Bailey reached up and shook his arm. "Snap out of it. What the hell are you talking about, four dimensions? Time is the fourth dimension; you can’t drive nails into that."
Teal shrugged him off. "Sure. Sure. Time is a fourth dimension, but I’m thinking about a fourth spatial dimension, like length, breadth and thickness. For economy of materials and convenience of arrangement you couldn’t beat it. To say nothing of the saving of ground space—you could put an eight-room house on the land now occupied by a one-room house. Like a tesseract—"
"What’s a tesseract?"
"Didn’t you go to school? A tesseract is a hypercube, a square figure with four dimensions to it, like a cube has three, and a square has two. Here, I’ll show you." Teal dashed out into the kitchen of his apartment and returned with a box of toothpicks which he spilled on the table between them, brushing glasses and a nearly empty Holland gin bottle carelessly aside. "I’ll need some plasticine. I had some around here last week." He burrowed into a drawer of the littered desk which crowded one corner of his dining room and emerged with a lump of oily sculptor’s clay. "Here’s some."
"What are you going to do?"
"I’ll show you." Teal rapidly pinched off small masses of the clay and rolled them into pea-sized balls. He stuck toothpicks into four of these and hooked them together into a square. "There! That’s a square."
"Obviously."
"Another one like it, four more toothpicks, and we make a cube." The toothpicks were now arranged in the framework of a square box, a cube, with the pellets of clay holding the corners together. "Now we make another cube just like the first one, and the two of them will be two sides of the tesseract."
Bailey started to help him roll the little balls of clay for the second cube, but became diverted by the sensuous feel of the docile clay and started working and shaping it with his fingers.
"Look," he said, holding up his effort, a tiny figurine, "Gypsy Rose Lee."
"Looks more like Gargantua; she ought to sue you. Now pay attention. You open up one corner of the first cube, interlock the second cube at the corner, and then close the corner. Then take eight more toothpicks and join the bottom of the first cube to the bottom of the second, on a slant, and the top of the first to the top of the second, the same way." This he did rapidly, while he talked.
"What’s that supposed to be?" Bailey demanded suspiciously.
"That’s a tesseract, eight cubes forming the sides of a hypercube in four dimensions."
"It looks more like a cat’s cradle to me. You’ve only got two cubes there anyhow. Where are the other six?"
"Use your imagination, man. Consider the top of the first cube in relation to the top of the second; that’s cube number three. Then the two bottom squares, then the front faces of each cube, the back faces, the right hand, the left hand—eight cubes." He pointed them out.
"Yeah, I see ’em. But they still aren’t cubes; they’re whatchamucallems—prisms. They are not quare, they slant."
"That’s just the way you look at it, in perspective. If you drew a picture of a cube on a piece of paper, the side squares would be slaunchwise, wouldn’t they? That’s perspective. When you look at a four-dimensional figure in three dimensions, naturally it looks crooked. But those are all cubes just the same."
"Maybe they are to you, brother, but they still look crooked to me."
Teal ignored the objections and went on. "Now consider this as the framework of an eight-room house; there’s one room on the ground floor—that’s for service, utilities, and garage. There are six rooms opening off it on the next floor, living room, dining room, bath, bedrooms, and so forth. And up at the top, completely enclosed and with windows on four sides, is your study. There! How do you like it?"
"Seems to me you have the bathtub hanging out of the living room ceiling. Those rooms are interlaced like an octopus."
"Only in perspective, only in perspective. Here, I’ll do it another way so you can see it." This time Teal made a cube of toothpicks, then made a second of halves of toothpicks, and set it exactly in the center of the first by attaching the corners of the small cube to the large cube by short lengths of toothpick. "Now—the big cube is your ground floor, the little cube inside is your study on the top floor. The six cubes joining them are the living rooms. See?"
Bailey studied the figure, then shook his head. "I still don’t see but two cubes, a big one and a little one. Those other six things, they look like pyramids this time instead of prisms, but they still aren’t cubes."
"Certainly, certainly, you are seeing them in different perspective. Can’t you see that?"
"Well, maybe. But that room on the inside, there. It’s completely surrounded by the thingamujigs. I thought you said it had windows on four sides."
"It has—it just looks like it was surrounded. That’s the grand feature about a tesseract house, complete outside exposure for every room, yet every wall serves two rooms and an eight-room house requires only a one-room foundation. It’s revolutionary."
"That’s putting it mildly. You’re crazy, bud; you can’t build a house like that. That inside room is on the inside, and there she stays."
Teal looked at his friend in controlled exasperation. "It’s guys like you that keep architecture in its infancy. How many square sides has a cube?"
"Six."
"How many of them are inside?"
"Why, none of ’em. They’re all on the outside."
"All right. Now listen—a tesseract has eight cubical sides, all on the outside. Now watch me. I’m going to open up this tesseract like you can open up a cubical pasteboard box, until it’s flat. That way you’ll be able to see all eight of the cubes." Working very rapidly he constructed four cubes, piling one on top of the other in an unsteady tower. He then built out four more cubes from the four exposed faces of the second cube in the pile. The structure swayed a little under the loose coupling of the clay pellets, but it stood, eight cubes in an inverted cross, a double cross, as the four additional cubes stuck out in four directions. "Do you see it now? It rests on the ground floor room, the next six cubes are the living rooms, and there is your study, up at the top."
Bailey regarded it with more approval than he had the other figures. "At least I can understand it. You say that is a tesseract, too?"
"That is a tesseract unfolded in three dimensions. To put it back together you tuck the top cube onto the bottom cube, fold those side cubes in till they meet the top cube and there you are. You do ll this folding through a fourth dimension of course; you don’t distort any of the cubes, or fold them into each other."
Bailey studied the wobbly framework further. "Look here," he said at last, "why don’t you forget about folding this thing up through a fourth dimension—you can’t anyway—and build a house like this?"
"What do you mean, I can’t? It’s a simple mathematical problem—"
"Take it easy, son. It may be simple in mathematics, but you could never get your plans approved for construction. There isn’t any fourth dimension; forget it. But this kind of a house—it might have some advantages."
Checked, Teal studied the model. "Hm-m-m— Maybe you got something. We could have the same number of rooms, and we’d save the same amount of ground space. Yes, and we would set that middle cross-shaped floor northeast, southwest, and so forth, so that every room would get sunlight all day long. That central axis lends itself nicely to central heating. We’ll put the dining room on the northeast and the kitchen on the southeast, with big view windows in every room. O.K., Homer, I’ll do it! Where do you want it built?"
"Wait a minute! Wait a minute! I didn’t say you were going to build it for me—"
"Of course I am. Who else? Your wife wants a new house; this is it."