Hypercube Slices Viewed Corner First
The movies below represent the four slicing sequences of the hypercube (cube first, square first, edge first, and corner first) that you already know from the Projections of Hypercube Slices section of our notes. In those movies, the projection of the hypercube corresponded to the slicing direction; so the slicing sequence for the hypercube square first was projected from a viewpoint that was also square first. This corresponded to the views of the cube being sliced in A Cube Falls Through Flatland. We also observed, however, that these slicing sequences can be viewed from many different directions, and we saw several of them in Projections of Sliced Cubes, both stereographic and orthographic.
As with the cube, we can view the slices of the hypercube in any of its projections. The projection that is looking directly at a corner is one of the most useful ones, at all the cubes are easily visible in this view (at least one pair of cubes is flattened out in all the other projections). The movies below show the four slicing sequences in this projection, and they correspond to the movies of the slices of the cube when viewed corner first.
In the first movie, we see the initial slice as one of the cubes, and it seems to move across the cube to the opposite cube. This is the "cube sweeping out a hypercube" idea that we first used to introduce the hypercube. The second movie shows the square-first sequence, an in it you can see the rectagular box grow and shrink, but instead of appearing to stay in place, as it did in the square-first projection, it now travels across the hypercube, since our viewpoint no longer flattens out any of its cubes. The third movie shows the edge-first sequence, and since the top and bottom cubes are now opened up, we see the prisms move upward as the slices pass through these cubes. Finally, the fourth sequence is the same one you know from before, since that was already in the corner-first projection.