As morning approached, Jill complained that its sonar vision was getting fuzzy. It had sent a portion of the Christmas Branch outside into the radome to investigate, but could find nothing wrong.
"I've checked all the wires and connectors," said Jill, "Since they were jury-rigged when we replaced the radar with the sonar, they were the first things to suspect, but they seem to be fine. I also had the Christmas Branch measure the sonic pressure from the transducers themselves, and the power seems to be getting out, but the returns are getting more blurred by the hour."
"I'll get suited up and go out for a look," said Shirley. "Perhaps I can see something the imp missed." She twirled out of the chair at the engineering console on the upper flight deck, hopped down to the main floor and wended her way back through the crowded central corridor of Dragonfly to the air lock at the rear of the plane. As she passed through the galley she noticed that the privacy curtain was pulled across the aisle. She slid the curtain open anyway and entered. As she passed by the shower, she was bumped by the door as Richard tried to get out.
"Excuse me! Excuse me! So sorry," she said in mock courtesy, as she bounced back and forth between the wall and the shower door, effectively preventing Richard from leaving.
Once Richard realized what was going on, he applied some steady muscle and with a little effort and a flurry of "pardon me"s, "after you"s, "excuse me"s, and "so sorry"s, the two super-sized humans were soon jostling each other in the narrow corridor like a couple of Great-Dane puppies, Richard getting the worst of the tussle since he had to hold onto his bath towel.
With a completely insincere "my sincerest apologies", Shirley sent Richard reeling with one last shoulder block, closed the privacy curtain between the work area and the sleeping quarters, and went to the suit locker, a huge grin on her face.
Up front Arielle looked over the back of her pilot seat at David, who was sitting at the computer console. The plane was still rocking slightly from the rumpus amidships. They didn't exchange any words, but a wry glance from one, followed by a resigned shake of the head from the other, spoke volumes.
Shirley was only halfway into her suit when the privacy curtain was pulled back and Richard appeared. He had his coveralls on, but no corridor boots. Shirley's stomach twinged in sympathy when she noticed the stubs on his feet where his little toes had once been.
"Why are you suiting up?" he asked, concerned.
"Jill said the sonar is acting up, but she and the Christmas Branch can't seem to find the problem."
"Need any help?" he offered.
"Not yet," she replied. "But you can give me a hand with the backpack."
Richard and the part of the Christmas Branch that was inside went over the checkout with Shirley, then cycled her through the air lock.
Shirley waited until the inrush of water into the lock subsided, then swam out the door to the front of the airplane. Jill had turned off the propellers, and the craft was drifting slowly forward. The Christmas Branch was waiting for her, and together they unlatched the radome and opened it up. As her suit imp relayed Jill's voice, the Christmas Branch pointed to various sections of the bank of sonar transducers and explained what the computer had checked previously. Shirley could see nothing obviously wrong, but had the computer take the Christmas Branch through the entire checkout again while she watched.
As the Christmas Branch was going through its programmed routine, the water began to get cloudy as if a glass of milk had been released in the water. Suddenly the entire cavity of the radome was milky white, and Shirley could hardly see the Christmas Branch through the murkiness.
«Hi! What is this thing?»
Shirley felt the high pitched tones of White Whistler through her suit as Jill provided the translation through her suit imp. She felt her hand being raised as the curious alien pushed a portion of its body under her glove to feel the equipment hidden beneath. Shirley waited patiently until the alien had finished feeling and tasting everything inside the dome.
«Bad!» came an explosive chirp.
"I think it just tasted some of the epoxy glue that we used to attach the sonar array," Jill's voice interjected. "It would still have a strong residual of hardener."
«Teach me.» came another whistle.
Shirley smiled at the childlike eagerness of the alien to learn something new, while she in turn was awed by someone who had a greater mental capacity than a dozen humans. She started in to explain how the sonar system worked. It turned out to be fairly easy, since she could have Jill operate it while she pointed, and White Whistle, having its own sonar system, could easily comprehend the purpose. Some of the components were bewildering to it, however, especially the concept of a "wire" to carry "electricity". White Whistle wanted to "feel" the electricity, but both Jill and Shirley didn't want to risk applying a voltage of any magnitude to such a highly sensitive creature, despite its immense size.
White Whistle quickly understood most of the operation and purpose of the sonar system, then asked, «Why bubbles? I not see well. Machine must not see well.»
"Yes!" said Shirley in surprise. "The sonar cannot see well. What bubbles?"
«These,» White Whistle chirped as a long snakelike tentacle brushed the inside of the radome and scraped a swarm of tiny bubbles off the inside surface of the dome, leaving a clear path of black plastic.
«See better,» said the alien through the imp. The tentacle whipped around the inside of the dome, clearing away the tiny bubbles that had been scattering the sonar waves as they entered and left the radome. «Now see lots better.»
"That did it," said Jill. "The sonar image is perfectly clear now. There must be a slow chemical reaction between the material in the dome and the ammonia-water of the ocean that creates microscopic bubbles on the inside of the dome. I will have the Christmas Branch wipe them off periodically."
"Thank you," said Shirley to the alien.
«What means thankyou?» asked the alien.
Shirley sighed, her breath whistling from between her lips, and started to try to explain the human practice of polite conversation to an alien whose whole social structure was based on directness. Jill, trying to translate between the two, included the sigh in the conversation without translation.
