Goals and Methods of Mathematical Visualization

If graphics is to contribute to mathematics, we must next ask: What are the principal tasks of mathematical visualization? Like other visualization domains, it must address several levels of problems:

• Information Content. Images, animations, and interactive systems involving displays of mathematical objects must possess intrinsically non-trivial information content and reveal this information in some way to the viewer. Since each viewer sees the world differently, the information display should also take into account the beliefs, assumptions, and perceptual preconceptions carried by the viewer.
• Teaching. To carry out effective research, one must master what has gone before. Worthwhile visualization techniques attempt to convey a useful and accurate knowledge of the subject material. Computer-assisted methods should enhance breadth, depth, and learning speed compared to standard teaching methods.
• Insight. Visual ways of representing objects that the viewer has only studied algebraically should offer previously unattained insights, help expose general principles, and should suggest fruitful conjectures.
• Results. The final objective is to prove interesting new results, thereby extending the corpus of known mathematics.