Beyond the Third Dimension
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Orthographic views of the cube and hypercube

This movie shows two projections of a cube: on the left, the view is in perspective, where parts that are farther away are smaller; on the the right is an orthographic view, where items are always the same size no matter how far away they are. The orthographic view is what you would see as a shadow cast from a light source that is infinitely far away (so that the light rays are parallel), while the perspective view comes from a light source that are finitely far away, so that the light rays are diverging.

We begin with a view that you recognize as a view of a cube, and then rotate so that you are looking directly at a face of the cube. In the perspective view, you see a square within a square (the front face is a large square, and the back face is a smaller square); in the orthographic view, however, the front and back squares are the same size, and are one on top of the other, so you seem to see only one square. The remaining four sides appear as trapezoids in the perspective view; but, in the orthographic view, they are flattened out to simple line segments, since these sides are parallel to our line of sight.

As the movie continues, the cube rotates. The view on the left shows the rotation clearly, but we have to think harder about the rotation on the right. If you track the colors carefully, you can see which parts of the two-dimensional shadow correspond to which faces of the cube. This is the "revolving door" illusion, in which we seem to see squares moving past each other. We stop rotating at a view where we are looking directly at an edge of the cube; a picture we might never have considered to be a view of a cube.

Before moving to the next rotation, we back up just a bit to make it easier to see the different faces of the cube, and then rotate the cube in a different direction, and end up looking directly at the corner of the cube (along its long diagonal). In the orthographic view, we see a hexagon crossed by six lines meeting at the center. This, too, is an unexpected view of the cube. Note that both the closest and farthest corners appear at the center of the hexagon.

Now we shrink the colored edges down to simple, black edges, and follow the rotations again in reverse. Although the orthographic views seem harder to understand, they will help us to interpret the views of the hypercube in the movie below.


 

This movie shows an orthographic view of the hypercube, so it corresponds to the pictures of the cube on the right of the previous movie (perspective views are possible, but get quite complex and hard to interpret; see the movies of cubes rotating for some samples of perspective views of the hypercube rotating).

We begin with a view of the hypercube looking straight at one of its cubical faces. Since the view is orthographic, both the front and back cubes are the same size, and we appear to see only one cube. The remaining six faces are flattened out (they are parallel to our direction of view) and each appears as one of the six faces of the shadow cube. We can see this by noting the different colors on the different faces; each is a flattened cube.

As we begin rotating the hypercube in four-space, we see the analog of the "revolving door" illusion in four dimensions, where cubes seem to move past each other. Note that on four of the six faces of the cube, the flattened cubes are performing the same revolving door illusion that we saw in the movie above; we will often be able to locate such two-dimensional sequences within the three-dimensional one. We end at a view where we are looking directly at one of the square faces of the hypercube (the one at the center of the image; compare this to looking at an edge in the movie above).

To continue, we first back up a little, as before, and then rotate in a new direction. Two of the cubes that were flattened out become opened up, and we see them forming the hexagonal view of the cube what we saw in the previous movie. The remaining six cubes each have one face on the hexagon above and one in the hexagon below. One such cube with blue edges can be seen clearly in the front of this view. This is what we see if we look directly at an edge of the hypercube.

One more rotation can be used to open up the two remaining flattened cubes. We see the top (purple) cube open up as this rotation occurs. The final result is a beautifully symmetric view of the hypercube. This time we are looking directly at one of its corners (along its long diagonal).

To help see the three-dimensional shape that is this view of the hypercube, we rotate the shadow in three dimensions. As in the movie of the cube above, we now shrink the colored segments down to blue ones, and view the rotations in reverse. Our initial view is toward a corner, then we look toward an edge, then a square face, and finally directly at one of the hypercubes cubical faces, and end with the view where we started.


 

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