«Stop!» interrupted the voice from the white alien. «Your pet talk?» A white cloud enveloped Shirley's visor, while another portion touched the sonar array that was Jill's vocal cords.
"An ideal time to make an important point," Jill whispered through Shirley's imp. "Repeat the following after me. It is a salutation plus the name of the individual that is surrounding your helmet." Jill whistled a short, but complex tune. It had a few triple-tongues in it, but it was easy for Shirley, who had been a trumpet player in her high school band.
«Your pet say hello!» The white cloud lifted from the helmet and then the sonar array, which had stayed silent while Shirley had whistled. Those two portions seemed to be absorbed into the interior of the alien, as if checking them out, then another arm of white jelly reached out from the alien to retouch Shirley's visor. There was a simple tone, a complex whistle, another simple tone, and a different complex whistle.
"Say TtWwOo," said the imp into Shirley's ear.
Shirley whistled a respectable imitation of the alien number. Her whistle was repeated by the alien twice with the same complex whistles in between.
Now this will show you are smarter than Loud Red's pets," said Jill. "Provided you can get your lips around this one. If not, you can fake it and I'll have the imp make the sound."
Shirley's pucker and pitch was up to the challenge, and a reasonable facsimile of the number "FfFfOoOoUuUuRrRr" vibrated out through the helmet into the sensitive body of the white alien.
«Smart pet!»
Jill dropped her bombshell.
"This element not pet. Other similar elements not pet. I am pet."
There was a long pause as the white alien thought through the statement. A large portion started to rock up and sink, but then redissolved. The alien formed a lens with part of its body and moved it up close to Shirley's visor to look in. Shirley put her hand up next to her face and raised her gloved fingers as she went through the addition tables up to five, her fingers adding counterpoint to the whistles coming from her lips. Fortunately she was a quick study in music.
«Stiff¤Movers intelligent. Not pets. But not talk correct.»
"Stiff¤Movers are humans. Not pets. Not made to talk. They think. I talk."
"I hope I got you out of that with minimum disruption to your superiority," said Jill through Shirley's imp.
"If we really are superior," said Shirley.
«What is human buzzing?»
"Human talk to me with buzz. I talk to you with whistle."
Shirley's imp whispered in her ear. "Do something while talking about it, so I can translate as you talk."
Shirley reached for her belt and pulled off her Mech-All. She set the handle for a large oval-bladed screwdriver and the blob at the end of the tool reconfigured.
"This is a tool." She ducked out of the radome to the outside.
"I leave." She reached out and pulled the radome shut.
"I put front of airplane back." Jill translated airplane as pet.
"I fasten front of airplane." Shirley fastened the screw holddowns with the screw-driver tool, deactivated the mechanism into a soft blob, and put it back on her tool belt. The white alien, ever curious, tried to make a small hard sliver similar in shape to the oval screwdriver and undo the hold-downs. Shirley decided this was time to assert authority.
"NO!" she shouted, and struck the white appendage with her gloved hand. There was some resistance, but her hand cut through the alien's appendage, leaving a liver-sized portion floating by itself in the ocean. There were strange whimpering noises from the blob, that were immediately quelled when the main body of the alien quickly sent out another appendage to make contact with the severed piece of flesh.
"OH! I'm sorry," said Shirley.
The alien reformed its clear lens and moved it in from of Shirley's helmet, while a substantial portion of its body lifted up to form a retina behind the lens so it could look at the strange "human" inside the hard metal suit.
«What means 'sorry'?» asked Clear«»White«»Whistle, ever curious.
Shirley sighed again.
"It's too bad that we always have to go through you," said Shirley to Jill. "With a lot of practice maybe I could whistle some of their names, but I certainly couldn't carry on a decent conversation even if I could learn the language in a hurry. What we need is a magic translation machine." She paused, then added, "Of course that's what you are. Too bad it takes up so much of your brainpower to do it."
"Actually, since they are very logical thinkers and we used Boolean logic to develop our communication, we ended up speaking in a very formalized manner that is quite different from the way that they talk between themselves. Most of the translation is automatically handled by a translation table and some simple rules for syntax. I only have to use my more general translation programs when new words or situations arise. The translation table and the syntax rules are too complicated to be programmed into your imps, but they could easily be stored in the miniprocessor in your suit chest-pack."
"The suit imp could stay outside as the transmitter and receiver," said Shirley, immediately grasping the concept. "How about programming my suit now and letting me try it?"
There was a significant increase in the brightness of the laser light passing between the transponder on the top of Shirley's helmet and a similar one blinking from the left eye of the Magic Dragonfly. Shirley heard the rustle of her suit imp making its way through the valves in the life support back pack to the outside. Soon the metallic green body of the suit imp with its red, yellow, and blue lights was perched on the shoulder of her suit.
Shirley watched the illuminated message board display at the bottom of her visor until it said: "TRANSLATION PROGRAM LOADED."
"Try it," said her personal imp. "But keep your sentences simple. The translator will ask you to rephrase a sentence if it gets too complicated for the syntax program to handle."
"Hello, White Whistle. My name is Shirley."
Shirley could hear the whistles from the waving cilia of her suit imp at the same time she heard a more complex whistle coming from Jill's sonar as she explained what had been done. The white cloud swirled up and paused in front of Shirley's helmet.
