The Girl Who Went Into the Cave

It was rather late, and our heroine was doing what she always did when she could not sleep.

She had the torch under the covers, you see, and the equations spread out in her mind, and she considered this a perfectly reasonable state of affairs for a girl of her particular disposition. She thought back to when she was at school and doing just such a thing, late at night. The pretty girls in the dormitory were dreaming of boys, one supposed. She was working out the implications of Euler's formula. She had always considered this the better use of everyone's time, and nothing in her considerable experience had given her reason to revise this opinion.

She was not, it must be said, the sort of girl that boys looked at. She had established this early, the way she established most things, by examining the evidence carefully and drawing the only reasonable conclusion. She had filed it away without particular distress, the way you file away any fact that is simply true and therefore not worth arguing with, and had got on with things, which she was, in the precise tradition of her upbringing, tremendously good at.

One simply did.

She was the clever one. She had, over time, come to consider this the superior arrangement.

She was, after all, exactly the sort of girl who arrived at the first day of term with a properly packed trunk, name tapes sewn in with military precision, who knew without being told that one's indoor shoes went on the left and one's outdoor shoes on the right, and who had strong opinions about the quality of the wool in one's regulation school jumper. She found comfort in equations, and in fabric. The weight of good linen. The particular authority of a well-cut wool. The way silk moves differently in different humidities, which she knew about because even before the school trunk there had been the houses, and the houses had been rather varied.

She was, you see, the sort of girl who had spent her childhood in the kinds of houses where things were done properly even when properly was a long way from anywhere, where one dressed for dinner in the middle of Africa because that was simply what one did, and where the combination of good manners and genuine wilderness had produced in her a particular kind of person: someone who could read the weight of a room the way she read the weight of a textile, by feel, within moments of arrival. She had grown up in enough countries that she had learned, early and permanently, that home was not a place but a frequency, and that if you were sufficiently attentive you could pick it up almost anywhere.

The pretty girls, in her experience, were often not very good at frequencies or mathematics. They were good at other things, no doubt, but frequencies were not among them.

She was very good at frequencies. She was very good at mathematics. She was very good at fabric, and lessons, and manners and conversation and rules, and at arriving in new places and at keeping her torch batteries fresh and her rucksack packed and her sensible shoes by the door. She had built, out of these materials and several others, a life of considerable richness and not inconsiderable adventure, and she had done it without once being the sort of girl that boys looked at.

She loved many equations, as the truly devoted always do, but she loved two above all others, and this is a story about both of them. About what they did to her, and what she did with them, and how she came to understand, finally and completely, why she had always needed both.

But we are getting ahead of ourselves. All good adventures begin at the beginning, and the beginning, for our girl, was always the mathematics.

She was, in short, our kind of girl. Tremendously good at getting on with things. One simply did.

Mathematics was the frequency that never changed. Whatever country she arrived in, whatever room she walked into, whatever the humidity was doing to the silk, the equations were the same. They waited for her with the patient reliability of something that has no reason to be otherwise. She returned to them the way she returned to nothing else: without luggage, without the particular alertness she carried everywhere, without the low-level navigation that living in many places requires of a person. In the mathematics she could simply be still.

She had found the first formula young, the way you find things that were always going to be yours. She had been returning to it ever since, in the small hours, in the difficult weeks, in the particular loneliness of being someone for whom home is a frequency rather than a place. It gave her peace the way very few things gave her peace: completely, and without asking anything in return.

The second formula she had also found young. That one she kept for a different kind of night altogether.

But we shall come to that.

Chapter One, in which our heroine encounters a most romantic equation, and knows perfectly well what it means, and decides to look at it from below anyway.

Now then. The first formula.

Euler's formula: e^{iωt} = cos(ωt) + i·sin(ωt).

It is, by any reasonable measure, one of the most beautiful things human beings have ever written down, and she had known this since she was young enough that knowing it had felt like a secret, like something the universe had leaned down and whispered specifically to her.

What it describes, in the language of those who speak such things, is rotation. Two quantities, cosine and sine, breathing in and out of each other as time moves forward, one rising as the other falls, bound together in a relationship so elegant it seems less like mathematics and more like an argument for the existence of something. She loved the way it felt in her mind, the way you could hold it and turn it and it would always resolve, always come back to itself, always tell the truth. In the difficult years, and there had been difficult years, she had returned to it the way other people return to prayer. It did not ask her to believe anything. It simply was.

