Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
In 1884 Edwin A. Abbott published a novel about the concept of higher dimensions entitled Flatland: A Romance of Many Dimensions. His novel chronicled the adventures of A Square (a play on the author's own name), who resides in a two-dimensional world called "Flatland." A Square is a plane figure, and as such has only two degrees of freedom. He recognizes the directions "left," "right," "forward," and "backward," but he has no concept of "up" or "down."
One day, A Square receives a visit from a visitor from the third dimension, A Sphere. A Sphere "lifts" A Square out of Flatland so that he can experience a three-dimensional world that was, up until that point, unthinkable. Abbott's book is a classic and is well worth reading, as its descriptions of how to think about higher dimensions are still quite useful.
Let's focus on one particular incident in the book, the part in which A Sphere first makes contact with A Square. A Sphere introduces himself in this way:
I am not a plane Figure, but a Solid. You can call me a circle; but in reality I am not a Circle, but an infinite number of Circles, of size varying from a Point to a Circle of thirteen inches in diameter, one placed on the top of the other. When I cut through your plane as I am now doing, I make in your plane a section which you, very rightly, call a Circle. For even a Sphere—which is my proper name in my own country—if he manifest himself at all to an inhabitant of Flatland—must needs manifest himself as a Circle.
A Sphere's appearance in Flatland is an example of how we can use lower-dimensional slices to get an idea of the structure of higher-dimensional objects. If you've ever seen a topographical map, you have some idea of how such "slices" are used to represent a 3-D landscape on a 2-D page.
The lines represent what are known as "level curves." They are what we would see were we to slice the landscape at different elevations. We can use a similar slicing process to get a sense of the structure of objects in four dimensions.
A sphere is a three-dimensional object, so it cannot be represented in two dimensions in the same way that it is in three dimensions. We could try to use an illusion, as we did when portraying the w-axis, or we could consider a series of slices taken at different positions on the sphere, as A Square encountered A Sphere in Flatland.
Note that any 2-D slice of a sphere is a circle. Let's take a moment to look at what this entails mathematically.
The equation for a sphere in three dimensions comes from its definition:
all the points in space that are a given distance from the center. Remember that distance in this space is calculated by using the 3-D version of the
If we designate a point on the sphere as (x, y, z), and if we set the center at the origin, this equation simplifies to:
d2 = x2 + y2 + z2
So, to our friend A Square, who has no notion of "z," this will look like d2 = x2 + y2, which is the equation for a circle in the 2-D world. What actually happened to the "z" dimension? Well, if we imagine that the size of the circle in the plane depends on where exactly the plane is slicing the sphere, then z must have something to do with the size of the circle.
Mathematically, we can see this by rearranging our sphere equation a bit to get:
d2 – z2 = x2 + y2
So, if z represents where the plane is slicing the sphere, the act of slicing equates to holding z constant. We can readily see that smaller absolute values of z will yield larger circles, assuming, of course, that z = 0 represents the slice that passes through the exact center of the sphere. These slices, also called "level curves," equivalent to the lines on a topographical map, are a useful way of thinking about how lower-dimensional slices "stack up" to make a higher-dimensional object.
Let's look at the case of the hypersphere, whose equation is just like that of the sphere, with an added variable:
D2 = x2 + y2 + z2 + w2
We can think of the hypersphere as a 4-D version of a sphere, just as a hypercube is a 4-D version of a cube. Before taking a slice of the hypersphere, let's just rearrange the equation, as before, to get:
D2 - w2 = x2 + y2 + z2
So, if we hold w constant, we will get a slice of the hypersphere.
C = x2 + y2 + z2, where C is (D2 – w2)
Notice that this is just the equation for a sphere in three dimensions. So, our "slice" is actually a three-dimensional object. To be precise, what we normally think of as a three-dimensional sphere is really a two-dimensional surface; we are not concerned with points on the interior.
To create a hypersphere, we would glue together all the slices from w = -d to w = +d. This gluing and the resulting form are a bit hard to imagine, but looking at the slices gives you some sense of the features of a hypersphere, such as the observation that its volume decreases as you approach extreme values of w.
Taking slices of a hypersphere is relatively straightforward. We don't need to worry about how it is situated in relation to the slicing plane because it appears the same from all angles—it exhibits radial symmetry. Might the same be true of the hypercube? To find out, let's first consider a regular cube.
Similarly to how a plane can be used to slice a circle, we can also use a plane to slice a cube. This time, however, the shape of the slice depends on the orientation of the cube as it passes through the plane.
All three of the cubes shown are the same z-distance from the plane, but notice that the slices are different! This is because the cube is positioned differently in each example. Imagine slicing a block of cheese; the shape of your slice depends on whether you are slicing a corner or a face and at what angle.
Imagine now a cube that is sliced perfectly through the middle by the xy-plane, thus creating a square in the plane. Rotations in the xy-plane still give a square and, were we to keep all other rotation angles constant, we could change the z value from − to positive , while rotating the cube and we would always have the same-sized square, albeit a rotated one. This kind of rotation would be fathomable for a Flatlander.
However, if we rotate the square in the xz- or yz-planes, the shape of the slice changes. The most extreme example of this would be to imagine what the slices of a cube would look like if it were to enter the plane vertex first. It might look like this:
In a similar way, the slice of a hypercube will depend on its orientation in the xw-, yw-, and zw-planes. Here is a sequence of images representing 3-D slices of a hypercube entering our space, vertex first:
So, we have seen that taking slices can help give us some idea of how four-dimensional objects behave. Because slices are often incomplete pictures, however, they necessarily miss many features of an object, depending on how the slice is taken.
If we extend this thinking to a four-dimensional being intersecting our 3-D world, we would perceive something like this:
This 4-D creature does indeed have a continuous body, but the connections are all situated outside of 3-D space, as with the preceding hand example. An extra dimension can provide connections and paths that are not available in lower dimensions. An interesting sidenote is that going into this fourth dimension does not somehow shrink the distance in 3-D space—it simply allows a being to circumvent 3-D barriers. So, although going into "hyperspace" to travel among the stars, as many a sci-fi character has done, does not necessarily mean you can get anywhere more quickly, it does mean you won't have to worry about running into any objects along the way.
An alternative way to view a higher-dimensional object in lower dimensions is through a projection. There are many different techniques of projecting, but the one that we will examine is probably the most intuitive—we'll simply ignore a dimension.
To project a square, a fundamentally 2-D object, onto a lower-dimensional space, the number line, we imagine a sort of transparent shadow that it casts on the line.
A similar process can be used to project a 3-D cube onto a 2-D plane.
We could also, if we wanted to, project a 3-D cube onto a 1-D line. To do this, we would first project the cube onto the plane, then project the resulting planar shape onto the line, as we did with the square.
Representing a hypercube on a flat page requires a similar double projection. First, we project the original 4-D object onto a 3-D object; then we project the 3- object onto the 2-D page. The result is quite different from what we would see were we somehow able to view the hypercube in four dimensions, but it does convey important information about its structure.
We can think of a projection as the flattening of an object. Consider how you can flatten a flower or leaf by placing it between the pages of a thick book. The result captures much about the essential shape of the object while, at the same time, distorting it in some fashion.
These techniques, slices and projections, can come in handy when trying to understand what higher-dimensional spatial objects are like. We said earlier, however, that dimensions need not necessarily be spatial. We will now turn our attention to some, possibly surprising, uses of dimension in our own, normal, three- (or four-, or five-, or more) dimensional experience.