More Directions Than There Are

The cube was easy. The four-dimensional cube was where the trouble started.

"You can't draw it," Soren said. He had pushpins between his teeth, which made him hard to understand. "You can only draw its shadow."

Maya had string wrapped around three fingers. "That's not the same as not being able to make it."

"It's exactly the same." He spat the pins into his palm. "We live in three directions. Up, sideways, forward. There's no fourth pin to put the string on. There's no place to point."

Maya pointed at the ceiling. Then at the wall. Then at Soren. "Up. Sideways. Forward." She turned her hand slowly, looking for somewhere else to aim it, and her hand ran out of room. "Okay. So where would the fourth one go."

"Nowhere. That's the whole problem. We're stuck with three."

She didn't answer. She was looking at the cube they had made, eight pins and twelve threads, casting a flat tangled shadow on the floorboards from the bare attic bulb.

"The shadow has fewer directions than the cube," she said.

"Sure. The shadow's flat. Two directions. The cube's three."

"So the shadow can't point where the cube points. There's a whole direction the shadow doesn't have. It's just down there being flat, and it has no idea."

Soren put the pins down. "What are you doing."

"I'm being the shadow." She crouched on the floor next to the dark tangle. "If I lived in the shadow I'd say, there's no third pin to put string on, there's no place to point up. And I'd be wrong. Up would be right there. I just couldn't get to it."

"That doesn't help us. We're not the shadow. We're the cube."

"How do you know."

He opened his mouth and closed it.

Maya stood up too fast and her hair caught the bulb's string and set the light swinging. The shadows on the wall lengthened and shortened, lengthened and shortened. "My brother does this thing," she said. "On his computer. For the physics class. He's got these arrows, and they're not pointing anywhere in the room. He swears they're pointing somewhere."

"Pointing where."

"That's what I asked him. He said the arrow that means the electron is here, and the arrow that means the electron is there, those two arrows are at right angles to each other."

Soren picked the cube up off the floor. He held two threads out, pulled them straight, made them cross at ninety degrees. "Right angles. Like these."

"Like those. Except here is at right angles to there, and there is at right angles to over there, and every single place the electron could be, that's its own direction, all of them square to each other." Then he counted on his fingers, and ran out of fingers, and kept counting anyway in the air. "How many places can the electron be."

"As many as you want."

"No. How many. Give me the number."

Maya grinned at him, because this was the good part, the part she'd been carrying around since Tuesday waiting for someone to drop it on. "There isn't one. It can be anywhere along a line. Anywhere. And between any two spots there's another spot, forever. So that's how many directions you need. One for every spot. More than the cube. More than four. More than you can count to."

"Infinite," Soren said. He said it the way you'd say a word in a language you were just learning.

"Infinite directions, all square to each other. And the arrow for the electron points somewhere in there. Not up. Not sideways. Not forward." She turned her hand again, that same slow searching turn, and again it ran out of room, and this time she let it run out on purpose. "Somewhere we don't have a wall for."

Soren set the cube down very carefully, like it might tip over into a direction he couldn't follow it into. He reached for his notebook on the floorboard and the swinging bulb threw his hand's shadow huge across the page before it landed. He drew two arrows crossing at a right angle. Then he tried to draw a third one square to both. It went diagonal on the flat paper, a lie, a shadow of a right angle.

"I can't draw it either," he said. "Same as the cube. I can only draw its shadow."

"But it's there."

"It has to be there. Because the math works." He stared at his three crooked arrows. "My brother. Your brother. The physics class. They're not pretending the directions are there. They're using them. They do sums in them and the sums come out and the machines work."

"The machines work," Maya agreed.

"So the space is real. The space with more directions than there are." He looked up. "That's a space. That's a real place mathematicians wrote down, and physics lives inside it, and it's got room for infinite right angles, and we're in here with three."

"Three and the shadow," Maya said. She crouched back down by the tangle on the floor. "We're the shadow, Soren. The whole room is. Everything we can point at is the flat part. And the actual thing is up out of the floor in a direction we don't have a name for."

Soren didn't say anything for a while. He kept turning his pencil, trying to find the angle that would let him draw three lines all square to each other on one flat page, and there wasn't one, there was never going to be one, and he knew it, and he kept turning the pencil anyway.

"Stop the light swinging," he said finally. "It's making me dizzy."

Maya reached up and caught the bulb's string. The swinging shadows on the wall slid to a stop. The tangled shadow of the cube held still on the floor, flat, certain, eight points and twelve lines, knowing nothing about the third direction it was made from.

She held the string in her fist and didn't let go.

"It can't feel us up here," she said. "The shadow. It can't tell we're standing over it in a direction it doesn't have."

Soren looked at the still flat shadow, and then up past the bulb at the dark sloped ceiling, and then at nothing, at the place where his hand kept wanting to point and couldn't.

He lifted his pencil off the page and held it straight up off the paper, into the one direction the drawing didn't have.