Factoring Humanity

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.

There was a water hole, thought Heather. There was a common ground between species. But it was nothing as simple as a set of radio frequencies. The common ground wasn’t related to ordinary physics, or the chemistry of atmospheres, or anything that mundane. And yet it was something that in many ways was even more basic, more fundamental, more a part of the very fabric of existence.

The water hole was dimensional. Specifically, it was the fourth dimension.

Except that one of them was more right than all the others.

Depending on sensory apparatus, scheme of consciousness, consensual agreement with others of its kind, and more, a life form could perceive the universe, perceive its reality, in one dimension, two dimensions, three dimensions, four dimensions, five dimensions, and on and on, ad infinitum.

But of all the possible dimensional frames, one is unique.

A four-dimensional interpretation of reality is special.

Heather didn’t understand it all—as a psychologist, she had an excellent grounding in statistics, but she wasn’t really up on higher mathematics. But it was clear from what she’d read that the fourth dimension did have unique properties.

Heather had found the Science News Website and read, astonished, an article from May 1989 by Ivars Peterson that began:

Heather didn’t pretend to understand all that Peterson went on to say, such as that only in four dimensions is it possible to have manifolds that are topologically but not smoothly equivalent.

But that didn’t matter—the point was that mathematically, a four-dimensional frame was unique. Regardless of how a race perceived reality, its mathematicians would be inexorably drawn to the problems and singular traits of a four-dimensional framework.

It was a water hole of a different sort—a gathering place for minds from all possible life forms.

Christ.

No—no, not just Christ.

Christus Hypercubus.

She could make three-dimensional cubes out of her pages. And with forty-eight pages, one could make a total of eight cubes.

Eight cubes, just like in the Dali painting on Kyle’s lab wall.

Just like an unfolded hypercube.

Of course, Cheetah had said there was more than one way to unfold a plain, ordinary cube; only one of eleven possible methods yielded the cross shape.

There were probably many ways to unfold a hypercube as well.

But the circular marks provided a guide!

There was probably only one way to align all eight cubes so that the imaginary hoops went through them at the right places to line up with the circular marks.

She’d tried arranging the pictures as cubes before, hoping that they’d line up in meaningful patterns. But now she tried mapping them on her computer screen onto the separate cubes of an unfolded tesseract.

U of T had site licenses for most software used in its various departments; Kyle had shown Heather how to access the CAD program that had been used to determine the way in which the individual tiles fit together.

It took her a while to make it work properly, although fortunately the software operated by voice input. Eventually she had the forty-eight messages arranged as eight cubes. She then told the computer she wanted it to arrange the eight cubes in any pattern that would make the circular registration marks line up properly.

Boxes danced on her screen for a time, and then the one correct solution emerged.

It was the hypercrucifix, just like in Dali’s painting: a vertical column of four cubes, with four more cubes projecting from the four exposed faces on the second cube from the top.

There was no doubt. The alien messages made an unfolded hypercube.

What, she wondered, would you get if you could actually fold the three-dimensional pattern kata or ana?

It was a typically hot, muggy, hazy August day. Heather found herself glistening with sweat just from walking over to the Computer-Assisted Manufacturing Lab; the lab was part of the Department of Mechanical Engineering. She didn’t really know anybody there and so just stood on the threshold, looking around politely at the various robots and machines clanking away.

“May I help you?” said a handsome, silver-haired man.

Heather approved of those who knew the difference between “can” and “may.”

“I certainly hope so,” she said, smiling. “I’m Heather Davis, from the Psych Department.”

“Somebody got a screw loose?”

“I beg your pardon?”

“A joke—sorry. See, a shrink coming to see an engineer. We tighten loose screws all the time.”

Heather laughed a little.

“I’m Paul Komensky,” said the man. He extended his hand. Heather took it.

“I do need some engineering help,” Heather said. “I need something built.”

“What?”

“I’m not sure exactly. A bunch of prefabricated panels.”

“How big are the panels?”

“I don’t know.”

The engineer frowned—but Heather couldn’t tell if it was a “dumb woman” or “dumb artsy” frown. “That’s a little vague,” he said.

Heather smiled her most charming smile. Today the various engineering schools had fifty-percent female undergrads, but Komensky was old enough to remember when engineers were all horny men who would go days without seeing a female. “I’m sorry,” she said. “I’m working on the alien radio messages, and—”

“I knew I knew you from somewhere! I saw you on TV—what show was that?”

Heather found the question embarrassing because she’d been on so many shows lately—but it sounded pompous to say that out loud.

“Something on Newsworld?” she offered tentatively.

“Yeah, maybe. So this has to do with the aliens?”

“I’m not sure—I think so. I want to make a series of tiles that represent the alien message grids.”

“How many messages are there?”

