Fractal Mode Read online

  The guard seemed taken aback at the unintelligible speech. Satisfied that these were indeed strangers, he pointed to the gate.

  "Go there?" Darius asked. He took a step, hesitantly.

  The guard turned and walked ahead, leading them in. Stave stood where he was, ignored. Go with them, he thought, his face impassive. They have no interest in me, and that is best. I will tell Ana, whom you know as Nona, that you are here. I will come if you call me, if you can mind-talk from a distance.

  We can, Colene thought. She wished Stave had come in with them, however, because then they would have had a far easier time understanding what was going on. As it was, she was nervous, despite Provos' assurance that they would get through satisfactorily. How accurate was the woman's memory of future events? It couldn't be perfect, because sometimes their actions changed their future.

  The gate did not lead straight in. Instead they had to mount a long, steep ramp which seemed to go to the top of the wall. But it didn't; perhaps three quarters of the way up it turned away from the wall and deposited them on a slightly sloping platform, an interior glacis. Apparently the wall surrounded a steep mountain ridge, and the castle proper was at the top of that. This would be some redoubt to storm!

  Inside, it looked even more like a city. The outer wall did not connect to the interior structures; there was a wide space between them. That way the inhabitants could defend against an enemy who breached the outer ramparts; he would have to expose himself to further fire before reaching the inner compound.

  They were led to a chamber just inside the wall. The head guard barked a command, but this time they really did not understand it.

  Then an old man appeared, also in black. He did not walk in, he appeared in the chamber. He smiled. He spoke in more gibberish, addressing Darius.

  "I do not understand what you are saying," Darius replied in his own language.

  The man lifted his hand, and a doll appeared in it. The doll looked much like Darius. The doll reached out, and in its hand appeared a cloak similar to the one the old man wore, but green. Then the cloak was on the doll.

  "An icon!" Darius exclaimed. "My kind of magic!"

  "Not necessarily," Colene said. "Guess something else, just in case." Because Nona had not mentioned that type of magic.

  "You want us to change clothing?" Darius asked.

  The doll was suddenly wearing the cloak. The implication was clear enough. Colene's caution had been justified. Darius might have given away his magic, if it worked here.

  "But we need a private place to change," Darius protested.

  "Oh, forget it," Colene said. "They don't care about our bodies." She extended her hand toward the man in white, and immediately a green cloak landed on it. She stepped out of her clothing, except for her bra and panties, and dropped the cloak over her head. It was light and silken, pleasant enough to wear, and had a green sash she tied around her waist. Green slippers appeared before her, and she donned them too. The cloak seemed designed for a larger person, as it reached right down to her ankles, but she was satisfied.

  Darius had averted his gaze as she changed; it was one of the little ways he had about him, both frustrating and endearing. That was more than could be said for the black-clad old man; he had stared at Colene's body. She wasn't sure whether to be angry or flattered.

  Now Darius changed too. On him the cloak reached to the knees. The man stared at him too, so "at least it wasn't sexual.

  Provos was in between, the tunic falling to her mid-calves. The man stared at her body too, or as much of it as showed around her long, loose green corset. What was it about their bodies the man found so odd?

  For Seqiro there was a double collar, yellow and green. Colene put it on him. They'll probably separate us, she thought. But we'll be in touch anywhere in the castle. For his telepathy could reach her anywhere on a planet; they were attuned to each other. That gave her great comfort, especially in a strange situation like this.

  Appropriately garbed, they were conducted across the open section to the inner gate. Here another black-cloaked man came to lead Seqiro away, surely to a stable. They had been through this before; it didn't bother the horse to be considered an animal, though it irritated Colene.

  The three were admitted to an interior court. Here several men and women sat in comfortable thronelike chairs. All the men were in black tunics, and the women in white tunics. Color coding, Colene realized. Stave had worn blue, and Nona red, which could be the colors for the theows. The green was probably reserved for visitors of either sex. It made sense, for a highly regimented society: nobody had to think about status.

