Episode 02 - Powers Of Ten

Tomlin Hawthorne

A quick one before we start. No calculating, just your gut.

Tomlin Hawthorne

A million seconds ago. Roughly when was that?

Tomlin Hawthorne

And a billion seconds ago?

Tomlin Hawthorne

A million seconds ago was about twelve days back. You can probably remember what you had for dinner.

Tomlin Hawthorne

A billion seconds ago was nearly thirty-two years. Depending on your age, you were at school, or you were not yet anybody.

Tomlin Hawthorne

If those two answers felt further apart than you expected, good. That gap, between how a billion sounds and how big it is, is today's subject.

Tomlin Hawthorne

Last time, a librarian measured the Earth with a stick. And I left you with two drawings, a round Earth with a distant Sun and a flat Earth with a near one, and asked what else you would look at to choose between them. If you said ships going over the horizon, or the round shadow on the Moon in an eclipse, you have the idea. One measurement, many witnesses.

Tomlin Hawthorne

Today's question is simpler to ask and harder to feel. How big is big, how small is small, and where in all of it are we?

Tomlin Hawthorne

I want to begin with Jonathan Swift, which is not where physics programmes usually begin.

Tomlin Hawthorne

In Gulliver's Travels, Gulliver washes up first in Lilliput, where the people are six inches tall, and then in Brobdingnag, where they are as tall as church steeples. And the joke, the deep one, is that it is the same society both times. Seen from above, human politics is ridiculous: tiny creatures going to war over which end of an egg to crack. Seen from below, human bodies are revolting: Gulliver is put on a giantess's lap and can't cope with the pores.

Tomlin Hawthorne

Swift understood something that physicists took a while to say out loud. Change the scale, and you change what you see. The world isn't just bigger or smaller at different magnifications. It's different.

Tomlin Hawthorne

Hold that. We'll come back to why Swift's giants couldn't actually exist, which would have annoyed him.

Tomlin Hawthorne

First we need a way to talk about size that doesn't break our heads.

Tomlin Hawthorne

The trouble with numbers like a billion is that we hear them as amounts, on a line. A million is a lot. A billion is, what, a lot more. The words are nearly the same length, so they feel like neighbours.

Tomlin Hawthorne

So physicists do something sensible. They stop counting how much and start counting how many times you multiplied by ten. A thousand is three steps. A million is six. A billion is nine. Each step is the same size of jump: ten times bigger.

Tomlin Hawthorne

If you play an instrument, you already think this way. Go up an octave and you have doubled the frequency. Up another, doubled again. The keyboard looks like even steps, and each step is a multiplication. Your ear does this on its own. So does the scale for earthquakes: a magnitude seven is not a bit worse than a six. It shakes the ground ten times harder.

Tomlin Hawthorne

So let's walk the ladder. Every rung is a factor of ten, and I'd like you to picture them.

Tomlin Hawthorne

Start with you. Call a person about a metre, give or take. That's rung zero.

Tomlin Hawthorne

Go down. A tenth of a metre is your hand. A hundredth is a fingertip. A thousandth, a millimetre, is a grain of coarse sand, and that's about where your eyes give up.

Tomlin Hawthorne

Three more rungs. A millionth of a metre. Now we're at the size of a bacterium. A red blood cell is a little bigger. This whole region was invisible to every human who ever lived until the 1670s, when a Dutch cloth merchant called Leeuwenhoek ground a lens good enough to see it, and found that a drop of pond water was crowded. He called them little animals. He wrote to the Royal Society about it, and at first they did not believe him.

Tomlin Hawthorne

Keep going. A billionth of a metre. We've passed viruses. We are at the width of a DNA strand, and a few rungs on we reach an atom: one ten-billionth of a metre. Ten rungs below you.

Tomlin Hawthorne

You'd think that's the bottom. It isn't close.

Tomlin Hawthorne

The atom is almost all outskirts. Its mass sits in a nucleus at the centre, and to get to the nucleus you go down another five rungs. Five. If you blew an atom up to the size of a cathedral, the nucleus would be a fly, somewhere near the altar. And inside that are protons, about a millionth of a billionth of a metre across. Fifteen rungs down.

Tomlin Hawthorne

Right. Back to the surface. Deep breath. Now go up.

