Many mathematicians have the feeling that using a computer is akin to cheating and say that computation is merely an excuse for not thinking harder.

By providing vivid images that suggest new questions, computers are helping to mend a rift that had developed between pre and applied mathematics. These changes are enriching a subject that many outsiders have regarded as a abstract, even useless, pursuit.

In general, the computer can be used in situations where physical experiments aren't possible. Computer graphics plays the same role for surfaces that soap films play for surface spanning a wire contour.

Although an explicit equation for a surface spanning a given contour is almost impossible, by making a much data more accurate than soap film, the visual exploration often furnishes clues that can be used later for nailing down the mathematical proof.

Computer graphics provides a way for researchers to study what the solution curves look like for various physical systems and sets of initial conditions. These pictures greatly enrich the study of dynamical systems - the way things change in time or space.

The theorist's job is to come up with a suitable mathematical model that captures an electrode's observed characteristics and its consequent behaviour. That mathematical model must be simple and regular enough to be mathematically solvable but not so far removed from an electrode's observed features that any mathematical results from applying the model would be suspect. The idea is not to paint a realistic portrait but to capture the spirit of the phenomenon with a caricature. In that way, researchers may catch a glimpse of what is actually happening at the interface.

(October 30, 1998)