But none of them divided evenly.
None, except 1 and 59.
Fifty-nine was a prime number.
And—a thought occurred to her. One itself was sometimes considered a prime. Two was definitely a prime. So was three. But in a way, all those numbers were trivial primes: every whole number lower than them was also divisible only by itself or one. In many ways, five was the first interesting prime number—it was the first one in sequence that had numbers lower than itself that weren’t primes.
So if you discounted one, two, and three as trivially prime, then in the table she’d produced, 59 was the only non-trivial prime that divided evenly into the total number of undecoded alien messages.
It was another arrow pointing at that figure. The alien transmissions could possibly be arrayed in 48 pages each consisting of 59 individual messages, or 59 pages each consisting of 48 messages.
Researchers had been looking for recurring patterns in the messages for years, but so far none had turned up that hadn’t seemed coincidental. Now, though, that they knew the total number of messages, all sorts of fresh analyses could be done.
She opened another window on her computer and brought up the file directory of alien messages. She copied the directory into a text file, where she could play with it. She highlighted the bit counts for the first 48 undecoded messages and tallied them up: they totaled 2,245,124 bits. She then highlighted the next twenty-four. The tally came to 1,999,642.
Damn.
She then highlighted the counts for messages 12 through 71—the first 59 undeciphered messages.
The total came to 11,543,124 bits.
Then she highlighted messages 72 to 131 and tallied their sizes.
The total was also 11,543,124 bits.
Heather felt her heart pounding; perhaps someone had noticed this before, but. . .
She did it again, working her way through the material.
Her spirits fell when she found the fourth group tallied only 11,002,997 bits. But after a moment, she realized she’d highlighted only 58 messages instead of 59. She tried again.
The tally was 11,543,124.
She continued on until she’d done all 48 groupings of 59 messages.
Each group totaled precisely 11,543,124 bits.
She let out a great whoop! of excitement. Fortunately, her office did have that sturdy oak door.
The aliens hadn’t sent 2,832 separate messages—rather, they’d sent 48 large ones.
Now, if only she could figure out how to tile the messages together. Unfortunately, they were of many different sizes, and there was no orderly repetition from page to page. The first message making up the first group of 48 was 118,301 bits long (the product of the primes 281 and 421), whereas the first message of page two was 174,269 bits long (the product of the primes 229 and 761).
Presumably, the individual tiles formed square or rectangular shapes when properly placed together. She doubted she could figure it out by trial and error.
But surely Kyle could write her a computer program that would do it.
After last night, she was hesitant. What would she say to him?
She steeled her courage and picked up her phone.
“Hello?” said Kyle’s voice.
He doubtless knew it was Heather calling; he could read it off the status line on his phone. But there was no particular warmth in his voice.
“Hi, Kyle,” said Heather. “I need your help.”
Frosty: “You didn’t need my help last night.”
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.
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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.
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 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:
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 space, 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 space has mathematical properties quite unlike those characterizing space 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 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.”