Tomlin Hawthorne

Ten metres, a house. A thousand, a short walk. Seven rungs up, ten thousand kilometres, and you have the Earth, which Eratosthenes did with a stick.

Tomlin Hawthorne

Eleven rungs: the distance from here to the Sun. Light, the fastest thing there is, takes eight minutes to cross it.

Tomlin Hawthorne

Sixteen rungs: the nearest other star. Light takes four years. There is a great deal of nothing between stars. If the Sun were a grapefruit in London, the next star would be another grapefruit in Newfoundland, and the space between would hold a few grains of grit.

Tomlin Hawthorne

Twenty-one rungs: our galaxy, the Milky Way. A few hundred billion stars in a slowly turning disc.

Tomlin Hawthorne

And about twenty-seven rungs up is the edge of what we can see at all. The observable universe.

Tomlin Hawthorne

So here's the whole thing. Fifteen rungs down to a proton. Twenty-seven up to the horizon. Forty-two steps of ten, top to bottom. And we are not at either end. We're in the middle of the ladder, a bit nearer the bottom.

Tomlin Hawthorne

I find that a strange piece of news, and I'm not sure what to do with it. It is tempting to feel either very small or secretly central. I'd resist both for now. Just notice it. We are mid-sized. There is as much world below us as above.

Tomlin Hawthorne

Now do it for time, because time is worse.

Tomlin Hawthorne

The universe is 13.8 billion years old. That sentence slides straight off the brain. So here's a device from the astronomer Carl Sagan. Squash the entire history of the universe into one calendar year. The Big Bang is the first instant of the first of January. Right now is midnight on the thirty-first of December.

Tomlin Hawthorne

On this calendar, our galaxy takes shape in the spring. Then a long quiet stretch. The Sun and the Earth don't form until early September. Two-thirds of the year is gone before our planet exists.

Tomlin Hawthorne

Life starts quite promptly, later in September. And then for months it's single cells. Nothing you could see without Leeuwenhoek's lens. The first animals worth the name turn up in mid-December.

Tomlin Hawthorne

Dinosaurs appear around Christmas. They're wiped out on the thirtieth.

Tomlin Hawthorne

Now it's New Year's Eve. All day, nothing human. Our species arrives at about eleven minutes to midnight.

Tomlin Hawthorne

The pyramids go up about ten seconds before midnight. Eratosthenes and his stick, five seconds. Shakespeare is writing in the last second. And every person you have ever met has lived their whole life in the final fraction of that second, a quarter of a second at most.

Tomlin Hawthorne

I've known that calendar for years and it still gets me. All of recorded history, every empire and every argument, fits between the last ten strokes of the clock.

Tomlin Hawthorne

Now. Swift's giants.

Tomlin Hawthorne

Here's the idea I most want you to take away today, because it comes back throughout the series: scale changes the rules. You can't take a thing and simply make it bigger.

Tomlin Hawthorne

Galileo worked this out in the 1630s, under house arrest, with time on his hands. Take an animal and double it in every direction. Twice as tall, twice as wide, twice as deep. How much heavier is it?

Tomlin Hawthorne

Not twice. Eight times. Two times two times two, because weight goes with volume.

Tomlin Hawthorne

But how much stronger are its leg bones? The strength of a bone depends on how thick it is across, its cross-section. That's an area, so it only goes up by two times two. Four.

Tomlin Hawthorne

Eight times the weight on four times the strength. Every bone is under double the stress. Do it again and it's four times the stress. Scale a man up to steeple height, as Swift did, sixty feet or so, and his thigh bones snap the first time he takes a step.

Tomlin Hawthorne

Galileo drew this, by the way. He drew a normal bone and beside it the bone a giant would need, and the giant's bone is a grotesque fat thing, more like a dumbbell. A giant wouldn't look like a big person. It would look like an elephant, which is, if you think about it, exactly what large land animals look like. Thick legs straight under the body, no jumping.

Tomlin Hawthorne

It works going down too. A British biologist, J. B. S. Haldane, wrote a lovely essay on this in the 1920s called "On Being the Right Size." He points out that you can drop a mouse down a thousand-yard mine shaft, and at the bottom it gets a slight shock and walks away. A rat is killed. A man is broken. A horse, he says, and I'm afraid this is his word, splashes.

