Teacher professional development and classroom resources across the curriculum
Teacher professional development and classroom resources across the curriculum
The idea that there are levels of reality that are normally inaccessible in our daily lives is an ancient one. Mathematicians of the mid-nineteenth century brought this ancient fascination into the modern age with their study of spaces of four dimensions and higher. There are a few ways to interpret what we mean by "the fourth dimension," but they all boil down to considering another degree of freedom that is independent of the three spatial dimensions that we have defined. After just a few years of running and jumping around, we all develop a pretty good intuitive sense of three dimensions, but imagining a fourth independent "direction" can pose somewhat of a challenge. Perhaps the most intuitive way to conceive of this dimension is to think about it as time.
Viewing time as the fourth dimension is appealing for a number of reasons. The first is that we naturally have experience with time coordinates. When we tell someone we will meet them for coffee at 3 P.M., we are specifying a point in time. However, to increase the odds that the meeting actually occurs, we also need to specify a place. So, establishing the meeting uniquely requires three spatial coordinates and one time coordinate. You might say, "Meet me at 3 P.M. on the fifth-floor terrace of the building on the northwest corner of 3rd Street and 4th Avenue," for example. Of course, it is possible for time to change independently of the spatial coordinates—all you have to do is sit relatively still and your time coordinate will change while your position will not. So, if your friend is late, you can maximize your chances of still meeting the person by waiting at the correct spatial coordinates as the time coordinate continues to change.
There are a couple of problems with considering time the fourth dimension, however. The first is that you aren't entirely free to "move around" in the time dimension. In fact, you are pretty much stuck moving forward at a rate that you cannot control (but that, according to Einstein, is not necessarily the same for everybody). So, time allows only a partial degree of freedom. The second problem is that, while you can change your time coordinate without changing your spatial coordinates, the reverse is not true: how could you move from point A to point B without a passage (i.e., change in "position") of time?
So, time's role as a fourth dimension may be debatable on some philosophical level, but for practical purposes, it works quite well. In fact, Einstein treated time as inseparable from the three dimensions of space and gave us the concept of "spacetime," which is the four-dimensional equivalent of a surface, something that we discuss in some depth in other units. This spacetime, however, is curved by massive objects, which suggests that there might be a fifth dimension that allows this curvature to take place. While this may seem mind-boggling, string theory, one attempt by physicists to unify the fundamental laws of the universe, is even more of a stretch. Depending on which version of string theory you adopt, you will be asked to envision a space with between 8 and 26 dimensions. At some point, this just seems like the stuff of science fiction, and a perfectly rational question would be: what are these higher dimensions? Are they spatial?
Before we get carried away by trying to comprehend a world of many dimensions, we can start by considering what a fourth spatial dimension would be like. Let's back up and think about how we expanded our thinking through the lower-dimension worlds that we introduced previously. Remember that we used familiar concepts from the 2-D world to understand the 3-D world, so perhaps we can use concepts from the 3-D world to understand the 4-D world.
First off, to specify a point in four-space, we need four numbers.
Consequently, a point such as (1, 2, 3) is not uniquely defined in four-space; it would, in fact, designate a line parallel to the fourth axis, which we'll call the w-axis. In four-space, the w-axis is perpendicular to the x, y, and z, axes.
Now we've created a visualization problem. Most people are not accustomed to thinking about a fourth axis in the space around us, and representing it poses a challenge. To produce a visual model, we have to rely upon an illusion. This should not overly concern us, however—we already do this when we depict a 3-D object on a 2-D piece of paper or computer screen. For example, to represent the third dimension, the z-axis, on a flat piece of paper (or a screen), the convention is to draw a diagonal, dashed line in the xy-plane—we then use our imaginations to view this line as "coming out of" the page.
To draw the fourth dimension, the w-axis, on a flat page also requires an illusion and our imaginations. Let's draw another line in the xy-plane and imagine that it is "coming out of" the 3-D space that we already have in mind. In some ways, we're creating an illusion within an illusion.
Before you are tempted to dismiss this as hocus-pocus, consider that the mathematics is rock solid; it is only our habitual perception that is troubling us. This is an interesting case of how techniques from mathematics can help us to think about things that are difficult for our natural faculties of perception.
