Factoring Humanity

Heather sighed. “I’m sorry about that. Really I am. This is a difficult time for all of us.”

Kyle was silent. Heather felt the need to fill the void. “It’s going to take time to sort all this out.”

“I’ve been gone for a year now,” said Kyle. “How much time do you need?”

“I don’t know. Look, I’m sorry I called; I didn’t mean to disturb you.”

“That’s all right,” said Kyle. “Was there something?”

Heather swallowed, then went on. “Yes. I’ve had a breakthrough, I think, with the Centauri transmissions. If you take them in groups of fifty-nine messages, each group is exactly the same size.”

“Really?”

“Yes.”

“How many groups are there?”

“Exactly forty-eight.”

“So you think — what? — you think the individual messages form forty-eight bigger pages?”

“Exactly. But the individual pieces are all different sizes. I assume they fit together into a rectangular grid of some sort, but I don’t know how to work that out.”

Kyle made a noise that sounded like a snort.

“There’s no need to be condescending,” said Heather.

“No — no, that’s not it. Sorry. It’s just funny. See, this is a tiling problem.”

“Yes?”

“Well, this

tiling problem — seeing if a finite number of tiles can be arrayed into a rectangular grid — is eminently solvable, just by brute-force computing. But there are other tiling problems that involve determining if specific tile shapes can cover an infinite plane, without leaving gaps that we’ve known since the nineteen-eighties fundamentally can’t be solved by a computer; if they’re solvable at all, it’s with an intuition that’s non-computable.”

“So?”

“So it’s just funny that the Centaurs would choose a message format that echoes one of the big debates in human consciousness, that’s all.”

“Hmm. But you say this is solvable?”

“Sure. I’ll need the dimensions of each message — the length and width in bits or pixels. I can write a program easily enough that will try sliding them around until they all fit together in a rectangular shape — assuming, of course, that there is such a pattern.” He paused. “There’ll be an interesting side effect, you know: if the individual tiles are not square and they all fit together only one way, you’ll know the orientation of each individual message. You won’t have to worry anymore about there being two possible orientations for each one.”

“I hadn’t thought of that, but you’re right. When can you do this?”

“Well, actually I’m too busy — sorry, but I am. But I can put one of my grad students on it. We should have an answer for you in a couple of days.

Heather tried to sound warm. “Thank you, Kyle.”

She could almost hear him shrug. “I’m always here for you, ” he said and clicked off.

13

It turned out to Heather’s delight that the fifty-nine tiles in each group did indeed make up a rectangular grid. In fact, they made up forty-eight perfect squares.

There were many circular patterns visible if the grids were rendered as black-and-white pixels. The circles had a variety of diameters — some were big, some were small. They, too, fell into size categories — no circle had a unique diameter.

Unfortunately, though, except for the circles — which seemed good supporting evidence that this was indeed the way in which the tiles were supposed to be arranged — still no meaningful patterns emerged. She’d been hoping for a picture book, with four-dozen leaves: Forty-Eight Views of Mount Alpha Centauri.

She tried arranging the forty-eight messages into even bigger groups: eight rows of six, three rows of sixteen, and so on. But still no pattern emerged.

She also tried building cubes. Some seemed to make sense — if she drew imaginary hoops through the cubes, in some configurations the circles on the cube faces were positioned just right to be the cross sections through those hoops.

But still she couldn’t get the whole thing to make sense.

She’s intelligent, but inexperienced. Her pattern suggests three-dimensional thinking.

Spock had said “he,” not “she,” of course.

And—

God.

In the film, he’d said two-dimensional, not three-dimensional. Why hadn’t she noticed that before?

Khan had been guilty of two-dimensional thinking; an attack through three dimensions defeated him.

Heather, perhaps, was being guilty of three-dimensional thinking. Would a four-dimensional approach help?

But why would the aliens use a four-dimensional design?

Well, why not?

No. No, there had to be a better reason than that.

She used her Web terminal to search for information about the fourth dimension.

And when she’d digested it all, she sagged back in her chair, stunned.