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Hypercubes
The most famous example of a higher dimensional object is the hypercube. Broadly speaking, the hypercube is to the cube as the cube is to the square. But, what does this mean? What can we determine about the geometrical properties of the hypercube?
The best way to grasp a picture of the hypercube is to work up to it dimension by dimension. We can start in zero dimensions and proceed onwards to produce a whole family of cubes. A point does not have any extension. It is the unique zero dimensional object. If we allow our point to sweep out a length of one unit, say along the x axis, we can produce a line segment. Next we can allow the line segment to sweep out a distance of one unit in a perpendicular direction, say one unit along the y axis. The figure that we will produce is a square. Next, our square can sweep out a distance of one unit in a direction perpendicular to the first two directions, say the direction along the z axis. The result will be a unit cube.
What happens next? We now seem to have run out of perpendicular directions in which we can continue this procedure. However, we can imagine that there are further perpendicular directions and envisage abstract higher dimensional space. There is no reason why we should restrict ourselves to the three dimensions of physical space; as long as we reason clearly and consistently, we can deduce the mathematical properties of higher dimensional objects, even if they cannot be physically realised in our three dimensional universe. We can simply declare that we are now working in four dimensional space and see where our exploration leads us. If we find ourselves in an interesting place, then that will be sufficient reward for our journey. Often the research of pure mathematicians is guided by the search for interesting abstract scenery rather than a quest for scientific utility. In our abstract four-dimensional space, there will be a fourth axis that is perpendicular to our other three axes. We can label this axis the w axis. Measuring the distance parallel to each of the four axes gives us four coordinates that will specify the position of any point within our four dimensional space.
With a fourth perpendicular direction that we might label the w axis, we can now take the cube and sweep it one unit along this direction. The object that we will generate is the four dimensional equivalent of the cube. It is the object of our dreams, our hypercube. So what does it look like? Is there any way that we can visualize it?
Our sections through a cube formed a sequence of squares. And similarly, our sections through the hypercube form a sequence of cubes. [We can imagine going down to two and one dimension to see how the analogy works in the case of the two-dimensional cube, which we usually call a square and the one-dimensional cube, which we usually call a line segment. (Our sequence of sections of a square are a sequence of line segments. Our sequence of sections of a line segment is a sequence of points.)]
What else can we infer from this description of a hypercube? If we draw the hypercube with an exaggerated perspective it will help us to understand the analogy with a cube. First we can draw a transparent cube face-on, so that we can see the back face within the front face and with edges connecting the corners of the front face to the corners of the back face. We can make a similar drawing of a transparent hypercube. If we position the hypercube in the same orientation as in our previous discussion, in our perspective drawing we will see a large cube, with a smaller cube within. Each of the corners of the outer cube are connected by edges to the corners of the inner cube. What we are looking at is the front cube (the outer cube) with the back cube, which is further away in four-dimensional space, appearing within it because its size appears diminished with distance. This is completely analogous to seeing the back face of the transparent cube within its front face.
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