The Hypercube: Projections and Slicing
A frame from the film "The hypercube: projections and slices", and a
modern version from the artwork "Iced cubes".
This film treats the convex hull of the sixteen points (±1, ±1, ±1, ±1) in 4-space, first
by orthogonal projection then by central projection from 4-space to
3-space. In each case we rotate in the coordinate planes xy,
yu, xw, yw, and zw, ending at the original
position. We then slice each figure by hyperplanes perpendicular to the
vectors (1,0,0,0) then (1,1,0,0) then (1,1,1,0) and finally (1,1,1,1). For
a more thorough description of this film, see Banchoff .
In these days, programming images of hypercubes is a beginning exercise in
introductory courses in computer graphics. Instead of demanding a
workstation, it is possible to realize scenes on a laptop computer. Even
then the topic had been treated by several researchers, most notably
A. Michael Noll at Bell Laboratories . Our main contribution was a
scripted tour of the 4-dimensional cube with three movements, orthographic
projections, central projections, and slicing by planes and hyperplanes.
After a quarter of a century, this film, now available in video, is still
in demand, especially in schools and colleges. For modern interactive
versions of the object, see for example the two cube sequences in the
interactive art exhibit site "Para Além da Terceira
Dimensão'' . The reference  is a short description in an
article where the illustrations came from Polaroid pictures taken directly
off the computer screen!