—And He Built a Crooked House

“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 square, 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.”