  The old man came to stand between them and the seated despots. He gestured. A picture appeared in the air over his head: himself. "Hobard," he said, and the figure glowed momentarily. He touched himself. "Hobard."

  That was clear enough. It was his name. Seqiro was making progress on getting into the man's mind. Colene depended on that; she would have been suicidally tense without the assurance of the horse's ability and support.

  A picture of Darius appeared. "Darius," Darius said.

  Then pictures of Provos and Colene, both of whom gave their names. This was an efficient introduction! But there was no picture of Seqiro. Evidently they didn't think the horse was important. How little they knew!

  A picture of the man on the largest throne appeared. "King Lombard," Hobard said. The word for king was foreign, but Seqiro translated it. Maybe it was dictator or monarch or muck-a-muck or chicken-manure; it didn't matter. This was the head despot.

  Lombard? Colene stifled a giggle. There was a special lexicon of colloquial acronyms, back in her subculture on Earth, and one of these was LOMBARD: Lots of Money but a Real Dickhead. Well, maybe that was the case here. Lombard was also a Germanic tribe that invaded Italy after the fall of the Roman Empire. That, too, might fit.

  Then the chief woman: "Queen Glomerula." Colene kept a straight face; this was the one who was going to try to seduce Darius! Was it her imagination, or was the woman a nymphomaniac? She enjoys playing with unfamiliar men, Seqiro clarified. This is the impression Hobard has. She seduced him some time ago, then lost interest. The king is tolerant, since his own interest is in helpless theow girls. The king and queen consider it bad form to be stuck with each other for entertainment. The horse was merely reporting Hobard's private assessment, but Colene found it hilarious, except that the queen's next target was Colene's man. She also found it funny that the queen's name in translation sounded something like an aspect of a kidney. She'd love to give the queen a kidney disease!

  Finally "Knave Naylor," the one who was going to try to rape Colene herself. Colene kept her face straight. He looked sinister to her, a knave indeed. But he was going to have a rude surprise when he tried to tackle this supposedly innocent visiting maiden.

  Hobard appeared again in the image. Beside him formed a picture of a neat house on a hill. The picture drew away, and landscape appeared between the house and the man. Hobard spoke, saying his name and several other words. He is from Hillside Acres, some distance west of here, Seqiro clarified.

  Now the group of them appeared. "Darius, Provos, Colene, from where?" Hobard asked.

  They pretended to be slow to understand, which was reasonable enough, while they consulted mentally with each other. This was a straightforward question, but difficult to answer, assuming they wanted to give such information. How could they clarify that they were not only from three different places but from three different realities, and the horse from a fourth? That they had traveled through the Virtual Mode, which was a kind of temporary reality anchored at each of their homes, crossing other realities at ten-foot intervals, so that things could change abruptly with a single step forward? That the people, geography, and fundamental natural laws changed with each reality, so that in some animals were telepathic while in others there was super-science that allowed gravity cancellation or travel at many times the velocity of light? That Darius could perform a kind of magic
in his reality, while Provos remembered the future and not the past? That each of them was associated with his/her anchor, which was both place and person, and that they could get off the Virtual Mode only through one of the five anchors that held it in place? That Nona and the place of the huge stone musical instruments had just become the fifth anchor?

  No, even if there were no language barrier, they could not blab all that to these grim strangers. So what could they say? They didn't really know what kind of a world this was, apart from the facts that it had magic and a strong upper-class/lower-class social structure. What would satisfy the despots without giving away too much?

  We have come from afar by virtue of a spell, and wish only to return, Darius decided. That did seem to be the best answer.

  So Darius used gestures and pointings to images to try to get that across. It turned out that the despots already knew that, they thought; they had a concept of interplanetary travel that was weird, and assumed that the party had somehow walked from one planet to another. They wanted to know the origin planet.

  Colene and the others were baffled by this. How could anyone walk between planets? Even if the force of gravity did not prevent this, Earth's moon was so far away that it would take thirty years to walk there, and the other planets were much farther away. Were there seven-league boots for this purpose?

