Perhaps the most well-known higher-dimensional object is the hypercube, the four-dimensional analog to the three-dimensional cube. Many were introduced to it in the story A Wrinkle in Time, by Madeleine L'Engle, or ``And He Built a Crooked House'', by Robert Heinlein. Anyone who wishes to understand four-dimensional objects would do well to study the hypercube as a first attempt.
One way to understand four dimensions is to look closely at how we can understand three dimensions using only images drawn in the plane. How we use these two-dimensional images to reconstruct in our heads the full three-dimensional objects can serve as a basis for trying to use three-dimensional pictures to construct four-dimensional objects in our heads in a similar way. A careful study of two and three dimensions, and in particular, how we can move from one to the other, is necessary for this to be successful.
To understand the four-dimensional hypercube, then, we should look at the three-dimensional cube, and its analog in two dimensions, the square. Indeed, we can go even further by looking also at the corresponding one-dimensional object, the line segment, and even the zero-dimensional version, the point. This family of objects--the point, segment, square, cube, and hypercube--form a wonderfully interconnected collection, whose relationships we now investigate.
The first thing to note is that the lower dimensional versions appear as parts of the higher dimensional ones. For example, the cube has faces that are squares, the square has edges that are segments, and even the segment has ends that are points. This suggests that the hypercube should have faces that are cubes, and indeed, this is the case. Just as the cube has square faces meeting at segments (its edges), and points (its corners), the hypercube has cubical faces that attach to each other along their square faces, and which meet at segments and points. But how many of these are there, and how do they connect together? To understand the hypercube better, we will try to count the number of pieces that make it up. In particular, we want to know how many cubical faces it has.
We begin by looking back at the lower-dimensional analogs. To make things easier, we will call all the objects ``cubes'', but also give the dimension; in this way, the 3-cube is the standard cube, the 2-cube is the square, the 1-cube the segment, and the 0-cube the point. The hypercube is then the 4-cube. This gives us the ability to talk about an n-cube without having to specify which one we mean; the goal is to be able to say things about the n-cube that are true no matter what the n is. These will be the statements that help us understand the hypercube.
Our first observation is that the n-cube can be formed my moving the (n-1)-cube in a direction perpendicular to itself. For example, if we move a segment (a 1-cube) a unit distance in a direction perpendicular to the segment, it will sweep out a square (a 2-cube) in doing so. Similarly, if we move a square a unit distance in a direction perpendicular to the square, it sweeps out a cube as it moves. The same is true even for the point; moving it in any direction a unit distance causes it to sweep out a line segment. The sequence is clear: a point moves to form a line segment; the segment moves to form a square; the square moves to form a cube. The natural conclusion, then, is that if we move a cube in a direction perpendicular to itself, the cube should sweep out a hypercube.
The difficulty is in finding this perpendicular direction. In our three-space, there is no such direction. It is in imagining such a direction that we take our first real step into the fourth dimension. Even though we don't have a physical example to point to, we can still try to conceptualize the properties such a collection of directions would have: there would be four directions, each perpendicular to all the others. In this space, we could construct a hypercube. Since we don't really have such a place, our mental pictures of it always will be incomplete or distorted, but what would be required for a four-dimensional space should be fairly clear.
We can use the ``(n-1)-cube moving through time'' mechanism to help count the parts that make up the n-cube. For example, suppose we want to count the number of edges of the square; then we can first look at how these are formed by the moving line segment as it sweeps out the square. One edge (the bottom edge, say) is the initial segment just before it starts moving. The opposite edge (the top) is the position of the segment when it stops moving. The other two edges are formed by the endpoints of the segment as it moves: each endpoint sweeps out a new segment (a 0-cube in motion sweeps out a 1-cube). This gives two copies of the original segment (one at the top and one at the bottom) and two new segments (one for each of the two endpoints of the moving segment) for a total of four edges.
Lets do the same thing for the square faces of the cube (formed by a square in motion). One of these (the bottom) is the original square before it moves, and one (the top) is the square when it stops moving. The remaining faces are formed by the edges of the moving square, each edge sweeping out a new square face for the cube (a 1-cube in motion sweeps out a 2-cube). This means the cube has two squares (top and bottom) plus four more squares (one for each edge of the original moving square), for a total of six.
