“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 rationale 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: R1 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 R1, 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.”