the scene. Here’s the Archbishop of York atop a stepladder, carefully piling grapefruit in neat pyramids in the North Transept. They tessellate nicely, and up to about 6 feet he has no problems. But the farther he goes, the harder it gets. The ones at the bottom are starting to compress under the weight of the ones above. The roof of the Minster is still way off. Outside, truckloads of grapefruit are souring in the midday sun. And the same goes for Minsters. How do we get Minsters into the Grand Canyon? Do we assemble them on-site? Does each one need an archbishop to actually be a Minster?! This is the horror and beauty of it all.

At its best, what starts out as a little harmless procrastination can end up inverting your entire worldview. The physicist Richard Feynman called it “the pleasure of finding things out.” Simple calculations, straightforwardly communicated, have the power to be understood by almost anyone. Like the very best of rabbit holes, thinking about the physical world is a potentially infinite adventure. The only hope you have of not falling deeper in is to simplify your assumptions, but where’s the fun in that?