Part 4 (1/2)

”Ryan's right,” said Regan, taking a sip from the plastic water bottle in her hand. ”There isn't a fourth dimension. It's a trick question. The universe ran out of dimensions after three.”

”You may be right,” said her father. ”But then again, one of the most popular theories in modern physics suggests there are as many as 10 or 11 dimensions.” He smiled. ”So I'll tell you the answer. You would have to move in a direction that no human has ever been able to visualize. A direction that no human has ever moved in. A direction that isn't north, south, east or west. Or up and down. Or anything in between.”

Ryan frowned. ”There is no such direction,” he said in annoyance. ”It's ridiculous.”

”Just because we can't imagine such a direction doesn't necessarily mean it doesn't exist. But even if we can't visualize the fourth dimension, there are still ways we can understand some of its properties. Understand how beings living there would interact with us-with poor humans that can only sense three dimensions.”

Ryan shook his head. ”I'm still going with the 'you lost your mind' thing,” he said.

Their father looked amused. ”The best way to grasp some of the possibilities,” he said, ”is to try a thought experiment.”

”A thought experiment?” repeated Ryan questioningly.

”Yes. An experiment you do in your mind only. Using nothing but your imagination. It's an enormously powerful tool in physics. Some of Einstein's greatest breakthroughs were the result of using this technique. The thought experiments we'll be doing first appeared in a book written by an English schoolmaster, Edwin Abbott, in 1884. A book called Flatland.

”Abbott figured the best way to understand how we would appear to fourth dimensional beings,” continued Mr. Resnick, ”is to think about how beings living in lower dimensions would appear to us. So he imagined a kingdom that existed in a universe with only one dimension. He called this Lineland. And he imagined a kingdom that existed in a universe with only two dimensions. He called this Flatland.”

Mr. Resnick raised his eyebrows. ”So if there were a kingdom of people that existed in a universe with only one dimension, what would that kingdom be like?”

The siblings looked at each other perplexed. ”I have no idea,” said Ryan for them both.

”Well, in Lineland, all the inhabitants would be line segments. And they could never change their order. Here, I'll show you what I mean.”

He picked up a black marker and pulled off its cap with a loud pop. He went to the whiteboard and began writing squeakily.

Ben Resnick gave his kids a few seconds to digest his drawing and then, pointing to the line segment labeled, ”The Queen,” he continued. ”For example, if you were the Queen, you'd be stuck between the Court Jester and the King. Forever. Without any width dimension you couldn't pa.s.s anyone-if you tried you would just slam into them. Like two trains trying to pa.s.s each other on the same track. Now if you could make use of the second dimension-move side to side-you could just move to a different track, so to speak, and easily get by. But a Linelander can't. Their entire universe exists on a single line and they have no awareness of anything outside of this line.”

Mr. Resnick capped the marker and slid it onto the tray at the bottom of the whiteboard. He motioned to his desk. ”I'm not a very good artist, so let's move to my computer,” he said.

They walked a few yards to his gla.s.s-topped desk, on which sat a sleek laptop computer connected to a thirty-six inch monitor. In less than a minute of searching he found a cartoon drawing that would demonstrate his point and put it up on the screen.

Both kids smiled broadly and barely managed not to laugh when they saw it.

”Dad couldn't draw that?” broadcast Ryan. ”Really?”

”We're lucky Dad has such a powerful computer,” replied Regan sarcastically.

”So this brings us to Flatland,” continued Mr. Resnick, unaware that his kids were teasing him telepathically. ”Flatland exists in a 2D universe. So think of Flatland as a giant piece of flat notebook paper. And Flatlanders-who appear as circle-people in the figure-are totally, well ... flat. Flatlanders have no idea there is such a thing as up or down. They can only look and move in sideways directions. So if a Flatland dad suggested to his kids there might be another direction to move in, other than back and forth and sideways, they would tell him he was crazy. Impossible, they would say. There is no such direction, they would say. To us 3D beings, the up direction is obvious. But to the poor 2D Flatlanders, no matter how much you told them about the up direction and described what it was like, they couldn't even begin to imagine it.”

”Like we can't even begin to imagine what direction the fourth dimension would be in,” said Regan.