«Hello, Shirley. My name is White Whistler. I touch your SoundoxMaker?» A long white pseudopod extended to within a few centimeters of the suit imp.
"Yes," said Shirley, sure that the wiry limbs of the imp were more than a match for the soft jellylike flesh of the alien.
The imp was engulfed in a white ball, which withdrew again after a few seconds of feeling.
«Interesting. Each subset of SoundoxMaker is like larger sub-set. The smallest subsets are very tiny. SoundoxMaker made like statement in recursive logic.»
"Do all the flouwen like mathematics?" Shirley asked, intrigued by the gigantic cloud with the Einstein brain.
«Yes!» piped White Whistler. «Tight premises, narrow conditions... surprise conclusions! Fun!!» The milky white cloud swirled as it spoke, forming a tight knot that almost condensed into a quartzine rock, then unfurled again to curl around Shirley's body, as close as it could, so as to feel what she was doing.
The presence of the alien was not disturbing physically, for Shirley had tested herself surreptitiously some time ago by closing her eyes and trying to tell whether it was clear water or curious alien that enveloped her. Without sight, she had as little luck distinguishing aliens from water as she had at college parties telling Heineken from Budweiser beer. Still, having an intelligent creature looking over your shoulder, around your waist, and between your crotch at the same time did affect her performance.
"What kind of problems do you solve?" she asked. "Pure logic or complicated mathematical ones?"
«Wha?» was the only reply from the imp in Shirley's ear, although she could hear the multitoned response of White Whistler's chirped response. The communications link that Jill so expertly supplied had broken down. Shirley rephrased her question.
"We study mathematical problems too," said Shirley.
«Fun?» said the alien.
"Most of them," said Shirley. "Some use real numbers, some use numbers that are not real, but still exist."
«Yes. When number squared is one—that number is number one. When some other number squared is negative one—that number is not real number, but it exists.»
Shirley was slightly surprised by the rapid response, especially by the clarity of its simple description. It was obvious that the minor mysteries of imaginary numbers were well known to these amorphous geniuses. She decided to test it with another problem. She would have to think it out carefully ahead of time and state it clearly if she were to be understood.
"Suppose you have a growing thing," started Shirley. "The growing thing gets bigger. The amount it gets bigger depends on how big it is..." She paused to let the idea of an exponentially growing organism sink in, then was rudely interrupted by Jill.
"They already know about exponential growth," said Jill. "I taught them the numerical value for pi as 3.14159..., but that was only because they were still hesitant to impose their language structure on me. They taught me the exponential growth factor e as 2.71828... in our language before I could figure out a way to define it in their language. Just say 'e', or 'pi', or 'i', and the imp will translate it for them."
Shirley paused a second to let Jill's revelation seep in, then a little more humbly she proceeded.
"The number e multiplied by itself one time is e," she said.
«Correct.»
"The number e multiplied by itself zero times is one," she said.
«Yes.» the white cloud whistled quietly.
"The number e multiplied by itself pi times is..." she paused for a second to listen to Jill through her imp, "23.1407 plus a slight bit more," she said. She took a deep breath, and then started her next question, only to have it interrupted by the excited squeal of White Whistler.
«And the number e multiplied by itself pi times i times is minus one!» said the white cloud with awe. «Isn't that fun! Isn't that exciting!! We wish we could find another one.»
Her wind taken out of her sails, Shirley gave up, "We wish we could find another one too."
There was a loudness and a rippling crackle and two large blobs appeared in their midst. One poked through the milky cloud that was White Whistle, and the other snaked its way under Shirley's left armpit. One was red, the other was purple, and both quivered with eagerness and questions.
*Another one!!* Roaring*Hot*Vermillion vibrated loudly.
‡Tell!‡ rasped Strong‡Lavender‡Crackle. ‡Tell about other!‡
"Excuse the buzzing one with the strong lavender color," said Jill. "It is an old one that just redissolved from a thinking rock, and as not quite picked up all the human language nuances from the others. I call it Deep Purple."
*Another one!?!*
«No. You heard wrong, Roaring*Hot*Vermillion, not another one. NO other one.»
*zzzzzzzzzzzzztttt!*
There was a shocked silence.
"Well, there are some other interesting problems," said Shirley. "Some in logic theory, some in number theory, and some in geometry. There is one famous problem that is part geometry and part number theory."
The wrinkled purple lobe expanded and with careful enunciation asked, ‡Logic—OK, Numbers—OK, Geometry??‡
Shirley was puzzled when she heard this. In her engineer's world, geometry was inextricably mixed with numbers. Yet these beings did not manipulate the external world, they just existed in it. Could it be that they had no idea of the relationship of the length of a line to the progression of numbers, and the area of a geometrical square with the product of a mathematical square of a number? She tried an experiment. Pulling a diamond scribe from her tool kit, she started to scratch a diagram in the duralloy wing-plate above her. Her motions were interrupted by a squirming feeling at her beltline as three inquisitive pseudopods imitated her entry into her tool pouch.
‡Lots of hard things,‡ said Deep Purple.
*Wheeoo! SHARP!!* said Loud Red, as it tested a hard acrylic handle on the point of the scribe that Shirley was trying to use.
«Too many things,» White Whistler admonished.
