The graphics computer is but the latest in a series of inventions that have enabled us to see in previously inaccessible directions. Four hundred years ago Galileo Galilei turned his newly constructed telescope toward Jupiter and saw moons, an unthinkable vision in a society convinced that all heavenly bodies revolved around the Earth. Now the descendants of Galileo's little optical device reveal the presence of quasars billions of light years away.

A century after Galileo, Anton van Leeuwenhoek's microscope allowed him to explore undreamed-of worlds of very small animals and plants literally within our own blood and tears. Today's powerful electron microscopes show us objects smaller by orders of magnitude, including the structure of genetic material.

Early in this century, Wilhelm Könrad Roentgen's discovery of X rays reveald the solid skeleton inside our bodies and gave evidence of the state of the organs functioning there. How much more powerful are today's CAT scans and magnetic resonance imaging, which literally expose to our view slices of our bodies.

These dramatic excursions into the almost unimaginably far away, the equally unfathomably small, and our hidden insides are evidence of our ability to go beyond ourselves, to see the previously unseeable. Equally dramatic is a present-day revolution in our ability to visualize phenomena in other dimensions.

A smoothly shaded and illuminated computer-generated image of a Klein bottle, a twisted surface that cannot be built in ordinary space without intersecting itself, but that can be constructed without self-intersection in four-dimensional space.

Thanks to striking developments in computer graphics, it is possible for us to have direct visual experience of objects that exist only in higher dimensions. As we watch images moving on the screen of a graphics computer, we are faced with challenges like those of the first scientists to make use of telescopes or microscopes or X rays. We are seeing things now that have never been before, and we are just learning how to interpret these images. It is literally true that we are in the first stages of a new era when it comes to visualizing dimensions.

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