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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 [4].

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 [24]. 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'' [17]. The reference [4] is a short description in an article where the illustrations came from Polaroid pictures taken directly off the computer screen!

Additional Reading:
[Link] Além 3D: Iced Cubes
[Link] Além 3D: A Rotation of Cubes
[Link] Course notes for "Visualizing the Fourth Dimension" (Math 53, Union College)
[Link] Math Awareness Month 2000: Math Spans all Dimensions

[HOME] Thomas F. Banchoff's ICMS 2002 Poster
Created: 12 Aug 2002
Last modified: 13 Aug 2002 16:45:39
Comments to: thomas_banchoff@brown.edu
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