 


137K 

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 cubical faces. The initial slice is
one complete cube, and then this slice seems to translate across the
projection to the opposite. 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.




138K 

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, and since this square is part of two cubes in the
hypercube, it is colored with two different colors. 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, the yellow face at the front of the slice has
grown fron an edge through a series of rectangles to the widest rectangle
possible within a cube as it sweeps out one of the cubical faces of the
hypercube (the one toward the front of the projection). Note at this
point 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; thus they change colors as they move from one cube to the
next. As the slice continues, the rectangular box shrinks again to a
single face. Its rectagular faces have each sweept out one of the cubical
faces of the hypercube, while its quare faces have sweept out two each; so
all eight cubes can be located within the sequence.




128K 

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. The
three rectangular faces are sweeping out the three cubes contining the
initial edge. Each vertex of the hypercube is a member of four cubes, so
at each endpoint of the initial line segment there is one additional cube;
the triangular faces come from these cubes are 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 faces 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 additional faces show
up forming the hexagonal prism. These faces grow larger as the original
ones grow smaller, and the prism again becomes triangular as the fslice
passes thorough three more of the vertices of the hypercube. Finally, the
prism shrinks back to an edge opposite the original edge.




143K 

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. In this case, the slice is
perpendicular to the long diagonal, so it begins as a tetrahedron at the
vertex at the center. 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
slive 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 begins to pass through the other four
cubes in the hypercube. Meanwhile the original triangular faces are
beginning to form hexagons as they slice 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 Achimedean 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.
