The next flex brought up 3 again. Then 2. And 1.
"Back to where we started," he said. And changed diagonals. A blank face appeared, which he marked 5. Then on through 2, 1, 3, and finally to the last blank one, 6.
"Those loops are only closed if you let them," he said with satisfaction. "I'd forgotten how much fun these hexes were! You can tell where you are because the faces change orientation."
Then the realization hit him.
"Hey, Tam!" he breathed.
He had spoken no louder than before, and the volume of the ambient music had not abated, but she opened her eyes immediately. "Yes?"
"Maybe this is a bum lead -- but I think I know why we're repeating worlds. And maybe how to snap out of the loop in controlled fashion."
She sat effortlessly, the muscles in her stomach tightening. "Speak."
He showed her his plastic construct, opaque because of its many layers. "You know what this is?"
"A doodle from plane-frame material."
"A hexa-hexaflexagon. See, I flex it like this and turn up new faces."
She took it and flexed it. "Clever. But to what point?"
"Well, they don't come up in order -- not exactly. Look at the face numbers as you go -- and at the composition of the repeats."
"One," she called off. She flexed. "Three... Two... One... Five... Two, inverted." She looked up. "It's a double triad. Intriguing, not remarkable."
"Suppose we numbered the worlds we've been going through -- and found a repeat that was backwards? I mean, the same, but like a mirror image?"
For the first time, he saw an agent do a double take. "The second blizzard was backwards!" she exclaimed. "Or rather, twisted sixty degrees. The igloo -- the irregularities in it and pattern of our prior tracks, what was left of them, the projector -- all rotated by a third!"
"Yeah. That's what I figured. Didn't make sense at first."
"Flexing alternates! Could be." Rapidly she flexed through the entire sequence, fixing the pattern in her mind. "It fits. We could be in a six-face scheme on this framework. In that case our next world will be -- the forest." She certainly caught on rapidly! "But we can't go home from there."
"No. The face will be twisted, part of a subtriad. But we would know our route."
"Yeah," he agreed, pleased.
She pondered momentarily. "There's no reason the alternates should match the hex faces. But there is a clear parallelism, and it may be a useful intellectual tool, in much the way mathematics is a tool for comprehending physical relations. Our problem is to determine the validity of our interpretation without subjecting ourselves to undue risk."
"You sound like Cal now!"
"No shame in that," she muttered. "Your friend has a freakishly advanced intellect. We could travel the loop again just to make sure -- but that would mean a delay of several hours, waiting for the projectors to recharge. In that time our competition could gain the advantage."
"So we just go ahead fast," Veg finished. "We can follow the flex route and see if it works. If it does, we've got our map of alternity."
"In your bumbling male-normal fashion, you may have helped me," Tamme said. "Come here." Veg knelt down beside her.
She put both hands to his head, pulled him to her, and kissed him. It was like the moment in free fall when a spaceship halted acceleration in order to change orientation. His whole body seemed to float, while his own pulse pounded in his ears.
She let him go. It took him a moment to regain composure. "That isn't the way you kissed me before."
"That was demonstration. This was feeling."
"You do feel? I thought -- "
"We do feel. But our emotions are seldom aroused by normals other than amusement or distaste."
Veg realized that he had been paid an extraordinary compliment. But that was all it was. He had helped her, and she was appreciative. She had repaid him with a professionally executed gesture. Case dismissed.
"We should have a choice here," she said. "Repeat the triad indefinitely -- or break out of it. Only way to break out is to project elsewhere than to the plane world. But how can we do that -- without interfering with the settings on the projector?"
Veg appreciated the problem. Touch those settings, and they could be thrown right out of this hex framework and be totally lost -- or dead. That would accomplish nothing worthwhile. They wanted to follow the existing paths wherever they led, and catch up to -- whom?
"These settings are built into the hexaflexagon," he said. "All you have to do is find them."
"Yes. Too bad alternity isn't made of folded plastic."
They remained in silence for a time, while the music swelled around them. And Veg had a second revelation. "The music!"
Again she caught on almost as fast as he thought of it and quickly outdistanced his own reasoning. "In phase with the music! Of course. Catch it during one type of passage, go on to Plane. Catch it during another -- "
"Now's the time!" Veg cried.
They ran to the projector. Tamme had it on instantly.
And they were in the forest.
"Victory!" Veg exclaimed happily. Then he looked about uncertainly. "But is it -- ?"
"Yes, it is rotated," Tamme said. "So it is part of a different triad. There'll be another odd-handed projector here."
They located it, and it was. "Hypothesis confirmed," she said. "Now if our interpretation is correct, we won't have to worry about being sent back to Blizzard because this inverted version is part of a different loop. The next one should be new. Brace yourself." She reached for the switch.
