In Davide Cervone's math course for nonmajors, he introduces the idea of a fourth spatial dimension, asking his students to try to imagine what it would look like.
“It takes ten weeks,” he laughs.
Do they really get it?
“On some level. It's never going to be as visceral as your understanding of three dimensions, but you can see that the right things are happening. It's a tool that people have known about for over one hundred years.”
For those of us who want to think about this, the way to do so is not through four dimensions at all but through two dimensions. “Consider a world that's flat-and what the inhabitants of a two-dimensional world would understand about our three-dimensional world,” he says.
Students spend the first two or three weeks of the course talking about this flat world because it's counterintuitive. Cervone starts by asking students to come to the board and draw pictures-a person and a house, mountains, a lake, the sun. Then he asks them questions-for example, can the person in the picture see the sun? “Yes, of course he can” is a typical response, to which Cervone replies, “'Well, let's find out.
“Remember, the entire world is on the surface of the board,” he explains. “Here's a ray of light coming from the sun, and the first thing it hits is the mountain. So no, the person in the picture can't see the sun, because the mountain is in the way. But even if the mountain weren't there, he still wouldn't see the sun, because the light would hit the side of his head, and his eyes are inside his head. He can't see anything 'out there.' He can see his skull, he can see his mouth-that's it.
“Everything in this world
is in the plane of the board. Nothing is 'in front of' or 'behind' anything else,” he continues. “The man in the drawing is actually in the mountain, and the lake is an underground lake.”
Then Cervone asks the
students to see if they can move the poor fellow's eyes
so he can see the sun.
“They'd have to put them on the edge of his head. And his mouth would have to be on the edge, too. But then where would the mouth lead to?”
They draw a throat and a stomach, and so on.
“But this is a problem,” says Cervone, “because it would split him in half. The digestive system-in effect, the idea of a tube-is a purely three-dimensional idea that does not exist in two dimensions. In a 2D world, you can't have a hole that goes through something without splitting it up.”
By now, the students are getting a sense of what 2D people would see and not see, know and not know. And they are beginning to understand that, as 3D beings, we see every point in the two-dimensional world-to us, nothing is hidden. But in Flatland there are no layers.
This is unlike a photo, because what we see in a photo is a 2D projection of something that's 3D. “Our photos are shadows of the real world,” says Cervone. “We're so used to interpreting them, we do it without even thinking how we use notions of forward and backward, and decreasing sizes, and so forth, to represent the missing dimensions.”
He then takes the students' imaginings to the next step: “How do I explain 3D stuff to a guy with this 2D sense of reality? How do I explain a cube to this poor fellow?
“Well, I could unfold the cube and lay it out flat in his 2D world, where he could see all of its parts. It's made out of six squares, and I could give him instructions for gluing them together. But, he would object, 'There's no way I could glue these pieces together without destroying the squares-I'd have to stretch and bend them, or rip or distort them.' And I would counter, 'Ah, but I have a third dimension in which I can do this!'
“To which he would say, 'You're full of beans! There's no way this can happen!' So I would show him shadows of this happening: I'd take my cube and lay it flat, unfolding it above him, under a light source that would project a shadow onto his 2D world.” *
As Cervone folds up the squares, some edges move closer to the light source, so their shadows get bigger. The shadows no longer appear as squares, but as trapezoids. As the sides of the cube come together in 3D, the trapezoids seem to meet in the 2D shadow. In the world of the Flatlanders, the picture looks like a little square inside a bigger square. We recognize this as looking down into a box, but the 2D man would see trapezoid shapes.
“Pointing to one of these, he might say, 'That's not a square! That doesn't have right angles.' You'd reply, 'I know that. But it's a shadow of one that does, and it's tilted in another dimension. Things closer to the light source have bigger shadows.'
“When the cube is completed, you'd say, 'See? There are all six squares, and they all fit together perfectly.' And he'd say, '(grumble, grumble) All right….' He wouldn't be as happy about it as he is about squares in his Flatland, but it would be one way for him to make sense of a cube.”
Cervone moves on to the next dimension: “Imagine we are like this poor flat guy, but in a 3D place, and there's a fourth dimension above us, where every point in our 3D space is available to it. Now, imagine that somebody up there has a hypercube, and it's been unfolded, and he tells us how many cubes there should be (eight) and how they should be glued together. We would see 3D shadows of this 4D thing as he folds it together.
“We have to build into our heads ways of doing that with 4D stuff-the shadows we'd see would not be flat but 3D. Because three dimensions are flat in 4D space, in the same way that a piece of paper is flat to us.”
Does Cervone believe that a fourth dimension actually exists? “Is there a physical one that we can get our hands on? Probably not. But do the concepts make sense? Absolutely. Is it valuable to think about? Yes, because it sheds light on our perceptions, on what we understand about our own dimension. We never would have thought about a tube being a three-dimensional idea otherwise.”
Understanding the workings of unfamiliar things can help us better understand the things we can actually get our hands on-such as taking a foreign language to understand English better. Or going to Mars to understand the Earth better.
The other reason to study the unfamiliar fourth dimension, says Cervone, is that you can represent three-dimensional things, and actions
particularly, in four dimensions as long as you have
Physicists, for example, use time as a fourth dimension, to describe an event happening at a particular position (three dimensions) and a particular time (fourth dimension); they hook together these coordinates and figure out how they relate to each other. This has to do with what they call spacetime-looking at points in a four-dimensional space, with time as one of the coordinates.
“Very beautiful and interesting things happen when they do that,” Cervone says.
The fourth dimension he talks about in his class, though, is a spatial dimension.
The course Cervone teaches is designed for students who are afraid of math. “They are adventurous enough to want to try something different,
but were probably afraid of taking calculus. So this is an alternative to fulfill their
“They're a little fragile, and I have to treat them carefully. They also have their preconceived ideas that they're reluctant to give up. But they tend to like the course and eventually get the concepts. At the end, I ask them, 'Remember the first two weeks, when you fought me tooth and nail about whether the beach ball was two-dimensional or three-dimensional? How hard that was? Think about it now-you understand it immediately!' So I can really say they learned something, their ideas have changed.”
Cervone believes that “you have to shake them up-kick the cobwebs out of what they think. But that's what we're here for.”
After all, math is about solving problems you've never been able to solve before. “It's not really about the answer, or about calculations-it's about what are the right questions to ask, no matter the topic. It's about taking something you know and using it in a situation you've never before faced.
“Most students have never been asked to do math this way before. I'm interested in their understanding of the questions, not so much in the answer. Looking in the back of the book for answers is training your self to fail in the future. In life, there is no back of the book!”