Illuminating the fourth dimension.

A photograph in a magazine or an image on a computer screen is inherently two-dimensional. Nonetheless, such representations often contain visual cues that help create an illusion of three dimensions. For example, by taking into account the size, relative placement, and shading of familiar objects in a scene, one can get a remarkably complete three-dimensional picture of a two-dimensional view.

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Analogous visual cues may also enable viewers to "see" four-dimensional objects in three-dimensional settings, says computer scientist Andrew J. Hanson of Indiana University at Bloomington. Hanson and graduate student Robert A. Cross have developed a scheme for viewing a knotted sphere -- the four-dimensional analog of a knotted string. In this case, the fourth dimension is simply an additional spatial coordinate.

Suppose one were creating a two-dimensional picture of a knot in a string. As a mathematical object, the knotted string would be invisible because it has no thickness. In drawing a mathematical knot, one is forced to thicken the line to make it visible. Moreover, because the knot itself is a three-dimensional object, one must use shading or breaks in the drawn line to show which parts cross over or under each other.

Visualization experts face similar problems in creating an image of a knotted sphere. The four-dimensional object's infinitely thin, three-dimensional shell must be thickened and crossings suggested by suitable markings. It's also possible to go a step further by making the shell shiny. Illuminated by a single light source and seen from different angles, various points on the shell reflect light in particular directions.

"The direction of the light tells us an amazing amount -- much more than a [knot] crossing diagram," Hanson says. "It gives us additional information about where the object is facing in the fourth dimension."

After developing techniques that allow the generation of such three-dimensional images in fractions of a second, Hanson and his coworkers displayed their knotted sphere in the Cave Automatic Virtual Environment (CAVE) at the National Center for Supercomputing Applications, located at the University of Illinois at Urbana-Champaign. Here, the viewer can don stereo glasses to interact with three-dimensional graphical images projected into the CAVE room.

But the knotted sphere visualization didn't work very well in the CAVE. "It's hard to see the thing unless you have some additional human context -- like shadows and familiar objects," Hanson says. Cross has now developed computer techniques for rapidly creating graphic representations of rich, three-dimensional environments to serve as nonthreatening settings for exploring the strange, unfamiliar realm of four-dimensional objects.