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I looked at Flapper and Raspberry in the looking glass, who were waiting for us to continue, and sighed. ‘You aren’t going to make this easy for us, are you?’

To be fair, the heptapods were completely cooperative. In the days that followed, they readily taught us their language without requiring us to teach them any more English. Colonel Weber and his cohorts pondered the implications of that, while I and the linguists at the other looking glasses met via videoconferencing to share what we had learned about the heptapod language. The videoconferencing made for an incongruous working environment: our video screens were primitive compared to the heptapods’ looking glasses, so that my colleagues seemed more remote than the aliens. The familiar was far away, while the bizarre was close at hand.

It would be a while before we’d be ready to ask the heptapods why they had come, or to discuss physics well enough to ask them about their technology. For the time being, we worked on the basics: phonemics/graphemics, vocabulary, syntax. The heptapods at every looking glass were using the same language, so we were able to pool our data and coordinate our efforts.

Our biggest source of confusion was the heptapods’ ‘writing.’ It didn’t appear to be writing at all; it looked more like a bunch of intricate graphic designs. The logograms weren’t arranged in rows, or a spiral, or any linear fashion. Instead, Flapper or Raspberry would write a sentence by sticking together as many logograms as needed into a giant conglomeration.

This form of writing was reminiscent of primitive sign systems, which required a reader to know a message’s context in order to understand it. Such systems were considered too limited for systematic recording of information. Yet it was unlikely that the heptapods developed their level of technology with only an oral tradition. That implied one of three possibilities: the first was that the heptapods had a true writing system, but they didn’t want to use it in front of us; Colonel Weber would identify with that one. The second was that the heptapods hadn’t originated the technology they were using; they were illiterates using someone else’s technology. The third, and most interesting to me, was that the heptapods were using a nonlinear system of orthography that qualified as true writing.

I remember a conversation we’ll have when you’re in your junior year of high school. It’ll be Sunday morning, and I’ll be scrambling some eggs while you set the table for brunch. You’ll laugh as you tell me about the party you went to last night.

‘Oh man,’ you’ll say, ‘they’re not kidding when they say that body weight makes a difference. I didn’t drink any more than the guys did, but I got so much drunker.’

I’ll try to maintain a neutral, pleasant expression. I’ll really try. Then you’ll say, ‘Oh, come on, Mom.’

‘What?’

‘You know you did the exact same things when you were my age.’

I did nothing of the sort, but I know that if I were to admit that, you’d lose respect for me completely. ‘You know never to drive, or get into a car if—’

‘God, of course I know that. Do you think I’m an idiot?’

‘No, of course not.’

What I’ll think is that you are clearly, maddeningly not me. It will remind me, again, that you won’t be a clone of me; you can be wonderful, a daily delight, but you won’t be someone I could have created by myself.

The military had set up a trailer containing our offices at the looking-glass site. I saw Gary walking toward the trailer, and ran to catch up with him. ‘It’s a semasiographic writing system,’ I said when I reached him.

‘Excuse me?’ said Gary.

‘Here, let me show you.’ I directed Gary into my office. Once we were inside, I went to the chalkboard and drew a circle with a diagonal line bisecting it. ‘What does this mean?’

‘“Not allowed”?’

‘Right.’ Next I printed the words NOT ALLOWED on the chalkboard. ‘And so does this. But only one is a representation of speech.’

Gary nodded. ‘Okay.’

‘Linguists describe writing like this’ – I indicated the printed words – ‘as “glottographic”, because it represents speech. Every human written language is in this category. However, this symbol’ – I indicated the circle and diagonal line – ‘is “semasiographic” writing, because it conveys meaning without reference to speech. There’s no correspondence between its components and any particular sounds.’

‘And you think all of heptapod writing is like this?’

‘From what I’ve seen so far, yes. It’s not picture writing, it’s far more complex. It has its own system of rules for constructing sentences, like a visual syntax that’s unrelated to the syntax for their spoken language.’

‘A visual syntax? Can you show me an example?’

‘Coming right up.’ I sat down at my desk and, using the computer, pulled up a frame from the recording of yesterday’s conversation with Raspberry. I turned the monitor so he could see it. ‘In their spoken language, a noun has a case marker indicating whether it’s a subject or object. In their written language, however, a noun is identified as subject or object based on the orientation of its logogram relative to that of the verb. Here, take a look.’ I pointed at one of the figures. ‘For instance, when “heptapod” is integrated with “hears” this way, with these strokes parallel, it means that the heptapod is doing the hearing.’ I showed him a different one. ‘When they’re combined this way, with the strokes perpendicular, it means that the heptapod is being heard. This morphology applies to several verbs.

‘Another example is the inflection system.’ I called up another frame from the recording. ‘In their written language, this logogram means roughly “hear easily” or “hear clearly.” See the elements it has in common with the logogram for “hear”? You can still combine it with “heptapod” in the same ways as before, to indicate that the heptapod can hear something clearly or that the heptapod is clearly heard. But what’s really interesting is that the modulation of “hear” into “hear clearly” isn’t a special case; you see the transformation they applied?’

Gary nodded, pointing. ‘It’s like they express the idea of “clearly” by changing the curve of those strokes in the middle.’

‘Right. That modulation is applicable to lots of verbs. The logogram for “see” can be modulated in the same way to form “see clearly,” and so can the logogram for “read” and others. And changing the curve of those strokes has no parallel in their speech; with the spoken version of these verbs, they add a prefix to the verb to express ease of manner, and the prefixes for “see” and “hear” are different.

‘There are other examples, but you get the idea. It’s essentially a grammar in two dimensions.’

He began pacing thoughtfully. ‘Is there anything like this in human writing systems?’

‘Mathematical equations, notations for music and dance. But those are all very specialized; we couldn’t record this conversation using them. But I suspect, if we knew it well enough, we could record this conversation in the heptapod writing system. I think it’s a full-fledged, general-purpose graphical language.’

Gary frowned. ‘So their writing constitutes a completely separate language from their speech, right?’

‘Right. In fact, it’d be more accurate to refer to the writing system as “Heptapod B”, and use “Heptapod A” strictly for referring to the spoken language.’