But here is the thing about Euler's formula, dear reader. If you watch the two points it describes from the ordinary innocent angle, from below, looking up, they appear to be circling each other in a perfect, closed curve.

From this angle, it looks like fate. It is a very convincing angle.

She knew, of course, what it actually described. She had always known. The formula, if you read it properly, past the first romantic glance, describes not a circle but a helix. Introduce the third axis, time, that uninvited guest, and the circle reveals itself to be two separate spirals, winding through the same space at different heights, sharing only a shadow on the floor below. They are not circling each other at all. They are chasing a meeting that the geometry decided, long before either of them arrived, could never happen.

This was, she had always thought, rather the pretty girl's equation. All romance and circular curves and the suggestion of fate when you looked at it from the flattering angle. The sort of equation that got you into trouble if you forgot to check the third axis.

She had never forgotten to check the third axis. She was not that sort of girl.

Until, of course, she was.

Now here is where our girl's story becomes interesting, and where we must pay close attention, because what she did next was not, as some might suppose, a failure of intelligence. Our girl was not the sort to fail at intelligence. It was something else altogether, something rather more complicated and considerably more brave.

The helix was a cave. She could see perfectly well that it was a cave. She had her torch. She had her ginger beer. She had her good rucksack and her sensible shoes on her sturdy little adventurer legs and she could read the geometry as clearly as she could read the weight of a room.

And she thought: well. Just this once. Just this one cave.

And in she went.

One simply did.

Chapter Two, in which there is a boy, as there always is, and our heroine makes a perfectly informed decision to look at things from the wrong angle, and finds the view rather wonderful.

There was a boy, as there is always a boy in stories like this.

He was not, it should be said, the sort of boy who looked at girls like her. In the ordinary way of things, in the ordinary geometry of such arrangements, boys like him looked at girls like the pretty ones in the dormitory, the ones who were good at being looked at, who had had considerable practice. Our girl had not had considerable practice. She had had considerable practice at other things, at frequencies and fabrics and arriving in new places, and she was superb at all of them, but being looked at was not among her accomplishments and she had made her peace with this long ago and got on with things.

And yet.

They found each other in the flat bright world of screens and became what you might generously call lovers on paper, or more accurately, lovers on screen, which amounts to the same flat geometry. Words, and the anticipation that collects around words when you let it. She was, it should be said, extremely good at words. She had grown up in the kinds of houses where language was taken seriously, where dinner conversation was considered a form of sport and you were expected to show up with something worth saying.

She showed up. She always did. One did.

And he, on his side of the screen, in his city, in his life she could only partly see, showed up too. For years they showed up for each other across the flat bright distance, and it looked, from the angle available to them, very much like something.

The thing about a screen is that it is resolutely two-dimensional. What exists there has x and y. It circles. It looks, from where you are standing, like it is heading somewhere, moving with the patient unhurried confidence of a thing that believes it has all the time in the world.

She watched it the way she watched all curves, with the practiced eye of someone who knew what convergence looked like. This is tending somewhere, she thought. This is a closed form.

She knew exactly what she was looking at. She had simply decided, for reasons that seemed entirely sufficient at the time, to look at it from below.

And from below, dear reader, it was rather wonderful. Even for a girl who had always known better. Perhaps especially for a girl who had always known better. There is a particular pleasure, it turns out, in choosing the romantic angle on purpose, with full knowledge of what you are doing. It is considerably more interesting than stumbling into it by accident, which is what the pretty girls tended to do.

She had her torch. She had her ginger beer. She tilted her head, looked up at the shadow on the floor, and thought: yes. Just this once. In we go.

Chapter Three, in which the universe obliges our heroine by placing her in the exact coordinates she requires, and she goes in, torch high, rucksack on, whole chest open, because that is simply what one does.

Then the universe moved the pieces, the way it does when it has decided to make a point.

It placed her, through the particular logic of a life perpetually in motion, in the same coordinates as he occupied. In the very city where he was. Now this was not, for our girl, an unusual situation in the broadest sense. She had always moved. She moved the way other people breathe, with the easy necessity of someone for whom staying still had never quite been an option.

New country, new frequency, new room to walk into and read within moments of arrival, because that is what you do when you have grown up everywhere and belong, in the technical sense, to nowhere in particular.

