“As for the nekronic matter itself, it may be symbiotic or vampiric. I wonder. Thought and matter are very similar. It may be that nekronic matter has the potential ability to embody itself provided it finds a suitable host. It’s significant that the creature itself is superficially manlike. Quite possibly it uses whatever prey it feeds on as a pattern from which to shape itself.”

“You think it feeds?’

“You know as much as I about that. Probably more if you were capable of thinking the thing through. We don’t know why the embodied nekronic entity kills. The most obvious solution is to replenish itself, to spread. Even a null-entropy organism might do that, in a sort of reverse pattern from the norm.”

He flashed a blue light thoughtfully and considered the results. So far as I could tell, there had been none but Belem seemed to fall into a minor trance for a few minutes, considering his work.

I was watching a rift like black lightning that ran across the light-wall outside. A red cloud puffed through but the gap healed swiftly and the cloud was dissipated.

Belem twisted a dial, bringing two lenses into sharper focus. “Very likely we’ll never know,” he said. “We can’t last much longer now. A War Council has taken command of this planet.”

“Not Paynter?”

“He’s one of them. That’s odd. They’ve outvoted him three times already on the question of attack. He doesn’t want us destroyed—which means he doesn’t want you destroyed.”

“Nice of him,” I said. “After he tried to kill me in the Subterrane.”

“Paralyze, not kill,” Belem corrected.

Silence after that, while Belem worked and I watched. “What would happen if you had time and material enough to make another of those marbles?” I inquired idly, after a while.

“A great deal. Both matrix-weapons—technically they’re electronic matrices—would be negatively charged, and would repel each other. Unfortunately we have neither time nor equipment for that.”

“What you need is a hacksaw to split that marble in two,” I said. “Then they’d both change from immovable bodies to irresistible forces and shoot each other out of the galaxy. Right?”

“Wrong. Besides being impossible it wouldn’t help. You wouldn’t have two electronic matrices of the same pattern as before. It’s exactly the same reason why the second-stage Mechandroid wouldn’t be dangerous to the social body. The whole is never larger than the sum of its parts, and the sum of the parts always equals the whole.”

“Then you never heard of Banach and Tarski,” I said.

“Who?”

“Once I was assigned to write a feature science story on their experiment. I did plenty of research, because I had to find human interest in it somewhere and it was pure mathematics. The Banach-Tarski paradox, it was called—a way of dividing a solid into pieces and reassembling them to form a solid of different volume.”

“I should remember that,” Belem said, “since I have all your memories. It was only theoretical, wasn’t it?” He searched my memory. I felt uncomfortable as though, under partial anaesthesia, I watched a surgeon investigating my digestive tract.

“Theoretical, sure,” I said. “But I did a repeat on the subject later. It took twenty-three years before somebody figured out how to apply the trick to a physical solid. I forget the details.”

“No you don’t,” Belem said, turning from his work and staring at me. “You have no control over your mind, that’s all. But the information is stored there. Apparently I didn’t get all the details when Paynter searched your memories. There’s a name—Robinson?”

“It could be. I don’t know.”

His face showed no change but I thought I sensed a growing excitement within him. “Cortland,” he said, “I want to enter your mind again. I think—”

20. Last Defense

Apparently he thought I might object—not that that would have made any difference—for the next thing I knew the quicksilver eyes were growing larger and the next instant they had changed and refocused so that I saw them, as it were, behind my own eyes. I could see the motionless body of Belem standing before me but his face was blanker than ever.

Within my head, he spoke to me. “Remember. It’s all there, in your memory. The right associations will recall it. The unconscious never forgets anything. Robinson. The University of—”

“California,” I thought and something clicked and swung open and I saw a page open before me—a page I had first read thousands of years ago—and the fine print swam into remembered visibility.

“Professor Raphael M. Robinson of the University of California now shows that it is possible to divide a solid sphere into a minimum of five pieces and reassemble them to form two spheres of the same size and the original one. Two of the pieces are used to form one of the new spheres and three to form the other.

“Some of the pieces must necessarily be of such complicated structure that it is impossible to assign volume to them. Otherwise the sum of the volumes of the five pieces would have to be equal both to the volume of the original sphere and to the sum of the volumes of the two new spheres, which is twice as great.”

That was all. It wasn’t as much as Belem would have liked—I could feel his impatience and the way he seemed to be shaking my mind over for more details but I couldn’t give him what I didn’t have. After awhile the metallic mind unlinked from mine and in a moment the motionless figure before me stirred, turned without a word and began making tentative drawings on the corner of a chart convenient upon the wall.

When I asked him questions he told me remotely to go away.

That was how it started. There’s no use in my trying to tell you how it ended. I didn’t understand. It would be ridiculous for me even to pretend I know how it was done in concrete fact before my eyes. But it was done.

Not easily. Not quickly. In fact it came dangerously close to not being done at all, simply because it took so long.

I was able to watch the first stages of Belem’s experiments. He knocked down the problem of lenses and lights upon which he’d spent so much time and began setting up theoretical paradoxes in three dimensions, following the Banach-Tarski geometric plan. I watched him playing with ghostly spheres and angles of light until my head began to ache from following the changing shapes. What he was attempting was clearly impossible. I wandered away after awhile and watched the play of lights outside. The display had recently become a lot more spectacular and more interesting to watch but that was not good. Even I could see that, though nobody would answer my questions. The methodical machine-men were not panicky but you could see they had accelerated their pace. They were recognizing the need for hurry.

The second-stage Mechandroid on its table had changed, too. The brilliant neural webbing above it had simplified. Light ran now only in the main channels, letting the finer nerve-wires run very pale, but the synapse-points glowed like stars along the major lines.

And there was a pale glow hanging like a cocoon of radiance low over the motionless figure.

I watched little groups of workers cluster around it, bending their heads together over the table, and I had the impression that they were communicating with their new-born super-kinsman. I even got the idea that he was advising them, for those who left the group went directly to work with a fresh impetus.

It was a little like what must go on in a hive as the workers cluster around the queen-mother.

They were very definitely working against time now—perhaps against hours, even minutes.

It was when the black lightning opened a second rift in the wall of shielding light that the last galvanic spasm of activity before the end stirred the workers to their final tasks.