Mathematical visualization is the art of creating a tangible experience with abstract mathematical objects and concepts. While this process has been a cornerstone of the mathematical reasoning process since the times of the ancient geometers, the advent of high-performance interactive computer graphics systems has opened a new era whose ultimate significance can only be imagined.
Typical geometric problems of interest to mathematical visualization applications involve both static structures, such as real or complex manifolds, and changing structures requiring animation. In practice, the emphasis is on manifolds of dimension two or three embedded in three or four-dimensional spaces due to the practical limitations of holistic human spatial perception - it is extremely challenging to construct intuitively useful images of anything more complicated! General approaches to visualizing -dimensional spaces are at best piecemeal, so that algebraic manipulations often remain our most powerful tool for high dimensions. Nevertheless, despite the apparent limitations of visual representations, their utility is far from being completely exploited; we may still gain significant intuitive value by pushing our visual understanding of relatively simple geometric objects as far as our imagination can take us.
Our goal is to show the nature of the interrelationship between mathematics and computer science, especially computer graphics. In this article, we adopt for the most part a computer scientist's perspective on the progress, techniques, and prospects of mathematical visualization, emphasizing those areas of 3D and 4D geometry where interactive paradigms are of growing importance. Without demanding detailed mathematical expertise of the reader, we present a selection of the domains with which we are familiar, describe some of the critical visualization problems involved, and discuss how various researchers have approached the solution of these problems.
The article begins with some general background and then turns its attention to some of the visualization methods that have been used to bring computer graphics technology to bear on mathematical problems of low-dimensional topology and geometry. The concluding sections discuss the system design philosophies of various research groups and prospects for the future. Examples of computer-generated images are supplied throughout, and separate sidebars are devoted to a brief glossary, sources of additional background information on visualizable mathematics, and an overview of selected film and video animations concerned with mathematical visualization.