Projecting the Hypercube onto 2 Dimensions

One way to visualize the hypercube is to look at a projection of its vertices into two space.

As usual, we begin by looking in 3 dimensions where we feel comfortable: we often depict the usual 3D cube in 2 dimensions by drawing the original square, the copy of it that's drawn forth or pushed back, and the edges connecting the two:

Since we create the hypercube similarly, by taking one cube, and moving a copy of it in a direction perpendicular to all three dimensions so that the copy ends up parallel to the original, we depict it similarly as well

Start out with two cubes, the original in red and the copy

in blue. Move the copy in a direction perpendicular to all of the faces. (This part, we can't see, of course, just as in two dimensions we can't really put one square in front of the other.)

Next, as we've discussed before, we connect each vertex of the original cube to the corresponding vertex of the copy. First we show that in green, so that you can see the two cubes still standing out, then we show the whole thing in black so you can see the lovely symmetry of it all (and see where the picture in your reading came from).

We've discussed that in addition to the original cube and the copy, which are in a sense the "top" and the "bottom" of the hypercube, when we connected each face of the original to the corresponding face of the copy, the six "side" cubes were formed. Let's see the projections of each of those six cubes:

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