She was superb at this. She had her good rucksack and her sensible shoes on her sturdy little adventurer legs, and the collected confidence of someone who has been the new girl in enough places that the novelty of it had worn off somewhere around the third continent.

But this was different. This was his city.

She walked into it the way she walked into rooms she already knew something about, with that particular attentiveness that was half instinct and half the lifelong habit of a girl who had always been paying close attention. He had given her a map of it, in his way, over years of conversation, and she followed it the way she followed all good instructions: thoroughly, with interest, stopping to examine things properly. She found what he had told her to find, one by one, and each finding felt like a small collaboration across the distance, like a treasure hunt laid out by someone who had been paying rather close attention to who she was.

She stepped onto what she believed was their shared coordinate and she was, it must be said, magnificent about it. Passport still warm in her pocket. The city opening around her like a proof she already half knew.

And there it was, that feeling. Not hope, precisely. Hope was too loud, too grabby, too much the sort of emotion that embarrasses itself in public. Not expectation either. She had packed very little of that, and quite deliberately. It was something quieter and more electric than either. The feeling of standing at the mouth of a very promising cave with a torch that works and a full bottle of ginger beer and the absolute certain knowledge that something is in there, something worth finding, and that she is exactly the right girl to go and find it.

She turned corners the way you turn corners when the map says there is something around the next one. Briskly. Hopefully. With her eyes very wide open. Nothing was demanded of the universe or of him. She simply walked, torch high, into the city that knew him, and let herself feel the full magnificent charge of it. The electricity of almost. The particular and wonderful feeling of a variable that has not yet resolved, which is, as any mathematician will tell you, one of the most exciting states a variable can be in.

The cave was right there. She could see it perfectly well.

In she went. Torch high. Whole chest open. Entirely herself.

What she had was bravery. The specific, clear-eyed, variety of bravery that does not make a fuss about itself but simply tightens the straps on its rucksack and gets on with it. The kind that knows perfectly well it is a cave and goes in anyway because that is the only way to find out what is inside, and our girl had always needed to know what was inside.

That is not foolishness, dear reader. That is the only way to live in three dimensions.

Chapter Four, in which the third axis appears, as it was always going to, and our heroine handles it with the grace of someone who packed for exactly this contingency.

And then the third axis appeared, as it had always been going to.

Reasons came, as reasons do in these stories. Perfectly reasonable reasons, in a perfectly reasonable register, delivered in the tone that makes questioning them feel like the unreasonable thing to do. She accepted them with grace, our girl, because grace was something she had also learned early, in those houses where one received disappointment as elegantly as one received everything else. She gave him an out. She told him she understood, that there was no pressure, that it was fine.

She meant it in the way you mean things when you are watching something end in real time and you would rather open the door for it than have it break down the wall.

What she did not say was: I can feel the z-axis splitting right now, and I cannot stop it.

She came out of the cave. A little muddier than she had gone in, perhaps. But still herself. Torch still working. Ginger beer finished. Rucksack on both shoulders, the way it should be.

She had known it was a cave. She had gone in anyway. She had come out the other side.

That, dear reader, is all there is to it. Anyone who tells you otherwise has never had a proper adventure.

Chapter Five, in which our heroine understands what she was looking at all along, and finds it neither sad nor surprising, merely the shape of things.

And here is what our girl understood, standing in the cave with her torch and her muddy shoes and her empty ginger beer bottle.

The formula had been correct from the very beginning. She had simply been reading it from the wrong angle, from below, looking up at a shadow on the floor and mistaking it for the thing itself.

Well. These things happen, even to the cleverest girls, and the important thing is not to make a fuss about it. She had walked his streets with her whole chest open and her eyes clear, and she would not apologise for that. Not even slightly. That is what it means to take mathematics seriously.

You go in. You see what is there. You come out the other side and you write it up accurately in your notebook and you have a biscuit.

One simply does.

The helix is not, it should be said, the villain of this story. It is not dramatic and it is not cruel and it is not even particularly sorry about itself. It is just a circle, seen from far enough away to finally tell the truth. And the truth, it turns out, requires no dramatic music. It is just the shape of things, and our girl, who had been reading the shape of things since she was small enough to need a torch under the covers to do it, found that she could look at it quite steadily and find it rather beautiful.

She will be where he is, and he will be where he is, and they will both be exactly where they were.

And that will be the proof.

Right then. Time for supper.