"Sure thing," Veg said. "I'm braced for one new world."
It was new, all right. Veg's first impression was of mist. They stood in a tangibly thick fog. He coughed as the stuff clogged his lungs. It wasn't foul, just too solid to breathe.
"Get down," Tamme said.
He dropped to the ground. There was a thin layer of clear atmosphere there, below the fog bank, like air trapped beneath river ice. He put his pursed lips to it and sucked it in.
"Crawl," she said, her voice muffled by the fog.
They crawled, shoving aside the fog with their shoulders. Suddenly the ground dipped -- but the bottom of the fog remained constant. It was too stiff to match the exact contour of the land. Now there was squatting clearance beneath, then standing room.
"That's some cloud!" Veg remarked, peering up. The stuff loomed impenetrably, a pall that blacked out all the sky. Wan light diffused through it. "Stuff's damn near solid!"
"You liked it better under the pine tree?" Tamme inquired. She was already looking for the next projector.
"Sure did!" He had the nagging feeling the fog bank could fall at any moment, crushingly.
A valley opened out before them. Tamme stared.
Veg followed her gaze. "A fog house?" he asked, amazed.
It was. Blocks of solidified fog had been assembled into something very like a cabin, complete with slanting, overlapping fog tiles on the roof. Beyond it was a fog wall or fence.
"This we have to look into," Tamme said. She moved toward the house.
A curtain of fog parted, showing a doorway and a figure in it. "Inhabited yet," Tamme murmured. Her hands did not move to her weapons, but Veg knew she was ready to use them instantly.
"Let's go ask directions," he suggested facetiously.
"Yes." And she moved forward.
"Hey, I didn't mean -- " But he knew that she had known what he meant since she could read his emotions. Awkwardly, he followed her.
Up close, there was another shock. The inhabitant of the house was a human female of middle age but well preserved -- with a prehensile nose.
Veg tried not to stare. The woman was so utterly typical of what he thought of as a frontier housewife -- except for that proboscis. It twined before her face like a baby elephant trunk. It made her more utterly alien than a battery of other nonhuman features might have -- because it occupied the very center of attention. It was repulsively fascinating.<
br />
Tamme seemed not to notice. "Do you understand my speech? she inquired sociably.
The woman's nose curled up in a living question mark.
Tamme tried a number of other languages, amazing Veg by her proficiency. Then she went into signs. Now the woman responded. "Hhungh!" she snorted, her nose pointing straight out for a moment.
"Projector," Tamme said. "Alternates." She shaped the projector with her hands.
The woman's nose scratched her forehead meditatively. "Hwemph?"
"Flex," Veg put in, holding out the hexaflexagon.
The woman's eyes lighted with comprehension. "Hflehx!" she repeated. And her nose pointed to the fog bank from which they had emerged, a little to the side.
"Hthankhs," Veg said, smiling.
The woman smiled back. "Hshugh."
Veg and Tamme turned back toward the fog. "Nice people," Veg remarked, not sure himself how he intended it.
"There have been others before us," Tamme said. "The woman had been instructed to play dumb, volunteering nothing. But we impressed her more favorably than did our predecessors, so she exceeded her authority and answered, after all."
"How do you know all that?" But as he spoke, he remembered. "You can read aliens, too! Because they have emotions, same as us."
"Yes. I was about to initiate hostile-witness procedures, but you obviated the need."
"Me and my flexagon!"
"You and your direct, naïve, country-boy manner, lucking out again." She shook her head. "I must admit: Simplicity has its place. You are proving to be a surprising asset."
"Shucks, 'taint nothin'," Veg said with an exaggerated drawl.
"Of course, our predecessors were the same: Tamme and Veg. That's why they obtained her cooperation."
"I noticed she wasn't surprised to see us. I guess our noses look amputated."
"Truncated. Yes."
He laughed. "Now she's punning. Truncated trunks!"
They were at the fog bank. "Stand here. I need another orientation point. The projector will be within a radius of twenty meters, or about sixty feet."
"You sure can read a lot from one nose-point!"
She plunged into the bank. The stuff was so thick that her passage left a jagged hole, as if she had gone through a wall of foam. It bled into the air from the edges, gradually filling in behind her. "Talk," Tamme said from the interior. "The sound will help me orient on it, by the echo."
It figured. She didn't ask him to talk because of any interest in what he might have to say! "This place reminds me of Nacre in a way. That was all fog, too. But that was thinner, and it was everywhere, made by falling particles. The real plant life was high in the sky, the only place the sun shone; down below was nothing but fungus, and even the animals were really fungus, like the mantas. So it wasn't the same."
