Colene saw that the ripples and swirls on the surface of the water were almost invisible, but they cast shadows which were quite clear against the white bottom of the aquarium. There were circular patterns with dark centers, the shadow forms of little whirlpools. These drifted outward from the region of the jet of water, becoming smaller and finally disappearing. But new and larger ones formed closer in.
"Note that the entire pattern is three-dimensional, but the shadows show it in two," Felix said. "We can not perceive the pattern as it truly is; we are as it were seeing a mere silhouette. Yet even that is instructive. There is a regular procession of typical shapes, and by observing it we can see the evolution of figures and derive insight into their nature. We can see that these are not fixed outlines, but moving boundaries, guided by specific rules. The currents of water move with certain amounts offeree, and friction with the stable water causes these currents to split and curl, forming vortices. We can photograph the shadows, but we know these are not genuine objects."
"But the universe I saw was genuine," Colene said.
"A universe," he said, disdaining her irrelevancy.
"With the equivalent of land and sea and stars and people and laws of nature, which are magical."
The professor continued as if he had not heard her. "Now consider the Mandelbrot set. This is a mathematical construction. It is obtained by plotting vector sums of points on an Argand plane—that is to say, with one real axis and one imaginary axis. It is a convenient way to graph a complex equation. That is, one with a component involving the square root of minus one. In this case—"
"This is more technical than I need," Colene said. Actually she understood him well enough, but she didn't need basic theory, she needed a way to count rads.
"My point is that this is not a physical object," the professor said. "In fact, the Mandelbrot set is not an ordinary graph. It is that portion of the plane for which the sequence of a mapped equation is bounded. So—"
"Professor, it may be just a mathematical construct to you," Colene said. "But it's pretty damned physical to me. All I want is a clear way to number the rads!"
He focused on her. "Would you try to explain color to a man who had been blind from birth?"
That set her back. "You're saying that first we have to understand the fundamentals before we get specific?"
"Yes. And to establish an analogy that will facilitate at least a partial comprehension."
She sighed. "Point made. I can't demand that you name that color if I don't know what color is. But you know, Prof, I haven't got time for a semester course on the nature of light."
"Agreed. Are you conversant with the concept of Julia sets?"
"I named that reality Julia. But all I know of Julia sets is that they're sort of squiggly shapes on the computer screen. I don't know what they mean. I figure that the Mandelbrot set is maybe one big Julia set."
"Not exactly. The Mandelbrot set helps define a particular family of Julia sets. Each point in the Mandelbrot set is a memory location for a distinct Julia set, which can be of any nature, generated by a fractal equation. But all Julia sets will be self-similar in detail, and a change of scale does not significantly affect the complexity of the figure. So it is possible to tell the general nature of a particular Julia set by knowing its placement on the Mandelbrot set."
"Say, I get it!" Colene exclaimed. "Each point on the Virtual Mode is a location for a distinct universe. And you can tell what that reality will be like, in general, if you know the region of the Virtual Mode you're on."
His brow furrowed. "The Virtual Mode?"
"We're on the same wavelength, Prof! The Virtual Mode is to each universe as the Mandelbrot set is to each Julia set. And the universe I'm talking about happens to look just like the Mandelbrot set, but I guess it's really just a Julia set."
Felix frowned. "If you can satisfy me as to your physical set, I will satisfy you as to the designation of its parts," he said. It was evident that he didn't believe her, and also that he was revising his estimate of her sanity downward. Colene had never been one to take that sort of thing without a fight. So she let him have it.