Tomlin Hawthorne

Same gravity, same fall. The mouse is small, so it has a lot of surface for its weight, so the air slows it down. For a mouse, gravity is a minor nuisance. What terrifies a mouse, or better, an insect, is something we never think about: surface tension. A wet fly is carrying many times its own weight in water and may never get free. We step out of the bath without a thought. A fly that falls in a puddle has had a very serious day.

Tomlin Hawthorne

That's why there are no ants the size of horses, which I promised you last time. An ant breathes through tiny tubes in its sides, which works when you're small and nothing is far from the surface. Scale it up and the middle of the ant suffocates, shortly before the legs give way. Every monster-insect film ever made is, to a physicist, a film about something that collapsed and died of asphyxiation in the first scene.

Tomlin Hawthorne

So each size of creature lives in its own physics. For us, gravity is the boss. For an insect, it's the stickiness of water. And for a bacterium, I'm told, water behaves like thick honey, and swimming is more like corkscrewing through tar.

Tomlin Hawthorne

Everything your intuition knows, it learned in the middle of the ladder. Rungs minus three to plus four, roughly. Sand grains to mountain ranges. Speeds between a snail and a jet.

Tomlin Hawthorne

And if the rules change between a mouse and a horse, a mere two rungs, why on earth would they stay the same across forty-two?

Tomlin Hawthorne

They don't. Go far enough down, to atoms, and things stop having definite positions. That's quantum mechanics, and we'll get there. Go far enough up, to stars and galaxies, and space itself bends. That's Einstein. Go fast enough and time runs at different rates for different people.

Tomlin Hawthorne

When we reach those episodes, part of you is going to object. It's going to say: that's ridiculous, things don't behave like that. And it'll be right, about the middle of the ladder. Your common sense is a first-rate guide to rungs minus three to plus four. Outside that range, it has never been anywhere.

Tomlin Hawthorne

That isn't a failing. A fish has excellent intuitions about water.

Tomlin Hawthorne

The myth of the week, then, is the one we opened with. I'll call it thinking in a straight line.

Tomlin Hawthorne

We hear million, billion, trillion as a staircase with even steps. They aren't even. If a million is a twelve-day holiday, a billion is a career, and a trillion seconds ago there were mammoths, and no writing anywhere on Earth.

Tomlin Hawthorne

This is not only a physics matter, as you'll know if you've ever watched a budget debate in which a sum in the millions and a sum in the billions were discussed with equal passion for equal lengths of time. One of those is a thousand times the other. If the small one is a step, the big one is a staircase the height of a skyscraper.

Tomlin Hawthorne

The habit worth building is this. When you meet a big number, ask how many rungs. Is this a factor of ten, or a factor of a thousand? It's often the only question that matters.

Tomlin Hawthorne

A question to take away, with no settled answer.

Tomlin Hawthorne

We found ourselves in the middle of the ladder. Is that a fact about us, or is it an illusion? Perhaps any creature that can think at all would find itself in the middle of what it can detect, because its instruments reach a certain distance up and a certain distance down from wherever it happens to be standing. A thinking bacterium, if you'll allow me one, might report that bacteria are mid-sized.

Tomlin Hawthorne

Or is there a real reason that something as complicated as a brain has to be about this big? Big enough to hold a very large number of atoms. Small enough not to be crushed by its own weight.

Tomlin Hawthorne

I lean toward the second, but not with much confidence.

Tomlin Hawthorne

And your challenge. You need a long strip of paper. A till receipt is fine, or a few sheets taped end to end.

Tomlin Hawthorne

Mark a line across the middle and label it "me, one metre." Then mark rungs going each way, one per factor of ten, evenly spaced. Fifteen down, twenty-seven up.

Tomlin Hawthorne

Now place ten things on it, your choice. A grain of rice. Your town. A virus. The Moon's distance. Your phone's screen pixels. Don't look anything up at first. Guess the rung, mark it in pencil, and then check.

Tomlin Hawthorne

I'll make a prediction: you will be most wrong about the things closest to everyday life. We're surprisingly bad at the distance to the Moon, and oddly good at galaxies, because nobody has intuitions about galaxies to get in the way.

Tomlin Hawthorne

Next time, a skill. How physicists get the answer roughly right before they've done any real work, using a method named after a man who once estimated the power of the first atomic bomb with a handful of torn paper. It also works on piano tuners.