by Brian Aldiss
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 unself-conscious, 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.’
‘And there are more of these variational principles?’
He nodded. ‘In all branches of physics. Almost every physical law can be restated as a variational principle. The only difference between these principles is in which attribute is minimized or maximized.’ He gestured as if the different branches of physics were arrayed before him on a table. ‘In optics, where Fermat’s principle applies, time is the attribute that has to be an extreme. In mechanics, it’s a different attribute. In electromagnetism, it’s something else again. But all these principles are similar mathematically.’
‘So once you get their mathematical description of Fermat’s principle, you should be able to decode the other ones.’
‘God, I hope so. I think this is the wedge that we’ve been looking for, the one that cracks open their formulation of physics. This calls for a celebration.’ He stopped his pacing and turned to me. ‘Hey Louise, want to go out for dinner? My treat.’
I was mildly surprised. ‘Sure,’ I said.
It’ll be when you first learn to walk that I get daily demonstrations of the asymmetry in our relationship. You’ll be incessantly running off somewhere, and each time you walk into a door frame or scrape your knee, the pain feels like it’s my own. It’ll be like growing an errant limb, an
extension of myself whose sensory nerves report pain just fine, but whose motor nerves don’t convey my commands at all. It’s so unfair: I’m going to give birth to an animated voodoo doll of myself. I didn’t see this in the contract when I signed up. Was this part of the deal?
And then there will be the times when I see you laughing. Like the time you’ll be playing with the neighbor’s puppy, poking your hands through the chain-link fence separating our back yards, and you’ll be laughing so hard you’ll start hiccuping. The puppy will run inside the neighbor’s house, and your laughter will gradually subside, letting you catch your breath. Then the puppy will come back to the fence to lick your fingers again, and you’ll shriek and start laughing again. It will be the most wonderful sound I could ever imagine, a sound that makes me feel like a fountain, or a wellspring.
Now if only I can remember that sound the next time your blithe disregard for self-preservation gives me a heart attack.
After the breakthrough with Fermat’s principle, discussions of scientific concepts became more fruitful. It wasn’t as if all of heptapod physics were suddenly rendered transparent, but progress was steady. According to Gary, the heptapods’ formulation of physics was indeed topsyturvy relative to ours. Physical attributes that humans defined using integral calculus were seen as fundamental by the heptapods. As an example, Gary described an attribute that, in physics jargon, bore the deceptively simple name ‘action,’ which represented ‘the difference between kinetic and potential energy, integrated over time,’ whatever that meant. Calculus for us; elementary to them.
Conversely, to define attributes that humans thought of as fundamental, like velocity, the heptapods employed mathematics that were, Gary assured me, ‘highly weird.’ The physicists were ultimately able to prove the equivalence of heptapod mathematics and human mathematics; even though their approaches were almost the reverse of one another, both were systems for describing the same physical universe.
I tried following some of the equations that the physicists were coming up with, but it was no use. I couldn’t really grasp the significance of physical attributes like ‘action’; I couldn’t, with any confidence, ponder the significance of treating such an attribute as fundamental. Still, I tried to ponder questions formulated in terms more familiar to me: what kind of world-view did the heptapods have, that they would consider Fermat’s principle the simplest explanation of light refraction? What kind of perception made a minimum or maximum readily apparent to them?
Your eyes will be blue like your dad’s, not mud brown like mine. Boys will stare into those eyes the way I did, and do, into your dad’s, surprised and enchanted, as I was and am, to find them in combination with black hair. You will have many suitors.
I remember when you are fifteen, coming home after a weekend at your dad’s, incredulous over the interrogation he’ll have put you through regarding the boy you’re currently dating. You’ll sprawl on the sofa, recounting your dad’s latest breach of common sense: ‘You know what he said? He said, “I know what teenage boys are like.”’ Roll of the eyes. ‘Like I don’t?’
‘Don’t hold it against him,’ I’ll say. ‘He’s a father; he can’t help it.’ Having seen you interact with your friends, I won’t worry much about a boy taking advantage of you; if anything, the opposite will be more likely. I’ll worry about that.
‘He wishes I were still a kid. He hasn’t known how to act toward me since I grew breasts.’
‘Well, that development was a shock for him. Give him time to recover.’
‘It’s been years, Mom. How long is it gonna take?’
‘I’ll let you know when my father has come to terms with mine.’
During one of the videoconferences for the linguists, Cisneros from the Massachusetts looking glass had raised an interesting question: Was there a particular order in which semagrams were written in a Heptapod B sentence? It was clear that word order meant next to nothing when speaking in Heptapod A; when asked to repeat what it had just said, a heptapod would likely as not use a different word order unless we specifically asked them not to. Was word order similarly unimportant when writing in Heptapod B?
Previously, we had focused our attention only on how a sentence in Heptapod B looked once it was complete. As far as anyone could tell, there was no preferred order when reading the semagrams in a sentence; you could start almost anywhere in the nest, then follow the branching clauses until you’d read the whole thing. But that was reading; was the same true about writing?