"Okay. Think of our universe as a series of diminishing spheres. There's the 'Big Bang' at the center, and clusters of galaxies flying out from it, forming the biggest sphere. Each cluster forms another sphere, if it hasn't fallen apart. Each galaxy is a cluster of stars and dust surrounding a ravenous black hole at its center. In the early days a lot of matter was being drawn into that black hole, and as it got torn apart at the edge of that maw it gave off a lot of energy, and we call that a quasar. Now that process has slowed, so we call them galaxies. They're still basically spheres with centers, only instead of flying out they're spiraling in. Meanwhile there's a sphere of dust and fragments around each star; those fragments we call planets. They're not flying out or being drawn in, they're in orbit, but it's the same idea on a smaller scale. Then consider the planets: each one seems to be a spherical conglomeration of solid materials, with a molten core. Same idea, again. Then look at the stuff the planet is made of, and we get down to molecules, which are like even smaller spheres, and then atoms, which seem to be spherical shells surrounding spherical nuclei. Down inside an atom we can get into baryons, made up of quarks: maybe more spheres. So each level of the reality we know is similar to each other level, only different too, never identical. Exactly as it is with fractals. This is a fractal universe, in essence."
She paused. She had gotten the professor's attention, and she could see his estimate of her rising again, as if the mercury in a thermometer had dropped with night and was moving up again with the heat of day. But she had only begun.
"Yet out of this assemblage of diminishing spheres comes the world we normally perceive, which consists of solid ground, liquid seas, and gaseous air. Of houses, cars, and next-door neighbors. Of life and death, love and hate, and parents and children, each similar to its origin yet never quite the same. We don't even think of the spheres, we just eat and drink and laugh and cry and wonder about the meaning of life. This is us. Even though we are so small, in terms of the universe as a whole, that someone viewing the universe from another dimension, seeing the whole thing, would never even notice us. We're just mold on a fragment circling a star on the fringe of one black hole among billions. We're not important at all, objectively speaking."
She met the professor's gaze. She could tell that he was on the verge of being impressed. Good; she wasn't done.
"Worse yet, the entire universe we know may be only one per cent of the whole thing. You've heard about the so-called Dark Matter, the stuff that no one can detect, yet it's supposed to make up ninety-nine per cent of everything. We can't see it, we can't touch it, we can't catch a sample of it; it just doesn't seem to exist, as far as we're concerned. But it has gravitational effect, and our galaxies are affected by it, so we know it's there. We just don't know what it is, or why there's so much of it."
"My friends in the physics department say much the same," the professor agreed. "But this hardly relates to fractals."
"Oh, yes, it does! You asked me for a physical set, and I'm setting up for it. Because the way I see it, it's not one per cent of the whole shebang we can see, it's more like a millionth of it. That gravity we see operating is just the trace that leaks through to our reality from the myriad other realities we can't see. Most of it stays in its own slice of reality, but nothing's perfect, and that tiny leakage may account for the special effects which so mystify our astronomers. From one reality there's hardly enough to make a difference here, but from millions of realities it adds up. So I figure they'll never find a particle to account for all of it, because some of it's coming from places that just don't exist for our scientists. Magic places."
"Magic," the professor said, frowning. "I really don't believe—"
"I'm just telling you how there can be a whole lot out there you never dreamed of, Prof," she said. "You don't have to believe it. Just accept it theoretically, as a r
ationale for how there can be a physical Mandelbrot set, and follow my lead. The way I guess you make your students do."
He nodded. "I think I would like to have you as a student. I can see that you are an unusually imaginative ninth grader."
"That's not the half of it, Prof!" Colene was aware that his comment was not necessarily a compliment. She marshaled her thoughts. "Now picture the Mandelbrot set not as a construct of the mapping of bounded sequences, but as an actual physical reality. With a monstrous central figure looking like a six-legged bug with hairs curling all around its body and a spike on its snout. That's like the sphere of galaxies surrounding the Big Bang. Each little satellite bug is a miniature of the original, like a galaxy. Each tiny satellite bug of a satellite bug looks much the same as its parent, but the pattern into which it fits is always a little different too. Right down to the quark level, and maybe beyond. Assume that in that reality there is a buglet way out on the fringe of nowhere significant that's the same size as Earth, and occupies the same place as Earth, if you superimposed the two realities. That has people on it who look just about like us. If you stood on that planet-bug and talked to those folk, you'd hardly know you weren't on Earth. Only if you had a microscope or a telescope would you be able to see that all the things of this reality, instead of being composed of diminishing spheres, are composed of diminishing iterations of the Mandelbrot set. And because of this fundamental difference, science wouldn't work well there, but magic would, with special rules of its own that might not make a lot of sense to folk of the spherical universe. And one of those rules was that to do just the right kind of magic, you had to find the ninth of the ninth rad. How would you find it?"