The other parts of the n-cube also can be counted in this way. For example, it is easy to count the number of corners in an n-cube by this means. There are twice as many as in the (n-1)-cube: the corners are the corners of the base (n-1)-cube before it moves plus the corners of the (n-1)-cube when it is at the top. This means that the point has 1 corner (itself), the segment has 2, the square has 4, and the cube has 8. In general, the n-cube has 2n corners. This means that the 4-cube, or hypercube, must have 24=16 corners. This is the first concrete information we have obtained about the hypercube!
In general, if we want to know the number of m-cubes in an n-cube, we get twice the number of m-cubes in an (n-1)-cube (the ones that are at the top and bottom), plus the number of (m-1)-cubes in the (n-1)-cube (each sweeps out an m-cube as the (n-1)-cube moves from the bottom to the top). This means that if we know the number of parts in a cube of a particular dimension, we can determine the number of parts in the cube that is one dimension higher. Carrying out this procedure generates the following table:
| n | m=0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 2 | 1 | 0 | 0 | 0 |
| 2 | 4 | 4 | 1 | 0 | 0 |
| 3 | 8 | 12 | 6 | 1 | 0 |
| 4 | 16 | 32 | 24 | 8 | 1 |
The bottom row represents the number of parts in the 4-cube. We see that there are 16 corners, 32 edges, 24 squares and 8 cubes in the hypercube! If we followed the sweeping cube carefully, we could even tell which of these were connected to which other ones. This gives us the combinatorial structure of the hypercube. (As an aside, we could continue to extend this table, and count the parts of the 5-cube, 6-cube, or any higher-dimensional cube we want.)
Of course, we really want to see the hypercube; how can we do that? Again, we move to the lower-dimensional case, and ask how could we get a two-dimensional person to ``see'' a 3-cube? If we imagine a world of two-dimensional people living in a plane, what methods could they use to understand a cube using only their two-dimensional world and things within it?
There are several approaches they could use, with our help. One method would be for us to shine a light on the three-dimensional cube and let it cast a shadow on their two-dimensional world. They could view this two-dimensional shadow and try to recognize it as a distorted or squashed view of the actual cube, and reconstruct the cube in their minds. This is exactly what we do when we look at a photograph, or a television screen. We see only a two-dimensional view of an object, but we are adept at building three-dimensional pictures out of this in our minds. Of course, we have three-dimensional experiences to help us with this, but our friends in the two-dimensional world would have a much harder time doing this.
Mathematically, this process is called projection, and is one of the key methods of viewing higher-dimensional objects. Projection is used throughout this exhibit; many of the images are actually three-dimensional ``shadows'' of the four-dimensional objects from which we must reconstruct the originals in hour minds.
Our minds are good at turning two-dimensional pictures into three-dimensional mental images. This process is made even easier and more concrete if we have a sequence of images that represent a series of views of a three-dimensional object in motion. For example, the television or a movie in a theater are quite effective in conveying a three-dimensional impression. This same idea is used in the movie sequences and some of the still images in this show to help us better understand the four-dimensional objects whose shadows we see.
Projections, and sequences of projections, form one important method of understanding higher dimensions. Returning to our friends in the two-dimensional world, we can develop other mechanisms to help them visualize three-dimensional objects. One would be to pass the object through their two-dimensional world. If they could see the portion of the object that intersects with their plane of existence, then they would see a series of ``slices'' of the three-dimensional object. This ``time sequence'' of shapes could be used to reconstruct the full three-dimensional shape in their minds. This is very much like how a doctor uses a CAT scan, which is a series of two-dimensional pictures, to develop a three-dimensional image of the human body being scanned.
Similarly, we can be called on to reconstruct four-dimensional objects in hour minds when we are presented with a sequence of three-dimensional slices of the object. Several images in this exhibit investigate the idea of slicing as a means of understanding a higher-dimensional object.
In conclusion, one's ability to visualize four-dimensional objects rests largely on a good understanding of two- and three-dimensions. Not only can these dimensions provide valuable information about higher-dimensional phenomena, but they also give us the power to produce beautiful images of them.