There was no response from the bank, so he continued. "You know, I read a story about a fog like this once. It was in an old science-fiction book, the kind they had in midcentury; I saw a replica printed on paper pages and everything. This thick fog came in wherever the sun didn't shine -- they spelled it 'phog' -- and inside it was some kind of predator you never saw that ate people. It never left the fog -- but nobody dared go in the fog, either. All they heard was the scream when it caught somebody -- "
"YAAAGH!"
Veg's mouth gaped. "Oh, no!" He plunged into the fog, knife in hand.
A hand caught his wrist and hauled him back out. "Next time don't try to tease an agent," Tamme said, setting him down. "I found the projector."
"Sure thing," he said, chagrined. Still, it was the first clear evidence of humor he had seen in her.
"Crawl under," Tamme said.
They crawled under the fog, snatching lungfuls of clear air from the thin layer on the ground. The projector was there.
"Not far from where we landed," Tamme said. "But the pattern is not consistent enough to be of much aid. We still have to search out the projector on each new world and figure out the mechanism for breaking out of loops. I don't like that."
Veg shrugged noncommittally. Except for Blizzard, he hadn't minded the searches. But of course if there were danger, they would not be able to afford much delay. "With the hexaflexagon, you can run through every face just by flexing the same diagonal as long as it will go. When it balks, you switch to the next. So maybe if we just keep going straight ahead, we'll get there, anyway."
Tamme sat up. She did not seem to be bothered by fog in her lungs. "We'll play it that way. If we get caught in a repeating loop, we'll look for something to change. Meanwhile, I want a concurring opinion."
"Another man-versus-tiger choice?"
She brought out a slip of paper. "Call off the order of your hexaflexagon faces."
Veg, hunched nose down to the ground to avoid the fog, was surprised at this request. Tamme knew the order that the faces appeared; she had flexed through them, and agents had eidetic recall. He could only confirm the obvious! But he brought out his toy and went through the whole pattern, calling off the numbers. "One. Five. Two. One. Three. Six. One. Three. Two. Four. Three. Two. One."
Tamme made a diagram of lines and numbers and little directional arrows. "This is triangular," she said. "A three-faced hexaflexagon would simply go around the central triangle. Your six-faced one adds on to the angles. Would you agree to the accuracy of this diagram?"
She showed him what she had drawn.
Veg traced around it, starting from the northwest face 1. "One, five, two, one, three -- yeah, that's the order. Makes sense of it finally!"
Tamme nodded. He could barely make out her gesture since her head was almost concealed by the fog. "As I make it, we actually started on Five, the City. That would make Two the Forest, One the Blizzard, Three the Orchestra, Six the Planes, and back to our first repeat, One/Blizzard. Then repeat Three/Orchestra. And repeat Two/Forest And now face Four/Fog."
"I guess so," Veg said, having trouble keeping up. "We're here now."
"Our next stop should be repeat Three/Orchestra -- this time twisted because it is on a separate loop. Then on to Two/Forest, One/Blizzard, and home to Five/City."
"It figures," he said. "We've used up all the faces."
"In which case we'll be back where we started -- closed loop, and nobody but ourselves."
"I guess so, right now. The others must have gotten off. Is that bad?"
"I can't buy it. Who set up all these other projectors?"
Veg shook his head. "Got me there! If they'd gotten off, they'd have taken back their projectors -- so they must be still on. And there can't be six Vegs and six Tammes." He sobered. "Or can there?"
"Suppose your hexaflexagon had twelve faces?"
"Sure. There can be any number of faces if you start with a long enough strip of triangles and fold it right."
"A twelve-face construction would merely add one new face to each of the six exterior angles," she said.
Veg shrugged. "I'll take your word. I'd have to make a live hexaflexagon to check it out myself."
"Don't take my word. Make your construction."
"Here? Now? Why not get to a better alternate to -- "
"No."
"I don't have anything to -- "
She took apart the six-faced hexaflexagon, straightened out the long folded strip of plastic, pried at the edge with a small knife that appeared in her hand, and peeled it into two layers lengthwise. She produced a little vial of clear fluid, applied it to the edges, and glued the strips together endwise. The result was a double-length strip.
Veg sighed. He took it and folded it carefully. He made a flat spiral so that the double length became the size of the original but with two layers instead of one. Then he fashioned a normal hexaflexagon.
"Run through it and number your faces," Tamme said.
"Okay." This was a more complicated process, involving thirty flexes, but in due course he had it. Meanwhile, Tamme had been making a new diagram.
"Now start at face One and flex," she said. "I wi
ll call off your numbers in advance. Five."
He flexed. "Five it is."
"Seven."
He flexed again. "Right."
"One."
"Right again. Hey, let me see that diagram!"
She showed it to him. It was an elaborated version of the prior one, with new triangles projecting from each of the six outer points. One angle of each of the outermost triangles carried the number of a new face, bringing the total to twelve.
Of Man and Manta Omnibus Page 59