It was a moment before the professor spoke. Then he found a new way to approach the problem. "Accepting such a theoretical construct, I would go to the most feasible nomenclature," he said. "Come here." He walked to a table and brought out a small sheaf of papers.
Colene went there. As she did, she saw Provos gesture to Slick. Slick was picking up on the woman's special ability, and joined her, and the two of them left the chamber. Colene wasn't sure what was going on, but she trusted Provos, whose mind she could read a little, and she didn't want to alarm Esta, who seemed bemused by the dialogue and the ongoing patterns in the water tank.
Felix unrolled a large picture of the Mandelbrot set. Every detail seemed to be there, and there were numbers all across it. The black center part of it was divided into sections, as if it were hollow with chambers ranging from huge to tiny. "I think for this you do not want computerized coordinates," he said. "You are not in the business of calculating the set itself, you merely want a way to identify the parts of it in a readily understood manner. As if you were standing on its edge and figuring out exactly where you are."
"Right," Colene said. "Actually it's more complicated than this. The one I'm on is, pardon the expression, spherical. That is, three-dimensional. The rads are on the front and back as well as the top and bottom."
"But there can be no front and back," he protested, "because the figure is in essence a silhouette, a mere shadow—"
"Of the reality," she finished. "The silhouette of a three-dimensional figure would look like this."
He nodded. "In that case there will be a problem of nomenclature. However, let's first define the existing designations." He lifted a stylus and pointed to the main part of the figure. "This is the Body of the Radical Master, or Rad Master, our primary figure." He pointed to the smaller disk on the left. "This is the Head." He pointed to the line extending to the left. "This is the Spike." He pointed to the depression on the right. "This is the East Valley." He pointed to the deep crevice between the body and the head. "And this is Seahorse Valley." He glanced at her. "Are you with me so far?"
"Right with you," she agreed. "I knew those terms. Those crevices are filled with water, where I've been. But it's the rads I need to know."
"We are coming to them. Now for convenience we always orient the Rad Master this way, with the Spike to the west, no matter which way it may be pointed as you see it. Thus the radicals, each of which is a miniature of the Rad Master, are North above and South below. To clarify the situation, we must assign Radical numbers: Rl for the Body, R2 for the Head, and the largest around the Body is the North Rad, which we designate R3. We descend from the larger to the next smaller for this purpose, never skipping down. Thus the only Rad larger than R3 is R2, which is the Head, and the only Rad larger than R2 is Rl, the Body. You remain with me?"
"I sure do! This is coming right onto what I need."
"I'm sure it is. Having proceeded east to reach R3, we continue east to reach R4, which is the largest of all the radicals between R3 and the East Valley, here. Then on to R4, R5, and so on, heading into that valley."
"Right down to the ninth, R9," Colene agreed. "But where is the ninth of the ninth?"
"That would be the ninth rad on that ninth rad," he said, pointing to an almost infinitesimally tiny bump on the small R9. "However, I'm not sure that is what you want. Hasty conclusions are often in error."
He was getting entirely too professorish for her taste. "Well, maybe. But I think that's it."
"But you see there are other R9's. For example, if you were to turn back at R3 and proceed west, you would in due course come to R3:R9, the colon indicating the change of direction. We don't bother to mark R1:R2, because every sequence starts with those two. Consider them implied."