”Exactly,” said Mr. Resnick happily. ”And since their universe exists in only two dimensions, Flatlanders are completely unable to lift themselves off the page. Not even a billionth of an inch. Just like we're unable to move even a billionth of an inch in the direction of the fourth dimension. Whatever direction that is. But unlike people living in the line universe, at least Flatlanders can move past each other.”

Regan raised her eyebrows. ”That must be a relief,” she said playfully.

Regan dropped her empty yogurt container and spoon into a small wastebasket nearby. Ryan decided he was done eating also and did the same with his spoon, s.c.r.e.w.i.n.g the lid closed on the jar he had been holding and setting it down on one of the few empty s.p.a.ces on his father's desk.

Ben Resnick walked to the whiteboard once again and motioned for his kids to follow. ”The key point is that the fewer dimensions you perceive and can operate in, the more limited you are.” He hastily drew another diagram on the board.

”So here you see a Flatlander eyeing a bank vault.”

”How is that a bank vault?” protested Ryan. ”It's just a square. And it's not enclosed.”

”Good point,” said Mr. Resnick. ”It doesn't have a roof. Why do you think that is?”

Regan's eyes widened. Could it be that this was finally beginning to make some kind of bizarre sense to her? ”Because it can't have a roof,” she said. ”Because you can't have any height in Flatland. A roof has to go over something. But there is no such thing as over or under in Flatland.”

”You've got it,” said her father. ”But it doesn't need a roof. Flatlanders have no way to see what's inside the vault. And they can't climb over the line blocking them from the money. So unless they can break through one of the walls there is no way they can get into this vault. But we live and move in the third dimension. So we could get the money. Easily,” he said. ”We could just reach in from above and grab it. Everyone in Flatland would think the theft was impossible. Like magic. But what seems impossible in one dimension can be laughably easy in a higher dimension.”

Ryan scratched his head. ”So everything you're saying about Flatland would be true for us also.”

His father nodded. ”Yes. That's exactly the point of this thought experiment.”

”So someone living in the fourth dimension could rob one of our locked vaults,” said Ryan. ”Just as easily as we could steal from a Flatland vault.”

”Very good,” said Mr. Resnick, nodding enthusiastically. ”Even if our vault had four walls, a floor and a ceiling that were all made of three-foot-thick steel. The Flatlanders can't possibly imagine that their vault is wide open from above. To them there is no above. Same with us. Our sealed vault seems absolutely closed and impenetrable to us. But it would be wide open when viewed from the fourth dimension. And when our money disappeared from a closed vault, we would think it was magic.”

”Are you understanding any of this?” asked Ryan.

”I don't know,” replied Regan. ”But Dad was right. This stuff is making my brain hurt.”

”So let's perform a thought experiment that wasn't in Abbott's book,” continued Mr. Resnick.

He scribbled on the board once again and said, ”Here I've drawn a three-dimensional object, shaped like a wedge of cheese.

”Notice how only its top edge is touching Flatland. So which part of this 3D object would the Flatlanders see?”

Regan rolled her eyes. ”Dad, you drew one of the lines thicker than the rest and you have an arrow pointing to it. So I'm going to say they'd only see that.” She pointed to her father's drawing. ”Are those little circles supposed to be Flatlanders?” she said in amus.e.m.e.nt.

Mr. Resnick smiled with his eyes. ”I'm afraid so. Now you see why I went into science and not art. Anyway, that's right, the Flatlanders would see only a line. Now if you and I were standing on Flatland, we could look down over the edge and see the cheese wedge. But not Flatlanders. They can't perceive anything above or below them. The wedge is there-they just have no way of looking in that particular direction. Are you with me?”

Both kids nodded.

”Let's suppose that one day a few Flatlanders stumble upon this line and measure it very, very carefully. Suppose they even name it.” His eyes twinkled. ”Let's say they call it Prometheus.”

He drew a second figure next to his first.

”So now they enter this line right where I have the arrow pointing, and spread out. Notice that the precise point where they enter is part of both Flatland and the surface of the cheese wedge that is facing us.”

Mr. Resnick capped the black marker and slid it onto the tray at the bottom of the whiteboard once again.

”So now, as far as the Flatlanders know, they're still on Flatland. They still perceive everything as being in two-dimensions. They have no idea they're now on the surface of a three-dimensional object. One that is touching their realm. All they know is that they're stepping onto a small line that they've measured over and over again. And when they do they're suddenly roaming around a surface that gives them hundreds of times more s.p.a.ce than they expected when they entered.”