by Neil Clarke
“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.”
“Hold on a second. Why use two languages when one would suffice? That seems unnecessarily hard to learn.”
“Like English spelling?” I said. “Ease of learning isn’t the primary force in language evolution. For the heptapods, writing and speech may play such different cultural or cognitive roles that using separate languages makes more sense than using different forms of the same one.”
He considered it. “I see what you mean. Maybe they think our form of writing is redundant, like we’re wasting a second communications channel.”
“That’s entirely possible. Finding out why they use a second language for writing will tell us a lot about them.”
“So I take it this means we won’t be able to use their writing to help us learn their spoken language.”
I sighed. “Yeah, that’s the most immediate implication. But I don’t think we should ignore either Heptapod A or B; we need a two-pronged approach.” I pointed at the screen. “I’ll bet you that learning their two-dimensional grammar will help you when it comes time to learn their mathematical notation.”
“You’ve got a point there. So are we ready to start asking about their mathematics?”
“Not yet. We need a better grasp on this writing system before we begin anything else,” I said, and then smiled when he mimed frustration. “Patience, good sir. Patience is a virtue.”
You’ll be six when your father has a conference to attend in Hawaii, and we’ll accompany him. You’ll be so excited that you’ll make preparations for weeks beforehand. You’ll ask me about coconuts and volcanoes and surfing, and practice hula dancing in the mirror. You’ll pack a suitcase with the clothes and toys you want to bring, and you’ll drag it around the house to see how long you can carry it. You’ll ask me if I can carry your Etch-a-Sketch in my bag, since there won’t be any more room for it in yours and you simply can’t leave without it.
“You won’t need all of these,” I’ll say. “There’ll be so many fun things to do there, you won’t have time to play with so many toys.”
You’ll consider that; dimples will appear above your eyebrows when you think hard. Eventually you’ll agree to pack fewer toys, but your expectations will, if anything, increase.
“I wanna be in Hawaii now,” you’ll whine.
“Sometimes it’s good to wait,” I’ll say. “The anticipation makes it more fun when you get there.”
You’ll just pout.
In the next report I submitted, I suggested that the term “logogram” was a misnomer because it implied that each graph represented a spoken word, when in fact the graphs didn’t correspond to our notion of spoken words at all. I didn’t want to use the term “ideogram” either because of how it had been used in the past; I suggested the term “semagram” instead.
It appeared that a semagram corresponded roughly to a written word in human languages: it was meaningful on its own, and in combination with other semagrams could form endless statements. We couldn’t define it precisely, but then no one had ever satisfactorily defined “word” for human languages either. When it came to sentences in Heptapod B, though, things became much more confusing. The language had no written punctuation: its syntax was indicated in the way the semagrams were combined, and there was no need to indicate the cadence of speech. There was certainly no way to slice out subject-predicate pairings neatly to make sentences. A “sentence” seemed to be whatever number of semagrams a heptapod wanted to join together; the only difference between a sentence and a paragraph, or a page, was size.
When a Heptapod B sentence grew fairly sizable, its visual impact was remarkable. If I wasn’t trying to decipher it, the writing looked like fanciful praying mantids drawn in a cursive style, all clinging to each other to form an Escheresque lattice, each slightly different in its stance. And the biggest sentences had an effect similar to that of psychedelic posters: sometimes eye-watering, sometimes hypnotic.
I remember a picture of you taken at your college graduation. In the photo you’re striking a pose for the camera, mortarboard stylishly tilted on your head, one hand touching your sunglasses, the other hand on your hip, holding open your gown to reveal the tank top and shorts you’re wearing underneath.
I remember your graduation. There will be the distraction of having Nelson and your father and what’s-her-name there all at the same time, but that will be minor. That entire weekend, while you’re introducing me to your classmates and hugging everyone incessantly, I’ll be all but mute with amazement. I can’t believe that you, a grown woman taller than me and beautiful enough to make my heart ache, will be the same girl I used to lift off the ground so you could reach the drinking fountain, the same girl who used to trundle out of my bedroom draped in a dress and hat and four scarves from my closet.
And after graduation, you’ll be heading for a job as a financial analyst. I won’t understand what you do there, I won’t even understand your fascination with money,
the preeminence you gave to salary when negotiating job offers. I would prefer it if you’d pursue something without regard for its monetary rewards, but I’ll have no complaints. My own mother could never understand why I couldn’t just be a high school English teacher. You’ll do what makes you happy, and that’ll be all I ask for.
As time went on, the teams at each looking glass began working in earnest on learning heptapod terminology for elementary mathematics and physics. We worked together on presentations, with the linguists focusing on procedure and the physicists focusing on subject matter. The physicists showed us previously devised systems for communicating with aliens, based on mathematics, but those were intended for use over a radio telescope. We reworked them for face-to-face communication.
Our teams were successful with basic arithmetic, but we hit a road block with geometry and algebra. We tried using a spherical coordinate system instead of a rectangular one, thinking it might be more natural to the heptapods given their anatomy, but that approach wasn’t any more fruitful. The heptapods didn’t seem to understand what we were getting at.
Likewise, the physics discussions went poorly. Only with the most concrete terms, like the names of the elements, did we have any success; after several attempts at representing the periodic table, the heptapods got the idea. For anything remotely abstract, we might as well have been gibbering. We tried to demonstrate basic physical attributes like mass and acceleration so we could elicit their terms for them, but the heptapods simply responded with requests for clarification. To avoid perceptual problems that might be associated with any particular medium, we tried physical demonstrations as well as line drawings, photos, and animations; none were effective. Days with no progress became weeks, and the physicists were becoming disillusioned.
By contrast, the linguists were having much more success. We made steady progress decoding the grammar of the spoken language, Heptapod A. It didn’t follow the pattern of human languages, as expected, but it was comprehensible so far: free word order, even to the extent that there was no preferred order for the clauses in a conditional statement, in defiance of a human language “universal.” It also appeared that the heptapods had no objection to many levels of center-embedding of clauses, something that quickly defeated humans. Peculiar, but not impenetrable.
Much more interesting were the newly discovered morphological and grammatical processes in Heptapod B that were uniquely two-dimensional. Depending on a semagram’s declension, inflections could be indicated by varying a certain stroke’s curvature, or its thickness, or its manner of undulation; or by varying the relative sizes of two radicals, or their relative distance to another radical, or their orientations; or various other means. These were nonsegmental graphemes; they couldn’t be isolated from the rest of a semagram. And despite how such traits behaved in human writing, these had nothing to do with calligraphic style; their meanings were defined according to a consistent and unambiguous grammar.
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 gav
e 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.”