Mathematical visualization, like other areas in the rapidly developing field of visualization science, is still defining itself. We have presented examples of interactive workstation systems for visualizing pure geometry as well as noting the importance of these systems for developing precomputed animations (see Sidebar B). Many of these techniques can be adapted to the emerging virtual reality medium as interactive performance continues to improve.
Furthermore, there are clearly many areas of development that are appropriate for the participation of computer scientists with skills in interactive interface design, computer graphics, efficient algorithms, and perhaps data management. One can also conclude from looking at the images we have presented that graphic arts and design skills also have a unique role to play in improving the quality of graphical communication. Yet clearly the active participation of the end-users, the mathematicians themselves, is just as critical as in other visualization problems, and possibly more so, since the subject material is so complex that research into visualization merges rapidly with the mathematics itself. Our purpose here has been to provide a look at some major developments and trends in this fascinating field, and to try to build a bridge between the computer science and mathematics cultures by summarizing problems of mutual interest, hopefully with the result of generating more activity in this style of research in both communities.