Shirley got annoyed. She reached down and slapped the offending tendrils away from her tool pouch, glared around angrily, then carefully sealed each centimeter of the rip-seal seam. She again reached up to the under part of the wing with her diamond scribe held firmly in her hand. For once she had the full attention of the frivolous multitude of alien's.
Carefully, she scratched a right-angled triangle on the skin of Dragonfly's wing. Then, just as carefully, she measured off the length of one of the shorter sides by laying the scribe along the side and pinching it carefully between her gloved fingers. She turned it at right angles and marked a point. In a few seconds she had constructed a square that used one of the triangle sides as one of its sides. She was half-way through constructing the square for the other side when the silent chorus broke into cacophony.
‡Yes! Three and four is five!‡
*HA! PYTH THEOREM!!* roared Loud Red.
"They mean the ancient Greek theorem ascribed to Pythagoras," said Jill. "I told them the human name for the theorem."
"Thanks," said Jill, trying to separate the alien responses from Jill's.
«We understand your diagram,» said White Whistle, «Even if you can not draw it because of others.»
There was a pause as the purple and red protuberances retracted under the seeming glare of the white cloud that they had intruded upon.
«Tell of problem.»
Shirley felt flustered. She was not a mathematician, and it was obvious that these creatures knew the fact that a right triangle had two short sides and one long side, and that the squares of the lengths of the two shorter sides equalled the square of the length of the longest side. She felt stupid as she tried to get across in simple language the idea of one of the most famous unsolved problems in human mathematics—Fermat's Conjecture.
She pointed her scribe at the triangle enclosed in squares inscribed about its perimeter.
"Three times three, plus four times four, equals five times five," she said.
‡OK!!‡ said the purple one enthusiastically.
*YES!?!* said Loud Red, with an inquisitive quiver in its tendril.
"One of the unsolved problems in human mathematics was conjectured by the human Fermat. There may be many solutions to x multiplied two times plus y multiplied two times equals z multiplied two times. But there is no solution to the problem of x multiplied three times, plus y multiplied three times, equals z multiplied three times... even if three is any number.
‡That not problem!‡ the purple one graveled.
*That's a DUMB problem!!* the red cloud exploded. *That problem not said right. I say right way. X squared times Y squared equals Z squared has many solutions. Is there a solution for U cubed plus V cubed plus W cubed equal Z cubed? That make more sense. You have two things X and Y. You multiply two times. You add two times. You get same as Z multiplied two times. Two things three times is DUMB!! If you multiply three times, then you should add three times!*
"But is there an answer?" persisted Shirley.
‡Answer?‡ echoed Deep Purple. There was a long pause. Then the other colored portions of the water retracted as the large purple blob condensed into a purple boulder some meters beneath Shirley's feet. She had not realized how massive Deep Purple was, for its cloud extended tideward as far as she could see. Like a motion picture of a vaporizing block of dry ice run backwards, the gigantic purple cloud condensed into a slippery purple rock; a thinking rock; thinking about a problem put forth many decades ago by a brilliant human mind many light-years away.
*I'm going surfing!* Loud Red swam up to the top of the wing and perched there, its weight causing a slight list to the plane, then as the next roller broke over the plane, Loud Red used the inertia of the wing to launch itself onto the forward surface of the wave.
*Wheeee!!!* came the cry of excitement through the water, fading into the distance as the wave carried the red alien off.
"I wish I could surf," said Shirley wistfully.
«You wrong shape to surf,» said White Whistle.
"I could surf if I had a surfboard," said Shirley, her thoughts going back over six light years and forty time years. Fortunately Jill was back in the translation loop. It took an extended discussion between the computer and the alien before it understood what a surfboard was.
«I be your surfboard!» said White Whistler. It swooped under Shirley and picked her up on its massive body, a cavity appearing in the top of the alien's body to cradle the human.
"What's going on?" came Richard's concerned voice over the imp link.
"I'm going to get a ride!" shouted a delighted Shirley. The white alien humped itself up on the plane wing.
"Jill! Make her stop! That could be dangerous!" Shirley could see Richard through the left cockpit window. He was yelling and waving his hands at her, trying to make her get down off the alien.
Shirley grinned and waved back at him.
"Cowabunga!" she cried as a roller lifted the wing and White Whistler launched its multiton body onto the wave.
The wave was traveling toward the center of an underwater volcanic vent field that had produced a long sloping ash and lava shield. As they moved toward the central region, the ocean became shallower and the wave steeper. It was a long surf, and after they had gone a kilometer they were out of range of Jill's sonar and laser beacon. Shirley had to depend upon her chest-pack translator for communication. White Whistler noticed the lack and switched to using simplified speech patterns. Although both enjoyed the ride, White Whistler's curiosity led to questions.
«Humans are strange elements. Not see before. I swim entire world many times. Never see humans. Where humans exist before now?»
"We came from lights in sky." Suddenly Shirley realized that the aliens had no eyes, so perhaps they didn't know about the stars, although they probably knew about Barnard.
«I know many lights in sky. There is Sky¤Rock, you name Roche. There is Hot, you name Barnard. There is Warm and little elements. There are many other lights. All tiny except one new one. They are my research.»
"An eyeless astronomer," mused Shirley. "It even has to make its telescope lenses from its own flesh." She started to think how to tell the alien that the stars were suns like Barnard, and that she came from a planet around one of those stars.
