

215K 

This movie shows a cube rotating in three dimensions. You should
think about this picture in two ways: first, as a thredimensional object
rotating in three dimensions; and second, as a twodimensional shadow of
a rotating threedimensional object. The connection between the two is
important: how the twodimensional image tells you about the
threedimensional obejct is what you need to try to grasp.
Note how the large red square is closer to you in threedimensions, and the
blue one which seems smaller in the shadow, is entirely within the red
one. As the cube rotates in three dimensions, we see the blue square
shift to the left and distort; then it puses through the side of the red
square, evetually pulling entirely through. As the cube rotates further,
the blue square becomes the side face of the cube, and we see it as a
trapezoid in two dimensions. You can fallow the modtino of the cube
further as the blue square moves to the front, and then to the other side,
and finally back to its starting location, noting the changes in shapes and
positions in the twodimensional shadow.



354K 

This movie is analogous to the previous one in that it shows the
threedimensional "shadow" of hypercube rotating in fourspace. To begin
with, we see a large red cube containing a smaller blue one. The red cube
is closer to the light source, so its shadow appears larger. As the
hypercube rotates, we see the blue cube move to the left and distort, just
as the saw the blue square do in the previous movie. Eventually, the blue
cube moves throught he side of the red cube, and becomes one of the side
subces of the hypercube. It looks to us like a truncated pyramid, but it
is actually a regular cube in four space. We just see it distorted
because one face of it is closer to the light source, so its shadow is
larger than that of the opposite side.
As the hypercube rotates further, we can see the blue cube move with it
and eventually become the larger "front" cube of the hypercube.
