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.
Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area. |