"You've been reading too much science fiction," I told him. "Cascade images are at least partly psychological, and they certainly have no visible substance. Besides, if you had to trace the proper path through a hundred universes every time you went through a cascade point, you'd lose ninety-nine ships out of every hundred that tried it."

"Actually, Mr. Raines is not being all that far out," Dr. Chileogu put in quietly. "It's occasionally been speculated that the branch cuts and Riemann surfaces that show up in Colloton theory represent distinct universes. If so, it would be theoretically possible to cross between them." He smiled slightly. "But it's extremely unlikely that a responsible captain would put his ship through the sort of maneuver that would be necessary to do such a thing."

"What sort of maneuver would it take?" I asked.

"Basically, a large-angle rotation within the cascade point. Say, eight degrees or more."

I shook my head, feeling relieved and at the same time vaguely disappointed that a possible lead had evaporated. "Our largest angle was just under four point five degrees."

He shrugged. "As I said."

I glanced around the table, wondering what avenue to try next. But Wilkinson wasn't ready to abandon this one yet. "I don't understand what the ship's rotation has to do with it, Dr. Chileogu," he said. "I thought the farther you rotated, the farther you went in real space, and that was all."

"Well... it would be easier if I could show you the curves involved. Basically, you're right about the distance-angle relation as long as you stay below that eight degrees I mentioned. But above that point there's a discontinuity, similar to what you get in the curve of the ordinary tangent function at ninety degrees; though unlike the tangent the next arm doesn't start at minus infinity." Chileogu glanced around the room, and I could see him revising the level of his explanation downward. "Anyway, the point is that the first arm of the curve—real rotations of zero to eight point six degrees—gives the complete range of translation distance from zero to infinity, and so that's all a star ship ever uses. If the ship rotates past that discontinuity, mathematical theory would say it had gone off the edge of the universe and started over again on a different Riemann surface. What that means physically I don't think anyone knows; but as Captain Durriken pointed out, all our real rotations have been well below the discontinuity."

Wilkinson nodded, apparently satisfied; but the term "real rotation" had now set off a warning bell deep in my own mind. It was an expression I hadn't heard— much less thought about—in years, but I vaguely remembered now that it had concealed a seven-liter can of worms. "Doctor, when you speak of a 'real' rotation, you're referring to a mathematical entity, as opposed to an actual, physical one," I said slowly. "Correct?"

He shrugged. "Correct, but with a ship such as this one the two are for all practical purposes identical. The Aura Dancer is a long, perfectly symmetrical craft, with both the Colloton-field generator and Ming-metal cargo shield along the center line. It's only when you start working with the fancier liners, with their towers and blister lounges and all, that you get a serious divergence."

I nodded carefully and looked around the room. Pascal had already gotten it, from the expression on his face; Wilkinson and Tobbar were starting to. "Could an extra piece of Ming metal, placed several meters off the ship's center line, cause such a divergence?" I asked Chileogu.

"Possibly." He frowned. "Very possibly."

I shifted my gaze to Lanton. His face had gone white. "I think," I said, "I've located the problem."

—

Seated at the main terminal in Pascal's cramped computer room, Chileogu turned the Ming-metal coil over in his hands and shook his head. "I'm sorry, Captain, but it simply can't be done. A dual crossover winding is one of the most complex shapes in existence, and there's no way I can calculate its effect with a computer this small."

I glanced over his head at Pascal and Lanton, the latter having tagged along after I cut short the meeting and hustled the mathematician down here. "Can't you even get us an estimate?" I asked.

"Certainly. But the estimate could be anywhere up to a factor of three off, which would be worse than useless to you."

I nodded, pursing my lips tightly. "Well, then, how about going on from here? With that coil back in the shield, the real and physical rotations coincide again. Is there some way we can get back to our universe; say, by taking a long step out from Taimyr and two short ones back?"

Chileogu pondered that one for a long minute. "I would say that it depends on how many universes we're actually dealing with," he said at last. "If there are just two—ours and this one—then rotating past any one discontinuity should do it. But if there are more than two, you'd wind up just going one deeper into the stack if you crossed the wrong line."

"Ouch," Pascal murmured. "And if there are an infinite number, I presume, we'd never get back out?"

The mathematician shrugged uncomfortably. "Very likely."

"But don't the mathematics show how many universes there are?" Lanton spoke up.

"They show how many Riemann surfaces there are," Chileogu corrected. "But physical reality is never obliged to correspond with our theories and constructs. Experimental checks are always required, and to the best of my knowledge no one has ever tried this one."

I thought of all the ships that had simply disappeared, and shivered slightly. "In other words, trying to find the Taimyr colony is out. All right, then. What about the principle of reversibility? Will that let us go back the way we came?" "Back to Earth?" Chileogu hesitated. "Ye-e-s, I think that would apply here. But to go back don't you need to know...?"

"The real rotations we used to get here," I nodded heavily. "Yeah." We looked at each other, and I saw that he, too, recognized the implications of that requirement.

Lanton, though, was still light-years behind us. "You act like there's still a problem," he said, looking back and forth between us. "Don't you have records of the rotations we made at each point?"

I was suddenly tired of the psychiatrist. "Pascal, would you explain things to Dr. Lanton—on your way back to the passenger area?"

"Sure." Pascal stepped to Lanton's side and took his arm. "This way, Doctor."

"But—" Lanton's protests were cut off by the closing door.

I sat down carefully on a corner of the console, staring back at the Korusyn 630 that took up most of the room's space. "I take it," Chileogu said quietly, "that you can't get the return-trip parameters?"

"We can get all but the last two points we'd need," I told him. "The ship's basic configuration was normal for all of those, and the Korusyn there can handle them." I shook my head. "But even for those the parameters will be totally different—a two-degree rotation one way might become a one or three on the return trip. It depends on our relation to the galactic magnetic field and angular momentum vectors, closest-approach distance to large masses, and a half-dozen other parameters. Even if we had a mathematical expression for the influence Lanton's damn coil had on our first two points, I wouldn't know how to reprogram the machine to take that into account."

Chileogu was silent for a moment. Then, straightening up in his seat, he flexed his fingers. "Well, I suppose we have to start somewhere. Can you clear me a section of memory?"

"Easily. What are you going to do?"

He picked up the coil again. "I can't do a complete calculation, but there are several approximation methods that occasionally work pretty well; they're scattered throughout my technical tapes if your library doesn't have a list. If they give widely varying results—as they probably will, I'm afraid—then we're back where we started. But if they happen to show a close agreement, we can probably use the result with reasonable confidence." He smiled slightly. "Then we get to worry about programming it in."