Stereographic Projections of Hypercube Slices

The movies in Projections of Hypercube Slices show orthographic views of the hypercube being sliced in various ways. One of the drawbacks to the orthographic view is that some of the cubical faces of the hypercube are flattened out and hard to see. (This is also one of its advantages, in that the 2D views of the 3D cube being sliced appear within these projections.) The movies below represent the slicing sequences in stereographic projection instead, so that every cube can be seen clearly in each view. You should compare these to the stereographic views available for the 3D cube at Projections of Sliced Cubes.

In the first movie, the hypercube is sliced cube first. The orthographic view showed a cube unchanging, since the far cube and close cube are both the same size in orthographic projection. Here, however, we see the smaller cube as the first slice, and it grows as the slice moves along the "vertical" edges that connect the farther (smaller) cube to the closer (larger) one.

The second movie shows the sequence of slices for the square-first sequence. The projection here looks like one of the views we saw in the rotations of the hypercube movies; it is the symmetric view at frame 16, but rotated in three-space slightly around the vertical axis in the image. The slice first hits the farthest away square (the small one at the center), and travels along the edges from there as it forms the larger and larger rectangular boxes. You should be able to see how the faces sweep out the various cubical faces of the hypercube. Can you locate all eight of these cubes in the dark-blue image of the hypercube?

The third movie shows the edge-first slicing sequence. Here, the farthest edge is shorter than the closer one (just as it is in the stereographic view of the 3D cube edge first), so we see a small triangular prism growing as the slice progresses. Notice how the top and bottom of the prism sweep out the top and bottom cubes of the hypercube (which are no longer flattened out, as they were in the orthographic view).

The stereographic view of corner-first is only slightly different from the orthographic view, since all eight cubes are already fully visible even in the orthograohic projection. The only difference is that four of the corners are slightly closer to the center and four slightly farther than the other six vertices (this is like the stereographic view of the cube corner first). Since there is nothing new to see, this movie is omitted.

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