Background for Further Reading

The exploitation of pictorial representations in mathematical problems is attracting new interest, as described in the main body of the paper. The reader who is interested in more general literature exhibiting the development of this subject area over the last 15 years is invited to consult other books and collections on the subject. Among these, the authors particularly recommend the following:

1. T.F. Banchoff, Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions, Scientific American Library, New York, 1990. This is general book, accessible to people with a moderate level of mathematical interest; the graphics are excellent, and the exposition very readable.

2. D.W. Brisson, Ed., Hypergraphics: Visualizing Complex Relationships in Art, Science and Technology, AAAS Selected Symposium 24, Westview Press, 1978. This collection contains many of the early seeds of the current work in visualizing geometry; this contains a considerable amount of mathematics as well as graphics.

3. Gerd Fischer, Mathematische Modelle/ Mathematical Models, Vols. I and II, Friedr. Vieweg & Sohn, Braunschweig/Wiesbaden, 1986. This book includes an exhaustive survey of classical models of mathematical shapes. It is worth noting that perhaps the most significant change in capability enabled by computer graphics is the new ability to animate models such as those in Fischer's book in response to a user's actions.

4. G.K. Francis, A Topological Picturebook, Springer-Verlag, New York, 1987. This book is primarily a mathematical survey that phrases its material in terms of ``descriptive topology'' with the goal of resurrecting our nineteenth century fascination with mathematical pictures.

5. J.R. Weeks, The Shape of Space, Marcel Dekker, New York, 1985. Weeks' brief book is a gem of clarity and mathematical insight, and yet is sufficiently complete that it has been used as the basis for courses on topology for secondary school teachers. Weeks also has developed an advanced computer program SnapPea, for creating and studying hyperbolic 3-manifolds, available by anonymous ftp from geom.umn.edu.

Finally, we remind the reader of the classic Geometry and the Imagination, by Hilbert and Cohn-Vossen [6], which has served to inspire generations of professional and amateur mathematicians.