Front View of Cube Slices
This movie shows stereographic and orthographic views of a cube seen face first while the cube is being sliced parallel to one of its faces. The first sequence shows slices parallel to the left and right faces. In the orthographic view, since our line of sight is parallel to these faces, the projected slices appear as line segments. We see these segments sweep out the square, and this represents the way that A Square used to explain himself to the King of Lineland, so the orthographic view appears as a lower-dimensional slicing sequence in this case. The second sequence slices the cube parallel to the face we are looking at: the slices start at the back face and move toward us. We can see this progression in the perspective view, but in the orthographic view, we see "a square for a while". This was our initial method of explaining a cube to the Flatlanders.
This movie shows the stereographic and orthographic view of a cube face first while the cube is being sliced starting at an edge. We start at the lower left-hand edge, and the slice begins as a thin rectangle, widens to a square and then widens still further to a rectangle that is as wide as a diagonal of the square face of the cube. Then it shrinks back down to an edge as the slice progresses through the last half of the cube. In the orthographic view, each slice appears as a line segment, since our view is parallel to the slices. This shows the slicing sequence of a square that is being cut corner first, a process that could be understood by the King of Lineland. The view is essientially the same when we slice from any of the four edges connecting the front face to the back face.
If we start at the back left edge and end at the front right edge, we get a different view of the slicing sequence. Even though the half-way point is a rectangle with width equal to the diagonal of one of the square faces, it appears as a square in the orthographic projection since it is slanted in comparison to our viewpoint. Starting at any of the edges that form the square in the projection of the cube will give this same sequence.
This movie shows the stereographic and orthographic views of a cube face first while the cube is being sliced starting at a corner. The sequence from any corner looks essentially the same, so there is only one progression shown. Here we see the slice start as a triangle, grow to a larger triangle, and when this hits the three corners of the cube, it becomes a truncated triangle. Half way through, the slice is a perfect regular hexagon (though it appears irregular in this projection). After this the sequence reverses itself, and the hexagon becomes a truncated triangle, then an equilateral triangle, and finally shrinks to a point.