“Two thousand, eight hundred and thirty-two—at least, that many undecoded ones; they’re the only ones I want to make into tiles.”

“That’s a lot of tiles.”

“I know.”

“But you don’t know how big they should be?”

“No.”

“What should they be made out of?”

“Two different substances.” She handed him her datapad. Its screen showed two chemical formulas. “Can you synthesize them?”

He squinted at the display. “Sure—nothing difficult about them. You’re certain they’re solid at room temperature?”

Heather’s eyes went wide. She’d read all the papers on the chemicals ten years ago, when they’d first been synthesized, but hadn’t really thought about them much since. “I have no idea.”

“This one will be,” he said, pointing at the top formula. “That one… well, we’ll see. Are these formulas from the alien messages?”

Heather nodded. “From the first eleven pages. People have synthesized these compounds before, of course, but no one ever figured out what they were for.”

Komensky made an impressed face. “Interesting.”

She nodded. “I want the zero bits to be made of one of these substances, and the one bits made of the other.”

“You want one painted onto the other?”

“Painted? No, no, I thought you’d build them out of the two materials.”

Komensky frowned again. “I don’t know. That formula looks to me like it’ll be a liquid, but it might dry into a hard crust. See those oxygens and hydrogens? They could evaporate out as water, leaving a solid behind.”

“Oh. Well, then, yes—and that answers the big question I’d been unable to solve.”

“Which is?”

“Well, I was trying to figure out which substance represented the one bits and which one represented the zero bits. The ones are ‘on’ bits, so the paint must represent the ones; it must go on the—the—”

“ ‘The substrate’ we call it in materials science.”

“The substrate, yes.” A pause. “How hard would it be to do that?”

“Well, again, it comes back to how big you want the tiles.”

“I don’t know. They’re not all the same size, but even the biggest shouldn’t be more than a few centimeters—I want to fit them together.”

“Fit them?”

“Yeah, you know—lay them side by side. See, if you arrange each group of fifty-nine tiles properly, they form a perfect square—there’s only one layout that’ll do that.”

“Why not just build the big panels instead of the individual tiles?”

“I don’t know—the tiling itself might be significant. I don’t want to make any assumptions.”

“Like the ‘on’ bits go ‘on’ the substrate?” His tone was one of gentle teasing.

Heather shrugged. “It’s as good a guess as any.”

He nodded, conceding the point. “So twenty-eight hundred tiles make up how many bigger squares?”

“Forty-eight.”

“And what are you going to do with the resulting squares?”

“Assemble them into cubes—and then assemble those cubes into an unfolded tesseract.”

“Really? Wow.”

“Yes.”

“Well, do you want the finished thing big enough so that you can crawl inside of one of the cubes?”

“No, that won’t be—”

She stopped dead.

No scale specified. Nowhere in any of the messages did there seem to be anything suggesting a size for the construct.

Make it any size, the aliens seemed to be saying.

Make it your size.

“Yes, yes—that would be perfect! Big enough to go inside.”

“Well, okay—sure. We can build the substrate tiles, no problem. How thick should they be?”

“I don’t know. As thin as possible, I guess.”

“I can make them one molecule thick if that’s what you want.”

“Oh, not that thin. They’ll have to hold together. A millimeter or two, maybe.”

“No problem. We’ve got a machine all set up to turn out plastic building panels for the School of Architecture; I could modify it easily enough to turn out the tiles you need. Do you want them to have smooth edges or would you like a tongue-and-groove arrangement, so they can snap together?”

“You mean so they form a big solid piece?”

Komensky nodded.

“That would be great.”

“What about the painting on of the other chemical?”

“I figured I’d have to do that by hand,” said Heather.

“Well, you could, but we’ve got programmable microsprayers that can do it for you, assuming the substance has a low enough viscosity. We use the sprayers to paint patterns onto the panels we make for the architecture students—you know, little outlines of bricks, or little dots to represent rivets, stuff like that.”

“That’d be perfect. How soon can you do it?”

“Well, during the school year, we’re usually pretty backed up. But in summer, we’ve got lots of free time. We can get at it right away. We’ve still got a couple of grad students hanging around; I’ll have one of them look into manufacturing those chemicals. As I say, at first glance they look simple enough, but we won’t know for sure until we actually try to synthesize them.” A pause. “Who’s going to pay for this?”

“What’ll it cost?” asked Heather.

“Oh, not much. Robots are so cheap these days, we no longer amortize their cost over manufacturing runs like we used to. Maybe five hundred dollars for the material.”

Heather nodded. She’d find some way to explain it to her department head later, once he got back from vacation. “That’s fine. Charge it to Psych; I’ll sign the requisition.”

“I’ll e-mail you the paperwork.”

“Terrific. Thank you. Thank you very much.”

“You’re very welcome.” He smiled and held her with his eyes.