We regularly asked the heptapods why they had come. Each time, they answered ‘to see,’ or ‘to observe.’ Indeed, sometimes they preferred to watch us silently rather than answer our questions. Perhaps they were scientists, perhaps they were tourists. The State Department instructed us to reveal as little as possible about humanity, in case that information could be used as a bargaining chip in subsequent negotiations. We obliged, though it didn’t require much effort: the heptapods never asked questions about anything. Whether scientists or tourists, they were an awfully incurious bunch.
I remember once when we’ll be driving to the mall to buy some new clothes for you. You’ll be thirteen. One moment you’ll be sprawled in your seat, completely unselfconscious, all child; the next, you’ll toss your hair with a practiced casualness, like a fashion model in training.
You’ll give me some instructions as I’m parking the car. ‘Okay, Mom, give me one of the credit cards, and we can meet back at the entrance here in two hours.’
I’ll laugh. ‘Not a chance. All the credit cards stay with me.’
‘You’re kidding.’ You’ll become the embodiment of exasperation. We’ll get out of the car and I will start walking to the mall entrance. After seeing that I won’t budge on the matter, you’ll quickly reformulate your plans.
‘Okay Mom, okay. You can come with me, just walk a little ways behind me, so it doesn’t look like we’re together. If I see any friends of mine, I’m gonna stop and talk to them, but you just keep walking, okay? I’ll come find you later.’
I’ll stop in my tracks. ‘Excuse me? I am not the hired help, nor am I some mutant relative for you to be ashamed of.’
‘But Mom, I can’t let anyone see you with me.’
‘What are you talking about? I’ve already met your friends; they’ve been to the house.’
‘That was different,’ you’ll say, incredulous that you have to explain it. ‘This is shopping.’
‘Too bad.’
Then the explosion: ‘You won’t do the least thing to make me happy! You don’t care about me at all!’
It won’t have been that long since you enjoyed going shopping with me; it will forever astonish me how quickly you grow out of one phase and enter another. Living with you will be like aiming for a moving target; you’ll always be further along than I expect.
I looked at the sentence in Heptapod B that I had just written, using simple pen and paper. Like all the sentences I generated myself, this one looked misshapen, like a heptapod-written sentence that had been smashed with a hammer and then inexpertly taped back together. I had sheets of such inelegant semagrams covering my desk, fluttering occasionally when the oscillating fan swung past.
It was strange trying to learn a language that had no spoken form. Instead of practicing my pronunciation, I had taken to squeezing my eyes shut and trying to paint semagrams on the insides of my eyelids.
There was a knock at the door and before I could answer Gary came in looking jubilant. ‘Illinois got a repetition in physics.’
‘Really? That’s great; when did it happen?’
‘It happened a few hours ago; we just had the videoconference. Let me show you what it is.’ He started erasing my blackboard.
‘Don’t worry, I didn’t need any of that.’
‘Good.’ He picked up a nub of chalk and drew a diagram:
‘Okay, here’s the path a ray of light takes when crossing from air to water. The light ray travels in a straight line until it hits the water; the water has a different index of refraction, so the light changes direction. You’ve heard of this before, right?’
I nodded. ‘Sure.’
‘Now here’s an interesting property about the path the light takes. The path is the fastest possible route between these two points.’
‘Come again?’
‘Imagine, just for grins, that the ray of light traveled along this path.’ He added a dotted line to his diagram:
‘This hypothetical path is shorter than the path the light actually takes. But light travels more slowly in water than it does in air, and a greater percentage of this path is underwater. So it would take longer for light to travel along this path than it does along the real path.’
‘Okay, I get it.’
‘Now imagine if light were to travel along this other path.’ He drew a second dotted path:
‘This path reduces the percentage that’s underwater, but the total length is larger. It would also take longer for light to travel along this path than along the actual one.’
Gary put down the chalk and gestured at the diagram on the chalkboard with white-tipped fingers. ‘Any hypothetical path would require more time to traverse than the one actually taken. In other words, the route that the light ray takes is always the fastest possible one. That’s Fermat’s principle of least time.’
‘Hmm, interesting. And this is what the heptapods responded to?’
‘Exactly. Moorehead gave an animated presentation of Fermat’s principle at the Illinois looking glass, and the heptapods repeated it back. Now he’s trying to get a symbolic description.’ He grinned. ‘Now is that highly neat, or what?’
‘It’s neat all right, but how come I haven’t heard of Fermat’s principle before?’ I picked up a binder and waved it at him; it was a primer on the physics topics suggested for use in communication with the heptapods. ‘This thing goes on forever about Planck masses and the spin-flip of atomic hydrogen, and not a word about the refraction of light.’
‘We guessed wrong about what’d be most useful for you to know,’ Gary said without embarrassment. ‘In fact, it’s curious that Fermat’s principle was the first breakthrough; even though it’s easy to explain, you need calculus to describe it mathematically. And not ordinary calculus; you need the calculus of variations. We thought that some simple theorem of geometry or algebra would be the breakthrough.’
‘Curious indeed. You think the heptapods’ idea of what’s simple doesn’t match ours?’
‘Exactly, which is why I’m dying to see what their mathematical description of Fermat’s principle looks like.’ He paced as he talked. ‘If their version of the calculus of variations is simpler to them than their equivalent of algebra, that might explain why we’ve had so much trouble talking about physics; their entire system of mathematics may be topsy-turvy compared to ours.’ He pointed to the physics primer. ‘You can be sure that we’re going to revise that.’
‘So can you build from Fermat’s principle to other areas of physics?’
‘Probably. There are lots of physical principles just like Fermat’s.’
‘What, like Louise’s principle of least closet space? When did physics become so minimalist?’
‘Well, the word “least” is misleading. You see, Fermat’s principle of least time is incomplete; in certain situations light follows a path that takes more time than any of the other possibilities. It’s more accurate to say that light always follows an extreme path, either one that minimizes the time taken or one that maximizes it. A minimum and a maximum share certain mathematical properties, so both situations can be described with one equation. So to be precise, Fermat’s principle isn’t a minimal principle; instead it’s what’s known as a “variational” principle.’