This movie shows the cube passing through a plane face first. On the left
is a view of the cube in perspective; on the right is a view from directly
above which represents what a two-dimensional person viewing the cube from
within the plane would be able to preceive. The portion of the cube that
is above the plane is in black, while the part that is below is in grey.
The slices are always squares, and so our two-dimensional person would see
"a square existing for a while". In this case, time is being used
at the third dimension.
This movie shows the cube passing through a plane edge first. Here, the
slice begins as an edge, then becomes a rectangle; the rectangle grows,
becomes a square for a moment, and then gets wider than it is tall. At its
widest, it is as wide as the diagonal of one of the square faces of the
cube. The rectangle then shrinks back to an edge at the top of the cube.
The most interesting slicing sequence for the cube is when it passes though
a plane corner first. In this case, the initial contact is a point, which
then becomes a small equilateral triangle. This triangle grows until it
touches three of the corners of the cube (the three edges are sweeping out
three of the faces of the cube, and are now half-way through each of these
faces). At this point, the corners of the triangles begin to be cut off by
the other three faces of the cube. The slice then becomes hexagonal, and at
the half-way point, the slice is a regular hexagon, with the slice cutting
each of the six faces of the cube in exactly the same way. As the cube
progresses through the plane, the slice turns again into a cut-off triangle
(but inverted with respect to the original one) and finally becomes an
equilateral triangle once again as three more vertices pass through
the plane. This triangle shrinks down to a point and disappears.