“Four of them, aren’t there?” she interrupted. “Dimensions, right?”
“Right. At least, physics has always dealt in terms of four. But in fact there are more . . . to be precise, six of them.”
“That’s the bit I thought was strange,” Sarah came in again. “Four I can visualize okay, but six . . . ? No way. Where are the other two?”
“That’s the whole point. There is no way anybody can perceive the higher ones . . . either by their senses or by instruments. We’ve got no way of knowing about them . . . no more than a shadow man on the wall can know about up or down out of his flat world. He not only can’t move out of it, he can’t even see out of it, so the words just don’t mean anything.”
Sarah held up her hand to prevent him from going any further and sipped her drink while she reflected on what he was saying. At last she put the glass down. “I don’t know if I’m missing something, but if all that’s as you’ve said, how do you know about them . . . the higher dimensions? I thought you just said nobody could.”
“Mmmm . . .” He studied the tabletop pensively, “that’s where the problem gets technical. If I just say that the mathematics of a lot of physical processes—down at the subatomic level—makes sense when the extra dimensions are assumed and don’t make sense when they aren’t, would that be good enough? You’d buy that?”
“Suppose I’ll have to,” she accepted. “But you said ‘assumed.’ That’s not good enough, surely. Aren’t you supposed to be able to prove things like that?”
“Absolutely right! And that’s what we’ve been trying to do, and that’s where we’re hitting problems.”
She rested her chin on her knuckles and said again:
“Well—I’m interested. Tell me.”
“Okay,” he agreed. He was beginning to enjoy the conversation. “Let’s play a game . . .”
“What, in public?”
“I’m serious. There’s a flat universe.” He indicated the top of the table. “Forget we’re solid 3-D people and imagine we’re shadow people that live in that universe—as we said a minute ago. Now . . .” he pointed at one of the coasters lying between them. “That’s an object that exists in our flat universe . . . it’s got no thickness at all, okay?”
“Okay,” she agreed.
He picked up the coaster and turned it at a right angle so that its edge rested on the table.
“Now I’ve rotated it so that, although it still exists, it now lies completely in the dimension that we—the shadow people—don’t know about. How much of it do we see?”
“It’s got no thickness at all, you said?” she checked.
“That’s right.”
Sarah shrugged and opened her fingers.
“We don’t see any of it,” she said. “It’s vanished.”
“Precisely. The tabletop is lo-order space . . . normal space. The up-down dimension is hi-space, and all of them together is k-space. Get it?”
A light of comprehension glowed in Sarah’s eyes.
“Just a second, before you say any more,” she said excitedly. “Let’s see if I can fill some of it in for myself. If you didn’t just rotate that, but spun it over and over all the time, the shadow people would see it disappearing and reappearing all the time, wouldn’t they? That’s the thing that Aub and you were getting worked up about when Aub was at Berkeley . . . those things you called k-space rotations. He showed us a picture of a particle doing just that.”
“Absolutely right,” Clifford confirmed. “It was doing just that. And that was the first concrete proof that it all really was real.” Sarah had nothing to add at that point and seemed eager for more, so Clifford went on. “Now suppose we have two objects, both of which exist purely in hi-space . . .” he picked up a second coaster and held it parallel to the first so that they were both standing edge-on to the table. “We don’t see anything in the shadow universe . . . normal space, right?”
“Right,” Sarah agreed.
“Now, if they collide and one or both of them flip over . . .” He went through the action and left her to complete the sentence.