Let’s continue our discussion about cubes from last week, Part 5.
Just as the 3-D cube can be generated with a square by sweeping it out of the plane into the third dimension, the projected 3-D cube shown in Figure 1 (from Part 4) can also be generated by sliding it sidewise in the plane. That is how, in fact, the cubes in Figure 1 and Figure 4 (from Part 5) were drawn. Notice that each of the cubes are formed out of the lower-dimensional cubes. For example, the line has two points, the square has four lines, and the cube has six squares. By the way, the six squares in the cube are also visible in its unfolded version shown in Figure 3 (in Part 5).
Now, the reason for going through all of this geometry is that it allows us to make pictures of higher-dimensional cubes. The generating process displayed in Figure 4 can be used to create the projection of, say, a 4-D cube, in two dimensions. How is this done? Simply slide the 3-D cube over. Its beginning position defines the first set of sides and endpoints (or vertices); then as it slides, each vertex sweeps out a line; and finally, the end position defines the last set of sides and vertices. In order to make such a drawing appear less confusing, the sliding is usually done by expanding the original 3-D cube in size rather than shifting it to one side. The result is essentially the same and can be seen in Figure 5.
This 4-D cube projected into two dimensions can be described as a cube within a cube, where corresponding vertices are connected. And the colors in the diagram make this connection more apparent. But in reality, we can extend the observation we made above about the lower-dimensional cubes and see that this cube, commonly called a 4-D hypercube, is constructed from eight 3-D cubes: namely the two already noted plus the six filling the space between them. At first glance, the latter six don’t appear as cubes because the angles are not right angles. But that is only because they have been projected into three dimensions. Their appearance is analogous to four of the six squares that make up the 3-D cube in Figure 1. That cube does not have right angles when the cube is projected into two dimensions.
Going back to Figure 5, the eight cubes making up the 4-D cube also can be seen in their unfolded version, shown at the right. The house described in the science fiction story in Part 1 was built in this shape: an unfolded 4-D cube. After it collapsed it appeared as a single cube. The folding took place in the fourth dimension. The reason it didn’t look like the object on the left is because it wasn’t projected into three dimensions; it simply intersected the 3-D space.
Figure 5: A 4-D “cube” projected into 2-D, and unfolded into 3-D
If I haven’t lost you in another dimension, I intend to wrap up this series next week.
|Part 1 | Part 2 | Part 3 | Part 4 | Part 5 | Part 6 | Part 7|