Chapter Six, in which our heroine comes home, and finds things waiting exactly where she left them, and finds them rather more wonderful than she had remembered.

Now then. The second formula.

It was rather late, and our heroine was back from her adventure.

She was lying in the dark thinking about it, the way you do when a thing has just resolved and you are not entirely sure what to do with the quiet it leaves behind. The cave had turned out to be, as she had known perfectly well it would be, a cave. She had gone in. She had looked around thoroughly. She had come out the other side with muddy shoes and a finished bottle of ginger beer and the particular feeling of someone who has done the thing they set out to do and found it to be exactly as advertised, which is both satisfying and not, in equal measure.

She could not sleep. Again.

So she did what she had always done, in the difficult years and the good ones and all the ordinary ones in between. She found the torch. She put it under the covers. She reached past the first formula, the romantic one, the helix, the cave, all of that, and she found the second one waiting for her exactly where she had left it, patient and solid and true, the way reliable things always are.

Not the helix. Not the two points circling through time toward a meeting that the geometry would not allow. Not the wildly romantic one, which she had always suspected was too much to ask for, even as she asked for it.

This one: e^{iπ} + 1 = 0.

Euler's identity.

It should not work. That is the first thing to understand about it, and the thing she had always loved most. Five constants from completely different corners of mathematics, discovered separately, over centuries, for entirely unrelated reasons. Five strangers who have no business being in the same room together, and yet there they are, and there they resolve, completely, without remainder, every single time.

Let us meet them properly.

e. The base of natural growth. The number that describes the rate at which things change when they are changing relative to themselves. She had changed, over the years of it. She had grown in the particular way you grow when someone has been paying close attention to you across a long distance, when someone knows which streets to send you down and which questions to ask in the small hours. That was e. That was the years of it.

i. The imaginary unit. The square root of negative one, which should not exist and does anyway, stubbornly, opening an entire extra dimension that turns out to describe rotation and waves and the behaviour of light. Mathematicians invented it rather reluctantly and then discovered it unlocked everything. He had been that. The thing that should not have been possible and was anyway. The dimension she had not known she was missing until she found herself moving through it, blinking, torch in hand, rather astonished.

π. Which lives in circles but turns up absolutely everywhere else too, in probability and rivers and the distribution of prime numbers, and nobody entirely knows why, which our girl found both maddening and deeply satisfying. He had been that too. Turning up. In her thinking, in her walking, in the particular way a city reveals itself when someone who loves it has told you where to look. Everywhere, and unreasonable, and not entirely explicable, and there nonetheless.

And then 1. And 0. The two numbers so simple they seem almost beneath the company they are keeping, sitting at the end of this remarkable guest list like the two most sensible people in the room.

She was going back. That was the 1. Still herself, entirely herself, the girl with the torch under the covers and the sensible shoes by the door and the rucksack packed and ready because one never knew.

Add her, the 1, to everything that had happened, to e and i and π, to the years of growth and the impossible imaginary and the unreasonable ubiquity of him, and you got zero.

Now. This is the part that matters, so do pay attention.

Not nothing. Zero.

These are entirely different things and it would be a great shame to confuse them now, at the end of such a very good equation.

Nothing is the absence of everything. Zero is what you get when everything that needs to resolve has resolved. When there is no remainder. When the equation has done what it came to do and closed, perfectly, completely, into silence.

That is not nothing. That is everything, finished.

Richard Feynman called it the most remarkable formula in mathematics. She had loved him for that, for the straightforwardness of it, for refusing to be coy about being astonished. The mathematician Benjamin Peirce had said, after deriving it: we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth. She had loved him too, for the honesty of that, for admitting that some true things remain past understanding and are not diminished by it.

She had always thought it was the better formula for a girl like her. Not the romantic one, the helix, the pretty girl's equation with its convincing angle and its shadow on the floor. This one. The one that needed nothing outside itself. The one that contained everything it required and closed, perfectly, into nothing and everything at once, which turn out, from the right angle, to be exactly the same.

It was, she had always felt, exactly the right formula for a girl with a torch under the covers and a properly packed trunk and strong opinions about wool. The ugly duckling's equation. The one that looked, at first glance, like it had no business being as beautiful as it was. And then you looked properly, and there it was.

The most remarkable thing in the room. It always had been.

She held it the way she always had, on the nights that needed it most. She let it resolve.

And she went to sleep.