Professor Felix frowned. “I suspect we have a confusion about the nature of the Mandelbrot set.”
“No, we don’t,” Colene said. “We just need to know how to number the rads.”
Provos got up and walked to the aquarium. Felix glanced at her, surprised. “You have heard about my analogy?”
“No,” Colene said. “Let’s have it, if it helps the numbering.”
The professor shrugged. “Perhaps it is best to begin at the beginning.” He went to the aquarium and turned on a light. It sent a strong beam down through the water. Then he turned on a submerged water jet, and the water began to circulate. “Note the shadow pattern,” he said.
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.”