Discouraged, I sat down and thought some more.
"What makes you think the second-stage Mechandroid can destroy the nekron?" I asked Belem.
He kept working on a cryptic device composed chiefly of vari-colored lenses. His placid face never changed.
"I can only hope so," he said. "He was designed expressly to solve that problem and he will have a fifty-five-power brain, compared to my twenty-power one. He'll be a tool – an extension of the social mechanism."
"With free will?"
"Yes – within obvious limits. He'll have to fulfill his purpose. He wouldn't be functional unless he did that."
"What is his purpose? Besides destroying the nekron?"
"I told you he was an extension. Like the specialized tool of your hand."
"But I can control my hand."
"Not always consciously," Belem pointed out. "If you suddenly found yourself falling your hand would seize the nearest grip. Extend that parallel a bit farther and imagine your hand has a brain of its own.
"It will do – within its limits – what a hand can do best and it would know its potentialities better than you could. And it wouldn't try to rebel, because it's part of the unit. The second-stage Mechandroid is a better hand for humanity – or a better brain in matters of intellect and logic."
He turned to his work again, flashing lights on and off at what looked like random. After a moment he went on speaking.
"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.