"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 a.s.sume 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 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." 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 is solvable?" 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-a.s.suming, 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. 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- G.o.d.

In the film, he'd said two-dimensional, two-dimensional, not three-dimensional. Why hadn't she noticed that before? 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 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 was a water hole, thought Heather. There a water hole, thought Heather. There was 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. 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. dimensional. Specifically, it was the fourth dimension. Specifically, it was the fourth dimension.

There are nine and sixty ways of constructing tribal lays And every single one of them is right!

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

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

A four-dimensional interpretation of reality is is special. 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 did have unique properties. have unique properties.

Heather had found the Science News Science News Website and read, astonished, an article from May 1989 by Ivars Peterson that began: Website and read, astonished, an article from May 1989 by Ivars Peterson that began: When mathematicians-normally cautious and meticulous individuals-apply adjectives like "bizarre," "strange," "weird" and "mysterious" to their results, something unusual is happening. Such expressions reflect the recent state of affairs in studies of four-dimensional s.p.a.ce, a realm just a short step beyond our own familiar, three-dimensional world.By combining ideas from theoretical physics with abstract notions from topology (the study of shape), mathematicians are discovering that four-dimensional s.p.a.ce has mathematical properties quite unlike those characterizing s.p.a.ce in any other dimension.

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 how a race perceived reality, its mathematicians would be inexorably drawn to the problems and singular traits of a four-dimensional framework. 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 kata or or ana? ana?

It was a typically hot, muggy, hazy August day. Heather found herself glistening with sweat just from walking over to the Computer-a.s.sisted 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 h.o.r.n.y 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 knew I knew you from somewhere! I saw you on TV-what show was that?" I knew you from somewhere! I saw you on TV-what show was that?"

Heather found the question embarra.s.sing because she'd been on so so many shows lately-but it sounded pompous to say that out loud. 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."