This movie shows the analogous sequence of threedimensional shadows of a
hypercube folding up in fourspace. Just as we can visualize the cube
folding in space using just its shadow (as is done at the end of the
previous movie), we must use these thredimensional shadows to try to
imagine the hypercube folding up in four dimension.
We begin with eight cubes forming a crosslike shape. Some faces are
partially removed to make the interior structure easier to see. The
central (yellow) cube will be the bottom of the hypercube, and the purple
one will be the top. The remaining six cubes form the faces of the
hypercube that join the bottom to the top. As these begin to fold up in the
fourth dimension, we see their shadows become distorted in three dimensions
(as one face of the cubes moves closer to the light source, its shadow
get larger). Eventually, the faces of the cubes come together and are
joined, just as the edges of the squares that form a cube are glued when
they are folded together. This leaves just the top remaining to fold into
place.
When the top folds over, it gets larger (closer to the light source), and
eventually seems to turn inside out (we go from seeing one side of it to
seeing the other). As it closes in to join the six other faces, we are
left with the wellknown "cube within a cube" view of the hypercube in
perspective. The small yellow cube is farthest from the light source, while
the large purple one is closest. The remaining six faces appear as
truncated pyramids joining these two; these are views of cubes in
(fourdimensional) perspective.