  The despots were skeptical of their confusion. It was almost as if the despots believed the visitors knew all about it, and were playing ignorant. "You are human," Hobard said, making a number of pictures to get the concept across. Because Colene and the others were genuinely curious about the nature of this world, they drew on Seqiro's telepathy to clarify it. The minds of the other despots remained opaque, but Hobard was trying to establish a liaison, and his mind was opening so that they could start to receive his more complicated thoughts. This was especially true when he focused on a specific thing. "You came from a human world." World was not exactly the concept, but the exact one was not quite fathomable. Planet? Aspect? Subdivision?

  "From a human world," Darius agreed, by indicating the correct pictures. "Far away."

  Now a diagram appeared, with what might be the wires and resistors of a weird radio set. There was something naggingly familiar about it, but Colene couldn't place it. "Which one?" Hobard demanded. One bug in the picture glowed, and then another.

  Maybe we had better try to tell a bit of the truth, Colene thought to Darius. They'll know if we claim a planet that is wrong; this is their territory.

  Darius agreed. "Far in a different way," he tried to clarify. "Not in distance, but in mode."

  But this was lost on the despots. Evidently they had no concept of the Virtual Mode or alternate realities. Indeed, their own reality seemed quite strange enough to hold their attention.

  Hobard generated another picture. In its center was a shape like a hairy roach, or perhaps a hairy fat-bodied spider, for it was at the center of a weblike structure of lines. Upon the main lines extending out were smaller bugs, with finer lines radiating from them. Yet it was not a spiderweb; the lines were jagged, and took funny turns.

  Then Colene placed the image. "The Mandelbrot set!" she exclaimed.

  The others looked at her. Then Seqiro's thought came, warningly: I have gotten farther into Hobard's mind. Beware asserting yourself.

  Colene was irritated at this interruption to her revelation. Why not assert myself? she demanded. I'm an assertive person.

  Because they judge men and women differently, he explained. Here men are dominant. Elsewhere women are. The two cultures are enemies.

  Oh. The last thing they needed was to be considered enemies before they knew their way around. Take it, Darius, she thought.

  But what is this Man's-brow set? he thought.

  I'll explain while you dominate. Tell me to shut up.

  The mental exchange had been swift, but things were getting somewhat strained. Darius frowned at Colene. "Silence, girl," he snapped.

  Colene hung her head, her gesture confessing that she had spoken out of turn. Darius faced Hobard. "Where?" he asked.

  The old man seemed to have lost some of his own concentration. It seemed that Colene's outburst had held considerable significance for the despots. Since they couldn't have understood the meaning of the words—Darius himself didn't understand them, and he had a working knowledge of her language—it had to be because of the thing the remarkable horse was warning them about. Men and women were not equal here. A woman who asserted herself was in trouble.

  Yet Nona the peasant girl had been assertive enough, and Stave had not taken offense. Did a different rule apply to the theows?

  The magic image had faded. Now it reappeared. The small bug on a webline above the upper leg of the bug glowed.

  Hobard pointed to the glowing bug, and tapped his foot on the floor. "Here," he said, using his word, but the translation came through. This was where they were.

  Meanwhile Colene's mind was racing through what she remembered of the Mandelbrot set. It was named after the man who had done a special computation involving a complex equation, and plotted the result on a graph. A simple equation was something like X + Y = 10, and if X was 10, then Y had to be 0 because there was nothing left for it. If X was 9, then Y was 1, and so on until X was 0 and Y was 10. Those answers could be plotted on graph paper, with X representing up and Y the side: go up ten and across none and place a point. Then up nine and across one, and set another point. A line of points formed, and all the possible answers to that equation were on that line. Simple.

  Colene had quickly gone beyond that, and used squares of X and Y to get curved lines. X2 + Y2 = Z? made a perfect circle with a radius of Z. Sine waves were trickier. But a complex equation was something else.