Remember that our conception of movement in the third dimension is "toward" and "away." If it helps you, think of this new, fourth degree of freedom as "in" and "out." Some mathematicians, however, prefer the terms "ana" and "kata," the Greek words for "up" and "down," respectively, to represent the directions one can move on the fourth axis.
Four-space has the capacity for all the configurations associated with lower dimensions—lines, angles, planar shapes, and solids. Also, in the Euclidean view of four-space, it's possible to find the distance between two points by using a straightforward extension of the Pythagorean Theorem.
You may recall that our "new" fourth dimension must introduce a quantifiable property that has not yet existed in any of the lower dimensions—this is simply a pre-requisite of a degree of freedom. Objects in four-space have a property, analogous to area and volume, that we call "hyper-volume." Possibly the most famous object with this property is the hypercube. To prepare to understand it, let's first look at how we formally construct "normal" squares and cubes.
First, to create a square in two dimensions, or a cube in three dimensions, we start with the analogous object from the dimension that is one lower. That is, we use parallel line segments, joined by perpendicular line segments, to create the square. To create the cube, we use parallel squares connected by perpendicular squares.
So, to create the hypercube, we start with a cube in 3-D space; then we create another cube at a distance equal to the side-length of the original cube along the w-axis. These two cubes can be thought of as being parallel in the same way that the opposite sides of a square or the opposite faces of a cube are parallel.
Think back: to make a square, we connected the endpoints of two parallel line segments using line segments of equal length; and to make a cube, we connected the edges of two parallel squares with squares of equal shape. So, to construct a hypercube, we will connect the faces of our parallel cubes with cubes of equal size. It should be clear that connecting all the faces of our two parallel cubes requires six "connector" cubes. Consequently, the hypercube is made up of eight regular cubes that are "glued together" such that all of their faces are attached to one another. Trying to visualize this can truly turn one's brain inside out, but here's a progression of images that might help:
If it helps, imagine constructing a cube from this 2-D plan, or pattern, which is called a "net":
To build the 3-D object from the 2-D net, you simply fold and glue the appropriate edges together.
We can think of the following shape as a 3-D net that can be folded up to make a hypercube:
To create the hypercube, we need to fold and glue faces to attach to one another. Obviously, this requires that we "smush" and stretch the cubes, but were we doing this in 4-D space, no deformation would be necessary.
Being in 4-D space has some rather strange properties. To imagine what some of these might be like, let's again use a lower-dimensional analogy. Let's say that a square in 2-D space has both a defined front and a defined back. If we were in the plane with the square, we would not be able to see its back if we were looking at its front.
However, if we raise ourselves up off of the plane, we can simultaneously see both the front and the back, as well as the interior, of the square. We may think this is no big deal, but the higher-dimensional extension of this thinking can be quite unnerving.
If a four-dimensional being were to look at us, they could see all sides of us simultaneously. Plus, they would be able to see our "interiors." Now, the interior part is a bit hard to visualize, but we can imagine seeing something from all angles simultaneously. Anyone who has constructed a 360-degree photo landscape has some idea of what a four-dimensional being would see in our 3-D world.
This idea of seeing something from multiple angles simultaneously, can be found in much of the art from the early twentieth century. The cubists, including Pablo Picasso and Marcel Duchamp, were very much influenced by the mathematical exploration of higher dimensions.
We have now seen how a fourth spatial dimension can exist in the mental realms of both mathematics and art. Whether or not it exists in the real world is a matter for science to settle. To prove it, we would have to observe phenomena that cannot be explained in the absence of a fourth spatial dimension. Regardless of whether a fourth spatial dimension is physically real, however, mathematical reasoning has shown that it is at least logically possible.
Mathematics provides tools with which we can explore and understand not only the world of our senses, but also worlds we can conceive of only in our minds. Higher-dimensional worlds are indeed possible for us to think about, but we need certain tools in order to be able to say anything meaningful about them. Analogies with lower-dimensional spaces represent one tool, the value of which we have already seen in our earlier discussions. In the next section we will learn about other mathematical techniques that we can use in our quest to achieve a broader comprehension of dimension.