During my most recent session with Flapper and Raspberry I had asked them if, instead of displaying a semagram only after it was completed, they could show it to us while it was being written. They had agreed. I insterted the videotape of the session into the VCR, and on my computer I consulted the session transcript.
I picked one of the longer utterances from the conversation. What Flapper had said was that the heptapods’ planet had two moons, one significantly larger than the other; the three primary constituents of the planet’s atmosphere were nitrogen, argon, and oxygen; and 15/28ths of the planet’s surface was covered by water. The first words of the spoken utterance translated literally as ‘inequality-of-size rocky-orbiter rocky-orbiters related-as-primary-to-secondary.’
Then I rewound the videotape until the time signature matched the one in the transcription. I started playing the tape, and watched the web of semagrams being spun out of inky spider’s silk. I rewound it and played it several times. Finally I froze the video right after the first stroke was completed and before the second one was begun; all that was visible onscreen was a single sinuous line.
Comparing that initial stroke with the completed sentence, I realized that the stroke participated in several different clauses of the message. It began in the semagram for ‘oxygen,’ as the determinant that distinguished it from certain other elements; then it slid down to become the morpheme of comparison in the description of the two moons’ sizes; and lastly it flared out as the arched backbone of the semagram for ‘ocean.’ Yet this stroke was a single continuous line, and it was the first one that Flapper wrote. That meant the heptapod had to know how the entire sentence would be laid out before it could write the very first stroke.
The other strokes in the sentence also traversed several clauses, making them so interconnected that none could be removed without redesigning the entire sentence. The heptapods didn’t write a sentence one semagram at a time; they built it out of strokes irrespective of individual semagrams. I had seen a similarly high degree of integration before in calligraphic designs, particularly those employing the Arabic alphabet. But those designs had required careful planning by expert calligraphers. No one could lay out such an intricate design at the speed needed for holding a conversation. At least, no human could.
There’s a joke that I once heard a comedienne tell. It goes like this: ‘I’m not sure if I’m ready to have children. I asked a friend of mine who has children, “Suppose I do have kids. What if when they grow up, they blame me for everything that’s wrong with their lives?” She laughed and said, “What do you mean, if?”’
That’s my favorite joke.
Gary and I were at a little Chinese restaurant, one of the local places we had taken to patronizing to get away from the encampment. We sat eating the appetizers: potstickers, redolent of pork and sesame oil. My favorite.
I dipped one in soy sauce and vinegar. ‘So how are you doing with your Heptapod B practice?’ I asked.
Gary looked obliquely at the ceiling. I tried to meet his gaze, but he kept shifting it.
‘You’ve given up, haven’t you?’ I said. ‘You’re not even trying anymore.’
He did a wonderful hangdog expression. ‘I’m just no good at languages,’ he confessed. ‘I thought learning Heptapod B might be more like learning mathematics than trying to speak another language, but it’s not. It’s too foreign for me.’
‘It would help you discuss physics with them.’
‘Probably, but since we had o
ur breakthrough, I can get by with just a few phrases.’
I sighed. ‘I suppose that’s fair; I have to admit, I’ve given up on trying to learn the mathematics.’
‘So we’re even?’
‘We’re even.’ I sipped my tea. ‘Though I did want to ask you about Fermat’s principle. Something about it feels odd to me, but I can’t put my finger on it. It just doesn’t sound like a law of physics.’
A twinkle appeared in Gary’s eyes. ‘I’ll bet I know what you’re talking about.’ He snipped a potsticker in half with his chopsticks. ‘You’re used to thinking of refraction in terms of cause and effect: reaching the water’s surface is the cause, and the change in direction is the effect. But Fermat’s principle sounds weird because it describes light’s behavior in goal-oriented terms. It sounds like a commandment to a light beam: “Thou shalt minimize or maximize the time taken to reach thy destination.”’
I considered it. ‘Go on.’
‘It’s an old question in the philosophy of physics. People have been talking about it since Fermat first formulated it in the 1600s; Planck wrote volumes about it. The thing is, while the common formulation of physical laws is causal, a variational principle like Fermat’s is purposive, almost teleological.’
‘Hmm, that’s an interesting way to put it. Let me think about that for a minute.’ I pulled out a felt-tip pen and, on my paper napkin, drew a copy of the diagram that Gary had drawn on my blackboard. ‘Okay,’ I said, thinking aloud, ‘so let’s say the goal of a ray of light is to take the fastest path. How does the light go about doing that?’
‘Well, if I can speak anthropomorphic-projectionally, the light has to examine the possible paths and compute how long each one would take.’ He plucked the last potsticker from the serving dish.
‘And to do that,’ I continued, ‘the ray of light has to know just where its destination is. If the destination were somewhere else, the fastest path would be different.’