"The other lights we call stars. They are suns like Barnard, but far away."
«Not all are like Barnard. Some have color of Barnard. Most have different color. Some yellow like spots on Barnard. Some white like light from storms. Some blue. Different kinds of suns.»
You may not have eyes, said Shirley to herself. But you have a marvelous color sense, if you've been able to deduce that stars are suns from their spectrum.
«One star not like others. Star yellow for long time. Then get brighter and color is green. Then after three thousand rotations star become yellow again.»
"That is star of humans," said Shirley, much relieved that the problem of pointing out which of the many stars was the Sun had been solved by their method of arrival.
«Human star far away. Your plane swim long time.»
"Humans not use plane. Humans use big...circle. You see big circle in sky?"
«Yes. Big circle not logical.»
"Not logical? I do not understand."
«Lights in sky are my research. I know stars are suns. I know Sky¤Rock is like world without water. I know Warm is like a big world with more clouds. I know little ones of Warm are almost equal to world. I can predict motions of all lights in sky except one. That one is big circle. Big circle not like anything. It is circle, not sphere. Its motion has no logic. I think hard to find logic of big circle motion. I can find no logic in motion.»
"The big circle is not heavy like other lights. Big circle swims in light from Barnard. Subtract big circle from set of lights in sky. They do not swim. They move by logic of gravity."
«Word is missing. Logic of...»
"Each sphere in sky is pulled by other lights in sky. A big sphere pulls more than a small sphere. If two spheres are near, pull is strong. If two spheres are far, pull is weak. Amount of pull varies as the inverse square of the distance. That is logic of gravity between two spheres."
«That is the hypothesis I was using! Then big circle came and its motion did not fit hypothesis. I rejected hypothesis and looked for a new hypothesis.»
Shirley felt the body of the alien stop its slow swim back to the airplane.
«I must think.»
Shirley then felt the body of the alien contract around her, getting harder and more rubbery as liquid was squeezed from the jellylike body. Suddenly she found herself floating in the water as a white rock sank beneath her to the bottom.
"White Whistler!" she cried through her outside imp. "Come take me back to the plane! I can't swim twenty kilometers in my suit, even if I knew which direction to swim!!!"
There was no answer.
Shirley tried again, and even sent the suit imp swimming down to the bottom in an attempt to revive the white rock.
"I guess once you have dived into the complexity of the three-body problem you don't want to come up until you have it solved," she said to herself. "I guess I'd better just relax and wait until White Whistler finishes thinking and comes to pick me up again."
Shirley set her suit heater on low to conserve power and closed her eyes to rest.
"I hope White Whistler isn't trying to solve the generalized n-body central force problem," she thought to herself as she drifted off to sleep.
A clear blue cloud strung through the cold water and wrapped itself around a red rock. There was a trilling sound as Dainty~Blue~Warble tried to attract the attention of the inactive elder.
~It's time for another lesson, Old-one Roaring*Hot*Vermillion! What'cha doing?~
*It is!?!* roared the red rock as it dissolved into a cloud. *Was just thinking of what to teach you next!*
Feathery red tendrils snaked out into the water, tasting each molecule.
*Let me taste. Know there was one around just a little bit ago.*
Suddenly there was a reaction, and like a bloodhound on the scent, the cloud concentrated into a long tendril that dragged the rest of the red body along. The blue cloud floated alongside. Suddenly the red cloud stopped, the tip of its tendril pointing stiffly off into the murky depths.
*There's a CreepyoxStink!*
*Sneak up on it! Ho! Ho!*
~Yes, Old-one,~ said the blue cloud, imitating the motions of the elder.
*Wait!* said Roaring*Hot*Vermillion as the eager blue youngster started to move ahead toward the slowly moving black slug plowing through the muddy bottom.
*First sneak up only half way! Then stop!*
The blue cloud obeyed and stopped in the water about half-way to the CreepyoxStink.
*Now half-way again! Stop!*
*Do it again!*
*Will you catch it?*
*Will you?*
Dainty~Blue~Warble followed the instructions of the elder, moving only half-way each time toward the slowly creeping blob of pungent food. As the gap lessened, the pauses at each half-way point became shorter. Dainty~Blue~Warble controlled its fluid body well with its eager, learning intellect, but finally, in a swirl that was too fast to follow, the CreepyoxStink was gone.
*Ho! Ho! Caught it, didn't you!*
~I calculated that even though I only went halfway each time, my velocity was greater, and eventually I would catch up. So I ate it!~
*Taste interesting?*
There was an unnaturally long pause.
~Stinky!~
*You'll get used to it. Hey! WAVE!!!*
~Wave!~ warbled the youngling. Then flowing smoothly into the elder's wake, it surfed across the surface of the shining sea.
"George? I'm worried," said Richard. "It's getting dark and Shirley and White Whistler aren't back yet."
George looked out the cockpit window at Barnard. The short day was almost over and the tall tail of the Magic Dragonfly was casting a long shadow up the water mountain toward Roche as Bernard set behind them.
"Are there any of the aliens around?" he asked Jill. George could hear the sonar in the nose go through its search scan.
"Loud Red and Little Blue are coming this way."
"Ask them if they have seen Shirley and White Whistler."
