Damon Knight - Orbit 16

This look about half an hour. The work went much slower than it had in the morning. He constantly found himself looking blankly at his own notes, confusing output statements with axioms, losing his tenuous grasp on the details of David’s proof.

When the whole thing was finally sent out to be punched up and compiler-checked on the A50 unit, it was ten minutes to two. He told Ruth to call him at five to three and gratefully laid his head on his folded arms on the desk, not even bothering to darken the room.

At first sleep would not come, only sharp-edged pictures, alternately threatening and soothing: his small house, his books, Mary as she was thirty years ago, when he had thought he might marry. Unrelated images swam about in his mind: a snatch of old rock music, a lemma from the Side Approximation Theorem, the smell of lilacs, and slowly one figure emerged from the mass and began to grow and dominate it all. At first it appeared to be only a smooth, featureless, somewhat metallic topological sphere. But this was only the bottom portion. Above, the figure split into two parts, not so much like a branching tree as like a squid with two plump arms, spread in a gentle flattened circle and coming together again—but not quite. Before the arms met, each of them split into two parts again, a thumb and an index finger, which linked together like a chain—but not quite. Before each finger reached its thumb, it was bifurcated, as was the thumb, and again the two arcs linked to form a chain—but not quite. This process continued infinitely, each step increasing the number of parts geometrically. The result was a figure simple in its construction, frighteningly complex in its final appearance.

“My God in heaven!” exclaimed Hans. “What in merciful hell is that? I never expected to find a book on demonology in your home, Donald.”

Don had been at the bar pouring brandy and didn’t know what Hans was referring to. By the time he got back to the table, Hans was standing silently, biting the side of his thumb and staring fixedly at the open copy of Hocking and Young’s Topology.

“That’s Alexander’s horned sphere,” explained Don evenly. For a moment he too felt an uncanny horror at the picture. But it passed, and what was to Hans the image of Satan became only a wild embedding of the two-sphere in Euclidean three-space.

“You see,” he went on, setting down the drinks, “it’s topologically a sphere, but its complement is not the same as the complement of an ordinary sphere. For instance, you could link a circle around it, just like you would around a torus, and it won’t come off. In fact, there are an infinite number of ways you can do it.”

Hans seemed not to be listening. He did not respond when Don sat down, but continued to bite his thumb rhythmically.

“The one thing that amazes me,” he said at length, “is that I have seen it—” He paused, running his fingers through his beard. “—that I have seen it . . . and I still live. What is it, Donald, and how has it found its way into your neat religious parlor?”

“I told you: it’s a wild sphere. It’s called Alexander’s horned sphere. It’s really not so extraordinary, Hans. There’s a wild arc on the next page. Here’s your brandy; now let’s get started with the game.” Don was impatient to forget the thing, for Hans to close the book and change the subject. He didn’t like this reaction to a simple mathematical object, as though it were something more than it actually was.

“One could imagine Dali painting it,” Hans continued, still fixing his gaze on the picture, “all ugly, bleeding lumps of flesh. That would be the obvious way. But I think I see it as a sculpture. It would have to be tremendous—say, thirty feet high—so that the branches start out fat as sequoias and end up—and end up microscopic—and never end. They should go on to the atomic level and beyond. Alexander’s horned sphere. Can you see it squatting in the sun in the middle of Chicago, like some horrid, slimy crab? Yeah! Come on along, come on along. A sculpture, yes, a sculpture, that’s the way I’d do it.”

He slammed the book abruptly and laid it on the floor.

“Donald,” he said, more in his natural voice. “Donald, if you have looked at that picture before and not felt the fear of the darkness in your veins, then all I can say is—you are hopelessly lost in your salvation. Have a Camel?”

Don shook his head, smiling, and Hans fitted a cigarette into his holder, continuing to talk between his teeth as he puffed life into it.

“Donald, the only man who is on such good terms with the devil that he can look him in the eye so casually is the satisfied theologian. You are a priest, not a prophet, and you must learn that even the bestest church what am will not protect you, your doctrinal orthodoxy will not save you, when the prophets begin to quake and wail outside the temple.”

“And you are the prophet?” asked Don, egging him on.

“Hah! No, not your prophet. No, honey lamb, you’ve missed my point. Or else my analogy doesn’t work out right. The prophets. Donald, I’m talking about mathematics, not art.”

“Well, then I wish you wouldn’t use a religious analogy. There is a fundamental difference in approach between religion and mathematics, and— No, let me finish. I know what you’re going to say: that mathematics is predicated on the worship of reason. Well, that’s wrong. Reason is only a tool to certain ends.”

“Well, I agree, of course, Don. Reason is a tool to certain ends, and in your case those ends are basically theological. It’s clear, you know, in this baroque fascination you have with the intricacies of your own proofs. You’re only interested in plastering over the cracks in the temple. You’ve grown too dependent on it; you’re afraid to worship in sunlight. Don’t hide behind reason, Don. Your enemy will use the same tool. It’s not reason that’s against you, sweetheart; it’s history.”

“Anyway,” Don interjected, annoyed at this turn in the conversation, “let’s start playing or we’ll never be done by ten.”

“All right. But, Donald, I am going to do that sculpture, that Alexander’s whatsit, someday. If ever I get a big commission. And you will be the only man who will understand all of its . . . all of its deeper meanings, my friend.”

It was to be another fifteen years before he would get that commission and carry out his threat. It would be only twenty feet tall, not thirty, and in Cleveland, not Chicago. But it would shock, amaze, and frighten thousands of art lovers/haters, just as the original conception had shocked, amazed, and frightened Hans Kaefig.

Hans Kaefig, whose thoughts enveloped, surrounded, like rows of black disks, moving, shifting, unpredictably, while Donald Lucus’ white disks coiled and struck, each move a step in a plan, each play a proverb. The patterns of black and white tesselations became too intricate to follow, and then there was no pattern at all, only the flashing black and white and a buzzing behind them, an insistent buzzing; as seconds stretched and expanded, he groped for his thoughts, sorting out the buzzing, reached for his glasses and the button on the intercom.

“Yes, Ruth?”

“It’s two fifty-five, sir.”

“Thank you, Ruth. Thank you.”

He sat another minute, not really awake. Then, both hands on the chair, he lifted himself to his feet and did his best to tidy his suit. The Director was twenty years his junior, and yet he felt like a truant student being sent to the principal’s office to explain himself, and knowing that the principal is never disposed to hear explanations.

He hadn’t prepared a lecture in years. He had spoken of math only with other mathematicians. He had lost the knack of translation. There were English words, phrases, similes, that could say the same thing as a few swift logical statements, but he had forgotten them. And so he did not prepare a lecture. He had no idea what he was going to say to the Director, whether to present him with a neat, clear proof or simply to shout “ Goit ist tot , “ and make his point loudly and emotionally. He knew the Director was not a believer in any of the fundamental truths that were at stake. He would view a breakdown in the fabric of logic the same as a breakdown of the subway system. It was a nuisance to him, but it was certainly not his job to address himself to the problem; that was what metro engineers were paid for. That was what mathematicians were paid for. Lucus could not approach him on that level. What level he should direct his strategy toward he was not certain. He had, however, foreseen his opponent’s moves well enough to expect the reception he received from the Director’s private secretary.