Beyond the Third Dimension
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An Introduction to "Surfaces Beyond the Third Dimension"

What is the best way to display a variety of surfaces in such a way as to encourage many people to interact with them? Stage an exhibit. In March of 1996, the Providence Art Club, one of the oldest such clubs in the country, hosted the show ``Surfaces Beyond the Third Dimension'' in their Dodge House Gallery. The first incarnation of that exhibition was for two weeks, and the gallery book includes the signatures of dozens of visitors, including artists, students, and mathematicians. The physical exhibit has long since been dismantled, yet the show lives on as a virtual experience and we still receive comments in the on-line guest book.

The Dodge House Gallery has a square base and one interior partition, providing well-lighted wall space for twelve large photographic reproductions of computer graphics images as well as an alcove for display of a continuous videotape featuring two three-minute videos. There was a guidebook that gave information about the nature of the objects, including technical descriptions of the software and hardware used in the design of the objects and the Ilfochrome process used in their reproduction. Additional pages gave mathematical descriptions of the various pieces, as well as references to places where they had appeared either in research articles or as illustrations in books and journals. There was also a well-attended afternoon gallery talk, describing the origins of the project and including a guided tour of the exhibit. All of these aspects of the physical exhibit are enhanced in the virtual counterpart. In a certain sense, the on-line version contains much more than the original. To what extent does it capture and augment the experience of those who visited the actual gallery opening, and came back to see and respond to the images on the walls? There are many questions raised by this means of portraying mathematical art and design, and we will address some of them now.

What is it that we are showing? Most of the images included were originally studied as abstract geometric constructions given by parametric surfaces in three- and four-dimensional space. In some cases, there is an elaborate theory behind the illustration, whereas in other cases, the phenomena are not yet well understood. In several instances, the display itself represents an innovation not only in the method of displaying a surface but in representing its mathematical properties in a way that suggests new results. Visitors to the virtual exhibit can read about the mathematics behind any of the images via a link from the main page for the image.

It is clear that this electronic gallery faithfully reproduces a great many of the aspects of the actual exhibit, while altering the experience in other ways. In some cases, the electronic version loses information, while in others it provides the opportunity for significant enhancement, particularly in satisfying the viewer's curiosity about the different parts of the mathematics and computer science that made it possible for us to produce these works. We are grateful for the opportunity to present our work in a way that will continue long after the physical exhibit has given way to the work of other artists, and we look forward to further responses from visitors to our virtual gallery.