~Floating¤Rock is calling us.~
*Hi! We come closer!* the loud voice roared through the water. The body of Roaring*Hot*Vermillion took on a more streamlined shape and zoomed off through the ocean, followed close behind by a pale blue arrow. The two drew to a halt in front of the chugging airplane.
"Where is White Whistler and the human Shirley?"
*Don't know. OrangeoxHunters will find them!* The red cloud gave a piercing whistle and soon three eager orange snakes came streaking through the water to dash themselves headlong into the red flesh of their master. After the obligatory free-for-all wrestling match was over, the OrangeoxHunters listened to the complex commands coming from Roaring*Hot*Vermillion, then they took off, making controlled crooning noises as they went.
"George," said Jill. "There's a message being relayed from Clete. The commsat says that the signal is very weak and broken up, but it's getting stronger as the sun sets. Here is what we have so far."
"Shirley calling commsats... Floating alone some twenty kil... ...Whistler rocked up and left me. Heater power getting low. Shirley calling commsats..."
"Can they get her position?" asked George.
"All they have are radio signals to work from," said Jill. "That only puts her somewhere in a hundred-kilometer circle. As soon as it gets darker the commsats will try to spot her helmet laser beacon. They've radioed to her suit imp to keep that going at high power."
"I'm going out to see if Loud Red will take me looking," said Richard, heading for the suit locker.
"It'll be pitch dark soon," said George. "You won't be able to see a thing."
"No it won't!" boomed Jinjur's voice through George's imp. "Prometheus is over the dark side and I'm going to illuminate the search area with the reflection from the sail. I won't be able to keep it up all night without knocking myself out of position, but I should be able to give you an hour's worth."
George peered upward through the cockpit window and found the fat ellipse in the sky. As he watched, the ellipse brightened and soon a tiny new sun was ablaze in the sky.
"I'm cycling through," said Richard. "Tell Loud Red that I'd like a ride."
"There's another message coming in from Shirley," said Jill.
"...heater power gone. Can't take cold much longer... Must have dozed off... Barnard has risen again... Nope... it's Prometheus. Good seeing you guys, but I'm afraid it's too late. Awfully cold..." There was a long stretch where they could hear digital chatter from Shirley's suit computer riding over the noise from Shirley's chattering teeth.
"The suit imp reports that Shirley's torso temperature has dropped to thirty-five degrees Celsius, with extremities well below that," said Jill.
"Wha... White Whistler, you're back... No... Orange blob... Stop bumping... Go 'way..." There were sounds of heavy breathing, then the chattering teeth suddenly stopped.
"I think she's lost consciousness," said Jill.
"I hope those pets of Loud Red are strong enough to bring her back."
"They each weigh twice what Shirley weighs," said Jill.
"I'm through the lock," said Richard. "Patch me through to Loud Red."
Jill stopped the engines and the Magic Dragonfly drifted to a stop in the ocean as Richard swam forward through the dimly lit seas toward the two aliens.
"Please carry me to Shirley," he asked through Jill's sonar.
*No. OrangeoxHunters come. They bring Shirley here.*
"But they'll take forever...a long time. She may die!"
*She not die. She too hard. Cannot eat Stiff¤Movers, so Stiff¤Movers not die!*
Suddenly Richard realized that the aliens had no concept of death except that of being eaten and assimilated in some predator's body. And since they were the dominant predator, with no natural enemies, they never died, just spent longer and longer times rocked up to think about more and more difficult logical problems until they gave themselves a problem that took an eternity to solve.
"Jill, convince that lazy red blob that we need to get Shirley here in a hurry."
"I don't think we really need to," said Jill. "My sonar can detect the pets up ahead, moving this way at a respectable speed. I think they have Shirley with them because the commsat trackers show a doppler velocity shift on their communication signal from her suit imp. Do you have your safety line on?"
"Of course," said Richard.
"Good," said Jill, starting the engines of the Magic Dragonfly. "See if you can catch the lock handholds as I go by, I'm going to save a few minutes by moving toward them."
"How's Shirley?" asked Richard, puffing as he swam a few quick stokes and swung into the open door of the lock as it moved by.
"Not good. Torso temperature down to thirty Celsius. The suit imp has dropped all power reserves and is running the last few grams of hydrogen through the fuel cell to get heat to the extremities before frostbite sets in."
Painful memories flooded back into Richard's mind as he recalled the six hours of agony walking in sock feet through Alpine snow carrying two unconscious tourists. He twitched his eight toes in his boots. Would he like Shirley less if she lost any of her toes? If she lived...
Stop that, you ass, he remonstrated himself. Get ready to cycle her through once those orange-colored hounds bring her in.
The smooth thump-thump of the large engine on the right and the pittidy-pittidy of the jury rigged fan on the left stopped, and the Magic Dragonfly drifted to a halt. The second the current was less than his space-suit swimming speed, Richard was through the door and breaststroking forward. The red alien was wrestling with his charges and loud roars, shrieks, and whistles sounded through the ocean. Bobbing just below the surface of the water was a limp figure in a space suit, the arms and legs hanging downward, jerking limply as the waves tossed the body to and fro. Near the head of the body was a blurred bundle of twigs that were waving frantically through the water in an attempt to drag the heavy carcass toward the airplane.
An orange blob hit Richard on the legs as it darted back into the fray, sending him tumbling in the water. Richard righted himself and stroked again to the distant limp form. He grabbed her by the belt and headed for the lock.