"Change of direction," Colene repeated, remembering the directions of magic indicated by the animus and anima. Her certainty faded.
"Perhaps you should explain why you want this particular designation."
"Okay, you asked for it. But you won't believe it."
"I don't need to believe it. I only need to understand exactly what you want."
"There's this woman, Nona, who can do magic because she's the ninth of the ninth. She needs to get to the ninth of the ninth rad to change things so that women can do the magic instead of the men, only she doesn't know where that is. So I have to find out, so I can tell her."
"She is the ninthborn child, of the ninthborn of her father's generation?"
"Not exactly. It's her mother, and her mother's mother. For nine generations back."
"That is quite a different matter. Nine generations! Those folk evidently run to large families."
"Actually they weren't all large. It was the secondborn girl, and then the thirdborn. I mean, if the secondborn was a girl, and then she had three children and the third was a girl, and then she had at least four, with the fourth a girl, and so on."
"Matrilineal, for this purpose. So your Nona is the ninth child in her family, the daughter of a woman who was the eighthborn in her family, and so on back through the seventh-born, sixthbom, and back to the firstborn."
"You got it. And they align magically with the Mandelbrot bugs, a chain of satellites nine layers deep."
The professor winced when she referred to the forms as bugs, but shook it off. "I believe I have it now. The ninth rad of the ninth rad would indeed be wrong. It would need to be the ninth rad of the eighth rad of the seventh rad, and so on. An entirely different address."
Colene's mouth fell open. "You're right, Prof! You do know where you're going!"
"It is my business to know," he said. He seemed to be better satisfied with Colene than before. "So let's proceed with the denouement. I believe I can give you a specific address that you can show your friend."
"I'm for that!"
He pointed to the Head. "You will note that the Head has a head, and so on ad infinitum. We now use a slash to designate a rad on a rad: R2/R2 for the head on the head, R2/R2/R2 for the head on that, and so on. Similarly the next largest rad on the head, here, is R2/R3."
"We can make a chain of rads on rads that way too!" she exclaimed.
"Precisely. And this chain more accurately reflects the numbers of the births."
"It sure does. So then we go to the fourth rad on that rad on the head—"
r /> "R2/R3/R4," he agreed. "And so on to the ninth on the eighth. Unfortunately my printed diagram does not have that level of definition. I can use my computer program to amplify it on the screen, if you wish, but this will take some time—"
The door opened. Slick and Provos entered. "Trouble," Slick said. "She put me on to it. The police must've located us. Do you have what you need, Colene?"
"Just about," Colene said. "But—"
"Take this," Felix said quickly, handing Colene an envelope. "This is an issue of Amygdala with a good discussion of nomenclature. You now understand the principles well enough to follow it."
"Right," Colene agreed. "You did the job, Prof."
"And your account is quit," Slick said. "I erased it last night. We don't know each other. If anyone asks you—"
"This encounter never occurred," the professor said. "I have spent this hour reviewing fractals alone." He looked relieved. "And I owe no one anything."
"Right," Slick said. He looked at Colene. "Come on." Provos was already hurrying Esta out the door.
Colene followed them out, pausing only long enough to wave goodbye to Professor Felix. He had in the end had what she wanted, and that was what counted. If she had helped him get out of some bad debt, maybe from gambling, she was glad.
Then she reconsidered. She couldn't just depart without more than a wave; anyone could wave. So she indulged her propensity for risk-taking, ran back into the room, caught the professor by the shoulders, and planted a passionate kiss on his surprised mouth. "You couldn't teach this ninth grader much, Prof!" she whispered, and stepped back.
He was still staring with satisfying stupefaction as she closed the door on him.
Provos was leading the way out of the building—but not the way they had entered. In fact they used a fire escape. Then she led them to an unfamiliar car.
It was locked. Slick brought out a tool and jimmied open the door. They piled in while he reached under the wheel to hot-wire the ignition. They were stealing a car!