MAM2000 (Essays/SciAm)

The fourth dimension has been a vehicle for physical and metaphysical speculation at least since the 19th century. The idea of a fourth, physical dimension culminated in Einstein's theories of special and general relativity; space and time together make up a four-dimensional continuum in which all real events are timelessly frozen. This view of the universe may be undergoing dimensional modifications; the so-called Kaluza-Klein theories introduce seven or more new dimensions in the form of miniature hyperbubbles attached to every point of spacetime [see "The Hidden Dimensions of Spacetime," by Daniel Z. Freedman and Peter van Nieuwenhuizen; Scientific American, March, 1985].

The fourth dimension that I have come to know and love is the child of mathematics. Readers in ordinary rooms have a three-dimensional coordinate system suspended overhead. Three walls meet in each corner of the room, and from that corner radiate three lines, each of which is the meeting place of a pair of walls. Each line is perpendicular to the other two lines. Can the reader imagine a fourth line that is perpendicular to all three lines? Probably not, but that is what mathematicians require in setting up the purely mental construct called four-dimensional space. You now have the chance to explore this space in a personal way and without danger to your person. You have only to write the program I call HYPERCUBE.

HYPERCUBE can trace its origins to a film produced in the mid-1960's by A. Michael Noll, then at Bell Laboratories, that depicts the two-dimensional shadows of four-dimensional objects moving in four-dimensional hyperspace. The program as it now stands, however, was developed by Thomas Banchoff and his colleagues in the Computer Graphics Laboratory at Brown University, and my inspiration for this column comes from the fascinating images it generates [see illustrations on pages 19, 21 and 22]. Banchoff, who is a professor of mathematics, directs the visual exploration of higher-dimensional surfaces and spaces as a complement to his writing and research as a geometer. In 1978 he and Charles Strauss produced a 9 1/2-minute computer-generated color film that has since become a classic in the mathematical underground: The Hypercube — Projections and Slicing. (The film can be obtained from the International Film Bureau, Inc., 332 South Michigan Avenue, Chicago, 111. 60604.) Banchoff is also probably the leading expert on the life and work of Edwin A. Abbott, the English clergyman and teacher who in 1884 wrote Flatland a tale of imagined life in two dimensions.