Once Richard had gotten Shirley inside, Katrina took charge. "Hold her by the waist while Arielle and I get her helmet off and her shoulders out!"
The helmet came off and a cold dank head dropped on Richard's shoulder as the subdued stink of a frozen tortured body arose from the enclosed space. Richard held tight to the waist until the stiff arms were extracted from their casings. He switched to a chest hold on the body while Katrina, Arielle and the Christmas Branch worked at detaching the bottom portion of the body from the plumbing. Richard's arms grew cold from the leaden breasts draped over his forearms.
"Turn her to the right so we can get her leg out," came the command from Katrina.
Richard shifted his grip and easily turned the large body in the weak gravity of Eau, holding Shirley by the waist and under the left arm. He looked down to see a deep depression in the soft underflesh of the breast, the fatty tissue turned to clay by the cold. The suit came off.
"OK! Out! We'll take it from here!" said Katrina, pushing him forward through the privacy curtain as Arielle and the Christmas Branch started to strip the cold soggy coveralls from the blue-cold body.
"How's she doing?" asked David as Richard made his way forward, closing the second curtain behind him.
"I don't like it!" said Richard, pounding one massive but ineffectual fist into the other. He paced four huge steps forward, his hips avoiding the backs of the console chairs like a halfback avoiding one linebacker after another. His fourth step landed solidly on the floor next to the flight deck, he pivoted, then turned to march back again.
~Lesson time again!~ piped a small blue cloud as it scurried up to a deep yellow rock with a cloud of amber water hanging softly around it.
§Certainly, Dainty~Blue~Warble,§ murmured the edges of the yellow cloud. §Just let me ingest my latest thinking.§
The small blue cloud waited patiently, only flickering a tendril or two while the large dark-yellow rock dissolved on the ocean bottom. The dark yellow dispersed in thin threads into the light yellow cloud above it. Soon you could not tell the difference between the threads and the cloud, and the rock was gone.
~What were you thinking about?~
§The fourth infinity.§
~Tell me about it!~
§Well... I will some day. But first you have to learn about the second infinity.§
~Tell me! Tell me!~
A yellow tendril poked a hole in the muddy bottom.
§Feel, youngling. There is a point.§
A delicate blue tendril felt into the murky bottom.
~That is a hole in the mud, Old-one Warm§Amber§Resonance.~
There was a long pause as the yellow cloud rippled in annoyance. However, the tone that resumed after the pause had all the warm patience that it had contained previously.
§Imagine it is a point, with no dimensions.§
~Yes, Old-one.~
The yellow tendril touched the surface of the soft mud again, leaving another tiny spot in the smooth surface close to the first one.
§Here is another point.§
§Here is another.§
§Here is another.§
The line of close-spaced spots grew.
§Imagine. Imagine points so close they make a line. A line infinitely long.§
There was a pause as the young blue cloud absorbed the sounds. Its blue cloud enveloped the motions of the yellow wisp making a long string of tiny dots in the ocean bottom.
~Infinite in both directions, Old-one?~
§Yes. Very good, youngling.§
§Now... Imagine a point not on the line.§
§Here is one.§
§Here is another.§
Soon a number of isolated spots were scattered above and below the dotted line on the muddy sea floor.
§Imagine an infinite number of them.§
There was a slight pause.
§Are there more points off the line than on the line?§
The youngling thought carefully before answering, its wisps of azure clumping and dissolving randomly. The elder waited patiently. Finally the youngling answered.
~No! They are the same.~
§Right! They both are infinities of the first order.§
~But that problem was the same as the one you gave me yesterday. That time you asked me if the irrational numbers were greater than the rational numbers. They too are both first infinities. Give me a harder one.~
§All right. Draw a line through any of those points I made.§
The blue cloud formed a tendril of its own and made a streak through one of the isolated spots in the mud.
§Draw another line through the same point. Make it wiggly if you want to.§
A wiggly line joined the streak.
§Draw more.§
Dainty~Blue~Warble concentrated and soon dozens of distinctly different lines were drawn through the same point. Then came the question.
§Imagine you did that to each point.§
§Are there more wiggly lines than points on the line?§
The blue cloud stopped moving as it started to think.
Warm§Amber§Resonance drifted off on the current as Dainty~Blue~Warble pondered the last question. The fluid body of the youngling slowly began condensing into a bright-blue rock as the difficulty of the tough logical question called for more and more concentrated neural matter.
§Huunnm!§ murmured the amber elder. §That will keep the blue menace quiet for a while.§
§Off to the waves! MMMMM!!!!§
Warm§Amber§Resonance vibrated through the cooling seas, leaving behind a tenuous blue cloud condensing into a thinking azure stone.
Two day cycles later Shirley was standing her shift. She still wasn't allowed to do anything strenuous, but Shirley had taken Arielle's place as pilot of the lumbering Dragonfly, while Arielle tried to take Shirley's place as Chief Engineer. Arielle did a respectable job. She equalled Shirley's performance in outship repairs and bettered her performance at the galley table.
"I don't see where she puts it," said Shirley to David. "If I ate that much I would be as fat as Richard!"
"I heard that!" said Richard. "How much do you weigh, Shorty?"
"Eighty-five kilos, fat man... and you?"
"One hundred kilos—but I'm taller than you."
"The last five centimeters must be all fat," said Shirley, sticking her tongue out at him.