  She had been fascinated by the new concept of fractals, which were like fractional dimensions. They enabled a person to take a simple figure and elaborate it infinitely, without taking up any more space. For example, one could start with an equilateral triangle, every angle and every side the same, then put a little triangle in the middle of each side, so that it became a six-pointed star:

  _/_

  / /

  /____

  /

  Then smaller triangles could be added on the twelve sides of that outline, and yet smaller triangles on the new sides. The figure got more complicated, yet sat in the same space. There was no end to the additions that could be made; there was always room for yet smaller triangles.

  Meanwhile the length of the outer line kept growing. If the initial triangle was three inches on a side, it was nine inches all the way around. The six-pointed star added an inch to each side, so was twelve inches around. The eighteen-pointed figure that resulted from the next round of additions was sixteen inches around. And so on; each step added more sides and points and length, yet the figure could fit on the same sheet of paper. It was an infinite process, with a finite boundary.

  Colene got hazy on the technical aspects beyond that. But she knew that a man named Benoit Mandelbrot had coined the term "fractal" for this type of figure, considering the process to be like a fractional dimension. A triangle was a two-dimensional figure; a fractal based on a triangle was a two-and-a-half-dimensional figure. There were implications for the ultimate nature of reality—and, it seemed, for the Virtual Mode. Because they seemed to have stepped into a fractal reality.

  Benoit Mandelbrot had plotted his complex equation, and come up with a fractal figure that was a good deal more complicated than a triangle or circle. In fact it was deemed to be the most complicated object in mathematics. It had buglike shapes, and shell shapes, and seahorse-tail shapes, and separate "floating molecules" connected by "devil's polymer," an intricate web of invisibly fine filament. No matter how much the magnification was increased, there were always more and smaller bugs and shells and tails. Colene had been fascinated by the Mandelbrot set, but had thought it had no immediate relevance to her life. So she had watched a video tape showing the Mandelbrot set and what
were called Julia sets, which she understood were two-dimensional aspects of the larger set, and let it fade from her thoughts.

  Well, that had changed. Because Hobard was telling them that this was not the planet Earth, but a Mandelbrot bug. That just might make this a Julia universe. The implications were mind-blowing.

  While she worked this out, Darius was coming at it hi a less theoretical manner. He was not burdened by her awareness of the mathematical aspects. "So we are here," he was saying. "On the planet Oria. And we are a satellite of this larger planet Jupiter. And you want to know whether we come from Mercury, Venus, or Mars." He was speaking in his own language, using different names for things, but this was the way the thoughts came to her.

  She focused on this confusing alignment. In this fractal universe, it seemed that Earth did not revolve around the sun, but around Jupiter, and so did the other small planets. Each planet was a Mandelbrot bug, and Jupiter was a big bug. The webwork of lines connected the four smaller bugs to the big one; apparently gravity didn't do the job here. Okay, if that was the way it was, that was the way it was. In some realities science worked, and in others pseudoscience like faster-than-light travel worked. Where Darius came from, a kind of sympathetic magic worked, buttressed by emotional telepathy. Here on Oria—their name for Earth—magic worked. And astronomy was weird. But at least she had a handle on it, because of her experience with the Mandelbrot set.

  "We come from none of these," Darius said. He illustrated the statement by pointing to each of the other three bug-planets in turn and shaking his head no.

  Colene was sent into another bypath of realization. The Mandelbrot set was portrayed two-dimensionally, but this was a three-dimensional world. Her pictures had shown small bugs of similar size to the north and south of a large one, because the south was the mirror image of the north. The north curlicues wound clockwise, the south ones counterclockwise. The Mandelbrot set was excruciatingly well organized, on its own terms. The image Hobard generated was three-dimensional. In fact it wasn't a picture, it was a hologram. It showed four orbiting bugs: North, South, Toward, and Away. The Mandelbrot bug was always represented as pointing its snout to the west, with its babylike bottom toward the east, so those directions weren't available for this. So Oria was north, and Mars south, with Venus and Mercury this way and that way. All of them tiny compared to Jupiter.