“Suppose,” he said, “you put a lot of points in a row, touching; you have a line—a one-dimensional figure. Put four lines together at right angles and on a plane; a square—two-dimensional. Six squares at right angles and extended into real space give you a cube—three dimensions. And eight cubes extended into four physical dimensions give you a tesseract, as it’s called—”

“And eight tesseracts make a pentaract,” I said. “Five dimensions.”

“Exactly. But naturally this is just a picture of a pentaract, in that sense. There probably isn’t any ifth and oofth at all.”

“I still don’t know what you mean by a picture,” I said, pushing the cubes around interestedly.

“You don’t?” he asked, pursing his lips. “It’s rather awkward to explain, but… well, on the surface of a piece of paper, you can make a very realistic picture of a cube—you know, with perspective and shading and all that kind of thing—and what you’d actually be doing would be illustrating a three-dimensional object, the cube, by using only two dimensions to do it with.”

“And of course,” I said, “you could fold the paper into a cube. Then you’d have a real cube.”

He nodded. “But you’d have to use the third dimension—by folding the flat paper up—to do it. So, unless I could fold my cubes up through ifth or oofth, my pentaract will have to be just a poor picture.”

“Well!” I said, a bit lost. “And what do you plan to use it for?”

“Just curiosity.” And then, abruptly, looking at me now, his eyes grew wide and he bumped up out of his chair. He said breathlessly, “What have you done to it?”

I looked down at my hands. I was holding a little structure of eight cubes, joined together in a small cross. “Why, nothing,” I said, feeling a little foolish. “I only folded most of them together.”

“That’s impossible! There were only twelve open ones to begin with! All of the others were six-sided!”

Farnsworth made a grab for it, apparently beside himself; the gesture was so sudden that I drew back. It made Farnsworth miss his grab and the little object flew from my hands and hit the floor, solidly, on one of its corners. There was a slight bump as it hit, and a faint clicking noise, and the thing seemed to crumple in a peculiar way. Sitting in front of us on the floor was a little one-inch cube, and nothing else.

For at least a full minute, we stared at it. Then I stood up and looked in my chair seat, looked around the floor of the room, even got down on my knees and peered under the chair. Farnsworth was watching me, and when I finished and sat down again, he asked, “No others?”

“No other cubes,” I said, “anywhere.”