Richard was glad. The brat must be OK if she could fight back like that. He was about to make some comment about squaws when Jill interrupted.
"White Whistler is returning."
"Ask why I was left!" demanded Shirley.
"Just a second," said Jill, almost impatiently. "White Whistler is saying something important to the other flouwen!"
«I have solved the motion of the lights in the sky!»
*Even the big circle?*
«All the lights except big circle. It is a swimmer of the light. It is like us. It's motions are not that of logic.»
§But you can know the motions of all the rest? You can know the arisings of Hot and the fadings of Warm and the tenacity of Sky¤Rock?§
«All,» said White Whistler with confidence.
*How can you be sure?*
«The humans gave me the logic of gravity for two spherical masses. The rule is very simple. Yet the mathematics seemed complex when the rule was used on more than two spheres. After some thinking, I found the simple rule for many spheres.»
§Was it difficult?§
«No. A simple variable substitution combined with an interesting coordinate transformation.»
§Let me taste.§
*Me too!*
~Me too!!!~
The white cloud sent out a tendril. The end was whitely concentrated with nerve tissue. The other three aliens crowded around to taste the essence of Clear«»White«»Whistle's thought.
§Subtle!§
*Tricky!!!*
~I don't understand the taste!~ said the little blue one.
§You will. Just savor the taste and recall it some thousand turns from now. It will be much clearer then when you can handle such complexities.§
~But I want to know now!~
§Later, Dainty~Blue~Warble,§ said Warm§Amber§Resonance. The yellow cloud expanded and contracted as it impressed on its memory the secret for the solution of the n-body central force problem. Warm§Amber§Resonance reveled in the cleanness of it. One complex variable transformation, and then just that one simple, yet unobvious, coordinate transformation! An nth-root dimension, indeed!§
During the next day, a vibration started coming from the left wing. Shirley's ears pricked up and she reached to turn off the fans just as Jill shut them down. The Dragonfly coasted to a stop, the lapping of the waves on the hull making the absence of the engine sounds even more ominous.
"The mounting bracket for the replacement fan is vibrating," said Jill.
"I'll go out and fix it," said Shirley. "Probably just a bolt coming loose."
"You are still weak," said Jill. "I'll wake up Richard."
"No!" exploded Shirley. "I can do anything that big ox can do. I'm going out! I'm tired of being cooped up in this tin can." She stormed down the corridor, bumping into every console chair along the way. David, twisted partially away from his console by Shirley's passage, gave a sigh, swiveled back, closed down his console, and went back to suit up with Shirley. He probably would only be a tool holder, but the sooner he got her back inside the better.
Together they cycled through the lock and swam out under the left wing. Shirley jiggled the bracing structure for the jury-rigged fan and found the loose bolt. She tightened it, and then for good measure, started to check the tightness of the rest. David pulled out his omniwrench and backed her up by holding onto the other side of each bolt.
Suddenly, they were all engulfed in an encompassing cloud of good will. The purple cloud was back.
"What's that!" said David in alarm.
"Take it easy," said Shirley. "That's just an old alien we call Deep Purple. Last time I saw him he was rocked up and trying to prove Fermat's Conjecture." She suddenly realized the importance of what she was saying; If not to their present problems, to the cadre of mathematicians six light years away on Earth.
"Welcome, Deep Purple," she said. "Did you solve the problem?"
‡Easy,‡ the lavender cloud responded. ‡Human named Fermat right.‡ There was a polite pause, then he continued, ‡DUMB problem!‡
This was the cue for Loud Red, *I told you! DUMB problem!! DUMB!!!*
David broke through over Shirley's imp. "Do you mean to say they have solved Fermat's conjecture? How did they do it? We have only proved it up to n equals 1,023,467."
The sharp sonar senses of the huge purple alien caught the careful reeling of numbers. It paused in its exodus, then came back to surround Shirley, as if it were really looking at a human for the first time.
‡You solve problems with numbers?!?‡ it asked, its incredulity rippling through its surface.
"Yes," said Shirley bravely, "We do quite often. If it works for many numbers then it must be true."
There was a long pause as the alien brain tried to absorb the alien thought. It rejected it.
‡There are many numbers. There are...too many numbers.‡
The alien was struggling to express the idea of an infinity of numbers. Shirley helped.
"No end to numbers," she said.
‡No end!‡ the alien rasped in relief.
‡No end, so no certainty. Not use numbers.‡
"...but how..." Shirley's eyes flitted from alien mounds to David's face.
"I think Deep Purple just proved Fermat's Conjecture for ANY number." said David. "Too bad we aren't smart enough to understand how it did it. We may think we're intelligent, but that alien equivalent of an aging surfer has us all beat."
While they were talking, the larger cloud had wrapped a purple tendril about two others, one red and one milky white. The contact had lasted for less than two seconds.
«Obvious.» said the white one.
*...and DUMB!!!* said the red one.
*Let's do something else! Like SURF!!*
‡WAVE!!!‡ said the old purple one, leading the group to the surface as they dropped their preoccupation with mathematics and enjoyed the caress of the waves on their fluid surface.
«Wave!!!» keened the voice of Clear«»White«»Whistle as the three of them caught a wave.
*DUMB!!!* came the final fading response from Roaring*Hot*Vermillion as they surfed off into the distance.