What's the Fourth Dimension?

As when we asked what a dimension was a few lessons ago, you're going to get a lot of different answers for this one depending on to whom you talk. For example, I'm sure you've heard somewhere that time is the fourth dimension. That's not technically true. Time isn't the fourth dimension. It can be considered to be a fourth dimension. You may think this is just splitting hairs (what? A mathematician splitting hairs?!) but it's actually an important distinction.

If you haven't read it recently, go back to the lesson on what a dimension is. You'll recall that for our purposes, there's the algebraic answer: the number of independent variables in a system, and the geometric answer: the number of axes that can be placed in the space that are all mutually perpendicular. Let's consider the algebraic definition first:

If the number of dimensions is simply the number of independent variables in a system, then a four dimensional system would simply be one with four independent variables. An example might be f(x, y, z, w) = x² + y²+z²+w². What does the w stand for? The mathematician answers clearly and unequivocally:

Who Cares?!

That's for the physicist or the chemist or the economist to worry about. Our job as algebraists is to study the properties and behavior of the equations. We don't particularly care to what (if anything!) they refer. If some scientist wants to come by and apply the equation to something, that's her problem.

Okay, perhaps I'm being a tad harsh, but the gist of what I'm saying is true. Mathematicians are interested in the properties of the equations themselves, independent of what they stand for -- if they stand for anything at all. As a mathematician, I could say that my function f(x, y, z, w) = x² + y²+z²+w² represents a hypersphere (a four dimensional sphere) and it does! But even if I didn't, I'd still be interested in how the equation behaves for its own sake.

Okay. I recognize that for many people, the algebraists' response is a less-than-satisfying answer. Let's consider how the geometer would answer the question what's the fourth dimension?

To begin with, as a geometer, our respondent would be more interested in where the fourth dimension is. The geometer must answer in a way consistent with the ways he answered where the first three are. You would take four axes, he would say, and place them all mutually perpendicular to each other. Voila. The fourth dimension.

"But wait!" I hear you cry. "You can't put four axes all at right angles to each other. Where does the fourth axes go?!"

The geometer isn't phased at all, and argues in the following way:

To a creature living in a plane -- not just living in the plane, but whose very universe was the plane, you couldn't explain where the third-dimension was. You couldn't say it is up, since the creature would think you meant north. If you explained that up was at right angles to north, the creature would think you meant east. If you tried to say that up was at right angles to both north and east at the same time, the creature would shrug, think you're nuts, and move on. There is no place, according to the creature, that is at right angles to north and east at the same time.

Now, here's an important point: just because the creature can't visualize where that direction is doesn't mean it can't exist. It just can't exist in the creature's two-dimensional universe. But the creature could spend some time trying to figure out what the universe would look like if it could travel in that direction. This is to say, just because you can't see something doesn't mean you can't study it. You'll never travel back into history. It's still instructive and interesting to study it for the shadows it casts on today.

So where is the fourth dimension? It is a direction that is at right angles to up, north, and east all at the same time. Can I visualize where that is? Nope. Can I ever go there? Nope. Do I have any hope of understanding it at all? Sure!

The main tools for studying higher dimensions are induction and analogy. Think of working by analogy this way:

Our two-dimensional creature might never be able to understand what a cube is, because a cube is a three-dimensional object. However our creature could get two very interesting views of a cube: if we were to shine a light on the cube, and cast a shadow of it onto the creature's two-dimensional plane, by moving the cube around, the creature could study how the shadows change.

A second way would be if we were to put the cube into our creature's planar universe, that is, make the cube and the plane intersect. Then the creature could investigate the cross-section of the cube that is in its space. By moving the cube in and out of the creature's plane, it could study how the cross-sections change.

Below is a picture of just that: it's a top view of a cube as it passes, vertex first, from above the plane to below. Before it intersects the plane, the cube is invisible to the creature. When it first intersects the plane, the creature would see a point come out of nowhere. The point would split, and grow to a triangle. The triangle would continue to grow, until its edges split to become a hexagon. By the time the cube is half way through, the cross-section is a regular hexagon (the middle picture in the series.) As the cube continues its passage, the hexagon collapses back into a triangle, shrinks down to a point, and again disappears.

In the figure, the plane is represented by dotted red lines. The cube is blue, and the cross-section between the cube and the plane is in green.

In the next lesson, we'll apply what we know about the regular three-dimensional polyhedra to the fourth dimension, and learn about the regular four-dimensional solids, called polytopes.