

588K 

This movie shows the orthographic view of a hypercube looking directly at a
corner of the hypercube (i.e., along its long diagonal, see the movies of projections of cubes for more about
the different views of the hypercube). In this movie, the hypercube is
being sliced parallel to one of its cubical faces. The initial slice is
one complete cube, and then this slice seems to translate across the
projection to the opposite cube. During the intermediate stages, each face of
the slice comes from one of the six cubes between the "top" and "bottom"
cubes in the hypercube. During the slicing sequence, these faces each
sweep out one of these six cubes.



548K 

This movie shows the orthographic view of a hypercube looking directly at a
corner of the hypercube (i.e., along its long diagonal) while the hypercube
is being sliced parallel to one of its square faces. The initial slice is
simply a square. As the slice progresses, it widens into a rectangular
box, and continues to widen until, at the halfway point, it is as long as
the diagonal of one of the faces of the cube. At this point, four of the
sides of the box have grown from an edge through a series of rectangles to
the widest rectangle possible within a cube as these sides sweep out
cubical faces of the hypercube. Note that the square sides of the
rectangular box have each sweep out one complete cube of the hypercube and
are about to enter and sweep out another set of cubes. As the slice
continues, the rectangular box shrinks again to a single face. Its
rectangular sides have each swept out one of the cubical faces of the
hypercube, while its square sides have swept out two each; so all eight
cubes can be located within the sequence.



692K 

This movie shows the orthographic view of a hypercube looking directly at a
corner of the hypercube (i.e., along its long diagonal) while the hypercube
is being sliced parallel to one of its edges. The initial slice is simply
an edge, but since this edge is a member of three cubical faces of the
hypercube, as the slice progresses it becomes a triangular prism. Its
three rectangular sides are sweeping out the three cubes containing the
initial edge. Each vertex of the hypercube is a member of four cubes, so at
each endpoint of the initial edge there is one additional cube; the
triangular faces come from these cubes as they are sliced corner first. As
the slice continues, the prism grows until the top and bottom triangles are
one third of the way through their respective cubes. At this point in the
slicing sequence for a cube, the triangle's corners become cut off and the
triangle becomes a hexagon. So the slice of the hypercube becomes a
hexagonal prism. The original three rectangular sides of the prism have
grown to their widest size as they slice through their respective cubes,
and then start to shrink in width. At this point, the slice begins to enter
three additional cubes (edge first) so three mores sides show up
forming the hexagonal prism. These sides grow larger as the original ones
grow smaller, and the prism again becomes triangular as the slice passes
thorough three more of the vertices of the hypercube. Finally, the prism
shrinks back to an edge opposite the original one.



700K 

This movie shows the orthographic view of a hypercube looking directly at a
corner of the hypercube (i.e., along its long diagonal) while the hypercube
is being sliced starting at a corner. The slice initially cuts off the
corner at one end of the long diagonal, so the slice begins as a
tetrahedron at the center of the figure. Each face of the tetrahedron is a
slice of one of the corners of one of the four cubes that meet at this
vertex. As the slice progresses, these faces grow and eventually pass
through four of the vertices of the hypercube. At this point, the corners
of the tetrahedron begin to be cut off as the slice starts to pass through
the other four cubes in the hypercube. Meanwhile the original triangular
faces are beginning to form hexagons as they slice farther through the
original four cubes. At a point halfway through these cubes (or onethird
of the way through the complete hypercube), these four cubes are sliced by
regular hexagons, and the complete slice forms the Archimedean solid known
as the truncated tetrahedron. It has four regular hexagonal faces and four
equilateral triangular faces. Further along, the hexagonal faces become
more triangular again, and the small triangular faces grow larger.
Halfway through the slicing sequence for the hypercube, all eight faces
are equilateral triangles forming a Platonic sold, the octahedron. At this
point, all eight of the cubes in the hypercube are sliced in exactly the
same way. As the slice moves further, the sequence repeats itself, but in
reverse; it passes through the truncated tetrahedron, then the tetrahedron,
which again shrinks to a vertex. This corner of the hypercube also appears
to be at the center of the projection; since we are looking along the long
diagonal, both the closest and farthest points on the hypercube are at the
same location in the projection. Note that the final tetrahedron is
inverted with respect to the one at the beginning of the slicing sequence.
